It has been more than a hundred years since the appearance of the landmark 1907 paper by
Erhard Schmidt where he introduced a method for finding an orthonormal basis for the span of a set of linearly independent vectors. This method has since become known as the classical Gram-Schmidt Process (CGS). In this talk we present a survey of the research on Gram-Schmidt orthogonalization, its related QR factorization, and the algebraic least squares problem.

We begin by reviewing the two main versions of the Gram-Schmidt process and the related QR factorization and we briefly discuss the application of these concepts to least squares problems. This is followed by a short survey of eighteenth and nineteenth century papers on overdetermined linear systems and least squares problems. We then examine the original orthogonality papers of both Gram and Schmidt.

The second part of the talk focuses on such issues as the use of Gram-Schmidt orthogonalization for stably solving least squares problems, loss of orthogonality, and reorthogonalization. In particular, we focus on noteworthy work by Ake Bjorck and Heinz Rutishauser and discuss later results by a host of contemporary authors.

*S. J. Leon, Ake Bjorck and Walter Gander are co-authors of the paper
Gram-Schmidt Orthogonalization: 100 years and more,
Numer. Linear Algebra Appl (2013)
This talk is to a large part based on that paper

We investigate the asymptotic behavior of the free path of a variable density random flight model in an external field as the initial velocity of the particle goes to infinity. The random flight models we study arise naturally as the Boltzmann-Grad limit of a random Lorentz gas in the presence of an external field. By analyzing the time duration of the free path, we obtain exact forms for the asymptotic mean and variance of the free path in terms of the external field and the density of scatterers. As a consequence, we obtain a diffusion approximation for the joint process of the particle observed at reflection times and the amount of time spent in free flight. This is based on joint work with Doug Rizzolo and Eric Wayman.

In many real-world statistical problems, we observe a large number of potentially explanatory variables of which a majority portion may be irrelevant. For this type of problems, controlling the false discovery rate (FDR) guarantees that most of the discoveries are truly explanatory and thus replicable. In this talk, we propose a novel method named SLOPE to control the FDR in sparse high-dimensional linear regression. This computationally efficient procedure works by regularizing the fitted coefficients according to their ranks: the higher the rank, the larger the penalty. This is in analogy with the Benjamini-Hochberg procedure, which compares more significant p-values with more stringent thresholds. Whenever the columns of the design matrix are not strongly correlated, we show empirically that SLOPE obtains FDR control at a reasonable level while offering substantial power. We also apply this procedure to a population cohort in Finland with the goal of identifying relevant genetic variants to fasting blood high-density lipoprotein levels.

Although SLOPE is developed from a multiple testing viewpoint, we show the surprising result that it achieves optimal squared errors under Gaussian random designs over a wide range of sparsity classes. An appealing feature is that SLOPE does not require any knowledge of the degree of sparsity. This adaptivity to unknown sparsity has to do with the FDR control, which strikes the right balance between bias and variance. The proof of this result presents several novel elements not found in the high-dimensional statistics literature.

The subject of Cauchy-Riemann Geometry (shortly: CR-geometry), founded in the research of Henri Poincare, is remarkable in that it lies on the border of several mathematical disciplines, among which we emphasize Complex Analysis and Geometry, Differential Geometry, and Partial Differential Equations. Recently, in our research, we have discovered a new face of CR-geometry. This is a novel approach of interpreting objects arising in CR-geometry (called CR-manifolds) as certain Dynamical Systems, and vice versa. It turns out that geometric properties of CR-manifolds are in one-to-one correspondence with that of the associated dynamical systems. In this way, we obtain a certain vocabulary between the two theories. The latter approach has enabled us recently to solve a number of long-standing problems in CR-geometry. It also has promising applications for Dynamical Systems. We call this method the CR (Cauchy-Riemann manifolds) - DS (Dynamical Systems) technique.

In this talk, I will outline the CR - DS technique, and describe its recent applications to Complex Geometry and Dynamics.

Let $\beta\equiv \beta^{(m)} = \{\beta_{i}\}_{i\in \mathbb{Z}_{+}^{n},
|i|\le m}$, $\beta_{0}>0$, denote a real $n$-dimensional multisequence of degree $m$
and let $K$ denote a closed subset of $\mathbb{R}^{n}$.
The
\textit{Truncated $K$-Moment Problem} concerns the existence of a
\textit{$K$-representing measure} for $\beta$, i.e.,
a positive Borel measure $\mu$, supported in $K$, such that
\begin{equation}
\beta_{i} = \int_{K} x^{i}d\mu ~~~~~~~~( i\in \mathbb{Z}_{+}^{n},~~|i|\le m).
\end{equation}
Let $\mathcal{P}_{m} := \{p\in \mathbb{R}[x_{1},\ldots,x_{n}]: ~~deg~p\le m\}$.
We associate to
$\beta$
the \textit{Riesz functional} $L_{\beta}:\mathcal{P}_{m} \mapsto \mathbb{R}$ defined by
$L_{\beta}(\sum a_{i}x^{i}) = \sum a_{i}\beta_{i}$. The existence of
a $K$-representing measure implies that $L_{\beta}$ is \textit{$K$-positive},
i.e., if $p\in \mathcal{P}_{m}$ satisfies $p|K\ge 0$, then $L_{\beta}(p)\ge 0$.
In the \textit{Full $K$-Moment Problem} for $\beta \equiv \beta^{(\infty)}$, a classical
theorem of M. Riesz ($n=1$) and E.K. Haviland $(n>1$) shows that $\beta$ has a
$K$-representing measure if and only if $L_{\beta}$ is $K$-positive. In the
Truncated $K$-Moment Problem, the direct analogue of Riesz-Haviland is not true.
We discuss the gap between $K$-positivity and the existence of $K$-representing measures,
with reference to Tchakaloff's Theorem, approximate $K$-representing measures,
a ``truncated" Riesz-Haviland theorem due to Curto-F., a ``strict" K-positivity
existence theorem of F.-Nie, and recent results concerning the \textit{core variety} of a
multisequence.

A well-known principle in symplectic geometry says that information about the smooth structure on a manifold should be captured by the symplectic geometry of its cotangent bundle. One prominent example of this is Nadler and Zaslow's microlocalization correspondence, an equivalence between a category of constructible sheaves on a manifold and a symplectic invariant of its cotangent bundle called the Fukaya category.

The goal of this talk is to describe a model for a relative version of this story in the simplest case, corresponding to Legendrian knots in the standard contact 3-space. This construction, called the
augmentation category, is a powerful invariant which is defined in terms of holomorphic curves but can also be described combinatorially. I will describe some interesting properties of this category and relate it to a category of sheaves on the plane. This is joint work with Lenny Ng, Dan Rutherford, Vivek Shende, and Eric Zaslow.

The Langlands program is an intricate network of conjectures, which are meant to connect different areas of mathematics, such as number theory, harmonic analysis and representation theory. One striking consequence of the Langlands program is the Ramanujan conjecture, which is a statement purely within harmonic analysis, about the growth rate of Fourier coefficients of modular forms. It turns out to be intimately connected to the Weil conjectures, a statement about the cohomology of projective, smooth varieties defined over finite fields.

I will explain this connection and then move towards a mod p analogue of these ideas. More precisely, I will explain a strategy for understanding torsion occurring in the cohomology of locally symmetric spaces and how to detect which degrees torsion will contribute to. The main theorem is
joint work with Peter Scholze and relies on a p-adic version of Hodge theory and on recent developments in p-adic geometry.

I will explain a new construction and characterization of the $p$-adic local Langlands correspondence for $\mathrm{GL}_2(\mathbb{Q}_p)$. This is joint work with Emerton, Gee, Geraghty, Pa\v{s}k\={u}nas and Shin and relies on the Taylor-Wiles patching method and on the notion of projective envelope.

Please note the unusual time for this week's seminar (10:00 am).

I will give a brief introduction to Hodge theory with some examples of how it is used to study algebraic varieties.

The interface between protein solute and aqueous solvent exhibits complex geometries, and can undergo conformational changes by combined influences from electrostatic force, surface tension, and hydrodynamic force. To understand the role of solvent Stokes flow in this process, we develop a Dynamic Implicit Solvent Model (DISM). Based on this model, we first analytically study the linear stability of a cylindrical solute-solvent interface, where the asymptotic dispersion relation reveals a power law. Moreover, we develop a computational method to simulate the solvent Stokes flow and the interface motion. The key components of our fluid solver are a virtual node method, a pressure Poisson equation, a specially designed boundary condition, the Schur complement method and the least square technique. Level set method is used for the interface motion. We present some 3D numerical results to demonstrate the accuracy and convergence of our method, and show interesting dynamics of protein conformational change.

We will discuss three stories revolving around convex programming. The first is of a new algorithmic framework for a century old problem in physics called phase retrieval, which involves recovering
vectors from quadratic measurements and naturally connects to questions in quantum mechanics and theoretical CS. The second is on recovering the 3D structure of a scene from a collection of images, a fundamental task in computer vision which requires algorithms that are robust to a large fraction of arbitrary corruptions in the input data. Lastly, we will present new non-convex guarantees for solving certain semidefinite programs quickly by exploiting parsimony in their solutions.

The dynamics of complex biological systems is often driven by multiscale, rare but important events. In this talk, I will introduce the numerical methods for computing transition states, and then give two examples in distinct biological systems: one is a multiscale stochastic model to investigate a novel noise attenuation mechanism that relies on more noises in different cellular processes to coordinate cellular decisions during embryonic development; the other is a phase field model to study the neuroblast delamination in Drosophila.

We will discuss some new density results about the $2$-primary part of
class groups of quadratic number fields and how they fit into the framework
of the Cohen-Lenstra heuristics. Let $\mathrm{Cl}(D)$ denote the class
group of the quadratic number field of discriminant $D$. The first result
is that the density of the set of prime numbers $p\equiv -1\bmod 4$ for
which $\mathrm{Cl}(-8p)$ has an element of order $16$ is equal to $1/16$.
This is the first density result about the $16$-rank of class groups in a
family of number fields. The second result is that in the set of
fundamental discriminants of the form $-4pq$ (resp. $8pq$), where $p\equiv
q \equiv 1\bmod 4$ are prime numbers and for which $\mathrm{Cl}(-4pq)$
(resp. $\mathrm{Cl}(8pq)$) has $4$-rank equal to $2$, the subset of those
discriminants for which $\mathrm{Cl}(-4pq)$ (resp. $\mathrm{Cl}(8pq)$) has
an element of order $8$ has lower density at least $1/4$ (resp. $1/8$). We
will briefly explain the ideas behind the proofs of these results and
emphasize the role played by general bilinear sum estimates.
\newline\newline

Note: The speaker will give a prep-talk for graduate students in
AP&M 7421 at 1:15pm. All graduate students interested in number theory
are strongly encouraged to attend.

Symplectic geometry has recently emerged as a key tool in the study of low-dimensional topology. One approach, championed by Arnol'd, is to examine the topology of a smooth manifold through the symplectic geometry of its cotangent bundle, building on the familiar concept of phase space from classical mechanics. I'll describe a way to use this approach to construct a rather powerful invariant of knots called "knot contact homology", and discuss its properties. If time permits, I'll also outline a surprising connection to string theory and mirror symmetry.

I will discuss Gromov's non squeezing theorem in symplectic topology,
the notion of Gromov radius, and the notion of monotonic symplectic invariant
due to Ekeland and Hofer. Then I will report on recent progress in the subject,
concerning the relations of these results to the existence of symplectic embeddings.

Starting from basic principles, we will present some recent developments in the theory of rough paths and in the theory of sub-Riemannian diffusions. The first part of the talk will be devoted to the theory of rough paths. This theory was developed in the 1990s by T. Lyons, and allows to give a sense to solutions of differential equations driven by irregular paths. The theory itself has nothing to do with probability theory but has had a tremendous impact on several recent developments in stochastic analysis; it served as an inspiration to Hairer’s regularity structure theory, for which he was awarded the Fields medal in 2014. In the second part of the talk, we will address several problems in the geometric analysis of some sub-Riemannian manifolds, which can (surprisingly) be solved using diffusion semigroups techniques.

Inspired by the recent developments and mature understanding of the notion of lower-boundedness of Ricci curvature in continuous settings (such as Riemannian manifolds), several independent groups of researchers have proposed intriguing analogs of such a curvature in discrete settings (such as graphs). The proposals depend on the perspective being probabilistic, analytical or combinatorial. In this talk, I will briefly mention a few of these approaches, consequences, and state some open problems.

I will present applications involving interfaces and consider how different points of view lead to numerical methods for capturing their motions.

We construct a functor from the category of $p$-adic local systems on a smooth rigid analytic variety $X$ over a $p$-adic field to the category of vector bundles with a connection on $X$, which can be regarded as a first step towards the sought-after $p$-adic Riemann-Hilbert
correspondence. As a consequence, we obtain the following rigidity theorem for $p$-adic local systems on a connected rigid analytic variety: if the stalk of such a local system at one point, regarded as a $p$-adic Galois representation, is de Rham in the sense of Fontaine, then the stalk at every point is de Rham. Along the way, we also establish some results about the $p$-adic Simpson correspondence. Finally, we give an application of our results to Shimura varieties. Joint work with Xinwen Zhu.

To every action, there is an equal and opposite reaction. However, there turn out to exist in nature situations where the reaction seems neither equal in magnitude nor opposite in direction to the action. We will see a variety of table-top demos and experimental movies, apparently in more and more violation of Newton's 3rd law, and give a full analysis of what is happening, discovering in the end that these phenomena are in a sense generic. The keys are shock, singular material property, and supply of `critical geometry'.

The interpretation of experiments is complicated when the outcome of an experimental unit depends not only on its assigned treatment but also on interference from other units. Here, we extend the potential outcomes framework of causal inference without such interference between units (Rubin, 1974) in order to define and assess causal effects. When two units cannot interfere with each other, then one unit's treatment assignment only affects that unit's outcome. However, when two units can interfere with each other, then one unit's treatment assignment generally affects both of their outcomes. Furthermore, the interference can depend on units' characteristics and the treatment assignment itself, and is often only partially revealed. Our analysis of data generated by such situations uses both Bayesian and frequentist ideas to test sharp null hypotheses about causal effects. In particular, to assess causal effects we model and estimate the interference between units as a network, and develop novel testing procedures that involve repeated sampling of the treatment assignment under constraints from the network topology and the tested hypothesis. We illustrate our causal framework in applications where such forms of interference are ubiquitous but currently not adequately addressed.

I will describe two classes of gluing constructions for minimal surfaces in the 3-sphere; both generate sequences of minimal embeddings converging to singular configurations of multiple Clifford tori. One class, an extension of the Clifford torus doubling of Kapouleas and Yang (2010), produces minimal embeddings each of which resembles a stack of approximate Clifford tori connected by many small catenoidal tunnels arranged doubly periodically on the tori. The other class (joint work with Kapouleas) yields examples which resemble multiple Clifford tori intersecting along certain great circles, except that neighborhoods of the intersection circles have been replaced by approximate Karcher-Scherk towers so that the resulting surfaces are smoothly embedded. All these constructions proceed by first building a surface that is only approximately minimal but possesses the other desired properties and then finding a graph over this initial surface which is exactly minimal. This last step is accomplished by solving the relevant PDE, whose linearization has small eigenvalues that require special attention.

Adaptive mesh refinement (AMR) algorithms are one of two necessary tools
for grand challenging problems in scientific computing. Reliability of computer
simulations is responsible for accurate computer predictions/designs. Efficient and reliable a posteriori error estimation are, respectively, the key for success of AMR algorithms and the reliability of computer predictions/designs.

Since Babuska's pioneering work in 1976, the a posteriori error estimation has been extensively studied, and impressive progress has been made during the past four decades. However, due to its extreme difficulty, this important research field of computational science and engineering remains wide open. In this talk, I will describe (1) basic principles of the a posteriori error estimation techniques for finite element approximations to partial differential equations and (2) our recent work.

The exponential growth of massive data streams is everywhere, and has been attracting increasing interest. In addition to size, the complexity is certainly an important issue. To handle this kind of datasets, of particular importance is an adaptive model, as well as innovative acquisition of intrinsic features/structure hidden in the massive data-sets. In this talk, I will discuss how to apply the knowledge from differential geometry to model and analyze massive datasets in
different fields. In particular, I will discuss algorithms like graph connection Laplacian and vector diffusion maps, and their theoretical justification based on the spectral geometry. I will also discuss at least one of the following applications: cryo-electron microscope, phase retrieval, vector nonlocal mean and F wave analysis.

The wrapping transformation is easily seen to intertwine convolutions of probability measures on the real line and the circle. It is also easily seen to not transform additive free convolution into the multiplicative one. However, we show that on a large class L of probability measures on the line, wrapping does transform not only the free but also Boolean and monotone convolutions into their multiplicative counterparts on the circle. This allows us to prove various identities between multiplicative convolutions by simple applications of the additive ones. The restriction of the wrapping to L has several other unexpected nice properties, for example preserving the number of atoms. This is joint work with Octavio Arizmendi.

I will introduce framework for computationally restrictive statistical analysis. I will focus on usefulness of non-asymptotic analysis and linearization techniques for quantifying uncertainty of estimation in complex datasets.

Jean-Marc Fontaine has shown that there exists an equivalence of categories between the category of continuous $\mathbb{Z}_p$-representations of a given Galois group and the category of \'{e}tale $(\phi,\Gamma)$-modules over a certain ring. We are interested in the question of whether there exists a theory of $(\phi,\Gamma)$-modules for the Lubin-Tate tower. We construct this tower via the rings $R_n$ which parametrize deformations of level $n$ of a given formal module. One can choose prime elements $\pi_n$ in each ring $R_n$ in a compatible way, and consider the tower of fields $(K'_n)_n$ obtained by localizing at $\pi_n$, completing, and passing to fraction fields. By taking the compositum $K_n = K_0 K'_n$ of each field with a certain unramified extension $K_0$ of the base field $K'_0$, one obtains a tower of fields $(K_n)_n$ which is strictly deeply ramified in the sense of Anthony Scholl. This is the first step towards showing that there exists a theory of $(\phi,\Gamma)$-modules for this tower.

