The classical Cauchy integral is a fundamental object of complex
analysis whose analytic properties are intimately related to the
geometric properties of its supporting curve. In this talk I will
begin by reviewing the most relevant features of the classical Cauchy
integral. I will then move on to the (surprisingly more involved)
construction of the Cauchy integral for a hypersurface in $\mathbb
C^n$. I will conclude by presenting new results joint with E. M. Stein
concerning the regularity properties of this integral and their
relations with the geometry of the hypersurface. (Time permitting) I
will discuss applications of these results to the Szeg\H o and Bergman
projections (that is, the orthogonal projections of the Lebesgue space
$L^2$ onto, respectively, the Hardy and Bergman spaces of holomorphic
functions).

It is well known that the extremal rays in the cone of effective curve classes on a K3 surface are generated by rational curves $C$ for which $(C,C)=-2$; a natural question to ask is whether there is a similar characterization for a higher-dimensional holomorphic symplectic variety $X$. The intersection form is no longer a quadratic form on curve classes, but the Beauville-Bogomolov form on $X$ induces a canonical nondegenerate form $(\cdot,\cdot)$ on $H_2(X;\mathbb{R} )$ which coincides with the intersection form if $X$ is a K3 surface. We therefore might hope that extremal rays of effective curves in $X$ are generated by rational curves $C$ with $(C,C)=-c$ for some positive rational number $c$. In particular, if $X$ contains a Lagrangian hyperplane $\mathbb P^n\subset X$, the class of the line $\ell\subset\mathbb P^n$ is extremal. For $X$ deformation equivalent to the Hilbert scheme of $n$ points on a K3 surface, Hassett and Tschinkel conjecture that $(\ell,\ell)=-\frac{n+3}{2}$; this has been verified for $n<4$. In joint work with Andrei Jorza, we prove the conjecture for $n=4$, and discuss some general properties of the ring of Hodge classes on $X$.

1. I'll show that Hunt's hypothesis (H) fails for Leonard Gross' infinite dimensional Brownian motion, by exhibiting a subset of the state space of the motion that is hit exactly once for certain starting points.

2. It is well known that a Lebesgue measurable additive function from R to R is necessarily continuous (and linear). I'll show how D. Stroock's recent proof of L. Schwartz's ``Borel graph theorem" can be adapted to show that a ``universally Gaussian measurable" and additive map from one Banach space to another is automatically continuous (and linear).

In this talk we consider the necessary and sufficient conditions for a
formal language to be represented by an infinite word. We extend our
results to the case of partial words and prove a partial uniqueness
result as well

Let X be a finite set, $C = <c>$ be a finite cyclic group acting on $X$, and $X(q) \in N[q]$ be a polynomial with nonnegative integer coefficients. Following Reiner, Stanton, and White, we say that the triple $(X, C, X(q))$ exhibits the $\emph{cyclic sieving phenomenon}$ if for any integer $d>0$, the number of fixed points of $c^d$ is equal to $X(\zeta^d)$, where $\zeta$ is a primitive $|C|^{th}$ root of unity. We explain how one can use representation theory to prove instances of the cyclic sieving phenomenon involving the action of tropical Coxeter elements on (complexes closely related to) cluster complexes. The representation theory involves cluster monomial bases of geometric realizations of finite type cluster algebras.

In broad terms this talk will discuss how much information about a f.g. residually finite group is carried by the collection of its finite quotients. For example a precise question in this direction (and which has been open for many years) is: Given a f.g. residually finite group G with the same collection of finite quotients as a free group of rank n, is G isomorphic to a free group of rank n?

In this talk I will discuss nonnegatively curved compact
Kahler manifolds and their classification. An overview of past results
will be given in the cases of bisectional and orthogonal bisectional
curvature. The more recent case of quadratic orthogonal bisectional
curvature will then be discussed along with recent results. The talk
is based on joint work with L.F. Tam.

In this talk, I will discuss my recent works on the
collapsing behavior of the Kahler-Ricci flow. The first work studies
the Kahler-Ricci flow on $P^1$-bundles over Kahler-Einstein manifolds.
We proved that if the initial Kahler metric is constructed by the
Calabi's Ansatz and is in the suitable Kahler class, the flow must
develop Type I singularity and the singularity model is $P^1 X C^n$. It
is an extension of Song-Weinkove's work on Hirzebruch surfaces. The
second work discusses the collapsing behavior in a more general
setting without any symmetry assumption. We showed that if the
limiting Kahler class of the flow is given by a holomorphic submersion
and the Ricci curvature is uniformly bounded from above with respect
to the initial metric, then the fibers will collapse in an optimal
rate, i.e. diam $\sim (T-t)^{1/2}$. It gives a partial affirmative answer to
a conjecture stated in Song-Szekelyhidi-Weinkove's work on projective
bundles.

The low-dimensional topology community has been energized in recent
years by the introduction of a wealth of so-called ``homology-type"
invariants. One associates to an object in low-dimensional topology
(e.g., a link or a 3-manifold) an abstract chain complex whose
homology is an invariant of the topological object. Such invariants
arise in two apparently different ways: ``algebraically," via the
representation theory of quantum groups and ``geometrically," via
constructions in symplectic geometry. I will discuss what is known
about the relationship between two such invariants: Khovanov homology,
an ``algebraic" invariant of links and tangles defined by Khovanov and
Heegaard-Floer homology, a ``geometric" invariant of 3-manifolds
defined by Ozsvath-Szabo. The portions of the talk describing my own
work are joint with Denis Auroux and Stephan Wehrli.

In this talk I will describe the limiting properties
Yang-Mills flow on a holomorphic vector bundle E, in the case where
the flow does not converge. In particular I will describe how to
determine the $L^2$ limit of the curvature endomorphism along the flow.
This proves a sharp lower bound for the Hermitian-Yang-Mills
functional and thus the Yang-Mills functional, generalizing to
arbitrary dimension a formula of Atiyah and Bott first proven on
Riemann surfaces. I will then show how to use this result to identify
the limiting bundle along the flow, which turns out to be independent
of metric and uniquely determined by the isomorphism class of $E$.

Modern material sciences focus on studies on the microscopic scale. This calls for mathematical understanding of electronic structure and atomistic models, and also their connections to continuum theories. In this talk, we will discuss some recent works where we develop and generalize ideas and tools from mathematical analysis of continuum theories to these microscopic models. We will focus on macroscopic limit and microstructure pattern formation of electronic structure models.

We will re-enact the story of proving a conjecture by Anders
Claesson and Svante Linusson. Along the way, we are naturally led to a
particular bijection between matchings and permutations; however, this
bijection is somewhat flawed. We will discover a general technique to
circumvent these flaws, leading to a new bijection that completes the
proof. Time permitting, we will also use this technique to prove a
recent conjecture by Miles Jones and Jeffrey Remmel. This talk should
be accessible to graduate students of all areas of math.

The existence of elliptic curves of large rank over number
fields is an open question, but it has been known for decades that
there exist elliptic curves of arbitrarily large rank over global
function fields. In this talk we will discuss some results of Ulmer
that showcase the ubiquity of large ranks over function fields, as
well as some newer work in the area.

I will introduce and motivate the Schroedinger Map problem. I will review the results obtain in the field. Then I will talk about the global regularity of equivariant maps in two dimensions with large data.

Compact Riemann surfaces are naturally divided into three types; the Riemann sphere, elliptic curves
and curves of higher genus.

We will explain the conjectural analogue of this classification in higher dimensions, recent progress towards
this classification and some open problems.

Given a curve (Riemann surface), one can construct an abelian variety: its Jacobian. Abelian
varieties are quotients of vector spaces by lattices. The classical Torelli theorem states that the Jacobian
determines the curve. We discuss some generalizations of this and their history.

We will use geometric Langlands theory to solve two problems
concerning number fields. One is Serre's question of whether there
exist motives over $\Bbb Q$ with motivic Galois group of type $E_8$ or $G_2$; the other is the concrete question of whether there are Galois extensions of $\Bbb Q$ with Galois group $E_8(p)$ or $G_2(p)$ (the finite simple groups of Lie type), for sufficiently large primes $p$.
The answer to both questions is ``YES".

Please note the change of day for this week's colloquium.

The new subject of "homotopy type theory" has been created by a fusion
of homotopy theory, higher category theory, and constructive type
theory. On one hand, it enables us to apply homotopical ideas in type
theory, giving new ways to deal with things like proof-irrelevance,
singleton elimination, type equivalence, and universes. On the other
hand, it gives us a formal language in which to do homotopy theory. A
proof written in this language will automatically be valid in many
different ``homotopy theories", and can also be formalized and checked
by a computer proof assistant. Taken to an extreme, the subject
offers the possibility of a new foundation for mathematics in which
the basic objects are homotopy types, rather than sets.

This is the beginning of a weekly seminar which will introduce the
subject and some of its highlights, assuming no background in either
homotopy theory or type theory. In addition to the mathematical
theory, we will learn to formalize it using the computer proof
assistant Coq.

In the first part of the talk I will discuss doubling
constructions. In particular I will discuss in some detail a recent
doubling construction for an equatorial two-sphere in the round
three-sphere, and also potential generalizations for self-shrinkers of
the Mean Curvature flow. In the second part of the talk I will briefly
discuss the current understanding of desingularization constructions
for minimal surfaces and self-shrinkers. In the third and final part I
will discuss open uniqueness questions for closed embedded minimal
surfaces in the round three-sphere inspired by the above
constructions.

In this talk, we will discuss a goal-oriented adaptive method for second order semilinear PDEs. In goal-oriented methods we are concerned with approximating a given quantity of interest, a function of the weak solution to the PDE. In linear problems, this is accomplished by defining a dual problem or formal adjoint and solving the two problems simultaneously. For the semilinear case, we will discuss the formation of the linearized and approximate dual problems. We will then review the standard contraction framework and discuss some additional estimates used to show convergence of the method. Finally, we introduce an appropriate notion of error to derive a strong contraction result.

The Shuffle Conjecture gives a Combinatorial setting to
the bi-graded Frobenius Characteristic of the Diagonal
Harmonic Module of $S_n$. We report here on the progress
in joint work with Angela Hicks in a three year effort to
prove this conjecture. Angela Hicks reduced a combinatorial side of
the problem to proving a deceptively simple property of a remarkable
family of polynomials in a single variable $x$ with coefficients
polynomials in $N[q]$. In this lecture and possibly following ones we
describe what remains to be done to resolve this decade old Algebraic
Combinatorial problem.

The largest eigenvalue of a rank one perturbation of random hermitian matrix is known to exhibit a phase transition. If the perturbation is small, one sees the famous Tracy-Widom law; if the perturbation is large, the result is simple Gaussian fluctuations. Further, there is a scaling window about a critical value of the perturbation which leads to a new one parameter family of limit laws. The same phenomena exists for random sample covariance matrices in which one of the population eigenvalues is "spiked", or takes a value other than one. Bloemendal-Virag have shown how this picture persists in the context of the general beta ensembles, giving new formulations of the discovered critical limit laws (among other things). Yet another route, explained here, is to go through the random matrix hard edge, perturbing the smallest eigenvalues in the sample covariance set-up. A limiting procedure then recovers all the alluded to limit distributions. (Joint work with Jose Ramirez.)

Let $K$ be a field. Suppose that the algebraic variety is given by the
set of common solutions
to a system of polynomials in n variables with coefficients in $K$.
Given a solution $P=(a_1,\dots,a_n)$ of this system
with coordinates in the algebraic closure of $K$, we associate to it
an integer called the degree of $P$,
and defined to be the degree of the extension $K(a_1,\dots,a_n)$ over
$K$. When all coordinates $a_i$ belong to $K$,
$P$ is called a $K$-rational point, and its degree is 1. The index of
the variety is the greatest common divisor of all possible degrees of
points on $P$. It is clear that if there exists a $K$-rational point
on the variety, then the index equals 1. The converse is not true in
general. We shall discuss in this talk various properties of the index,
including how to compute it when $K$ is a complete local field using
data pertaining only to a reduction of the variety. This is joint work
with O. Gabber and Q. Liu.

Building on work of Jordan from 1870, in 1979 Harris showed that a geometric monodromy group associated to a problem in enumerative geometry is equal to the Galois group of an associated field extension. Vakil gave a geometric-combinatorial criterion that implies a Galois group contains the alternating group. These Galois groups are difficult to determine, yet they contain subtle geometric information.

Exploiting Harris's equivalence, Leykin and I used numerical homotopy continuation to compute Galois groups of problems involving mostly divisor Schubert classes, finding all to be the full symmetric group. (This included one problem with 17589 solutions.) With Brooks and Martin del Campo, we used Vakil's criterion to show that all Schubert problems involving lines have at least alternating Galois group.

My talk will describe this background and sketch a current project to systematically determine Galois groups of all Schubert problems of moderate size on all small classical flag manifolds, investigating at least several million problems. This will use supercomputers employing several overlapping methods, including combinatorial criteria, symbolic computation, and numerical homotopy continuation, and require the development of new algorithms and software.