In this talk we will introduce the notions of formal modules and their deformations, strictly deeply ramified towers of fields, and $(\phi,\Gamma)$-modules, and sketch the proof that the Lubin-Tate tower is strictly deeply ramified.

I'll explain Kontsevich's approach, using Gelfand-Fuchs cohomology, to constructing characteristic classes of flat symplectic bundles. This can be used to talk about Rozansky-Witten invariants of holomorphic symplectic manifolds and construct cohomology classes on the moduli space of curves.

I will discuss several recent results on abstract homomorphisms between the groups of rational points of algebraic groups. The main focus will be on a conjecture of Borel and Tits formulated in their landmark 1973 paper. Our result settle this conjecture in several cases; the proofs make use of the notion of an algebraic ring. I will conclude by discussing several applications to character varieties of finitely generated groups and group actions.

For the pre-talk: I will recall some basic concepts from the theory of algebraic groups and outline a general philosophy for the study of rigidity phenomena between the groups of rational points of algebraic groups.

During my talk I will concentrate on the current status and ongoing work on the following:

1) Doubling constructions for minimal surfaces: I will first discuss the general motivation and framework and I will mention briefly the earlier work in [Kapouleas-Yang: 2010] and work of David Wiygul. I will then concentrate on the ideas in [Kapouleas: arXiv:1409.0226, 2014] and in particular I will present in some detail the Linearized Doubling methodology. I will also mention ongoing work and future possibilities.

2). Desingularization constructions for minimal surfaces: I will first discuss the $O(2)$-invariant initial configuration case as in [Kapouleas: JDG 1997]. I will then briefly discuss some recent constructions with more symmetry in various settings, including a current construction with Martin Li for free boundary minimal surfaces in the unit three-ball. I will outline and discuss extensions to less symmetric settings or settings without any symmetries as discussed in [Kapouleas: Clay proceedings, Vol.2, 2005], [Kapouleas: ALM, vol 20, 2011], and further ongoing work.

3). I will discuss potential constructions for Einstein four-manifolds and some related ancient solutions for the Ricci flow as in [Brendle-Kapouleas: arXiv:1405.0056, 2014].

4). Finally to the extent that time permits I will briefly discuss gluing constructions for Special Lagrangian cones as in [Haskins-Kapouleas: Inventiones 2007] and [Haskins-Kapouleas: ALM, vol.7, 2008]
and ongoing work, and also for CMC (hyper)surfaces in early work of mine and recent work with Christine Breiner [Breiner-Kapouleas: Math. Annalen 2014] and [Breiner-Kapouleas: Preprint close to completion].

This talk introduces a two-grid and a bootstrap multigrid finite element approximations to the Laplace-Beltrami eigenvalue problem on closed surfaces. The latter can be viewed as a special case of the BAMG (Bootstrap Algebraic Multi-Grid) framework applying on surface finite element method. Nonlinear eigenvalue problems are solved in the enriched finite element space on coarse mesh, while on the fine mesh only linear problems are approximated. Several interesting
phenomena for approximating eigenvalues with high multiplicity are shown comparing conventional two-grid/multigrid ideas with the new bootstrap multigrid methods. Then some a posteriori error estimation technique for the multigrid iterate will be discussed which considers
how accurate the linear problems need to be approximated to guarantee the overall optimal rate of convergence.

The concept of the autoregressive (AR) sieve bootstrap is investigated for the case of spatial processes in $Z^2$. This procedure fits AR models of increasing order to the given data and, via resampling of the residuals, generates bootstrap replicates of the sample. The paper explores the range of validity of this resampling procedure and provides a general check criterion which allows to decide whether the AR sieve bootstrap asymptotically works for a specific statistic of interest or not. The criterion may be applied to a large class of stationary spatial processes. As another major contribution of this paper, a weighted Baxter-inequality for spatial processes is provided. This result yields a rate of convergence for the finite predictor coefficients, i.e. the coefficients of finite-order AR model fits, towards the autoregressive coefficients which are inherent to the underlying process under mild conditions. The developed check criterion is applied to some particularly interesting statistics like sample autocorrelations and standardized sample variograms. A simulation study shows that the procedure performs very well compared to normal approximations as well as block bootstrap methods in finite samples.
[Joint work with Carsten Jentsch and Jens-Peter Kreiss.]

There are many important and striking differences between classical complex analysis in one variable and complex analysis in several variables. In this talk, we will illustrate this by discussing just one such difference, the Hartogs extension phenomenon. For example, if $D$ denotes the annular domain in $\mathbb C^n$ consisting of the unit ball $B$ minus the closed ball of radius ½, then any holomorphic function in $D$ extends holomorphically to the whole unit ball $B$ ... provided $n\geq 2$; it is clearly not true when $n=1$. This particular result can be proved by using the Cauchy integral formula, but a proof that works in more general situations leads to a study of the $\bar \partial$ equation.

For any Lie group $G$, we construct a $G$\--equivariant analogue of symplectic capacities and give examples when $G=\mathbb{T}^k\times\mathbb{R}^{d-k}$, in which case the capacity is an invariant of integrable systems. Then we study the continuity of these capacities, using the natural topologies on the symplectic $G$\--categories on which they are defined.

This work is joint with Alvaro Pelayo and Alessio Figalli.

Regularization and stabilization are vital tools for resolving the numerical and theoretical difficulties associated with ill-posed or degenerate optimization problems. Broadly speaking, regularization involves perturbing the underlying linear equations so that they are always nonsingular. Stabilization is designed to provide a sequence of iterates with fast local convergence, even when the gradients of the constraints satisfied at a solution are linearly dependent.

We discuss the crucial role of regularization and stabilization in the formulation and analysis of modern active-set and interior methods for nonlinear optimization. In particular, we establish the close relationship between regularization and stabilization and propose some new methods based on formulating an associated "simpler" optimization subproblem defined in terms of both the primal and dual variables of the original problem.

The notion of stable weak equivalence for actions on probability spaces has been introduced by Kechris and studied by many other authors including Abert, Bowen, Burton, Elek, Tucker-Drob, and Weiss. Particularly, it was shown by Bowen and Tucker-Drob that the space of stable weak equivalence classes of actions on the standard probability space of a fixed countable group is a metrizable simplex. In the amenable case, such a simplex coincides with the simplex of invariant random subgroups (IRS) of the group. In joint work with Burton, we initiated the study of stable weak equivalence for actions on the hyperfinite II1 factor and, more generally, tracial von Neumann algebras. In my talk I will provide an introduction to this subject, and present some of our new results.

Abstract notions of measurement -- Hilbert spaces, metric spaces, measure spaces -- most commonly use the real numbers as the associated measuring stick. In this talk I'll give an overview of some fun applications of more general ordered fields, namely in the areas of fractal geometry and integration theory, and discuss a few notable obstacles and moral guidelines for future use.

Google lives on data. Search, Ads, YouTube, Maps - they all live on data. The talk will recount how Google uses data and statistics, how Google is always experimenting to make improvements (yes, this includes your searches!), and how Google adapts statistical ideas to do things that have never been done before. (This is a nontechnical talk, also suitable for undergraduates.)

In this talk, we present some results originating from a new general inequality obtained by combining the Chen–Stein method with Malliavin calculus on the Poisson space, such as multidimensional Poisson approximations, mixed limit theorems, as well as a characterization of asymptotic independence for U–statistics. Applications to stochastic geometry through limit theorems involving the joint convergence of vectors of subgraph–counting statistics exhibiting both a Poisson and a Gaussian behavior will also be discussed.

At some point in graduate school, everyone encounters the rather abstract theorem that every compact topological group possess a unique translation-invariant probability measure, known as the Haar measure. Rarely, if ever, is the concrete problem of computing integrals under this measure addressed. I will explain an analogue of the Fundamental Theorem of Calculus in this setting: a theorem which reduces the computation of a large class of group integrals to a symbolic problem. This symbolic problem can, in some important cases, be cleverly solved using techniques from algebraic combinatorics.

Let $C/\mathbb{Q}$ be a curve of genus 3, given as a double cover of a
conic with no $\mathbb{Q}$-rational points. Such a curve is hyperelliptic
over the algebraic closure of $\mathbb{Q}$ but does not have a
hyperelliptic model of the usual form over $\mathbb{Q}$. We discuss an
algorithm that computes the local zeta functions
of $C$ simultaneously at all primes of good reduction up to a given
bound $N$ in time $(\log N)^{4+o(1)}$ per prime on average. It works
with the base change of $C$ to a quadratic field $K$, which has a
hyperelliptic model over $K$, and it uses a generalization of the
``accumulating remainder tree'' method to matrices over $K$. We briefly
report on our implementation and its performance in comparison to
previous implementations for the ordinary hyperelliptic case.

Joint work with David Harvey and Andrew V. Sutherland.

In the pre-talk, we will introduce some of the objects that the talk is
about, such as curves and their models, the zeta function and how it
relates to point counting, and the particular type of genus 3 curves
that we are interested in.

Solvation plays a fundamental role in many biological processes including biomolecular recognition, protein-protein interactions, membrane assembly and many others. The variational implicit solvation method (VISM) is currently developed to predict solvation free energies for systems of very complex topology, such as proteins. VISM theoretical foundation makes it unique in that i) it couples hydrophobic, dispersion interactions and electrostatic effects into one functional, and ii) it produces the solvation surface as an output of the theory. This allows VISM to capture more subtle solvation effects than do other implicit solvation methods. As we plan to expand VISM applications to more challenging scenarios, coarse-graining the protein seems a good strategy to keep the computational cost low. In this work, we adapted VISM to work with a well established coarse-grain forcefield for proteins and other biomolecules, MARTINI. We then tested how coarse-grained MARTINI-VISM compares with (atomistic) VISM for a set of six proteins that differ in shape, size and charge distribution. Promising results suggest that coarse-graining the protein is indeed the right step to broaden VISM applications in the near future.

In this talk, I will propose a refinement of the Langlands correspondence in the case of curves, or more generally symplectic Galois representations. Langlands conjectures that to such a representation one can attach a generic automorphic representation of an odd orthogonal group. I will show that every such representation has a distinguished line. The hope is that the properties of the resulting new form will make it easier to test the general conjecture computationally.

A classical result of Skolem, Mahler, and Lech asserts that a linearly recurrent sequence taking values in a field of characteristic zero has the property that its zero set is a finite union of one-way infinite arithmetic progressions along with a finite set. In positive characteristic, examples due to Lech show that this conclusion does not hold. For years the problem of finding a positive characteristic analogue was open until it was solved by Derksen in 2005. We describe extensions of Derksen's work involving finite-state machines and explain how these extensions allow one to effectively solve many classes of Diophantine problems in positive characteristic.

Tia is an aspiring globetrotter with a simple method of exploration: she walks straight ahead. Suppose Tia is randomly dropped on the surface of a unknown planet, which she investigates using her simple strategy. What will Tia get to see as she travels along the geodesic she is on? The answer to this question depends on the geometry of the surface she is on, and it can be partially addressed by studying the relationship between the curvature of a surface and the dynamics of its geodesic flow. In this talk, I will state a motivating set of results concerning the relationship between geometry and dynamics, discuss more recent work in this area, and show pictures of some surfaces whose geodesic flows have surprising dynamical properties.

Twisted Calabi-Yau algebras are a class of algebras with nice behavior regarding their Hochschild cohomology. They include many classes of examples of recent interest, for example Artin-Schelter regular algebras. We discuss in particular the theory of twisted Calabi-Yau algebras of low global dimension which are factors of path algebras of quivers Q. For example, we have preliminary results regarding the following question: for which quivers Q does there exists a twisted Calabi-Yau algebra of dimension 3 which is a factor of the path algebra of Q?

For the pre-talk: We will give an introduction to some techniques from homological algebra, in particular Hochschild cohomology, which are relevant for the talk.

Halmos once described problems and their solutions as “the heart of mathematics”. Following this line of thinking, one might naturally ask: “What, then, is the heart of problems?” In this talk, I attempt to answer this question using techniques from statistics, information visualization, and machine learning. I begin the journey by cataloging the features of problems delineated by the mathematics and mathematics education communities. These dimensions are explored in a large data set of students working thousands of problems at the Art of Problem Solving, an online company that provides adaptive mathematical training for students around the world. To increase the number of features of mathematical problems that can be studied, this quantitative exploration is partnered with a qualitative analysis that involves human scoring of 105 problems and their solutions. Using correlation matrices, principal components analysis, and clustering techniques, I explore the relationships among those features frequently discussed in mathematics problems (e.g., difficulty, creativity, novelty, affective engagement, authenticity). Along the way, I define a new set of uncorrelated features in problems and use these as the basis for a New Mathematical Problem Typology (NMPT). Grounded in the terminology of classical music, the NMPT works to quickly convey the essence and value of a problem, just as terms like “etude” and “mazurka” do for musicians.

In this talk we will review the paper "Linear Algebra Algorithms as Dynamical Systems" by Moody T. Chu. Many iterative methods in numerical linear algebra can be re-interpreted from a dynamical systems viewpoint. This vantage point can provide new insight into old algorithms and potentially guide the construction of new methods. In addition to some numerical demonstrations, we will ponder what impact structure-preserving algorithms can have on the field of numerical linear algebra.

I will talk about my recent work with Erik Carlsson in which we studied certain symmetric functions associated to Dyck paths. For each Dyck path from (0,0) to (n,n) the corresponding symmetric function is a generating function of labelings of the positions 1,2,...,n by positive integers. Each labeling is counted with a weight which depends on whether labels on positions i,j are in the correct or reversed order and whether (i,j) is above or under the path. While studying these symmetric functions we discovered an interesting algebraic structure that controls them. This ultimately led us to a proof of the shuffle conjecture by Haglund, Haiman, Loehr, Remmel, and Ulyanov.

As in the Riemannian setting, a subRiemannian heat kernel is controlled by the geometry of the underlying manifold. In particular, the asymptotic behavior of the kernel can reveal certain geometric and topological data. We study the logarithmic derivatives of subRiemannian heat kernels in some cases and show that, under appropriate scaling, they converge to their analogues on stratified groups. This gives one quantification of the now standard idea that stratified groups play the role of the tangent space to subRiemannian manifolds.

This is joint work with Joshua Campbell.

I will give a brief introduction to orbit equivalence theory. This aims at studying the orbit structure of measure preserving actions of countable groups.

A crowning achievement of Number theory in the 20th century is a theorem of Wiles which states that for an elliptic curve $E$ over $\mathbb{Q}$ of conductor $N$, there is a non-constant map from the modular curve $X_0(N)$ to $E$. For some curve isogenous to $E$, the degree of this map will be minimal; this is the modular degree. The Jacquet-Langlands correspondence allows us to similarly parameterize elliptic curves by Shimura curves.
In this case we have several different Shimura curve parameterizations for a given isogeny class. Further, this generalizes to elliptic curves over totally real number fields. In this talk I will discuss these degrees and I compare them with $D$-new modular degrees and $D$-new congruence primes. This data indicates that there is a strong relationship between Shimura degrees and new modular degrees and congruence primes.

I will discuss the special properties of differential forms on symplectic manifolds. I will show how the presence of a symplectic structure leads naturally to A-infinity algebra structures and hence also cohomology rings of differential forms. These cohomology rings are novel symplectic invariants and I will describe some of their interesting properties and applications.

Rivin conjectured that the formal conjugacy growth series of a hyperbolic group is rational if and only if the group is virtually cyclic. In this talk, I will present a proof of Rivin's conjecture and supporting evidence for the analogous statement for acylindrically hyperbolic groups. The class of acylindrically hyperbolic groups is a wide class of groups that contains (among many other examples) the outer automorphism groups of free groups and the mapping class groups of hyperbolic surfaces. This is a joint work with Laura Ciobanu.

For the pre-talk: Hyperbolic groups will be defined and it will be explained why the generating function of the sequence counting the number of elements of length n is rational.

In this talk, we compare and contrast a few finite element h-adaptive and hp-adaptive algorithms. We test these schemes on three example PDE problems and we utilize and evaluate an a posteriori error estimate.

In the process, we introduce a new framework to study adaptive algorithms and a posteriori error estimators. Our innovative environment begins with a solution u and then uses interpolation to simulate solving a corresponding PDE. As a result, we always know the exact error and we avoid the noise associated with solving.

Using an effort indicator, we evaluate the relationship between accuracy and computational work. We report the order of convergence of different approaches. And we evaluate the accuracy and effectiveness of an a posteriori error estimator.

Let $K$ be a field with a valuation $v$. Given a projective variety $X$ over $K$, we can associate an analytification $X^{an}$ with respect to $v$ called the Berkovich space. These spaces appear in various contexts, such as tropical geometry and number theory. More recently there have appeared surprising connections between Berkovich geometry and birational geometry. I will give a brief overview of Berkovich spaces with examples, and describe how the birational geometry of $X$ is reflected in the geometry of the associated Berkovich space.

Let G be a finitely generated subgroup of GL(n,Q). Under certain algebraic conditions, strong approximation describes the
closure of G with respect to its congruence topology. Super-approximation essentially tells us how dense G is in its closure! Here is my plan for this talk:

1. I will start with the precise formulation of this property.

2. Some of the main results on this subject will be mentioned.

3. Some of the (unexpected) applications of super-approximation will be mentioned, e.g. Banach-Ruziewicz problem, orbit equivalence rigidity, variation of Galois representations.

4. Some of the auxiliary results that were needed in the proof of super-approximation will be mentioned: sum-product phenomena, existence of small solutions

This talk will begin with an introduction to one of the fundamental stochastic processes in probability theory, Brownian motion. Next comes a constrained version, called reflected Brownian motion, a process arising as a diffusion limit of stochastic networks and of interacting particle models in statistical physics. This leads to many mathematical questions, some of which we shall see in the talk.

Quaternion algebras contain quadratic field extensions of the center. Given two algebras, a natural question to ask is whether they share a common field extension. This gives us an idea of how closely related those algebras are to one another. If the center is of characteristic 2 then those extensions divide into two types - the separable type and the inseparable type. It is known that if two quaternion algebras share an inseparable field extension then they also share a separable field extension and that the converse is not true. We shall discuss this fact and its generalization to p-algebras of arbitrary prime degree.

This talk is intended for those who, like the speaker, have at some point wondered whether there is a theory of three- or higher-dimensional matrices that parallels matrix theory. A $d$-dimensional hypermatrix may be viewed as a coordinate representation of an order-$d$ tensor but we will explain why it is not quite the same. We discuss how notions like rank, norm, determinant, eigen and singular values may be generalized to hypermatrices. We will see that, far from being artificial constructs, these notions have appeared naturally in a wide range of applications and can be enormously useful.

Let $p_{\hbar}$ and $q_{\hbar}$ be momentum and position operators
respectively. In 1973, Hepp showed the classical limit and quantum
correction of an observable $e^{i(rq_{\hbar}+sp_{\hbar})}$ under
the evolution generated by a Hamiltionian
\[
H_{\hbar}=-\frac{\hbar}{2m}\partial^2+V\left(\sqrt{\hbar}x\right)
\]
on all state $\psi\in L^{2}\left(\mathbb{R}\right)$ in The Classical
Limit for Quantum Mechanical Correclation Functions. In contrast to
the Hepp' s result, in our talk, we are interested in unbounded ``observables''
and more general Hamiltionians. Motivated by the idea in Quantum Fluctuations
and Rate of Convergence towards Mean Field Dynamics by Rodnianski
and Schlein in 2009. Classical mechanics can be recovered from quantum
mechanics by studying the asymptotic behavior of quantum expectations
relative to $\sqrt{\hbar}.$

A semitoric helix is a generalization of the notion of a toric fan for the case in which the acting torus is lower dimensional. The goal of this talk is to characterize the minimal models that can be obtained from a semitoric helix by a finite sequence of blowdowns. As a tool towards this end we produce a standard form for elements of $\mathrm{PSL}_2(\mathbb{Z})$.

This is an expository talk focused on the ideas involved in Moser's theorem for volume forms, and its subsequent generalizations
by Greene and Shiohama.

For a non-degenerate integral quadratic form $F(x_1, \dots , x_d)$ in 5 (or more) variables, we prove an optimal strong approximation theorem. Fix any compact subspace $\Omega\subset\mathbb{R}^d$ of the affine quadric $F(x_1,\dots,x_d)=1$. Suppose that we are given a small ball $B$ of radius $0<r<1$ inside $\Omega$, and an integer $m$. Further assume that $N$ is a given integer which satisfies $N\gg(r^{-1}m)^{4+\epsilon}$ for any $\epsilon>0$. Finally assume that we are given an integral vector $(\lambda_1, \dots, \lambda_d) $ mod $m$. Then we show that there exists an integral solution $x=(x_1,\dots,x_d)$ of $F(x)=N$ such that $x_i\equiv \lambda_i \text{ mod } m$ and $\frac{x}{\sqrt{N}}\in B$, provided that all the local conditions are satisfied. We also show that 4 is the best possible exponent. Moreover, for a non-degenerate integral quadratic form $F(x_1, \dots , x_4)$ in 4 variables we prove the same result if $N\geq (r^{-1}m)^{6+\epsilon}$ and some non-singular local conditions for $N$ are satisfied. Based on some numerical experiments on the diameter of LPS Ramanujan graphs, we conjecture that the optimal strong approximation theorem holds for any quadratic form $F(X)$ in 4 variables with the optimal exponent $4$.

We prove that the volume of a free boundary minimal surface $\Sigma^k \subset B^n$ where is a geodesic ball in Hyperbolic space $H^n$ is bounded from below by the volume of a geodesic k-ball with the same radius as $B^n$. More generally, we prove analogous results for the case where the ambient space is conformally Euclidean, spherically symmetric, and the conformal factor is nondecreasing in the radial variable. These results follow work of Brendle and Fraser-Schoen, who proved analogous results for surfaces in the unit ball in $R^n$. This is joint work with Brian Freidin.

This talk gives an overview of recent progress made in the design and analysis of algebraic multigrid methods. The focus is on the setup algorithm that automatically constructs the multilevel hierarchy used in the solve phase. A sharp two-grid theory is introduced and then used to derive various quality measures of the coarse spaces constructed by the setup algorithm, based on the ideas of compatible relaxation, a related identity that assumes the use of the so-called ideal interpolation operator, and an optimal form of classical algebraic multigrid interpolation that gives the best possible two-grid convergence rate. Various numerical results are presented to illustrate these theoretical results. As a test problem, we focus on a finite volume discretization of a scalar diffusion problem with highly varying (discontinuous) diffusion coefficient.

Heat kernel measure $K(t, I, A) dA$ on positive Hermitian $NxN$ matrices is a probability measure whose large $N$ limit is important for several different types of problems in mathematical physics. My talk introduces a new application: to random Kahler metrics on any Kahler manifold. The pair correlation function of random metrics is explicitly calculated for each $N$. The large $N$ asymptotics are closely related to zero sets of random holomorphic functions.

The goal of the talk is to give some idea of what the all-important Langlands program is all about, and it is aimed
at a general audience with no or limited experience in number theory. For $GL(2)$ we will illustrate these ideas by discussing the case of elliptic curves and modular forms, which led to a proof of Fermat's Last Theorem! We will then try to give a flavor of what's expected for $GL(n)$, and point to how the so-called p-adic Langlands program was envisaged. This ties up with the number theory topics class offered in the Spring quarter.

Let $A$ be an abelian variety defined over a number field $k$ that is isogenous over an algebraic closure to the power of an
elliptic curve $E$. If $E$ does not have CM, by results of Ribet and Elkies concerning fields of definition of $k$-curves, $E$ is isogenous to an elliptic curve defined over a polyquadratic extension of $k$. We show that one can adapt Ribet's methods to study the field of definition of $E$ up to isogeny also in the CM case. We find two applications of this analysis to the theory of Sato-Tate groups of abelian surfaces: First, we show that 18 of the 34 possible Sato-Tate groups of abelian surfaces over $\mathbb{Q}$, only occur among at most 51 $\overline{\mathbb{Q}}$-isogeny classes of abelian surfaces over $\mathbb{Q}$; Second, we give a positive answer to a question of Serre concerning the existence of a number field over which abelian surfaces
can be found realizing each of the 52 possible Sato-Tate groups of abelian surfaces. This is a joint work with Xevi Guitart.

Preparatory talk: In the preparatory talk I plan to review very briefly basic definitions concerning abelian varieties necessary to
introduce (in the main talk) the notion of abelian $k$-variety. I will also present the (general) Sato-Tate conjecture and show how it
motivates the problem considered in the main talk.

The numerical solution of high frequency wave propagation has been a longstanding challenge in computational science and engineering. This talk addresses this problem in the time-harmonic regime. We consider a sequence of examples with important applications, and for each we construct an efficient preconditioner (approximate inverse) that allows one to solve the system with a small number of iterations. From these examples emerges a new framework, where sparsity, geometry of wave phenomenon, and highly accurate discretizations are combined together to address this challenging topic.

Alvarez-Gaumé gave arguments based on path integrals in supersymmetric quantum mechanics for index theorems including the Gauss-Bonnet-Chern, Hirzebruch, and Atiyah-Singer theorems. In this talk, I summarize these heuristic arguments, and describe a new construction of path integrals. This is also a new construction of the heat kernel for a generalized Laplacian, and leads to rigorous path-integral proofs of these index theorems.

Supercurves are the simplest class of complex super manifolds, "one-dimensional" in some sense and thus analogs of Riemann surfaces. I will describe a remarkable duality between pairs of super curves that generalizes Serre duality for Riemann surfaces. Self-dual super curves are precisely the "super Riemann surfaces" introduced by physicists in connection with string theory. I'll suggest connections between this duality and the classical duality between points and hyperplanes in projective space. No prior knowledge of supergeometry is required.

It was shown by Bessis that the Hurwitz action is transitive on minimum-length reflection factorizations of a Coxeter element in a finite Coxeter group. In this talk, I'll explain how to extend Bessis's result to longer factorizations, showing that two factorizations of a Coxeter element into an arbitrary number of reflections lie in the same orbit under the Hurwitz action if and only if they use the same multiset of conjugacy classes. The proof makes use of a surprising lemma about the acuteness structure of minimal dependent sets in root systems. This work is joint with Vic Reiner.

We will discuss the common submanifolds of two Hermitian symmetric spaces. In particular, we proved that the Euclidean space and a bounded symmetric domain cannot share a common submanifold. This is based on the joint work with Professor X. Huang.

The problem of understanding the local geometry of CR embedded submanifolds in CR manifolds arises naturally in several complex variables analysis, e.g., in the study of isolated singularities of analytic varieties (via their links). Significant work has been done on this in connection with rigidity phenomena and the classification of proper holomorphic mappings between balls. We develop from scratch a CR invariant local theory based on the CR tractor calculus associated to the Chern-Moser-Cartan connection. This produces the tools for constructing local invariants and invariant operators in a way parallel to the classical Gauss-Codazzi-Ricci calculus for Riemannian submanifolds.

Turan-type problems in graph theory ask how many edges a graph may have if a certain subgraph is forbidden. One can think of this as an optimization problem, as one is maximizing the global condition of number of edges subject to the local constraint that there is no forbidden subgraph. Problems in combinatorial number theory ask one to deduce properties of a set of (for example) integers while knowing only how large the set is. We study the connection between these two seemingly disjoint areas. Graphs coming from finite projective planes are intimately related to both areas.

I will survey some aspects of the theory of noncommutative completely positive kernels, which are the generalization of usual positive kernels to the setting of free noncommutative function theory. I will then use the language of noncommutative completely positive kernels to discuss the noncommutative generalizations of the interpolation and realization theorems for the Schur-Agler class. This is a joint work with J. Ball and G. Marx.

The complex Hessian equation is a class of important fully nonlinear geometric elliptic equations which can be viewed as an intermediate case between the Laplacian equations and the complex Monge-Ampere equations. In this talk, we will talk about some a priori estimates for a complex Hessian equation motivated from Fu-Yau’s generalization of the Strominger system. This is a joint work with D. Phong and S. Picard.

In this talk we discuss the possible closures of geodesic planes in a hyperbolic 3-manifold M. When M has finite volume Shah and Ratner (independently) showed that a very strong rigidity phenomenon holds, and in particular such closures are always properly immersed submanifolds of M with finite area. Manifolds with infinite volume, however, are far less understood and are the main subject of this talk. This is based on a joint ongoing work with C. McMullen and H. Oh.

Integrable systems are, roughly, dynamical systems with many conserved quantities.
Recently, Pelayo-V\~{u} Ng\d{o}c classified semitoric integrable
systems, which generalize toric integrable systems in dimension four,
in terms of five symplectic invariants. Using this
classification, I construct a metric on the space of
semitoric integrable systems.
By studying continuous paths in this space produced via symplectic blowups
I determine its connected components.
This uses a new algebraic technique in which I
lift matrix equations from $\mathrm{SL}(2,\mathbb{Z})$ to
its preimage in the universal cover of $\mathrm{SL}(2,\mathbb{R})$ and
I further use this technique to completely classify all semitoric
minimal models.
I also produce invariants of integrable systems by constructing
an equivariant version of the Ekeland-Hofer
symplectic capacities
and, as a first step towards a metric on general integrable systems,
I provide a framework to study convergence properties of families of maps
between manifolds which have distinct domains.

This work is partially joint with \'Alvaro Pelayo, Daniel M. Kane, and Alessio Figalli.

Graduate Student Career Advisor, Giulia Hoffmann, will provide information on the variety of career options for Math/CSME students, as well as the resources and events available through the Career Services Center to help graduate students find the right career match.

Statistical latent class models are widely used in social and psychological researches, yet it is often difficult to establish the identifiability of the model parameters. This talk will focus on a class of restricted latent class models with binary responses. This
class of models have recently gained great interests in psychological and educational measurement, psychiatry and many other research areas, where a classification-based decision needs to be made about an individual’s latent traits, based on his or her observed responses. The model parameters are usually restricted via a pre-specified matrix to reflect the diagnostic assumptions on the latent traits. In this talk, I will first give an introduction to such restricted latent class models, followed by discussions on key issues and challenges. I will then present some fundamental identifiability results and specify which types of the restriction matrices would ensure the estimability of the model parameters. These identifiability conditions not only lead to the consistency and asymptotic normality of the maximum likelihood estimators, but also provide a guideline for the related experimental design, whereas in the current applications the design is usually experience based and identifiability may not be guaranteed.

Drinfeld level structures are a key concept in the arithmetic study of the moduli of elliptic curves. They also play an important role in the moduli of 1 dimensional p-divisible groups, and related Shimura varieties studied by Harris and Taylor. I'll explain why Drinfeld level structures (and the related "full set of sections" defined by Katz and Mazur) are not adequate for studying more general Shimura varieties. I'll discuss two examples of a satisfying theory of level structure outside the Drinfeld case: i) full level structures on the group $\mu_p x \mu_p$; ii) $\Gamma_1(p^r)$-type level structures on an arbitrary p-divisible group (joint work with R. Kottwitz).

This is a talk for a general audience. Let $G$ be a complex Lie group and let $Q$ be a Stein manifold (closed complex submanifold of some $\Bbb C^n$). Suppose that $X$ and $Y$ are holomorphic principal $G$-bundles over $Q$ which admit an isomorphism $\Phi$ as topological principal $G$-bundles. Then the famous Oka principle of Grauert says that there is a homotopy $\Phi_t$ of topological isomorphisms of the principal $G$-bundles $X$ and $Y$ with $\Phi_0=\Phi$ and $\Phi_1$ biholomorphic. We prove generalizations of Grauert's Oka principle in the following situation: $G$ is reductive, $X$ and $Y$ are Stein $G$-manifolds whose (categorical) quotients are biholomorphic to the same Stein space $Q$.

We give an application to the Holomorphic Linearization Problem. Let $G$ act holomorphically on $\Bbb C^n$. When is there a biholomorphic map $\Phi\colon \Bbb C^n \to \Bbb C^n$ such that $\Phi^{-1} \circ g \circ \Phi \in \rm{GL}(n,C)$ for all $g \in G$? We describe a condition which is necessary and sufficient for ``most" $G$-actions.

This is joint work with F. Kutzschebauch and F. Larusson.

Let $K$ be a compact Lie group and $V$ a unitary $K$-module. Let $\mu\colon V\to \frak k^*$ be the associated moment mapping and let $M_0$ denote the quotient of $\mu^{-1}(0)$ by $K$. This is the (symplectic) quotient associated to the $K$-action. Now $K$ is a real algebraic subgroup of the unitary group of $V$ and its complex points are a complex reductive subgroup $G$ of $\rm{GL}(V)$. We recall the invariant theory quotient $V{/\!\!/} G$ associated to the $G$-action, and the fact that $V{/\!\!/} G$ is homeomorphic to $M_0$. This fact is enormously useful.

The simplest kinds of symplectic quotients are those of the form $W/H$ where $W$ is a unitary $H$-module and $H$ is finite. Let $\dim K>0$. For $K$-modules $V$ which are ``small,'' there are examples of isomorphisms of $M_0$ with some $W/H$. We show that for most $K$-modules, there can be no such isomorphism. We give necessary and sufficient conditions for such isomorphisms for $K=S^1$ and $K=\rm{SU}(2,\Bbb C)$.

This is joint work with H.-C. Herbig and C. Seaton.

Phase retrieval is a non-convex inverse problem arising in many practical applications, such as diffraction imaging and speech recognition. More precisely, we seek to recover a signal of interest from its intensity measurements with respect to some measurement frame. In practice, phaseless measurements with respect to a Gabor frame are relevant for many applications. We are going to describe the idea of a reconstruction algorithm for the case of Gabor measurements, and then show how geometric properties of the measurement frame, such as projective uniformity and flatness of the vector of frame coefficients, are related to the robustness of the presented algorithm.

The Circular Beta Ensemble is a family of random unitary matrices whose eigenvalue distribution plays an important role in statistical physics. The spectral measure is a canonical way of describing the unitary matrix that takes into account the full operator, not just its eigenvalues. When the matrix is infinitely large (i.e. an operator on some infinite-dimensional Hilbert space) the spectral measure is supported on a fractal set and has a rough geometry on all scales. This talk will describe the analysis of these fractal properties. Joint work with Raoul Normand and Balint Virag.

In 2012 it was conjectured by Emerton and Helm that the local Langlands correspondence for $GL(n)$ of a $p$-adic field should interpolate in $\ell$-adic families, where $\ell$ is a prime different from $p$. Recently, Helm showed that the conjecture follows from the existence of an appropriate map from the integral Bernstein center to a Galois deformation ring. In this talk we will present recent work (joint with David Helm) showing the existence of such a map and describing its image.