Computing, counting, or even deciding on the existence
of real solutions to a system of polynomial equations
is a very challenging problem that is important in many
applications of mathematics. There is an emerging
landscape of structure in the possible numbers of
real solutions to systems of polynomial equations.
These include fewnomial upper bounds, gaps or congruences,
and lower bounds. My talk will survey what is known
about these bounds, focussing on lower bounds---which
are existence proofs of solutions---and open problems,
including some concrete challenges.

Eynard and Orantin have recently defined invariants of any
compact Riemann surface equipped with two meromorphic functions, as a
tool for studying enumerative problems in geometry. I will give a brief
introduction to these invariants and describe a particular example that
encodes the stationary Gromov-Witten invariants of the two-sphere. This
brings new insight into the well-studied problem of the Gromov-Witten
invariants of the two-sphere. Conversely, we gain insight into the
Eynard-Orantin invariants showing that in this example they are related
to the Landau-Ginzburg model dual to Gromov-Witten invariants.

We give a bijection between Parking Functions and Crossed Bar Diagrams and use it to derive properties of Parking Functions that play a crucial role in our attack on the Shuffle Conjecture. This is Bijective Combinatorics at its best, in that a simple bijective weight preserving correspondence between
two families of objects allows us to prove difficult results about one family by working with the other.
The talk will be completely self contained.

We discuss the detailed nature of the geometry of
rotationally symmetric degenerate neckpinch singularities which
develop in the course of Ricci flow.

Inspired by Prof. Jeff Rabin's talk ``What is a Supermanifold?" we will
discuss the analogous algebraic question: What is a Superalgebra? We
will start with basic definitions and examples, and then focus in
particular on Lie Superalgebras and their connections to other areas
of mathematics.

In this talk, a coupling interface method (CIM) is proposed for solving complex interface problems, especially for Poisson-Boltzmann Equation. The coefficients, the source terms, and the solutions may be discontinuous or singular across the interfaces. The method uses adaptive-order strategy and is extended to high dimensions through a dimension-by-dimension approach. To connect information from each dimension, a coupled equation for the principal derivatives is derived through the jump conditions in each coordinate direction. The cross derivatives are approximated by one-side interpolation. The method is easy to implement and flexible to integrate with other approaches. We compare our method with some existing methods. Numerical tests are carefully performed to show the efficiency, robustness and accuracy of our method.

Compressed sensing refers to a growing body of techniques that `undersample' high-dimensional signals and yet recover them accurately using efficient non-linear reconstruction algorithms. Instead of sampling a signal at a rate proportional to its frequency bandwidth, such techniques use a sampling rate proportional to the `information content' of the signal. There exist several useful theories in the literature that promise improvements over ordinary sampling rules in recovering sparse signals. However, most questions regarding the fundamental performance limits of the recovery algorithms are widely open. Such questions are of particular interest in the applications where we need to design the parameters of the systems in advance. In this talk, I present a new framework that settles such questions for a large class of algorithms including the famous $\ell_1$-penalized least squares (LASSO). As a special case of our result, we will derive tight bounds on the noise sensitivity of the LASSO. Furthermore, I will explain the implications and contributions of the new framework for some applications.

This talk is based on a joint work with David Donoho, Iain Johnstone and Andrea Montanari.

In 1983, Makar-Limanov showed that the quotient division algebra of the complex Weyl algebra contains a copy of the free algebra on two generators. This results shows that, unlike in the commutative case, noncommutative localization can behave very pathologically. Stafford and Makar-Limanov conjectured that the following general dichotomy should hold: if a division ring is not finite-dimensional over its center (essentially commutative) then it must contain a free algebra on two generators. We show that for division algebras with uncountable centers a weaker dichotomy holds: such a division ring must either contain a free algebra on two generators or it must be in some sense algebraic over certain division subalgebras. We use this to show that if $A$ is a finitely generated complex domain of Gelfand-Kirillov dimension two then the conjectured dichotomy of Stafford and Makar-Limanov holds for the quotient division ring of $A$; that is, it is either finite-dimensional over its center or it contains a free algebra on two generators. This is joint work with Dan Rogalski.

Abstract: I will talk about a computational problem inspired by the desire to improve the tables of curves over finite fields with many points (http://www.manypoints.org). Namely, if $q$ is a large prime power, how does one go about producing a genus-4 curve over $\mathbb F_q$ with many points? I will discuss the background to this problem and give a number of algorithms, one of which one expects (heuristically!) to produce a genus-4 curve whose number of points is quite close to the Weil upper bound in time $O\left(q^{3/4 + \epsilon}\right).$

Mesh patterns are a generalization of vincular patterns. Mesh patterns
were introduced by Branden and Claesson to provide explicit expansions
for certain permutation statistics as, possibly infinite, linear
combinations of (classical) permutation patterns.

We introduce the notion of a boxed mesh pattern and study avoidance of
these patterns on permutations. We prove that the celebrated former
Stanley-Wilf conjecture is not true for all but eleven boxed mesh
patterns; for seven out of the eleven patterns the former conjecture is
true, while we do not know the answer for the remaining four
(length-four) patterns. Moreover, we show that an analogue of a
well-known theorem of Erdos and Szekeres does not hold for boxed mesh
patterns of lengths larger than 2. Finally, we discuss enumeration of
permutations avoiding simultaneously two or more length-three boxed mesh
patterns, where we meet generalized Catalan numbers.

This is joint work with Sergey Avgustinovich and Alexander Valyuzhenich.

Assuming the natural compactification X of a hypersurface in $(C^*)^n$ is
smooth, it can exhibit any Kodaira dimension depending on the size and
shape of the Newton polyhedron of X. In a joint work with Ludmil
Katzarkov, we give a construction for the expected mirror symmetry
partner of a complete intersection X in a toric variety which works for
any Kodaira dimension of X. The mirror dual might be reducible and is
equipped with a sheaf of vanishing cycles. We give evidence for the
duality by proving the symmetry of the Hodge numbers when X is a
hypersurface. The leading example will be the mirror of a genus two
curve. If time permits, we will explain relations to homological mirror
symmetry and the Gross-Siebert construction.

There is a close relation between some aspects of algebraic
geometry (in particular, birational geometry) and symplectic geometry via
Gromov-Witten theory. For example, Kollar made some conjectures about the
symplectic topology of rationally connected varieties, and Ruan speculated
the existence of the so-called symplectic birational geometry. A theorem of
Graber-Harris-Starr states that sections of rationally connected fibrations
over a curve always exist, which has many important consequences in the
theory of uniruled and rationally connected varieties. In this talk I will
discuss the symplectic analogues of their result and how these results
might be used to understand the conjectures of Kollar and Ruan.