The Boolean Pythagorean Triples problem has been a long-standing open problem in Ramsey Theory: Can the set $N = {1,2,3,…}$ of natural numbers be partitioned into two parts, such that neither part contains a triple $(a, b, c)$ with $a^2 + b^2 = c^2$ ? A prize for the solution was offered by Ron Graham over two decades ago. We show that such a partition is possible for the set
of integers in $[1,7824]$, but that it is not possible for the set of integers in $[1,M]$ for any $M > 7824$. Of course, it is completely infeasible to attempt prove this directly by examining all $2^M$ possible partitions of $[1,M]$ when $M = 7825$, for example. We solve this problem by using the Cube-and-Conquer paradigm, a hybrid SAT method for hard problems, employing both look-ahead and CDCL solvers. An important role is played by dedicated look-ahead heuristics, which indeed allowed us to solve the problem on a cluster with 800 cores in about 2 days. Due to the general interest in this mathematical problem, our result requires a formal proof. Exploiting recent progress in unsatisfiability proofs of SAT solvers, we produced and verified a proof in the DRAT format, which is almost 200 terabytes in size. From this we extracted and made available a compressed certificate of 68 gigabytes, that allows anyone to reconstruct the DRAT proof for checking. These techniques show great promise for attacking a variety of similar computational problems arising in combinatorics and computer science.

Toric varieties are ubiquitous in algebraic geometry. They have a rich combinatorial structure, and give the simplest examples of log
Calabi-Yau varieties.

We give a simple criterion for characterizing when a log Calabi-Yau pair is toric, which answers a case of a conjecture of Shokurov.

This is joint work with James McKernan, Roberto Svaldi, and Runpu Zong.

Many recent results in the theory of measure preserving group actions and equivalence relations that do not {\it a priori} involve von Neumann algebras were obtained using von Neumann algebraic techniques. This talk will focus on further examples of this phenomenon in two directions. First, we study unique factorization for products of measure preserving equivalence relations $\mathcal{R}_1 \times \mathcal{R}_2 \times \cdots \times \mathcal{R}_k$. Second, we will discuss a joint work with Lewis Bowen and Adrian Ioana, which concerns properties of general non-amenable groups and equivalence relations that can deduced from properties of a prototypical non-amenable group $\mathbb{F}_2$, the free group on two generators.

In this talk I would like to discuss the global rigidity property for local holomorphic maps from an open piece of a Hermitian symmetric space $M$ into a Cartesian product of $M$. This study has been related to problems in number theory in classifying the modular correspondences, as initated by the work of Clozel-Ullmo. We will
discuss the work of Mok-Ng on the rigidity pehomenon when the map is local area preserving and $M$ is of non-compact type. We then focus on our recent joint work with H. Fang and M. Xiao when $M$ is of compact type.

Consider the defocusing quintic nonlinear Schrodinger equation on $R^3$ with initial data in the energy space. This problem is "energy-critical" in view of a certain scaling invariance, which is a main source of difficulty in the analysis of this equation. It is a nontrivial fact that all finite-energy solutions scatter to linear solutions. We show that this remains true under small compact deformations of the Euclidean metric.

A solution to the unnormalized Ricci flow equation is called immortal if it exists for all times $t > 0$. The asymptotic behavior of these solutions is in general much less understood than in the case of a finite time singularity. For instance, they might be collapsed, and they might also converge locally to non-gradient solitons, which cannot be detected using Perelman's entropy functional. In this talk, we will show that for immortal homogeneous solutions of arbitrary dimension and isometry group, the flow subconverges (after parabolic rescaling) to an expanding homogeneous Ricci soliton. We will also give further results in the special case of solvable Lie groups. This is joint work with Christoph Bohm.

Molecular transport and interaction are of fundamental importance in biology and medicine. The spatiotemporal diffusion map can reflect the regulation of molecular interactions and their intracellular functions. To construction subcellular diffusion maps based on bio-imaging data, we explore a general optimization framework with diffusion equation constraints (OPT-PDE) and finite element discretization to enable the solution of a spatiotemporal and anisotropic diffusion map. We characterize the wellposedness and convergence property of the OPT-PDE solver and demonstrate its applicability in re-covering heterogeneous and anisotropic diffusion maps in bio-imaging. Furthermore, to study molecular interactions based on live-cell imaging, we developed a correlative FRET imaging microscopy (CFIM) approach for the quantitatively analysis of subcellular coordination between the enzymatic activity and the structural focal adhesion (FA) dynamics. By CFIM, we found that different FA subpopulations have distinct regulation mechanisms controlled by local kinase activity. Therefore, our work highlights the importance of computational model-based analysis and its integration with bio-imaging.

This will be a panel discussion on the academic job search given by current 5th year graduate students. The panel will be interactive, so come prepared with questions!

A free spectrahedron D is the matricial solution set of a linear matrix inequality (LMI). Thus , for some positive integer g, D is a subset of the union, over n, of g-tuples of n by n matrices. Free spectrahedra arise naturally, in several contexts including model engineering problems and the the theory of operator systems and completely positive maps. We consider the problem of classifying, up to affine linear equivalence, free bianalytic mappings from one free spectrahedra D to another E. Under some irreducibility inspired hypotheses on D, there are few such maps, D must support an underlying algebra/module structure and the map itself has a particularly pleasing form arising from this algebra/module structure. The work is joint with Meric Augat, Bill Helton and Igor Klep.

Our variational implicit solvent model (VISM) improves upon the conventional implicit-solvent model by implementing a physical molecular surface that is generated from minimizing the free energy between the solute and the solvent. The free energy includes both the solute-solvent electrostatic and the van der Waals interactions, thus coupling the two types of interactions during the minimization process. Here we apply VISM to the calculation of the solvation free energy of realistic proteins. The VISM results are compared to both the conventional implicit-solvent and the molecular dynamics (MD) free energy perturbation results. It is shown that, without any parameter fitting, VISM solvation free energy is closer to that of MD comparing to the conventional implicit-solvent model.

I will review the quantization procedure of compact Kahler manifolds (geometric quantization), and I will describe a way to associate quantum states to certain submanifolds. If time permits, I will explain how to compare two such states and give semiclassical estimates when the two submanifolds are Lagrangians intersecting transversally.

Quasisymmetric functions and symmetric functions are important tools in algebraic combinatorics, especially in representation theory. On the other side of combinatorics, pattern avoidance is a subject in enumerative combinatorics which focus on counting the number of combinatorial structures avoiding a certain collection of patterns. It turns out that those two not-so-related subjects have interesting connections. In this talk, we will first introduce symmetric functions and, quasisymmetric functions, and discuss known results. Then, we will move on to basic definitions on pattern-avoiding permutations, as well as their known results. Finally, we will explore the relationship between the two subjects.

Random walks on groups and harmonic functions on groups are intimately related to stationary group actions, which are a generalization of measure preserving group actions. An important invariant of stationary group actions is their Furstenberg Entropy. The Furstenberg Entropy realization problem is the question of determining the range of possible entropy values realizable for a given random walk. The talk will include an introduction to this field, an overview of what (little) is known, and some new results.

Tensor, or multiarrays with $d>=3$ indices, are ubiquitous in modern applications, mainly due to data explosion. While matrices, $d=2$, are well understood and widely used, tensors pose theoretical and numerical challenges. Tensors also arise naturally in quantum physics, when dealing with d-particle systems. In this talk, for general mathematical audience, we will describe several fundamental results and problems in tensors: tensor ranks, low rank approximation of tensors, spectral and nuclear norm of tensors, and their relation to the entanglement and nonseparability in quantum information theory.

I'll start by explaining enough of the theory of supermanifolds to show that the De Rham complex of an ordinary manifold should most naturally be interpreted as the ring of functions on its odd tangent bundle. (Just this fact, to me, is enough to justify taking supergeometry very seriously!)

I'll then try to talk about some symplectic aspects of supergeometry - for example the quantization of odd symplectic vector spaces.

The first Weyl algebra $A = k\langle x,y \rangle/(xy - yx - 1)$ is a well-studied noncommutative $\mathbb {Z}$-graded ring. Generalized Weyl algebras, introduced by Bavula, are a class of noncommutative $\mathbb {Z}$-graded rings which generalize the Weyl algebra. In this talk, we investigate the category of graded modules over certain generalized Weyl algebras and construct commutative rings with equivalent graded module categories. Along the way, we will learn about graded rings, noncommutative projective schemes, and how to do geometry without a geometric space.

We prove global well-posedness of the $(d+1)$-dimensional $(d\geq 4)$ massless Maxwell-Dirac equation in Coulomb gauge for data with small scale-critical Sobolev norm. A key step is to exploit (and justify) a deep analogy between Maxwell-Dirac and Maxwell-Klein-Gordon (for which an analogous result was proved earlier by Krieger-Sterbenz-Tataru), which says that the most difficult part of Maxwell-Dirac takes essentially the same form as Maxwell-Klein-Gordon. This is a joint work with C. Gavrus.

A new class of derivative-free optimization algorithms is developed to solve, with remarkable efficiency, a range of practical nonconvex optimization problems whose function evaluations are computationally (or experimentally) expensive. These new algorithms, which are provably convergent under the appropriate assumptions, are response surface methods which iteratively minimize metrics based on an interpolation of existing datapoints and a synthetic model of the uncertainty of this interpolant, which itself is built on the framework of a Delaunay triangulation over the existing datapoints. Unlike other response surface methods, our algorithms can employ any well-behaved interpolation strategy.

We read misspelled words all the time, yet somehow we often manage to make sense of them. Coding theory deals with the design of error-correcting codes that do precisely this: take corrupted data and recover the original message. In this talk, we'll go on an adventure through the basics of codes, the types of errors that can be introduced, and some cool techniques to correct them. My talk will be mostly accurate, and by the end of it you should be able to correct any mistakes I've made.

Over the past 25 years, catastrophe analytics has made a remarkable imprint on the insurance and greater financial markets. By applying probabilistic modeling practices to an ever-advancing technological ecosystem, catastrophe modelers can provide the means to efficiently trade risk, which facilitates an acceleration of capital into risk-bearing markets. The study of Hurricane Andrew, Northridge Earthquake, 9/11, Hurricane Katrina, Joplin Tornado and other recent disasters both inform model calibration and steer the economy toward increased resilience in an uncertain world. Graduates in mathematics are currently in high-demand in this industry to fill positions in what is a young and growing profession.

In 2013, Kobayashi proved an analogue of Perrin-Riou's p-adic Gross-Zagier formula for supersingular primes. In this talk, we will explain an extension of Kobayashi's result to the Lambda-adic setting. The main formula is in terms of plus/minus Heegner points up the anticyclotomic tower, and its proof, rather than on calculations inspired by the original work of Gross-Zagier, is via Iwasawa theory, based on the connection between Heegner points, Beilinson-Flach elements, and different p-adic L-functions. (Joint work in progress with Xin Wan.)

In recent years, there have been major breakthroughs on the topic of super-approximation for algebraic groups, which is a qualitative version of strong approximation. Super-approximation has proven to be incredibly useful in many areas of both pure and applied Mathematics. We discuss the difficulties of super-approximation in positive characteristic, as well as recent new results for absolutely almost simple groups over $k(t)$, where $k$ is a finite field.

We study the spectrum of complete non-compact manifolds with bounded curvature and positive injectivity radius. We give general conditions which imply that their essential spectrum has an arbitrarily large finite number of gaps. As applications, we construct metrics with an arbitrarily large finite number of gaps in its essential spectrum on non-compact covering of a compact manifold and complete non-compact manifold with bounded curvature and positive injectivity radius.This is a joint work with Richard Schoen.

In this talk I will present a recent proof of a conjecture of C. Villani, namely the exponential convergence of solutions of spatially inhomogeneous Boitzmann equations, with hard sphere potentials, to some equilibriums, called Maxwellians.

Uncertainty quantification seeks to provide a quantitative means to understand complex systems that are impacted by uncertainty in their parameters. The polynomial chaos method is a computational approach to solve stochastic partial differential equations (SPDE) by projecting the solution onto a space of orthogonal polynomials of the stochastic variables and solving for the deterministic coefficients. Polynomial chaos can be more efficient than Monte Carlo methods when the number of stochastic variables is low, and the integration time is not too large. When performing long-term integration, however, achieving accurate solutions often requires the space of polynomial functions to become unacceptably large. This talk will introduce alternative approach, where sets of empirical basis functions are constructing by examining the behavior of the solution for fixed values of the random variables. The empirical basis functions are evolved over time, which means that the total number of basis functions can be kept small, even when performing long-term integration.

We will prove the Cayley-Hamilton theorem for commutative rings in an elementary/remarkable way. This is an exposition of a paper by Howard Straubing.

I will talk about the conjecture of Hausel, Letellier and Villegas, which gives precise predictions for mixed Hodge polynomials of character varieties. In certain specializations this conjecture also computes Hurwitz numbers, Kac's polynomials of quiver varieties, and zeta functions of moduli spaces of Higgs bundles. The subject is at an exciting intersection of number theory, algebraic geometry, combinatorics and mathematical physics, and is an area of active research. The talk is deemed as an introduction for a general audience. If time permits, I will explain my recent results in this area.

K3 surfaces are two dimensional Calabi-Yau manifolds. Their moduli space is
of interest in algebraic geometry, but also has connections with number
theory and string theory. I will discuss ongoing joint work with Alina
Marian and Rahul Pandharipande aimed at studying the tautological ring of
the moduli space of K3 surfaces. In particular, I will discuss different
notions of tautological classes. Next, I will explain a method of deriving
relations between tautological classes via the geometry of the relative
Quot scheme.

The proof of Fermat's Last Theorem established a deep relation
between elliptic curves and modular forms, mediated by an equality of
L-functions (which are analogous to the Riemann zeta function). The
common ground is Galois representations, and Wiles' overall strategy
was to parametrize their deformations via algebras of Hecke
operators. In higher rank the global Langlands conjecture posits a
correspondence between n-dimensional Galois representations arising
from the cohomology of algebraic varieties and certain so-called
automorphic representations of $GL(n)$, which belong in the realm of
harmonic analysis. There is a known analogue over local fields (such
as the p-adic numbers $Q_p$) and one of the key desiderata is
local-global compatibility. This naturally leads one to speculate
about the existence of a finer "p-adic" version of the local Langlands
correspondence which should somehow be built from a "mod p" version
through deformation theory. Over the last decade this picture has been
completed for $GL(2)$ over $Q_p$, and extending it to other groups is a
very active research area. In my talk I will try to motivate these
ideas, and eventually focus on deformations of smooth representations
of $GL(n)$ over $Q_p$ (or any p-adic reductive group). It seems to be an
open problem whether universal deformation rings are Noetherian in
this context. At the end we report on progress in this direction
(joint with Julien Hauseux and Tobias Schmidt). The talk only assumes
familiarity with basic notions in algebraic number theory.

This presentation will focus on (1) teaching positions in community colleges and (2) tenure-track faculty positions in universities. The tenure system (as practiced at UCSD) will also be explained.

BONUS: A guest has been invited to talk firsthand about the experience of recently becoming a tenure-track faculty member at a research university.

We classify reflection groups in riemannian manifolds with non negative sectional curvature. This is a joint work with Karsten Grove.

Nonlinear constrained optimization problem can be effectively solved by
minimizing a sequence of unconstrained or linearly constrained
subproblems, where the augmented Lagrangian function plays a vital role.
This talk introduces a generalized Hestenes-Powell augmented Lagrangian
function, which can be seen as a continuum of many well-known methods as
specific cases. A new primal dual sequential quadratic programming (pdSQP)
method will be given for minimizing the given augmented Lagrangian

The symmetric function operator, nabla, introduced by Bergeron and Garsia (1999), has many astounding combinatorial properties. The (recently proven) Shuffle Conjecture of Haglund, Haiman, Loehr, Remmel, and Ulyanov (2005) relates nabla en to parking functions. The rational Compositional Shuffle Conjecture of the author, Bergeron, Garsia, and Xin (2015) relates a whole family of operators (closely linked to nabla) to rational parking functions. In (2007), Loehr and Warrington conjectured a relationship between nabla pn and preference functions. We prove this conjecture and provide another combinatorial interpretation in terms of parking functions. This new formula reveals a connection between nabla pn and an operator appearing in the rational Compositional Shuffle Conjecture at $t = 1/q$.

The Gaussian distribution---which may be the center of a majority of statistical applications and research---will be introduced. We will focus on an interesting phenomenon on the maximum of a sequence of Gaussian variables, and a direct application to human genome research will be displayed.

In 2008, Oded Schramm asked the following question: For what values of k and q does there exist a stationary, k-dependent sequence of random variables with values in {1,2,...,q} assigning different values to consecutive integers? Schramm proved a number of results related to this question, and speculated about what the answer might be in general. As it turns out, the truth is quite different from his informed guess. This is joint work with A. E. Holroyd.

Given a graph, perform the following randomized coloring procedure to it: pick an edge at random and with probability $p$ color the end vertices of that edge blue; otherwise color them red. Repeat indefinitely, choosing the edges with replacement. This induces a random walk in the space of all red/blue colorings of the graph, which converges to a stationary distribution. In this talk we will study this stationary distribution for the complete graph with the goal of getting Jay a PhD.

Calcium is critical to a wide range of physiological processes, including neurological function, immune responses, and muscle contraction. Calcium-dependent signaling pathways enlist a variety of proteins and channels that must rapidly and selectively bind calcium against thousand-fold higher cationic concentrations. Frequently these pathways further require the co-localization of these proteins within specialized subcellular structures to function properly. Our lab has developed multi-scale simulation tools to elucidate how protein structure and co-localization facilitate intracellular calcium signaling. Developments include combining molecular simulations with a statistical mechanical model of ion binding, a homogenization theory to upscale molecular interactions into micron-scale diffusion models, and reaction-diffusion simulations that leverage sub-micron microscopy data. In this seminar, I will describe these tools and their applications toward molecular mechanisms of calcium-selective recognition and cross-talk between co-localized calcium binding proteins inside the cell.