The KdV equation is one of the most important equations in
soliton theory. It can be generalized to Gelfand-Dikii hierarchy and
there have been a lot of work related to it. In this talk, I will give
a geometric interpretation of the equations in Gelfand-Dikii hierarchy
as curve flows in $R^n$. I will also discuss Backlund transformation and
Hamiltonian structures for these curve flows.

Title says it all. ``Alex" will provide math-related clues and you will provide math-related questions. Come have some food and fun!

A poset is called (2+2)-free if it does not contain an induced subposet that is isomorphic to the union of two disjoint 2-element chains. In 1970, Fishburn proved that (2+2)-free posets are in one-to-one correspondence with intensively studied interval orders. Recently, Bousquet-Melou et al. (M. Bousquet-Melou, A. Claesson, M. Dukes, and S. Kitaev, (2+2)-free posets, ascent sequences and pattern avoiding permutations, J. Combin. Theory Ser. A 117 (2010) 884-909.) invented so-called ascent sequences which not only allowed to enumerate (2+2)-free posets (and thus interval orders), but also to connect them to other combinatorial objects, namely to Stoimenow's diagrams (also called regular linearized chord diagrams which were used to study the space of Vassiliev's knot invariants), to certain upper triangular matrices, and to certain pattern avoiding permutations (a very popular area of research these days). Several other papers appeared following the influential work by Bousquet-Melou et al. Among other results, two conjectures, of Pudwell and Jovovic, were solved while dealing with (2+2)-free posets and ascent sequences.

In my talk, I will overview relevant results and research directions.

Last spring, we presented Spectral Variational Integrators, a class of variational integrators that had both excellent conservation properties and exhibited geometric convergence. This talk will present extensions to this work, including Spectral Variational Integrators on lie groups, pseudospectral variational integration of the one dimensional wave equation, and several conjectures about the behavior of Spectral Variational Integrators based on observations of numerical examples.

Triangle groups, the symmetry groups of tessellations of the
hyperbolic plane by triangles, have been studied since early work of
Hecke and of Klein--the most famous triangle group being $\textrm{SL}_2(\mathbb Z).$ We
present a construction of congruence subgroups of triangle groups
(joint with Pete L. Clark) that gives rise to curves analogous to the
modular curves, and provide some applications to arithmetic. We
conclude with some computations that highlight the interesting
features of these curves.

In this talk, I will introduce new cohomologies and elliptic
operators on symplectic manifolds. Their construction follows from a
simple decomposition of the exterior derivative into two first-order
symplectic differential operators, which are analogous to the
Dolbeault operators in complex geometry. These symplectic cohomologies
encode new geometrical invariants especially for non-Kahler symplectic
manifolds. This is joint work with S.-T. Yau.

In this talk I will report on progress on the following two
questions, the first posed by
Cassels in 1961 and the second considered by Bashmakov in
1974. The first question is
whether the elements of the Tate-Shafarevich group are
infinitely divisible when considered
as elements of the Weil-Chatelet group. The second question
concerns the intersection of
the Tate-Shafarevich group with the maximal divisible subgroup
of the Weil-Chatelet group.
This is joint work with Jakob Stix.

The standard playground for a lot of analysis is $L^{p}$ spaces. These function spaces have great global properties (in terms of their relationships with each other and inequalities that connect them) but typically have very bad local properties (most of their constituent functions are extremely rough).

Instead, we will look at some $L^{p}$ spaces of holomorphic (aka complex analytic) functions. These spaces have extremely nice local properties: their elements are as smooth as can be, and they moreover satisfy universal growth estimates you might not expect. By contrast, their global properties are not as nice: for example, they are not related to their dual spaces in the way one might expect.

We'll discuss some of these dichotomies and try to give the flavor of modern research in holomorphic spaces. And we'll discover the truth about the delta function...

By definition, an affine variety is a closed subvariety of some
affine space. A classical result asserts that every smooth affine variety
of dimension n is isomorphic to a closed subvariety of a $2n+1$-dimensional
affine space. Given a fixed smooth affine variety X it is natural to ask
when X can be realized as a closed subvariety of affine space of dimension
$n+d$ for $d < n+1$. In general, there are cohomological obstructions to the
existence of such embeddings, and we will discuss such obstructions in the
context of homotopy theory of varieties (no prior knowledge of this theory
will be assumed).

In this talk I will discuss a combinatorial calculation of the polynomial that counts the number of indecomposable representations of a certain quiver and dimension vector. I will start by introducing quivers, their representations and Kac's results and conjectures on such counting polynomials in general. The combinatorial calculation involves the reliability polynomial of alternating graphs. I will end with the main motivation for the calculation: its relation to the geometry of character varieties.

The (general) Sato-Tate Conjecture for an abelian variety A of
dimension g defined over a number field k predicts the existence of a
compact subgroup ST(A) of the unitary symplectic group USp(2g) that is
supposed to govern the limiting distribution of the normalized Euler
factors of A at the primes where it has good reduction. For the case
g=1, there are 3 possibilities for ST(A) (only 2 of which occur for
k=Q). In this talk, I will give a precise statement of the Sato-Tate
Conjecture for the case of abelian surfaces, by showing that if g=2,
then ST(A) is limited to a list of 52 possibilities, exactly 34 of
which can occur if k=Q. Moreover, I will provide a characterization of
ST(A) in terms of the Galois-module structure of the R-algebra of
endomorphisms of A defined over a Galois closure of k.
This is a joint work with K. S. Kedlaya, V. Rotger, and A. V. Sutherland

Landau damping is relaxation without dissipation. For more than a half century it has been considered as a key phenomenon in plasma physics, and studied both in physics and mathematics, however mainly at the linear level. In this lecture I explain about the physical and mathematical theory of Landau damping, and the recent progress by Mouhot and myself about Landau damping in the nonlinear, close-to-equilibrium regime.

This introductory talk will cover several basic elements of convex
analysis, with particular attention paid to Fenchel conjugation and
infimal convolution. The coverage will begin with an introduction to
Fenchel conjugation and a consideration of its basic properties.
Examples of convex conjugate function pairs will be followed by the
introduction of infimal convolution. General properties of infimal
convolution will be considered, along with some particular properties
of the Moreau envelope case of infimal convolution.