Schreieder recently introduced a class of smooth projective varieties that have unexpected Hodge numbers. We investigate the geometry of these varieties and show how, in dimension two, these varieties may in fact be realized as elliptic modular surfaces.

Semitoric integrable systems, or semitoric manifiolds, have been completely classified in terms of five symplectic invariants, but there are properties of these systems which, while encoded in the invariants, are not easy to extract. In this talk I introduce a new invariant of semitoric systems which we call a semitoric helix and I outline its constrction. Using this invariant we are able to see clearly the effect of a semitoric blowdown/blowup on the system. We are then able to completely classify minimal semitoric manifolds; those that do not admit a semitoric blowdown. This work is joint with Alvaro Pelayo and Daniel M. Kane.

Finite element methods are numerical methods that approximate solutions to
PDEs using piecewise polynomials on a mesh representing the problem
domain. Adaptive finite element methods are a class of finite element
methods that selectively refine specific elements in the mesh based on
their predicted error. In order to establish the viability of an AFEM, it
is essential to know whether or not that method can be proven to converge.
In this talk I will present a general framework that would establish
convergence for an AFEM and apply the framework to specific problems.

Special values of zeta functions have been around for a very long time, and yet are still quite mysterious. Once the basic theory of the Riemann zeta function has been established, I will introduce the Dedekind zeta function along with ties between special values and algebraic K-theory.

Processor sharing is a mathematical idealization of round-robin scheduling algorithms commonly used in computer time-sharing. It is a fundamental example of a non-head-of-the-line service discipline. For such disciplines, it is typical that any Markov description of the system state is infinite dimensional. Due to this, measure-valued stochastic processes are becoming a key tool used in the modeling and analysis of stochastic network models operating under various non-head-of-the-line service disciplines.

In this talk, we discuss a new approach to studying the asymptotic behavior of fluid model solutions (formal functional law of large numbers limits) for critically loaded processor sharing queues. For this, we introduce a notion of relative entropy associated with measure-valued fluid model solutions. This approach is developed with idea that similar notions involving relative entropy may be helpful for understanding the asymptotic behavior of critical fluid model solutions for stochastic networks operating under protocols naturally described by measure-valued processes.

We show that the $\phi$-Selmer ranks of twists of an elliptic curve $E$ with a point of order two are distributed like the ranks of random groups in a manner consistent with the philosophy underlying the Cohen-Lenstra heuristics.

If $E$ has a point of order two, then the distribution of $dim_{\mathbb{F}_2} \operatorname{Sel}_\phi{(E^d/{\mathbb{Q}})} - dim_{\mathbb{F}_2} \operatorname{Sel}_{\hat\phi}{(E^{\prime d}/{\mathbb{Q}})}$ tends to the discrete normal distribution $\mathcal{N}(0,\frac{1}{2} \log \log X)$ as $X \rightarrow \infty$. We consider the distribution of $dim_{\mathbb{F}_2} \operatorname{Sel}_\phi{(E^d/{\mathbb{Q}})} - dim_{\mathbb{F}_2} \operatorname{Sel}_{\hat\phi}{(E^{\prime d}/{\mathbb{Q}})}$ has a fixed value $u$.

We show that for every $r$, the limiting probability that $dim_{\mathbb{F}_2} \operatorname{Sel}_\phi{(E^d/{\mathbb{Q}})}= r$ is given by an explicit constant $\alpha_{r,u}$ introduced in Cohen and Lenstra's original work on the distribution of class groups.

Symmetry, and the study of invariant and equivariant objects, is a deep and unifying principle underlying a variety of mathematical fields. In particular, geometric mechanics is characterized by the application of symmetry and differential geometric techniques to Lagrangian and Hamiltonian mechanics, and geometric integration is concerned with the construction of numerical methods with geometric invariant and equivariant properties. Computational geometric mechanics blends these fields, and uses a self-consistent discretization of geometry and mechanics to systematically construct geometric structure-preserving numerical schemes.

In this talk, we will introduce a systematic method of constructing geometric integrators based on a discrete Hamilton's variational principle. This involves the construction of discrete Lagrangians that approximate Jacobi's solution to the Hamilton-Jacobi equation. Jacobi's solution can be characterized either in terms of a boundary-value problem or variationally, and these lead to shooting-based variational integrators and Galerkin variational integrators, respectively. We prove that the resulting variational integrator is order-optimal, and when spectral basis elements are used in the Galerkin formulation, one obtains geometrically convergent variational integrators.

We will also introduce the notion of a boundary Lagrangian, which is analogue of Jacobi's solution in the setting of Lagrangian PDEs. This provides the basis for developing a theory of variational error analysis for multisymplectic discretizations of Lagrangian PDEs. Equivariant approximation spaces will play an important role in the construction of geometric integrators that exhibit multimomentum conservation properties, and we will describe two approaches based on spacetime generalizations of Finite-Element Exterior Calculus, and Geodesic Finite-Elements on the space of Lorentzian metrics.

Originally, groups emerged in mathematics as structures describing the symmetries of all kinds of objects. Nowadays we of course also have an abstract definition of a group. It is this abstract description that recently has been exploited to generalize things to the notion of so-called quantum groups. Once a good definition of a quantum group was established, people tried to reverse the process and see quantum groups as describing “symmetries”of various objects.

In this talk we will 1) try to understand the definition of a (compact) quantum group, 2) discuss several examples, and 3) see how we can define quantum symmetry groups via actions of quantum groups on various spaces.

Everybody is welcome! Necessary background beyond undergraduate analysis and algebra will be provided during the talk.

Perfectoid spaces were introduced to provide a geometric framework to the field of norms isomorphism from $p$-adic Hodge Theory, however, have proven their value well beyond this old result. For this reason, the geometry of perfectoid spaces is worth
studying, for its own sake. In this talk, we explicitly compute the Picard group for the projectivoid line.

With the desire to generalize this result to other perfectoid spaces, as well as to classify higher-rank vector bundles on these $p$-adic analytic spaces, we ask whether an appropriate GAGA Theorem holds for perfections of proper schemes over a base nonarchimedean field of characteristic $p >0$.

The log-Sobolev inequality (LSI) is a very useful tool for analyzing high-dimensional situations. For example, the LSI can be used for deriving hydrodynamic limits, for estimating the error in stochastic homogenization, for deducing upper bounds on the mixing times of Markov chains, and even in the proof of the Poincaré conjecture by Perelman. For most applications, it is crucial that the constant in the LSI is uniform in the size of the underlying system. In this talk, we discuss when to expect a uniform LSI in the setting of unbounded spin systems.

Accurate simulations of elasticity properties can be constructed by solving second order elliptic boundary value problems which have been approximated using finite elements. This talk will examine the process of converting the given PDE into a weaker form and applying the Galerkin method. In addition, novel MATLAB programs will be introduced, which will display a visual depiction of an object after force is applied, given a subdivision of the shape into regular or irregular triangles, a Dirichlet boundary condition, and a two dimensional force function.

While the sequence of primes is very well distributed in the reduced residue classes (mod q), the distribution of pairs of
consecutive primes among the permissible pairs of reduced residue classes (mod q) is surprisingly erratic. We propose a conjectural explanation for this phenomenon, based on the Hardy-Littlewood conjectures, which fits the observed data very well. We also study the distribution of the terms predicted by the conjecture, which proves to be surprisingly subtle. This is joint work with Kannan Soundararajan.

A tensor is a vector that lies in a tensor product of vector spaces. The rank of a tensor T is the smallest integer r such that T can be written as a sum of r pure tensors. Finding such low rank decompositions of a tensor T is known as the PARAFAC model. This model has many applications: Algebraic Complexity Theory, Chemometrics, Neuroscience, Signal Processing to name a few. An alternative is the Convex Decomposition (CoDe) model. It uses the nuclear norm of tensors and is more numerically stable. We will discuss upper and lower bound for the rank and nuclear norm of some tensors of interest, and some applications.

Supercurves are the simplest class of complex supermanifolds, “one-dimensional” in some sense and thus analogs of Riemann surfaces. I will describe a remarkable duality between pairs of supercurves that generalizes Serre duality for Riemann surfaces. Self-dual supercurves are precisely the “super Riemann surfaces” introduced by physicists in connection with string theory. I’ll suggest connections between this duality and the classical duality between points and hyperplanes in projective space. No prior knowledge of supergeometry is required.
(This is a continuation of the talk I gave earlier this quarter, and will begin with a review of that talk.)

During the last 20 years, it was realized that certain combinatorial objects (combinatorial Hopf algebras, to be precise) underly many mathematical theories. The talk will survey some of these developments, and then focus largely on two of the most emblematic and universal such objects, namely the higher algebraic structures that can be constructed out of permutations, and out of surjections.

The link is made through the notion of special poset (equivalent to labelled poset of Stanley): linear extensions of a poset can be seen as bijections, while the generating function of a poset P with respect to Stanley's classical definition of P-partitions associated to a special poset is a quasi-symmetric: in fact, it is a homomorphism between the Hopf algebra of labelled posets and that of quasi-symmetric functions; while linearisation is a homomorphism onto the Hopf algebra of permutations.

The aim is to generalize this frame to preorders, which are in one-to-one correspondence with finite topologies: the objects corresponding to bijections are surjections: they can be seen as linear extensions of a preorder and are encoded by packed words. We can hence define the notion of T-partitions associated to a finite topology T, and deduce a Hopf algebra morphism from a new Hopf algebra on topologies to the Hopf algebra of packed words.

This is joint work with L. Foissy and F. Patras.

Crystals are colored directed graphs encoding information about Lie algebra representations. Kirillov-Reshetikhin (KR) crystals correspond to certain finite-dimensional representations of affine Lie algebras. I will present a combinatorial model which realizes tensor products of (column shape) KR crystals uniformly across affine types. Some computational applications are discussed. A corollary states that the Macdonald polynomials (which generalize the irreducible characters of semisimple Lie algebras), upon a certain specialization, coincide with the graded characters of tensor products of KR modules. The talk is largely self-contained, and is based on a series of papers with A. Lubovsky, S. Naito, D. Sagaki, A. Schilling, and M. Shimozono.

Explicit constructions of Einstein metrics and various generalizations have long been a central problem in differential geometry. I will present a unified description, in the toric framework, for constructing the Page Einstein metric, the Koiso-Cao Ricci soliton, and the Lu-Page-Pope quasi- Einstein metrics on the first del Pezzo surface. The existence of quasi-Einstein metrics on toric Fano manifolds is in general open; our constructions yield some insight into the potential form of such metrics. Numerical evidence for the existence of such metrics on the second del Pezzo surface will also be outlined, as well as future plans to construct a rigorous proof. This is all joint with S. Hall, and partly joint with W. Bataat and A. Jizany.

We will prove that an Einstein manifold with positive isotropic curvature has constant sectional curvature.

This study revisits the Canadian Traveller Problem (CTP), which finds applications to dynamic navigation systems. Given a road network $G=(V,E)$ with a source $s$ and a destination $t$ in $V$, a traveller knows the entire network in advance, and wishes to travel as quickly as possible from $s$ to $t$, but discovers online that some roads are blocked (e.g., by snow or accidents) once reaching them. The objective is to derive an adaptive strategy so that its competitive ratio, which compares the distance traversed with that of the static shortest $s,t$-path in hindsight, is minimized. This problem was initiated by Papadimitriou and Yannakakis in 1991. They proved that it is PSPACE-complete to obtain an algorithm with a bounded competitive ratio. Furthermore, if at most $k$ roads can be blocked, then the optimal competitive ratio for a deterministic online algorithm is $2k+1$, while the only randomized result known is a lower bound of $k+1$.

In this study, we show for the first time that a polynomial time randomized algorithm can beat the best deterministic algorithms, surpassing the $2k+1$ lower bound by an $o(1)$ factor. Moreover, we prove the randomized algorithm achieving a better $(1.7k +1)$-competitive ratio in pseudo-polynomial time.

We will present a version of Bony's strong maximum principle for degenerate PDEs. We will mainly discuss the proof and its applications in studying rigidity problems in geometry.

The talk mainly concerns the structure at infinity for complete gradient shrinking Ricci solitons. We show that for such a soliton with bounded curvature, if the round cylinder $$R\times S^{n-1}/\Gamma$$ occurs as a limit for a sequence of points going to infinity along an end, then the end is asymptotic to the same round cylinder at infinity.

A basic result from complex function theory states that the logarithmic residue (i.e., the contour integral of the logarithmic derivative) of a scalar analytic function can only vanish when the function has no zeros inside the contour. Question: does this result generalize to the vector-valued case?

Assuming that the functions in question take values in a Banach algebra, the answer depends on which Banach algebra. Positive results have been obtained for large classes of algebras, among them that of the polynomial identity Banach algebras. Instrumental in this context is what is called non-commutative Gelfand theory involving the use of families of matrix representations.

There is a close connection between logarithmic residues and sums of idempotents. Pursuing this connection, negative answers to the above question have come up via the construction of non-trivial zero sums of a finite number of idempotents. It is intriguing that only five idempotents are needed in all known examples. The idempotent constructions relate to deep problems concerning the geometry of Banach spaces and general topology. In particular a novel approach to the construction of Cantor type sets plays a role.

The talk - accessible to non-specialists - reports on joint work with Torsten Ehrhardt (Santa Cruz, California) and Bernd Silbermann (Chemnitz, Germany).

The talk mainly concerns the structure at infinity for complete gradient shrinking Ricci solitons. We show that for such a soliton with bounded curvature, if the round cylinder $$R\times S^{n-1}/\Gamma$$ occurs as a limit for a sequence of points going to infinity along an end, then the end is asymptotic to the same round cylinder at infinity.

The talk mainly concerns the structure at infinity for complete gradient shrinking Ricci solitons. We show that for such a soliton with bounded curvature, if the round cylinder $R\times S^{n-1}/\Gamma$ occurs as a limit for a sequence of points going to infinity along an end, then the end is asymptotic to the same round cylinder at infinity.

The problem says that if $M$ is a smooth complete embedded self-shrinker with polynomial volume growth in Euclidean space and the squared norm of the second fundamental form $|A|^2 =$ constant, then $M$ is a generalized cylinder. It has been verified in dimension 2 without the assumption of polynomial volume growth. Cao and Li had proved if $M$ is an n-dimensional complete self-shrinker with polynomial volume growth in $R^n+q$, and if $|A|^2 \leq 1$, then $M$ is must be one of the generalize cylinders. But for the case $|A|^2 >1$, they don't know what it is. Therefore, Qingming Cheng and Guoxin Wei proved if the squared norm of the second fundamental form $|A|^2$ is constant and $|A|^2 \leq 10/7$,
then $M$ is must be one of the generalize cylinders. So we guess that it may be true if the squared norm of the second fundamental form $|A|^2$ is constant.

The problem says that if $M$ is a smooth complete embedded self-shrinker with polynomial volume growth in Euclidean space and the squared norm of the second fundamental form $|A|^2 =$ constant, then $M$ is a generalized cylinder. It has been verified in dimension 2 without the assumption of polynomial volume growth. Cao and Li had proved if $M$ is an n-dimensional complete self-shrinker with polynomial volume growth in $R^n+q$, and if $|A|^2 \leq 1$, then $M$ is must be one of the generalize cylinders. But for the case $|A|^2 >1$, they don't know what it is. Therefore, Qingming Cheng and Guoxin Wei proved if the squared norm of the second fundamental form $|A|^2$ is constant and $|A|^2 \leq 10/7$,
then $M$ is must be one of the generalize cylinders. So we guess that it may be true if the squared norm of the second fundamental form $|A|^2$ is constant.

This will be a continuation of the talk given on August 26th.

I will discuss recent joint work with Ben Elias, in which we develop a combinatorial method for computing the triply graded homology (equivalently, the superpolynomial) of certain links, especially torus links. Conjecturally, this link homology has deep connections with Hilbert schemes and symmetric functions. For instance the superpolynomial of the $(n,nk)$ torus link is conjecturally an evaluation of $nabla^k (p_1)^n$. I will discuss some evidence for this conjecture, including recent work of Andy Wilson.

In this talk I will try to prove De Giorgi’s Theorem on the regularity of minimal Caccioppoli Sets(Theorem 8.4). He used a measure theoretic method closely related to properties of function of bounded variation, which is technical and powerful. I will focus on Chapter 5-8 in E. Giusti’s book. It is based on M. Miranda’s simplification of De Giorgi’s original proof.

In this talk I will try to prove De Giorgi’s Theorem on the regularity of minimal Caccioppoli Sets(Theorem 8.4). He used a measure theoretic method closely related to properties of function of bounded variation, which is technical and powerful. I will focus on Chapter 5-8 in E. Giusti’s book. It is based on M. Miranda’s simplification of De Giorgi’s original proof.

This will be a continuation of the talk given on September 9th.

Given two actions of a countable, discrete group $G$ on probabilty space $X,Y$ there is a notion of when the action on $X$ is weakly contained in the action on $Y$ (analogous to weak containment of representations) due to Kechris: it roughly says that any finitary piece of the action of $G$ on $X$ can be approximated by some finitary piece of $G$ on $Y$ (equivalent the measure on $X$ is a weak* limit of the factors of the measure on $Y$). We then say that two actions are weakly equivalent when each is weakly contained in the other. We study when algebraic actions of $G$ (i.e. an action by automorphisms on a compact, metrizable, abelian group) are weakly equivalent to Bernoulli shifts and find a natural class of actions related to invertible convolution operators on $G$. As part of our work, we also give conditions under which such actions are free.