A Calabi-Yau algebra is a noncommutative analogue of the coordinate ring of a Calabi-Yau variety. It is well-known that if $G$ is a group acting on a Calabi-Yau algebra $A$, then the smash product $A \#G$ remains Calabi-Yau under sufficiently good conditions. However, there are cases in which a smash product $A \# G$ may become Calabi-Yau even if $A$ is not Calabi-Yau. We will explain how this can occur by studying the more general notion of a \emph{skew Calabi-Yau algebra}. This is joint work with D.~Rogalski and J.J.~Zhang.

Uncertainties in the Assessment and Detection of Climate Changes

Near absolute zero, a gas of quantum particles can condense into an unusual state of matter, called Bose-Einstein condensation (BEC), that behaves like a giant quantum particle. The rigorous connection has recently been made between the physics of the microscopic many-body dynamics and the mathematics of the macroscopic model, the cubic nonlinear Schrodinger equation (NLS). I'll discuss recent progress with Gerard Ben Arous and Benjamin Schlein on a central limit theorem for the quantum many-body systems, a step towards large deviations for Bose-Einstein condensation.

The paradox of the falling a cat is a famous example in
geometric mechanics. Simply stated, a cat in free fall can execute a
180 degree turn of its body, even though has zero angular momentum
throughout the entire maneuver. In this talk I will discuss how this
seeming paradoxical behavior can be explained through differential
geometry and holonomy, which in turn can offer insights into the
behavior of other mechanical systems. This talk is meant for a general
audience and no knowledge of differential geometry will be assumed.

Spontaneous directed motion, a hallmark of cell biology, is unusual in classical statistical physics. Here we study, using both numerical and analytical methods, organized motion in models of the cytoskeleton in which constituents are driven by energy-consuming motors. Although systems driven by small-step motors are described by an effective temperature and are thus quiescent, at higher order in step size, both homogeneous and inhomogeneous, flowing and oscillating behavior emerges. Motors that respond with a negative susceptibility to imposed forces lead to an apparent negative temperature system in which beautiful structures form resembling the asters seen in cell division.

The signature operator of a Riemannian metric is an
important tool for studying topological questions with analytic
machinery. Though well-understood for smooth metrics on compact
manifolds, there are many open questions when the metric is allowed to
have singularities. I will report on joint work with Eric Leichtnam,
Rafe Mazzeo, and Paolo Piazza on the signature operator on stratified
pseudomanifolds and some of its topological applications.

In recent work with Martin Kolar, a complete classification of ``ruled" infinite type hypersurfaces was found. We discuss this result as well as some work in progress with Ebenfelt and Zaitsev for the general $1$-nonminimal case.

Define a graph G by taking the vertices as $\mathbb{R}^2$ and the edges to be any pair of vertices that are distance 1 apart. The Hadwiger-Nelson Problem asks the chromatic number of this graph, written $\chi(\mathbb{R}^2)$. It is known that either $4\leq \chi(\mathbb{R}^2)\leq 7$ or $5\leq \chi(\mathbb{R}^2)\leq 7$. We explore some approaches to solving this problem, encountering along the way the Axiom of Choice (or lack thereof) and other infinite oddities.

Students of all levels of education tend to have ‘justification schemes’ (Harel & Sowder, 1998) that are inconsistent with conventional validation methods. Yet, there is limited research knowledge about how mathematics instruction can support progressions in students’ justification schemes so that they better approximate conventional validation methods. In this talk, I will draw on findings from a four-year design experiment in an undergraduate mathematics course to present and exemplify an instructional intervention that has been successful in supporting progressions in students’ justification schemes. The notion of ‘cognitive conflict’ featured prominently in the theoretical framework that underpinned the design of the intervention.

Mathematical optimization is still dominated by deterministic models and corresponding algorithms. But many engineering and industrial optimization challenges demand for more realistic modelling including stochastic effects. Common Monte-Carlo methods are too expensive for engineering applications. Polynomial chaos expansions have found to be an efficient mathematical approach for several industrial applications, like turbomachinery design and production failure reduction.

Competitive adsorption of counterions of multiple species to charged surfaces is studied by a size-effect included mean-field theory and Monte Carlo (MC) simulations. The mean-field electrostatic free-energy functional of ionic concentrations, constrained by Poisson's equation, is numerically minimized by an augmented Lagrangian multiplier method. Unrestricted primitive models and canonical ensemble MC simulations with the Metropolis criterion are used to predict the ionic distributions around a charged surface. It is found that, for a low surface charge density, the adsorption of ions with a higher valence is preferable, agreeing with existing studies. For a highly charged surface, both of the mean-field theory and MC simulations demonstrate that the counterions bind tightly around the charged surface, resulting in a stratification of counterions of different species. The competition between mixed entropy and electrostatic energetics leads to a compromise that the ionic species with a higher valence-to-volume ratio has a larger probability to form the first layer of stratification. In particular, the MC simulations confirm the crucial role of ionic valence-to-volume ratios in the competitive adsorption to charged surfaces that had been previously predicted by the mean-field theory. The charge inversion for ionic systems with salt is predicted by the MC simulations but not by the mean-field theory. This work provides a better understanding of competitive adsorption of counterions to charged surfaces and calls for further studies on the ionic size effect with application to large-scale biomolecular modeling. This is joint work with Shenggao Zhou, Zhenli Xu, and Bo Li.

A famous problem of Murray and von Neumann (1943)
asks whether the II$_1$ factors $L(\Bbb F_n)$ associated with free groups
with $n$ generators, $\Bbb F_n$, are non-isomorphic for distinct $n$'s.
While this problem is still open, its ``group measure space'' version,
showing that
the II$_1$ factors $L^\infty(X)\rtimes \Bbb F_n$ arising from
free ergodic probability measure preserving actions $\Bbb
F_n\curvearrowright X$ are non-isomoprphic for $n= 2, 3, ...$,
independently of the actions, has been recently settled by Stefaan Vaes
and myself. I will comment on this, as
well as on related results by Gaboriau, Ozawa, Ioana, Peterson.

This thesis makes contributions to extremal combinatorics, specifically extremal set theory questions and their analogs in other structures. Extremal set theory studies how large or small a family of subsets of a finite set $X$ can be under various constraints. By replacing the set $X$ with another finite object, one can pose similar questions about families of other structures. Remarkably, a question and its analogs essentially have the same answer, regardless of the object. Despite these similarities, not much is known about analogs because standard techniques do not always apply. Our main results establish analogs of extremal set theory results for structures such as vector spaces and subsums of a finite sum. We also study intersecting families and shadows in their classical context of sets by researching a conjecture of Frankl and F\"{u}redi."

Sequential Quadratic Programming (SQP) methods are a popular and successful class of methods for minimizing a generally nonlinear function subject to nonlinear constraints. Under a standard set of assumptions, conventional SQP methods exhibit a fast local convergence rate. However, in practice, a conventional SQP method involves solving an indefinite quadratic program (QP), which is NP hard. As a result, approximations to the second-derivatives are often used, slowing the local convergence rate and reducing the chance that the algorithm will converge to a local minimizer instead of a saddle point. In addition, the standard assumptions required for convergence often do not hold in practice. For such problems, regularized SQP methods, which also require second-derivatives, have been shown to have good local convergence properties; however, there are few regularized SQP methods that exhibit convergence to a minimizer from an arbitrary initial starting point. My thesis considers the formulation, analysis and implementation of: (i) practical methods that use exact second-derivative information but do not require the solution of an indefinite QP, (i) a regularized SQP method with global convergence and (iii) a rigorously defined version of a conventional SQP method with features that have been observed to work in practice for degenerate problems.

The Schramm-Loewner evolution is a one-parameter family of
random growth processes in the complex plane introduced by Oded
Schramm in 1999. In the past decade, SLE has been successfully used to
describe the scaling limits of various two-dimensional lattice models.
One of the first proofs of convergence was due to Lawler, Schramm, and
Werner who gave a precise statement that the scaling limit of
loop-erased random walk is SLE with parameter 2. However, their result
was only for curves up to reparameterization. There is reason to
believe that the scaling limit of loop-erased random walk is SLE(2)
with the very specific natural time parameterization that was recently
introduced by Lawler and Sheffield, and further studied by Lawler,
Zhou, and Rezaei. I will describe several possible choices for the
parameterization of the discrete curve that should all give the
natural time parameterization in the limit, but with the key
difference being that some of these discrete time parameterizations
are easier to analyze than the others. This talk is based on joint
work in progress with Tom Alberts and Robert Masson.

We give the basic definitions of invariants for classifying von
Neumann subfactors, as well as more recent formulations in terms of tensor categories. This will be illustrated with some examples.

The talk gives an introduction to the method for the calculation of electrostatic interactions put forward by Maggs, both in its Monte Carlo and in its Molecular Dynamics version. It is shown that the latter can be viewed as a straightforward application of the Car-Parrinello approach to the coupled dynamics of charges and electromagnetic fields, which is equivalent to a Galilei-invariant form of Maxwell theory. The talk then focuses on more recent developments, where the same idea is applied to solving the Poisson-Boltzmann equation. It is shown that the resulting algorithm is rather simple and intrinsically stable.

We find optimal strategies for a pursuit and evasion game which, when pitted against each other, solve the problem of constructing a small area in the plane in which a unit-length line segment can be rotated. Joint work with Y. Babichenko, Y. Peres, R. Peretz and P. Sousi.

In this talk, we will discuss goal-oriented adaptive methods for second order elliptic PDEs. In particular, we will look at linear nonsymmetric and semilinear problem classes. In goal-oriented methods we are concerned with approximating a given quantity of interest, a function of the weak solution to the PDE. The adaptive algorithm is driven by estimating the error in both the primal and a dual problem, which involves the quantity of interest. We will discuss the formation of an appropriate dual for each type of problem, and how the errors in the primal and dual problems relate to the error in the goal function. Finally, we will look at the contraction framework in each instance and address the appropriate notion of error to show convergence.

Join us in several interactive, hands-on stations ranging from preschool through high school level, formal and informal math and science education. Then take part in a discussion surrounding the implementation of hands-on learning in various environments and grade levels. Presenters include: Ricardo Nemirovsky, SDSU Center for Research in Mathematics and Science Education (CRMSE); Molly Kelton, Doctoral Student SDSU/UCSD Mathematics and Science Education (MSED); Nan Renner, SD Natural History Museum; Sandy Silverman, SD County Office of Education, and more! (See the flyer for details and to register.)

We discuss a new method for obtaining a local limit theorem
(LLT) from a known distributional limit theorem. The method rests on a
simple analytic inequality (essentially due to Hardy, Landau, and
Littlewood) which can be applied directly after quantifying the
smoothness of the distribution of interest. These smoothness terms are
non-trivial to handle and so we also provide new (probabilistic) tools
for this purpose. We illustrate our approach by showing LLTs with
rates for the magnetization in the Curie-Weiss model at high
temperature and for some counts in an Erdos-Renyi random graph. This
is joint work with Adrian Roellin.

Radiation therapy aims at delivering a prescribed dose to cancerous targets using high-energy radiation beams, while sparing dose to surrounding normal tissues and organs at risks. For this purpose, a treatment plan is customized for each individual patient, where parameters in a treatment plan, e.g. beam direction and fluence, are adjusted. Such a problem is mathematically formulated as an optimization problem and is solved with numerical algorithms. This talk will first give an introduction to the treatment plan optimization problem in radiotherapy, including intensity-modulated radiation therapy (IMRT) and volumetric modulated arc therapy (VMAT). It will then focus on a particular problem in IMRT, beam orientation optimization (BOO), which tries to find a solution that contains nonzero fluence map at only a small number of beam angles to achieve a dosimetric objective. We noticed that the objective of the BOO problem is equivalent to finding a fluence map that is sparse at the beam angle level. As such, we introduce a sparsity energy into the total energy function, which takes an L2 norm of beamlet intensities within each angle and then takes a weighted L1 norm over angles. Such an energy term favors solutions with nonvanishing fluence map at only a few beam angles. During optimization, the weighting factors in the L1 norm are adaptively adjusted. Starting with all candidate angles, the optimization process identifies unimportant orientations gradually and removes them without largely sacrificing the dosimetric objective. The whole process terminates when a target number of beams is achieved. The developed BOO algorithm is found to be effective for identifying important beam angles, which leads to better plan qualities than unoptimized beam configurations.

I will present some structural results for $II_1$ factors of products of
hyperbolic groups and their ergodic actions. Applications will be given to
the measure equivalence theory of such groups. This is joint work with
Ionut Chifan and Bogdan Udrea.

We will discuss the recent advances in the theory of profinite
groups and their verbal subgroups.

The first "non-prime" Jones indices are 4, 3 + $\sqrt{5}$ and 6. All
hyperfinite subfactors with index 4 are known, and it follows from work of
Nicoara, Popa and myself that the set of subfactors with composite integer
index is wild. I'll explain some of the beautiful structures appearing
here and will make some comments about the situation of hyperfinite
subfactors with index 3 + $\sqrt{5}$.

Talk time starts at 3:45 PM.