(joint with Asif Shakeel, Jiri Lebl & David Meyer)

The content of my previous ''Fun with gerbes" talk (of which the
current talk is wholly independent) was etale homotopy 2-types via
Grothendieck's theory of champs (the translation stack is awful), and the
content of this talk will be applications to Lefshetz theorems. A
particular feature of having the right definition of higher homotopy
groups via n-categories is that the Lefschetz theorem becomes a more or
less tautological induction, and at the very least I'll explain the
initial $\pi_0$ step in the induction- typically all problems in this area
are, when correctly understood, are problems about $\pi_0$. A manuscript
containingi the material from the "Fun with gerbes" series is available
here: http://arxiv.org/abs/1507.00797

The Holy Grail of Artificial Intelligence, true “deep” AI, has been 10 years away ever since the Dartmouth Conference in 1956, in the same way that fusion reactors have been 20 years out for the past 50 years. In the meantime, we’ve gone through 2 “AI Winters” and 3 “AI Summers,” as the expectations of investors get lowered to meet the rising actual capabilities of neural nets. The talk will be a brisk tour through the history of neural networks, with particular emphasis on the intuition of how neural networks work.

We will discuss the Makeenko--Migdal equation (MM equation) which relates variations of a "Wilson loop functional" (relative to the Euclidean Yang--Mills measure) in the neighborhood of a simple crossing to the associated Wilson loops on either side of the crossing. We will begin by introducing the 2d -- Yang-Mills measure and explaining the necessary background in order to understand the theorem. The goal is to describe the original heuristic argument of Makeenko and Migdal and then explain how these arguments can be made rigorous using stochastic calculus.

We discuss traveling front solutions $u(t,x) = U(x-ct)$ of reaction-diffusion equations $u_t = Lu + f(u)$ in 1d with ignition reactions $f$ and diffusion operators $L$ generated by symmetric Levy processes $X_t$. Existence and uniqueness of fronts are well-known in the cases of classical diffusion (i.e., when L is the Laplacian) as well as some non-local diffusion operators. We extend these results to general Levy operators, showing that a weak diffusivity in the underlying process - in the sense that the first moment of $X_1$ is finite - gives rise to a unique (up to translation) traveling front. We also prove that our result is sharp, showing that no traveling front exists when the first moment of $X_1$ is infinite.

Crouzeix's conjecture is among the most intriguing developments in matrix theory in recent years.
Made in 2004 by Michel Crouzeix, it postulates that, for any polynomial $p$ and any matrix $A$,
$||p(A)|| <= 2 max(|p(z)|: z$ in $W(A))$, where the norm is the 2-norm and $W(A)$ is the field
of values (numerical range) of $A$, that is the set of points attained by $v*Av$ for some
vector $v$ of unit length. Remarkably, Crouzeix proved in 2007 that the inequality above
holds if 2 is replaced by 11.08. Furthermore, it is known that the conjecture holds in a
number of special cases, including $n=2$. We use nonsmooth optimization to investigate
the conjecture numerically by attempting to minimize the “Crouzeix ratio”, defined as the
quotient with numerator the right-hand side and denominator the left-hand side of the
conjectured inequality. We present numerical results that lead to some theorems and
further conjectures, including variational analysis of the Crouzeix ratio at conjectured global minimizers.
All the computations strongly support the truth of Crouzeix’s conjecture.

This is joint work with Anne Greenbaum and Adrian Lewis.

Work by McQuillan and Brunella demonstrates the existence of a
Mori theory for rank 1 foliations on surfaces. In this talk we will
discuss an extension of some of these results to the case of rank 2
foliations on threefolds, as well as indicating how a complete Mori theory
could be developed in this case.

We will discuss Lefschetz theorems on the homotopy groups of
hyperplane sections in the arithmetic setting, i.e. for a divisor, ample
in the Arakelov sense, over a projective scheme defined over the ring of
integers in a number field. An interesting corollary is that the integral
model of a generic complete intersection curve of big height, is a simply
connected arithmetic surface. Joint work with Michael McQuillan.

In 1942 Kelly conjectured that any graph having at least 3 vertices is uniquely determined by the multiset of all its subgraphs obtained by deleting a vertex and all edges adjacent to it. In 1964 Harary conjectured analogously that any graph having at least 4 edges is uniquely determined by all its subgraphs obtained by deleting a single edge, which is known as the edge reconstruction conjecture. As of today, both conjectures are still open.

In the talk I will discuss some of the classical results about the conjectures and some evidence in favor of them. Also I will explicitly show that the edge reconstruction conjecture holds for a specific type of graphs.

In this talk, we compare and contrast a few finite element h-adaptive
and hp-adaptive algorithms. We test these schemes on three example PDE
problems and we utilize and evaluate an a posteriori error estimate.

In the process, we introduce a new framework to
study adaptive algorithms and a posteriori error estimators. Our innovative
environment begins with a solution u and then uses interpolation to
simulate solving a corresponding PDE. As a result, we always know the
exact error and we avoid the noise associated with solving.

Using an effort indicator, we evaluate the relationship between accuracy
and computational work. We report the order of convergence of different
approaches. And we evaluate the accuracy and effectiveness of an
a posteriori error estimator.

Differential Harnack estimates, also called Li-Yau-Hamilton estimates, play an important part in geometric analysis, especially geometric flows. Firstly, I will review the developments of differential Harnack estimates. Then I will talk about the Constrained matrix differential Harnack estimates for the heat equation on Kaehler Manifolds.

In a series of papers, E. Hecke described the representation of the group
$SL(2,p)$ on the regular differentials of the modular curve $X$ of level $p$. This was one of the
first applications of character theory outside of finite group theory, and one of the first
constructions of representations using cohomology. I will review Hecke's results, and
interpret them in the modern language of automorphic representations.

Topological data analysis (TDA) emerged as a branch of applied topology about twenty years ago and has produced some of the most valuable tools for the study of Big Data since then. TDA can be used to detect topologically significant features (holes, connected components, etc) of high-dimensional data sets, without any a priori knowledge of the data. It’s even been used to analyze basketball games, and found that basketball players displayed thirteen statistically distinct playing styles, despite there being only five official positions.

In this talk I will introduce the basics of persistent homology, a fundamental tool for TDA, and describe some of the recent achievements in this field. I'll also touch on some current areas of active research and why people like NASA are trying to get in on the action.

While the long-time behavior of small amplitude solutions to nonlinear dispersive and wave equations on Euclidean spaces $(R^n)$ is relatively well-understood, the situation is marked different on bounded domains. Due to the absence of dispersive decay, a very rich set of dynamics can be witnessed even starting from arbitrary small initial data. In particular, the dynamics in this setting is characterized by out-of-equilibrium behavior, in the sense that solutions typically do not exhibit long-time stability near equilibrium configurations. This is even true for equations posed on very large domains (e.g. water waves on the ocean) where the equation exhibits very different behaviors at various time-scales.

In this talk, we shall consider the nonlinear Schr\"odinger equation posed on a large box of size $L$. We will analyze the various dynamics exhibited by this equation when $L$ is very large

Initial data in general relativity must satisfy certain underdetermined differential equations called the constraint equations. A natural problem is to find a parameterization of all possible initial data. A standard method for this is called the conformal method. In the relatively simple ``constant mean curvature" (CMC) case, this method provides a good parameterization of initial data. However, the far-from-CMC case has resisted analysis. In part this is because researchers were trying to prove theorems that are false. In this talk, I'll introduce the problem and known results, and talk about our numerical results that show that the standard conjectures about solvability were all wrong. Numerical investigations can play an important part in informing conjectures about purely analytical questions.

Research mathematics is about asking questions. When's the last time you did? Bring your cell phone or computer, and be prepared to change the way you approach everything in mathematics, and life.

Mori introduced a program for classifying projective varieties by using algebraic surgeries, Jian Song and Gang Tian
introduced Analytical Minimal Model Program for the classification of Kahler varieties by using PDE surgeries. For the intermediate
Kodaira dimension they proved that there exists a unique generalized Kahler-Einstein metric which twisted with Weil-Petersson metric.
I extended their result in my PhD thesis on pair $(X,D)$ where $D$ is a snc divisor with conic singularities and I showed that there exists a generalized Kahler-Einstein metric which twisted with logarithmic Weil-Petersson metric plus additional term which we can find such additional term by using higher canonical bundle formula of Fujino and Mori. Moreover I extended Song-Tian program for Sasakian varieties in my PhD thesis. In fact when the basic first Chern class of a
Sasakian variety is not definite then the question is how can we find generalized Kahler-Einstein metric for such varieties. I gave a positive answer to this question in my thesis. Moreover I will explain how the Lei Ni method which later improved by V.Tosatti for the classification of the solution of Kahler-Ricci flow could be extended to conical Kahler-Ricci flow and I finally will explain how the classification of the solutions of relative Kahler-Ricci flow is related to the Gromov invariant of Ruan-Tian.

In a series of papers, E. Hecke described the representation of the group
$SL(2,p)$ on the regular differentials of the modular curve $X$ of level $p$. This was one of the
first applications of character theory outside of finite group theory, and one of the first
constructions of representations using cohomology. I will review Hecke's results, and
interpret them in the modern language of automorphic representations.

We prove that cohomology classes in $H^{2,2}(S_1\times S_2)$
of Hodge isometries
$$\psi \colon H^2(S_1,\mathbb Q)\rightarrow H^2(S_2,\mathbb Q)$$ between
any two
projective complex $K3$ surfaces $S_1$ and $S_2$
are polynomials in Chern classes of coherent analytic sheaves.

Consequently, the cohomology class of $\psi$ is algebraic
This proves a conjecture of Shafarevich announced at ICM in 1970.

Enumerative combinatorics is an area in combinatorics that mainly focuses on enumerating the number of ways to form certain configurations. With the help from computers and the internet, researches in enumerative combinatorics has developed dramatically in the past decades. In this talk, we will introduce signed permutations and their structures. Then we will use signed permutations as an example of how one might use a computer to come up with interesting research problems to solve. Then, we will end the talk by presenting some open problems in enumerative combinatorics.

Fusion categories appear in many areas of mathematics. They are realized by topological quantum field theories, representations of finite groups and Hopf algebras, and invariants for knots and Murray-von Neumann subfactors. An important numerical invariant of these categories are the Frobenius-Schur indicators, which are generalized versions of those for finite group representations. It is thought that these indicators should provide a complete invariant for a fairly wide class of fusion categories; in this talk we will discuss new families of so-called near-group fusion categories (i.e. those with only one non-invertible indecomposable object) which satisfy this property.

In many specialized imaging systems, an unknown signal $x$ $C^d$ produces
measurements of the form
$y_i = | a_i , x |2 + \eta_i$ , where ${a_i}\subset C^d$
are known measurement vectors and is an arbitrary noise term.
Because this system seems to erase the phases of the entries of $x \in C^d$ , the
problem of reconstructing $x$
from $y$ is known as the phase retrieval problem. The first approaches to this
problem were ad hoc iterative
methods which still have no theoretical guarantees on convergence. Recent
advancements including
gradient descent and convex relaxation have supplied some theoretical promises,
but often require such
conditions on the system ${a_i}$ that they cannot be used by scientists in
practice. In particular, they tend
to require some randomness to be used in the choice of ${a_i}$ that does not
reflect the physical systems
that typically yield the phase retrieval problem. Our work contributes a
solution to this problem which
features (a) a deterministic, practicable construction for ${a_i}$ (b) numerical
stability with respect to noise
(c) a reconstruction algorithm with competitive runtimes. Our most recent
result is an improvement on
the robustness gained by leveraging the graph structure induced by our
measurement scheme.

This lecture will be about a remarkable law of nature discovered by George Polya. Consider a particle initially situated at a given point of the d-dimensional integer lattice. Suppose that, at each tick of the clock, the particle jumps to a neighboring lattice site, with equal probability of jumping in any direction. Polya's law states that the particle returns to its initial position with probability one in dimensions d = 1,2, but with probability strictly less than one in all higher dimensions. Thus, a drunk person wandering a city grid will always return to their starting point, but if the drunkard can fly s/he might never come back.

Suppose $\Gamma_1,\ldots,\Gamma_n$ are hyperbolic ICC groups and denote by $\Gamma =\Gamma_1\times \cdots \times \Gamma_n$. We show whenever $\Lambda$ is an arbitrary discrete group such that $L(\Gamma)\cong L(\Lambda)$ then $\Lambda =\Lambda_1\times \cdots \times \Lambda_n$ and up to amplifications $L(\Gamma_i)\cong L(\Lambda_i) $ for all $i$; in other words the von Neumann algebra $L(\Gamma)$ completely remembers the product structure of the underlying group. In addition, we will show that some of the techniques used to prove this product rigidity result can also be successfully applied to produce new examples of prime factors. In particular, we significantly generalize the primeness results obtained earlier by I. Chifan, Y. Kida and S. Pant for the factors arising poly-hyperbolic and surface braid groups. These are joint works with I. Chifan and T. Sinclair, and S. Pant, respectively.

Conformal and CR geometries are among the class of "parabolic geometries" which posses a canonical Cartan connection characterizing the geometry. Replacing the Levi-Civita connection with the Cartan connection we develop submanifold theory in parallel with the classical Riemannian case. This allows us to apply tools developed for conformal and CR invariant theory to develop a theory of submanifold invariants and invariant operators, relevant to the study of conformally or CR invariant boundary value problems and other problems in geometric analysis involving submanifolds. The technical details of the theory are substantial (especially in the CR case). I will try to emphasize some of the concrete geometric ideas behind the approach, giving insight into the original work of Elie Cartan.

Critical lattice models are believed to converge to a free field in the scaling limit, at or above their critical dimension. This has been (partially) established for Ising and $Phi^4$ models for $d \geq 4$. We describe a simple spin model from uniform spanning forests in $\mathbb{Z}^d$ whose critical dimension is 4 and prove that the scaling limit is the bi-Laplacian Gaussian field for $d\ge 4$. At dimension 4, there is a logarithmic correction for the spin-spin correlation and the bi-Laplacian Gaussian field is a log correlated field. The proof also improves the known mean field picture of LERW in d=4, by showing that the renormalized escape probability (and arm events) of 4D LERW converge to some "continuum escaping probability". Based on joint works with Greg Lawler and Xin Sun.

Dynamic Conditional Beta (DCB) is an approach to estimating regressions with time varying parameters. The conditional covariance matrices of the exogenous and dependent variable for each time period are used to formulate the dynamic beta. Joint estimation of the covariance matrices and other regression parameters is developed. Tests of the hypothesis that betas are constant are non-nested tests and several approaches are developed including a novel nested model. The methodology is applied to industry multifactor asset pricing and to global systemic risk estimation with non-synchronous prices.

Free registration is required to attend. Registration information is available at

http://www.math.ucsd.edu/$\sim$rosenblattconf/rosenblattlecture.html

Gross showed that to every Hermitian symmetric tube domain we
may associate a canonical variation of Hodge structure (VHS) of Calabi-Yau
type. The construction is representation theoretic, not geometric, in
nature, and it is an open question to realize this abstract VHS as the
variation induced by a family of polarized, algebraic Calabi-Yau
manifolds. In order for a geometric VHS to realize Gross's VHS it is
necessary that the invariants associated to the two VHS coincide. For
example, the Hodge numbers must agree. The later are discrete/integer
invariants. Characteristic forms are differential-geometric invariants
associated to VHS (introduced by Sheng and Zuo). Remarkably, agreement of
the characteristic forms is both necessary and sufficient for a geometric
VHS to realize one of Gross's VHS. That is, the characteristic forms
characterize Gross's Calabi-Yau VHS. I will explain this result, and
discuss how characteristic forms have been used to study candidate
geometric realizations of Gross's VHS.

In the late 1980's, after Jones' define his polynomial, there is a
revolution in this area, followed by Witten's reinterpreting Jones
polynomial by using Chern-Simons theory and predicting new quantum
invariants. Finally Reshetikhin-Turaev was the first one to define a
mathematically rigorous theory of such complex-valued invariant for closed
3-manifolds. More importantly, Reshetikhin-Turaev define their invariant
not only at roots of unity $q(1)$ originally considered by Witten but also at
other roots of unity. Later Turaev-Viro defined a real valued invariants
for closed 3-manifolds by triangulation both at $q(1)$ and other roots of
unity.

In 1997, Kashaev discover his invariants of hyperbolic knots will become
exponentially large as $N->infinity$ and he further conjectured that the
growth rates corresponds to hyperbolic Volume of complement of that knot in
$S^3$. In 2001, H. Murakami-J. Murakami extend Kashaev's Volume Conjecture
from hyperbolic knots to all knots and hyperbolic volume to simplicial
volume by using colored Jones polynomials.

For many years, Witten-Reshetikhin-Turaev invariants evaluated at $q(1)$ was
considered to be only polynomial growth and its asymptotic expansion is
called WAE Conjecture (Witten's Asymptotic Expansion). Last year, in a
joint work with T. Yang, we first define a real valued Turaev-Viro type
invariant for 3-manifold with boundary by using ideal triangulation. Then
we discovered this Turaev-Viro type invariant and Reshetikhin-Turaev
invariant evaluated at other roots of unity (especially at $q(2)$) will have
exponentially large phenomenon and the the growth rates corresponds to
Volume of 3-manifold with boundary and Volume of closed 3-manifold
respectively.

Thankful to the new tool developed by Ohtsuki recently, asymptotic
expansion of Kashaev invariants (including Volume Conjecture) up to 7
crossing has been solved. This new tool can also be used to attack my
volume Conjecture with Tian Yang. I will give a brief introduction for all
these new developments.

Finally we expect Reshetikhin-Turaev at roots of unity other than $q(1)$
could have a different Geometry/Physics interpretation than original
Chern-Simons theory given by Witten in 1989.