Taking the Long View examines the life of a remarkable mathematician whose classical
Chinese philosophical ideas helped him build bridges between China and the West.
Shiing-shen Chern (1911-2004) is one of the fathers of modern differential geometry. His
work at the Institute for Advanced Study and in China during and after World War II led to
his teaching at the University of Chicago in 1949. Next came Berkeley, where he created a
world-renowned center of geometry, and in 1981 cofounded the Mathematical Sciences
Research Institute. During the 1980s he brought talented Chinese scholars to the United
States and Europe. By 1986, with Chinese government support, he established a math
institute at Nankai University in Tianjin. Today it is called the Chern Institute of Mathematics.

A connected Riemannian manifold $M$ has constant vector
curvature $\epsilon$, denoted by cvc$(\epsilon)$, if every tangent
vector $v \in TM$ lies in a 2-plane with sectional curvature
$\epsilon$. By scaling the metric on $M$, we can always assume that
$\epsilon = -1, 0$, or $1$. When the sectional curvatures satisfy an
additional bound sectional curvature $\leq \epsilon$ or sectional
curvature $\geq \epsilon$, we say that $\epsilon$ is an {\it extremal}
curvature.

In this talk we first motivate the definition and then describe the
moduli spaces of cvc$(\epsilon)$ metrics on three manifolds for each
case, $\epsilon = -1, 0$, or $1$, under global conditions on $M$. For
example, in the case $\epsilon = -1$ is extremal, we show, under the
assumption that $M$ has finite volume, that $M$ is isometric to a
locally homogeneous manifold. In the case that $M$ is compact,
$\epsilon = 1$ is extremal and there are no points in $M$ with all
sectional curvatures identically one, we describe the moduli space of
cvc$(1)$ metrics in terms of locally homogeneous metrics and the
solutions of linear elliptic partial differential equations. Solutions
of some nonlinear elliptic equations arise in the proof.

Abstract: In the continuity method to Kahler-Einstein problem, Tian
conjectured the Bergman kernels of solution metrics are uniformly
bounded below away from 0. I will show that Tian's partial $C^0$
estimate holds on any toric Fano manifold. This allows us to calculate
the multiplier ideal sheaf for certain toric Fano manifolds with large
symmetry. This is an corollary of my earlier study on the limit
behavior of solutions to continuity method on toric Fano manifolds.

This will be a hands-on class on LaTeX typesetting on May 5 in from 10:00 AM to 11:30 AM. The material covered is appropriate for beginners and those with intermediate knowledge of the markup language. Individual questions are welcome and will be answered as time permits.

What is categorification? If you de-categorify Vector-Spaces, you replace isomorphism classes of objects with natural numbers (their dimensions), replace direct sum with addition of those numbers, and replace tensor product with multiplication. To categorify is to undo this process--for instance, one might start with the ring of symmetric functions and realize it has replaced the representation theory of the symmetric group.

In this talk, I will discuss how Khovanov-Lauda-Rouquier (KLR) algebras categorify quantum groups. I will discuss their simple modules, and in particular that they carry the structure of a crystal graph. This is joint work with Aaron Lauda.

Cancer radiotherapy, together with chemotherapy and surgery, form the basis of modern day cancer treatment. Its treatment pro- cess generally involves directing a high energy radiation beam at an identied cancerous growth from dierent directions and with varying beam shapes, durations, and intensities in order to kill the cancerous tissues while preserving nearby healthy ones. Volumetric modulated arc therapy comprises a recently developed setup using a full-rotation trajectory of the beam about the patient along with a multi-leaf collimator for beam shape sculpting. We introduce a variational model in this setup for the optimization of beam shapes and intensities while preserving certain constraints imposed by the equipment used. We apply a binary level-set strategy to represent beam shapes and a fast sweeping technique to satisfy beam intensity variation limits. The result is a ow-based shape optimization algorithm that guarantees constraint satisfaction and energy decrease for the generation of improved treatment plans in volumetric modulated arc therapy.

In his 1985 monograph Interval Orders and Interval Graphs, Fishburn noted the dearth of enumerative results for interval orders and labelled semiorders, standing in contrast to the well-understood case of interval graphs and unlabelled semiorders. (The latter are enumerated by the Catalan numbers.) Recently, work by Bousquet-Mélou et al. linked certain integer sequences termed ascent sequences to unlabelled interval orders. This allowed for an asymptotic enumeration of unlabelled interval orders through earlier work by Zagier involving the same generating function that enumerates ascent sequences. Building on subsequent work by Khamis, this talk develops an asymptotic enumeration of the labelled interval orders on an $n$-element set. This is joint work with Graham Brightwell (LSE).

We present a method to enhance the quality of a video sequence captured through a turbulent atmospheric medium. Enhancement is framed as the inference of the radiance of the distant scene, represented as a latent image," that is assumed to be constant throughout the video. Temporal distortion is thus zero-mean and temporal averaging produces a blurred version
of the scene's radiance, that is processed via a Sobolev gradient flow to yield the latent image in a way that is reminiscent of the lucky region" method. Without enforcing prior knowledge, we can stabilize the video sequence while preserving ne details. We also present the well-posedness theory for the stabilizing PDE and a linear stability analysis of the numerical scheme.
This is a joint work with Sung Ha Kang, Stefano Soatto and Andrea Bertozzi.

We will discuss recent developments
in the probabilistic and spectral approaches for graph limits.
In particular, we will extend the notion of quasi-randomness,
which concerns a class of equivalent properties that random graphs satisfy.
For example, we will give several necessary and sufficient conditions for a graph to be the union of two or more quasi-random graphs.
One of these characterizations involves eigenvalues and scalable eigenspaces,
that we call "graphslets", which dictate the behavior of graph limits
for both dense and sparse graphs.

The interplay between geometry and electrostatics contributes significantly to hydrophobic interactions of biomolecules in an aqueous solution. With an implicit solvent, such a system can be described macroscopically by the dielectric boundary that separates the high-dielectric solvent from low-dielectric solutes. This work concerns the motion of a model cylindrical dielectric boundary as the steepest descent of a free-energy functional that consists of both the surface and electrostatic energies. The effective dielectric boundary force is defined and an explicit formula of the force is obtained. It is found that such a force always points from the solvent region to solute region. In the case that the interior of a cylinder is of a lower dielectric, the motion of the dielectric boundary is driven initially by the surface force but is then driven inward quickly to the cylindrical axis by both the surface and electrostatic forces. In the case that the interior of a cylinder is of a higher dielectric, the competition between the geometrical and electrostatic contributions leads to the existence of equilibrium boundaries that are circular cylinders. Linear stability analysis is presented to show that such an equilibrium is only stable for a perturbation with a wavenumber larger than a critical value. Numerical simulations are reported for both of the cases, confirming the analysis on the role of each component of the driving force. Implications of the mathematical findings to the understanding of charged molecular systems are discussed. This is joint work with Li-Tien Cheng, Bo Li, and Shenggao Zhou.