Direct observations of the modern geomagnetic field enable us to understand its role in protecting us from the depredations of the solar wind and associated space weather, while paleomagnetic studies provide geological evidence that the field is intimately linked with the history and thermal evolution of our planet. In the past the magnetic field has reversed polarity many times: such reversals occur when its overall strength decays, and there are departures from the usual spatial structure which at Earth's surface predominantly resembles that of an axially aligned dipole. Reversals are one element of a continuum of geomagnetic field behavior which also includes geomagnetic excursions (often viewed as unsuccessful reversals), and paleosecular variation. The fragmentary and noisy nature of the geological record combined with distance from the field's source in Earth's liquid outer core provide a limited view, but one that has been partially characterized by time series analysis, and development of stochastic models describing the variability. Analyses of changes in the dipole moment have revealed distinct statistical characteristics associated with growth and decay of field strength in some frequency ranges. Paleomagnetic studies are complemented by computationally challenging numerical simulations of geomagnetic field variations. Access to details within the numerical model allow the evolution of large scale physical processes to be studied directly, and it is of great interest to determine whether these computational results have Earth-like properties. The parameter regime accessible to these simulations is far from ideal, but their adequacy can be assessed and future development guided by comparisons of their statistical properties with robust results from paleomagnetic observations. Progress in geomagnetic studies has been greatly facilitated by the application of statistical methods related to stochastic processes and time series analysis, and there remains significant scope for continued improvement in our understanding. This is likely to prove particularly important for understanding the scenarios that can lead to geomagnetic reversals.

Free registration is required to attend. Registration information is available at

http://www.math.ucsd.edu/$\sim$rosenblattconf/rosenblattlecture.html

Are you tired of graphs, paths, and flags on staffs? Have all those Dyck paths finally crossed the line? Are you ready to make a full binary tree and "leaf" the Catalan numbers behind? Then this is the talk for you! We are going to talk about concepts in combinatorics that have connections to areas of applied mathematics such as compressed sensing, quantization, and convex optimization; in particular, we're going to discuss some magnificent ways in which we can either solve (or approximately solve) certain problems in combinatorics by applying techniques from these areas. Conversely, we present cases where we can use ideas from combinatorics to prove results in applied math!

Let $q$ be a non-degenerate indefinite quadratic form over $ \mathbb{R}$
in $n \ge 3$ variables. Establishing a longstanding conjecture of Oppenheim, Margulis proved in 1986 that if $q$ is not a multiple of a rational form, then the set of values $q( \mathbb{Z}^n)$ is a dense subset of $ \mathbb{R}$.
Quantifying this result, Eskin, Margulis, and Mozes proved in 1986 that unless $q$ has signature $(2,1)$ or $(2,2)$, then the number $N(a,b;r)$ of integral vectors $v$ of norm at most $r$ satisfying $q(v) \in (a,b)$ has the asymptotic behavior $N(a,b;r) \sim \lambda(q) \cdot (b-a) r^{n-2}$.

Now, let $S$ is a finite set of places of $ \mathbb{Q}$ containing the Archimedean one, and $q=(q_v)_{v \in S}$
is an $S$-tuple of irrational isotropic quadratic forms over the completions $ \mathbb{Q}_v$. In this talk I will discuss the question of distribution of values of $q(v)$ as $v$ runes over $S$-balls in $ \mathbb{Z}[1/S]$.
This talk is based on a joint work with Seonhee Lim and Jiyoung Han.

The field of geometric numerical integration(GNI) seeks to exploit the
underlying (geometric)structure of a dynamical system in order to
construct numerical methods that exhibit desirable properties of
stability and/or preservation of invariants of the flow. Variational
Integrators are built for Hamiltonian systems by discretizing the
generating function of the symplectic flow, rather than discretizing
the differential equations directly. Traditionally, the generating
function considered is a type I generating function.
In this talk we will discuss the properties and
advantages/disadvantages of discretizing the type II/III generating
function of the flow. After establishing error analysis and adjoint
results, we consider the possible numerical resonance properties
corresponding to the different types of generating functions.

In Math 20, we learned how to differentiate and integrate functions defined on Euclidean spaces. There is a much wilder world of smooth spaces (manifolds) where a generalization of this calculus is possible, but it requires a steep learning curve and a lot of new language to understand. There is a class of manifolds, however, that is both large and interesting, and also retains enough Euclidean-like structure to do calculus almost the same way as in Math 20. These are called Lie groups.

I will discuss (with two or three guiding examples) how do to calculus on Lie groups, which can usually be realized as groups of square matrices. I will then discuss the most important differential equation in the world -- the heat equation -- in the context of matrix Lie groups, and the beautiful interplay between geometry and heat flow. Finally, I'll talk about my research into the heat flow of eigenvalues in matrix Lie groups -- and there'll be lots of cool pictures.

In this talk, we discuss results on gravitational perturbations of black holes by evaluating quasi-local mass on surfaces of fixed size at the null infinity in a gravitational radiation. In particular, a general theorem regarding the decay rate of the quasi-local energy-momentum at infinity is proved and is applied to study the gravitational perturbation of the Schwarzschild solution. The theorem associates a 4-vector to each loop near null infinity, which encodes the distinctive features of a gravitational wave.

Variational integrators provide a systematic way to derive geometric numerical methods for Lagrangian dynamical systems, which preserve a discrete multisymplectic form as well as momenta associated to symmetries of the Lagrangian via Noether's theorem. An inevitable prerequisite for the derivation of variational integrators is the existence of a variational formulation for the considered dynamical system. Even though this is the case for a large class of systems, there are many interesting examples which do not belong to this class, e.g., equations of advection-diffusion type like they are often found in fluid dynamics or plasma physics.

We propose the application of the variational integrator method to so called formal Lagrangians, which allow us to embed any dynamical system into a Lagrangian system by doubling the number of variables. Thereby we are able to derive variational integrators for arbitrary systems, extending the applicability of the method significantly. A discrete version of the Noether theorem for formal Lagrangians yields the discrete momenta preserved by the resulting numerical schemes.

The theory is applied to dynamical systems from fluid dynamics and plasma physics like the vorticity equation, the Vlasov-Poisson system and magnetohydrodynamics, including numerical examples. Recent attempts of applying this method also to noncanonical Hamiltonian ODEs will be sketched.

In the theory of ``motives", algebraic cycles are central objects. For instance, the so-called ``motivic cohomology groups”, that give the universal bigraded ordinary cohomology on smooth varieties, are obtained from a complex of abelian groups consisting of certain algebraic cycles.

In this talk, we discuss how one can go beyond it, and we show that an infinitesimal version of the above complex of abelian groups of algebraic cycles can be identified with the big de Rham-Witt complexes after a suitable Zariski sheafification. This in a sense implies that the crystalline cohomology theory admits a description in terms of algebraic cycles, going back to a result of S. Bloch and L. Illusie in the 1970s. This is based on a joint work with Amalendu Krishna.

In symplectic manifolds, isotropic, coisotropic, and lagrangian submanifolds play a central role, and their study leads to deep problems in symplectic geometry and topology. It turns out that the linearized version of this study is already quite non-trivial.
The classification of pairs of isotropic subspaces in a symplectic vector space turns out to be rather simple, but for isotropic triples, it is much more complicated. In particular, there are families of inequivalent indecomposable isotropic triples depending on one parameter (but no more).

In these talks, I will report on progress on this problem in ongoing work with
Christian Herrmann (University of Dartmstadt) and Jonathan Lorand (University of Z\"urich).

In symplectic manifolds, isotropic, coisotropic, and lagrangian submanifolds play a central role, and their study leads to deep problems in symplectic geometry and topology. It turns out that the linearized version of this study is already quite non-trivial.
The classification of pairs of isotropic subspaces in a symplectic vector space turns out to be rather simple, but for isotropic triples, it is much more complicated. In particular, there are families of inequivalent indecomposable isotropic triples depending on one parameter (but no more).

In these talks, I will report on progress on this problem in ongoing work with
Christian Herrmann (University of Dartmstadt) and Jonathan Lorand (University of Z\"urich).

If R is the free associative algebra in two variables, say over the complex numbers, then the ring of two by two matrices over R is also a quotient of R by a differential ideal I. Playing off the two descriptions of R/I leads naturally to the Schwarzian Korteweg-DeVries equation, in which a function of x evolves in time driven by the Schwarzian derivative. I will present this abstract setup and then explain how ruled surfaces (appropriately chosen) over hyperelliptic curves provide solutions of the equation. I will also describe several commuting flows.

No spoilers!

Let $\Lambda<SL(2,\mathbb{Z})$ be a finitely generated, non-elementary Fuchsian group of the second kind, and $\bf{v},\bf{w}$ be two primitive vectors in $\mathbb{Z}^2-\bf{0}$. We consider the set $\mathcal{S}=\{\langle \bf{v}\gamma,\bf{w}\rangle_{\mathbb{R}^2}:\gamma\in\Lambda\}$, where $\langle\cdot,\cdot\rangle_{\mathbb{R}^2}$ is the standard inner product in $\mathbb{R}^2$. Using Hardy-Littlewood circle method and some infinite co-volume lattice point counting techniques developed by Bourgain, Kontorovich and Sarnak, together with Gamburd's $5/6$ spectral gap, we show that if $\Lambda$ has parabolic elements, and the critical exponent $\delta$ of $\Lambda$ exceeds $0.995371$, then a density-one subset of all admissible integers (i.e. integers passing all local obstructions) are actually in $\mathcal{S}$, with a power savings on the size of the exceptional set (i.e. the set of admissible integers failing to appear in $\mathcal{S}$). This supplements a result of Bourgain-Kontorovich, which proves a density-one statement for the case when $\Lambda$ is free, finitely generated, has no parabolics and has critical exponent $\delta>0.999950$.

Several long-standing open problems in the theory of C*-algebras reduce to whether for a given class of C*-algebras there is a locally universal one among them with certain nice properties. I will discuss how techniques from model theory, in particular model-theoretical forcing, can be used to shed light on these problems. This is joint work with Isaac Goldbring.

We introduce a general framework for studying a class of randomized load balancing models in a system with a large number of servers that have generally distributed service times and use a first-come-first serve policy within each queue. Under fairly general conditions, we use an interacting measure-valued process representation to obtain hydrodynamics limits for these models, and establish a propagation of chaos result. Furthermore, we present a set of partial differential equations (PDEs) whose solution can be used to approximate the transient behavior of such systems. We prove that these PDEs have a unique solution, use a numerical scheme to solve them, and demonstrate the efficacy of these approximations using Monte Carlo simulations. We also illustrate how the PDE can be used to gain insight into network performance.

I will present a joint work with Martin Li. Minimal surfaces with free boundary are natural critical points of the area functional in compact smooth manifolds with boundary. In this talk, I will describe a general existence theory for minimal surfaces with free boundary. In particular, I will show the existence of a smooth embedded minimal hypersurface with free boundary in any compact smooth Euclidean domain. The minimal surfaces with free boundary were constructed using the min-max method. I will explain the basic ideas behind the min-max theory as well as our new contributions.

We prove that any quasi-isometry between hyperbolic manifolds is homotopic to a harmonic quasi-isometry.

I will discuss joint work with Gabriele Di Cerbo on boundedness
of Calabi-Yau pairs. Recent works in the minimal model program suggest
that pairs with trivial log canonical class should satisfy some
boundedness properties. I will show that Calabi-Yau pairs which are not
birational to a product are indeed log birationally bounded, if the
dimension is less than 4. In higher dimensions, the same statement can be
deduced assuming the BAB conjecture. If time permits, I will discuss
applications of this result to elliptically fibered Calabi-Yau manifolds.

The Lerch zeta function is a three variable zeta function,
with variables $(s, a, c)$,
which generalizes the Riemann zeta function and has
a functional equation, but no Euler product. We discuss its properties.
It is an eigenfunction of a linear partial
differential equation in the variables $(a, c)$
with eigenvalue $-s$, and it is also preserved under a
a commuting family of two-variable Hecke-operators $T_m$
with eigenvalue $m^{-s}$. We give a characterization
of it in terms of being a simultaneous eigenfunction of
these Hecke operators.
We then give an automorphic interpretation of the Lerch zeta
function in terms of Eisenstein series taking
values on the Heisenberg nilmanifold, a
quotient of the real Heisenberg group modulo
its integer subgroup. Part of this work is joint with
W.-C. Winnie Li.

Have you always wondered how the Banach-Tarski paradox - namely cutting a ball into pieces and assembling those pieces into two balls - actually works? Or what the mathematical reason behind it is? In this talk we will answer those questions by 1) reformulating the Banach-Tarski paradox, 2) explaining what paradoxical decompositions of various objects are, 3) constructing one for a ball, and 4) motivated by all those things introducing the concept of amenability, which has its origin in the aforementioned paradox and is ubiquitous in many areas of mathematics nowadays.

I will talk about the problem of separating multiple signals from each
other when we only have access to a few linear (or non-linear)
combinations of them. An example of this type of problem is at a
cocktail party when you are trying to have a conversation with a
friend but there are several conversations happening around you. Your
ears provide you with a superposition of all the voices, and your
brain does remarkably well at focusing on your friend's voice and
drowning out all the others. We will talk about one computer algorithm
(or time permitting, more) that does such a task (reasonably)
successfully. Along the way, we will talk about important tools in
mathematical signal processing, including the Fourier transform and
sparsity.

A ``Higgs bundle" is essentially a family of matrices, and if you
try to diagonalize one you get a "spectral cover." They arise in the
study of moduli of vector bundles. I will introduce Higgs bundles and
spectral covers, and then I will describe their cohomological
invariants.

We shall discuss the recent work of Brendle, Choi and Daskalopoulos on the classification via a Pogorelov's type estimate.

Fusion categories appear in many areas of mathematics. They are
realized by topological quantum field theories, representations of
finite groups and Hopf algebras, and invariants for knots and
Murray-von Neumann subfactors. An important numerical invariant of
these categories are the Frobenius-Schur indicators, which are
generalized versions of those for finite group representations. Using
these categorical indicators we are able to distinguish near-group
fusion categories, that is those fusion categories with one
non-invertible object, and obtain some realizations of their tensor
equivalence classes.

We present a general framework for approximating the component structure of random combinatorial assemblies when both the size $n$ and the number of components $k$ is specified. The approach is an extension of the usual saddle point approximation, and we demonstrate near-universal behavior when the rank $r := n-k$ is small relative to $n$ (hence the name `low rank’).

In particular, for $\ell = 1, 2, \ldots$, when $r \asymp n^\alpha$, for $\alpha \in \left(\frac{\ell}{\ell+1}, \frac{\ell+1}{\ell+2}\right)$, the size~$L_1$ of the largest component converges in probability to $\ell+2$. When $r \sim t\, n^{\ell/(\ell+1)}$ for any $t>0$ and any positive integer $\ell$, we have $P(L_1 \in \{\ell+1, \ell+2\}) \to 1$. We also obtain as a corollary bounds on the number of such combinatorial assemblies, which in the special case of set partitions fills in a countable number of gaps in the asymptotic analysis of Louchard for Stirling numbers of the second kind.

This is joint work with Richard Arratia

\noindent
Convex cones over a real unital algebra, which in addition are closed under
inversion, may seem peculiar. However, Convex Invertible Cones (CICs)
naturally appear in stability analysis of continuous-time physical systems.
\vskip 0.2cm

\noindent
With this motivation, in this talk we explore examples
of CICs over some algebras and establish interconnections among them.
\vskip 0.2cm

\noindent
This indicates at the importance of the study of
rational functions, of non-commuting variables, with
certain positivity properties.
\vskip 0.2cm

\noindent
This talk is based on an ongoing research for many years.
Some of it in collaboration with Daniel Alpay, Chapman University,
California,
Nir Cohen, Natal, Brazil and the late Leiba Rodman, from the College
of William and Mary, Virginia.

Given a rectangular board B with any set of forbidden states, the rook
number $r_B(k)$ is the number of ways of placing k non-attacking rooks
on board $B$ which avoid the set of forbidden states. When non-attacking
only means no two rooks lie in the same column, the number of such
configurations is called the file number, $f_B(k)$. When the board $B$ is
the staircase board, with row lengths $n-1, n-2, ..., 1$, then the rook
number coincides with the Stirling numbers of the second kind, and the
file number coincides with the unsigned Stirling numbers of the first
kind.

We will demonstrate, using Poisson approximation and an explicit
coupling, how one can obtain quantitative bounds on rook and file
numbers under certain conditions. In the case of Stirling numbers, our
results fill a gap in a recent asymptotic expansion which uses
explicitly defined parameters due to Louchard.
This is joint work with Richard Arratia.

After offering some quick answers to the questions "What is learning?" and "What is a quantum computer?", I'll explain how you can improve your learning with a quantum computer. We measure the efficiency of a learning algorithm by its query complexity, and in this field one tries to find upper bounds for the query complexity (by creating algorithms) as well as lower bounds (by proving the optimality of certain algorithms). Many of the first quantum algorithms are in fact learning algorithms, and we'll discuss two important ones: Grover's search algorithm and the Bernstein-Vazirani algorithm. These offer amazing speedups in query complexity over classical computers. If time permits, I'll describe recent research which introduces a huge family of learning problems with nonabelian symmetries for which little is known but the (upper and lower bounds for the) query complexity for any given problem can be easily computed on computer algebra software such as GAP or SAGE. This work is in collaboration with Orest Bucicovschi, Hanspeter Kraft, David Meyer and Jamie Pommersheim.

In this talk, I focus on factor analysis of high-dimensional time series
data, in which the dimension of data is allowed to be even larger than the
length of data. Analysis of high-dimensional data suffers from the curse
of dimensionality. Factor analysis is considered as an effective way for
dimension reduction. Factor models presume that a few common factors can
explain most of the variation/dynamics of an observed process in high
dimensions. In the models, factor loadings are introduced to reflect the
percentages of variations explained and contributions made by these common
factors. Based on real data analysis, it has been discovered that the
loadings may vary in different situations/regimes. To interpret this
observation and capture the regime-switching mechanism often encountered
in practice, we propose a threshold factor model for high-dimensional time
series data, in which a threshold variable is introduced to distinguish
different regimes. Loadings controlled by the threshold variable vary
across regimes. The theoretical properties of the procedure are
investigated.

The affine sieve is a technique first developed by Bourgain, Gamburd, and Sarnak in 2006 and later completed by Salehi Golsefidy and Sarnak in 2010 to study almost-primality in a broad class of affine linear actions. The beauty of this is that it gives us effective bounds on the saturation number for thin orbits coming from $GL_n$ - in particular, producing infinitely many $R$-almost primes for some $R$. However, in practice this value of $R$ is often far from optimal. The case of thin Pythagorean triangles has been of particular interest since the outset of the affine sieve, and I will discuss recent progress on improving bounds for the saturation numbers for their hypotenuses and areas using Archimedean sieve theory.

We give a geometric characterization of mean ergodic convergence in the Calkin algebras for certain Banach spaces. (Joint work with William B. Johnson)

Shrinking gradient Ricci solitons (shrinkers) are models for the local geometry of singular regions of solutions to the Ricci flow and their classification is critical to the understanding of singularity formation under the flow. Growing evidence suggests that the asymptotic geometry of complete noncompact shrinkers may be particularly constrained; in fact, all examples currently known which do not split locally as products are smoothly asymptotic to a regular cone at infinity. I will present some results from a joint project with Lu Wang, in which we study the uniqueness of shrinkers asymptotic to such structures as a problem of parabolic unique continuation, and discuss the applications of these results to a conjectured classification in four dimensions.

We fix a monic polynomial $\bar f(x) \in \mathbb{F}_q[x]$
over a finite field of characteristic $p$, and consider the
$\mathbb{Z}_{p^{\ell}}$-Artin--Schreier--Witt tower defined by $\bar
f(x)$; this is a tower of curves $\cdots \to C_m \to C_{m-1} \to
\cdots \to C_0 =\mathbb{A}^1$, whose Galois group is canonically
isomorphic to $\mathbb{Z}_{p^\ell}$, the degree $\ell$ unramified
extension of $\mathbb{Z}_p$, which is abstractly isomorphic to
$(\mathbb{Z}_p)^\ell$ as a topological group.
We study the Newton slopes of zeta functions of this tower of curves.
This reduces to the study of the Newton slopes of L-functions
associated to characters of the Galois group of this tower. We prove
that, when the conductor of the character is large enough, the Newton
slopes of the L-function
asymptotically form a finite union of arithmetic progressions. As a
corollary, we prove the spectral halo property of the spectral variety
associated to the $\mathbb{Z}_{p^{\ell}}$-Artin--Schreier--Witt tower.
This extends the main result of Davis--Wan--Xiao from rank one
case $\ell=1$ to the higher rank case $\ell\geq 1$.

Motivated by Kolmogorov's theory of hydrodynamic turbulence, we considerdissipative weak solutions to the 3D incompressible Euler equations and the 2D surface quasi-geostrophic equations. We prove that up to a certain regularity threshold weak solutions are not unique. In the case of the Euler system this is the threshold determined by the Onsager conjecture.
For SQG, this answers an open problem posed by De Lellis and Szekelyhidi Jr.

Following ideas of Marian and Oprea, finite quot schemes can be
used to investigate Le Potier's strange duality conjecture for surfaces. I
will discuss recent work with Aaron Bertram and Drew Johnson in which we
prove the existence of a large class of finite quot schemes on the
projective plane. We use nice resolutions of general stable vector
bundles, which also yield an easy proof that these bundles are globally
generated whenever their Euler characteristic suggests that they should
be.

Synthetic aperture radar (SAR) uses relative motion to produce fine resolution images from microwave frequencies and is a useful tool for regular monitoring and mapping applications. Unfortunately, if target distance is estimated poorly, then phase errors are incurred in the data, producing a blurry reconstruction of the image. In this talk, we introduce a multistatic methodology for determining these phase errors from interferometry-inspired combinations of signals. To motivate this, we first consider a more general problem called phase retrieval, in which a signal is reconstructed from linear measurements whose phases are either unreliable or unavailable. We apply certain ideas from phase retrieval to resolve phase errors in SAR; specifically, we use bistatic techniques to measure relative phases and then apply a graph-theoretic phase retrieval algorithm to recover the phase errors. We conclude by devising an image reconstruction procedure based on this algorithm, and we provide simulations that demonstrate stability to noise.

In the classification of (commutative) projective surfaces, one first classifies minimal models for a given birational class, and then shows that any surface can be blown down at a finite number of curves to obtain a minimal model.

Artin has proposed a similar programme for noncommutative surfaces (that is, domains of $GK$-dimension 3). In the generic ``rational'' case of rings birational to a Sklyanin algebra, the likely candidates for minimal models are the Sklyanin algebra itself and Van den Bergh's quadric surfaces. We show, using our previously developed noncommutative version of blowing down, that these algebras are minimal in a very strong sense: given a Sklyanin algebra or quadric $R$, if $S$ is a connected graded, noetherian overring of $R$ with the same graded ring of fractions, then $S=R$.

This is a joint work with Rogalski and Stafford.

2015 was the Centinnial year of Einstein's General Theory of Relativity, and fittingly concluded with the discovery of gravitational waves, which he had predicted. Despite knowing the key physical principles, Einstein was only able to formulate his theory after learning differential geometry from mathematician Marcel Grossmann in 1912. In a sense, General Relativity simply $is$ applied differential geometry. This talk will sketch the key ideas of differential geometry and how they apply to Einstein's theory of gravity. The presentation will emphasize ideas and pictures, rather than equations.

In this talk we will try to discuss the
following questions:

1. What is the space of distributions? What are its key
properties? Why do we need it? How do we use it?

2. What is a function space? What are the nice properties that
we would like our function spaces to possess?

3. Why is the Gagliardo seminorm defined the way it is?

4. How do interpolation theory and Littlewood-Paley theory come
into play in the study of Slobodeckij spaces?

5. For what values of $s$ and $p$, $\partial^\alpha: W^{s,p}(\Omega)\rightarrow
W^{s-|\alpha|,p}(\Omega)$ is a well defined bounded linear
operator for all $\alpha\in \mathbb{N}_0^n$? Why do we care about
this question?

The asymmetric simple exclusion process (ASEP) is a model from statistical physics that describes the dynamics of particles hopping right and left on a finite 1-dimensional lattice. Particles can enter and exit at the left and right boundaries, and at most one particle can occupy each site. The ASEP plays an important role in the study of non-equilibrium statistical mechanics and has appeared in many contexts, for instance as a model for 1-dimensional transport processes such as protein synthesis, molecular and cellular transport, and traffic flow. Moreover, it displays rich combinatorial structure: one can compute the stationary probabilities for the ASEP using fillings of certain tableaux. In this talk, we will discuss some of the combinatorial results from the past decade as well as recent developments, including combinatorial formulae for a two-species generalization of the ASEP and a remarkable connection to orthogonal polynomials. This talk is based on joint works with X. Viennot and separately with S. Corteel and L. Williams.

This talk will introduce the space of $id{e}les$ for a global field $K$. For convenience, the construction given in the talk will use the rational numbers. In the process, completions, additive/multiplicative valuations, and connections to ideals will be discussed. If time permits, we will generalize to an arbitrary Galois number field, ultimately ending with some Galois-theoretic properties of the $id{e}les$.

We will discuss rigidity results for holomorphic mappings in CR and complex geometry, emphasizing the connections between the two types of rigidity. We discuss in more detail rigidity of volume-preserving maps between Hermitian symmetric spaces, based on the work of Mok-Ng and my recent joint work with Fang and Huang.

Mean curvature flow is the negative gradient flow of volume, so any closed hypersurface flows in the direction of steepest descent for volume and eventually becomes extinct in finite time. In most cases, the flow develops singularities before its extinction time. It is known that the asymptotic behaviors of the flow near a singularity are modeled on a special class of solutions to mean curvature flow, which are called self-shrinkers. In this talk, we will outline a program on the classification of noncompact two-dimensional self-shrinkers, and report some recent progress with an emphasis on the geometry at infinity of these self-shrinkers.

In this talk, I will start with an introduction to the Einstein 4-manifold. Then I will discuss some earlier result on classification of the positive case. Finally I will mention some recent development in this area.

A common problem in modern statistical applications is to select, from a large set of candidates, a subset of variables which are important for determining an outcome of interest. For instance, the outcome may be disease status and the variables may be hundreds of thousands of single nucleotide polymorphisms on the genome. For data coming from low-dimensional ($n \ge p$) linear homoscedastic models, the knockoff procedure recently introduced by Barber and Cand\'es solves the problem by performing variable selection while controlling the false discovery rate (FDR). In this talk I will discuss an extension of the knockoff framework to arbitrary (and unknown) conditional models and any dimensions, including $n < p$, allowing it to solve a much broader array of problems. This extension requires the design matrix be random (independent and identically distributed rows) with a covariate distribution that is known, although the procedure appears to be robust to unknown/estimated distributions. No other procedure solves the variable selection problem in such generality, but in the restricted settings where competitors exist, I will demonstrate the superior power of knockoffs through simulations. Finally, applying the new procedure to data from a case-control study of Crohn’s disease in the United Kingdom resulted in twice as many discoveries as the original analysis of the same data.

The classic 1935 paper of Erdos and Szekeres entitled ``A combinatorial problem in geometry" was a starting point of a very rich discipline within combinatorics: Ramsey theory. In that paper, Erdos and Szekeres studied the following geometric problem. For every integer $n \geq 3$, determine the smallest integer $ES(n)$ such that any set of $ES(n)$ points in the plane in general position contains n members in convex position, that is, n points that form the vertex set of a convex polygon. Their main result showed that $ES(n) \leq {2n - 4\choose n-2} + 1 = 4^{n -o(n)}$. In 1960, they showed that $ES(n) \geq 2^{n-2} + 1$ and conjectured this to be optimal. Despite the efforts of many researchers, no improvement in the order of magnitude has been made on the upper bound over the last 81 years. In this talk, we will sketch a proof showing that $ES(n) =2^{n +o(n)}$.

In joint work with Daniel Hast and Joseph we count degree 3 and 4 covers of the projective line over finite fields. This is a geometric analogue of the number field side of counting cubic and quartic fields. We take a geometric approach, by using a vector bundle parametrization of these curves which is different from the recent work of Manjul Bhargava, Arul Shankar, Xiaoheng Wang "Geometry of numbers methods over global fields: Prehomogeneous vector spaces" in which the authors extend the geometry of numbers methods to global fields. Our count is just for $S_3$ and $S_4$ covers, and we put the rest of the curves in our error term.

We start by discussing Bochner's canonical decomposition of negative definite functions on the space of infinite sequences of real numbers and then look at extensions of this theorem within the framework of more general infinite-dimensional groups like the infinite symmetric group. The chosen approach relies on the theory of spherical functions developed by G. Olshanski.

In classical Hodge theory, variations of Hodge structure and their associated period mappings play a crucial role. In the $p$-adic world, it turns out there are *two* natural kinds of period maps associated with variations of $p$-adic Hodge structure: the ``Grothendieck-Messing" period maps, which roughly come from comparing crystalline and de Rham cohomology, and the ``Hodge-Tate" period maps, which come from comparing de Rham and $p$-adic etale cohomology. I'll discuss these period maps, their applications, and some new results on their construction and geometry. This is partially joint work with Jared Weinstein.

I will describe a construction which associates a canonical $p$-adic L-function with a refined cohomological Hilbert modular form $(\pi, \alpha)$ under some mild and natural assumptions. The main novelty is that we do not impose any hypothesis of “small slope” or “noncriticality” on the allowable refinements. Over $\mathbb{Q}$, this result is due to Bellaiche. Our strategy for dealing with critical refinements is roughly parallel to his, and in particular relies on a careful study of the local geometry of eigenvarieties at classical (but possibly critical) points. This is joint work with John Bergdall.

Data assimilation or filtering of nonlinear dynamical systems combines numerical models and observational data to provide the best statistical estimates of the systems. Ensemble-based methods have proved to be indispensable filtering tools in atmosphere and ocean systems that are typically high dimensional turbulent systems. In operational applications, due to the limited computing power in solving the high dimensional systems, it is desirable to use cheap and robust reduced-order forecast models to increase the number of ensemble for accuracy and reliability. This talk describes a multiscale data assimilation framework to incorporate reduced-order multiscale forecast methods for filtering high dimensional complex systems. A reduced-order model for two-layer quasi-geostrophic equations, which uses stochastic modeling for unresolved scales, will be discussed and applied for filtering to capture important features of geophysical flows such as zonal jets. If time permits, a generalization of the ensemble-based methods, multiscale clustered particle filters, will be discussed, which can capture strongly non-Gaussian statistics using relatively few particles.

The Verlinde formula is a celebrated explicit computation of the
dimension of the space of sections of certain positive line bundles over
the moduli space of semistable vector bundles over an algebraic curve. I
will describe a recent generalization of this formula in which the
moduli of vector bundles is replaced by the moduli of semistable Higgs
bundles, a moduli space of great interest in geometric representation
theory. A key part of the proof is a new ``virtual non-abelian
localization formula" in K-theory, which has broader applications in
enumerative geometry. The localization formula is an application of the
nascent theory of Theta-stratifications, and it serves as a new source
of applications of derived algebraic geometry to more classical questions.

Large amounts of data now stream from daily life; data
analytics has been helping to discover hidden patterns, correlations and
other insights. This talk introduces the mode decomposition problem in
the analysis of oscillatory data. This problem aims at identifying and
separating pre-assumed data patterns from their superposition. It has
motivated new mathematical theory and scientific computing tools in
applied harmonic analysis. These methods are already leading to
interesting and useful results, e.g., electronic health record analysis,
microscopy image analysis in materials science, art and history.

The development of the theory of mirror symmetry in high energy
physics has led to deep conjectures regarding the geometry of a
special class of complex manifolds called Calabi-Yau manifolds. One of
the most intriguing of these conjectures states that various geometric
invariants, some classical and some more homological in nature, agree
for any two Calabi-Yau manifolds which are ``birationally equivalent"
to one another. I will discuss how new methods in equivariant geometry
have shed light on this conjecture over the past few years, leading to
the first substantial progress for compact Calabi-Yau manifolds of
dimension greater than three. The key technique is the new theory of
``Theta-stratifications," which allows one to bring ideas from
equivariant Morse theory into the setting of algebraic geometry.

In this talk I will talk on our recent work on asymptotically hyperbolic Einstein manifolds. I will present a proof for a sharp volume comparison theorem for asymptotically hyperbolic Einstein manifolds, which will imply not only the rigidity theorem for hyperbolic space in general dimension but also curvature estimates for asymptotically hyperbolic Einstein manifolds. In particular, as a consequence of our curvature estimates, one now knows that the asymptotically hyperbolic Einstein metrics with conformal infinities of sufficiently large Yamabe constant have to be negatively curved.

This talk focuses on two instances in which scientific fields outside mathematics benefit from incorporating the geometry of the data. In each instance, the applications area motivates the need for new mathematical approaches and algorithms, and leads to interesting new questions. (1) A method to determine and predict drug treatment effectiveness for patients based off their baseline information. This motivates building a function adapted diffusion operator for high dimensional data X when the function F can only be evaluated on large subsets of X, and defining a localized filtration of F and estimation values of F at a finer scale than it is reliable naively. (2) The current empirical success of deep learning in imaging and medical applications, in which theory and understanding is lagging far behind. By assuming the data lies near low dimensional manifolds and building local wavelet frames, we improve on existing theory that breaks down when the ambient dimension is large (the regime in which deep learning has seen the most success).

The spectra of signed matrices have played a fundamental role in social sciences, graph theory and control theory. They have been key to understanding balance in social networks, to counting perfect matchings in bipartite graphs, and to analyzing robust stability of dynamic systems involving uncertainties. More recently, the results of Marcus et al. have shown that an efficient algorithm to find a signing of a given adjacency matrix that minimizes the largest eigenvalue could immediately lead to efficient construction of Ramanujan expanders.
Motivated by these applications, this talk investigates natural spectral properties of signed matrices and address the computational problems of identifying signings with these spectral properties. There are three main results we will talk about: (a) NP-completeness of three signing related problems with (negative) implications to efficiently constructing expander graphs, (b) a complete characterization of graphs that have all their signed adjacency matrices be singular, which implies a polynomial-time algorithm to verify whether a given matrix has a signing that is invertible, and (c) a polynomial-time algorithm to find a minimum increase in support of a given symmetric matrix so that it has an invertible signing.

Quantum computers are known to outperform classical
computers in a variety of computational tasks. This includes search on
various networks or databases, which can be encoded as graphs. The
search is performed using a quantum walk--the quantum mechanical
analogue of a random walk--and quantum walks often search
quadratically faster than random walks. Despite this success, we show
that certain graphs and arrangements of marked vertices cause
difficulties for quantum walks, causing them to perform worse than
classical random walks. On the other hand, some of these difficulties
are successes in disguise, and we use them to construct
greater-than-quadratic speedups for spatial search by quantum walk.

This is joint work with Krisjanis Prusis, Jevgenijs Vihrovs, and
Raqueline Santos in http://arxiv.org/abs/1608.00136 and
http://arxiv.org/abs/1610.06075.