Open Gromov-Witten invariants are essential ingredients of Lagrangian-Floer intersection theory, and they serve as quantum corrections in mirror symmetry from SYZ viewpoint. They are difficult to compute in general due to non-trivial obstructions in the moduli. In this talk, I will illustrate by examples how to compute open Gromov-Witten invariants of toric manifolds, by relating them to closed Gromov-Witten invariants which are better understood. This also gives an enumerative meaning of mirror maps. This is joint work with Kwokwai Chan, Naichung Leung and Hsian-Hua Tseng.

The classical result of H.Poincare states that a local
biholomorphic mapping of an open piece of the 3-sphere in
$\mathbb{C}^2$ onto another open piece extends analytically to a global holomorphic automorphism of the sphere. A big stream of further publications was dedicated to the possibility
to extend local biholomorphic mapping between real hypersurfaces in complex space. The most general results were obtained by D.Hill, R.Shafikov and K.Verma who generalized Poincare's extension phenomenon for the case of an essentially finite hypersurface in the preimage and a quadric in the image, and also for the case of a minimal hypersurface (in the sense of Tumanov) in the preimage and a sphere
in the image.
In this joint work with R.Shafikov we consider the - essentially new - case where a hypersurface $M$ in the
preimage contains a complex hypersurface, i.e. where $M$ is nonminimal. We demonstrate that the above extension results fail in this case, and prove the following analytic continuation phenomenon: a local biholomorphic mapping of $M$ onto a non-degenerate hyperquadric in $\mathbb{CP}^n$ extends
to a punctured neighborhood of the complex hypersurface $X$, lying in $M$, as a multiple-valued locally biholomorphic mapping. The extension phenomenon is based on the properties of Segre sets introduced by Baouendi, Ebenfelt and Rothschild
near the complex hypersurface $X$. We also establish an interesting interaction between nonminimal spherical real hypersurfaces and linear differential equations with an isolated singular point.

We are exploring the viability of a novel approach to cardiocyte contractility assessment based on biomechanical properties of the cardiac cells, energy conservation principles, and information content measures. We define our measure of cell contraction as being the distance between the shapes of the contracting cell, assessed by the minimum total energy of the domain deformation (warping) of one cell shape into another. To guarantee a meaningful vis-a-vis correspondence between the two shapes, we employ both a data fidelity term and a regularization term. The data fidelity term is based on nonlinear features of the shapes while the regularization term enforces the compatibility between the shape deformations and that of a hyper-elastic material. We tested the proposed approach by assessing the contractile responses in isolated adult rat cardiocytes and contrasted these measurements against two different methods for contractility assessment in the literature. Our results show good qualitative and quantitative agreements with these methods as far as frequency, pacing, and overall behavior of the contractions are concerned. We hypothesize
that the proposed methodology, once appropriately developed and customized, can provide a framework for computational cardiac cell biomechanics that can be used to integrate both theory and experiment. For example, besides giving a good assessment of contractile response of the cardiocyte, since the excitation process of the cell is a closed system, this methodology can be employed in an attempt to infer statistically significant model parameters for the constitutive equations of the cardiocytes.

Toric varieties have rich connections to plane geometry, which allows questions about algebraic geometry (hard) to be reformulated into questions about combinatorics (easy). In this talk, we will introduce toric varieties, and we will discuss several examples. This talk is intended to be fun, and should be generally accessible.

Among the various ideas put forward in the search for a theory
of quantum gravity, the causal set hypothesis is distinguished
by its logical simplicity and by the fact that it incorporates
the assumption of an underlying spacetime discreteness
organically and from the very beginning. After presenting the
problem of quantum gravity in general, I will precis the
causal set programme and touch on some old and some recent
developments.

Stochastic networks are used as models for complex processing systems involving dynamic interactions subject to uncertainty. Applications arise in high-tech manufacturing, the service industry, telecommunications, computer systems and bioengineering. The control and analysis of such networks present challenging mathematical problems. In this talk, a concrete application will be used to illustrate a general approach to the study of stochastic processing networks based on deriving more tractable approximate models. Specifically, we will consider a model of Internet congestion control in which processing can involve the simultaneous use of several resources (or links), a phenomenon that is not well understood. Elegant fluid and diffusion approximations will be derived and used to study the performance of this model. A key insight from this analysis is a geometric representation of the consequences of using a "fair" policy for the sharing of resources. The talk will conclude with a summary of the current status and description of open problems associated with approximate models for general stochastic processing networks.

In 1961 G.E.Wall conjectured that the number of maximal subgroups in a finite group is bounded by the order of group. In this talk I will discuss a generalization of this conjecture in the setting of subfactors and recent progress on related problems.

The talk will describe old and new(er) results on the structure of modules for the Virasoro algebra. Joint work of Rocha and Goodman with the speaker will be the old work. A newer result to be is a description of the annihilators of Verma modules (analogous to Dixmier's result for semi-simple Lie algebras) and its relationship with Small's albatross.

The human brain connectome is an ambitious project to provide a complete map of neural connectivity and a recent source of excitement in the neuroscience community. Just as the human genome is a triumph of marrying technology (high throughput sequencers) with theory (dynamic programming for sequence alignment), the human connectome is a result of a similar union. The technology in question is that of diffusion magnetic resonance imaging (dMRI) while the requisite theory, we shall argue, comes from three areas: PDE, harmonic analysis, and convex algebraic geometry.

The underlying mathematical model in dMRI is the Bloch-Torrey PDE but we will approach the 3-dimensional imaging problem directly. The main problems are (i) to reconstruct a homogeneous polynomial representing a real-valued function on a sphere from dMRI data; and (ii) to analyze the homogeneous polynomial via a decomosition into a sum of powers of linear forms. We will focus on the nonlinear approximation associated with (ii) and discuss a technique that combines (i) and (ii) for mapping neural fibers.

This is joint work with T. Schultz of MPI Tubingen.

Cancer is currently viewed as an evolutionary process. In an organ there is a population of cells that give birth, die and mutate according to population dynamics that are determined by the types of cells under consideration. If certain cell mutations are acquired then the cells can become cancerous. In this manuscript we consider two evolutionary models that may each be viewed as a model of cancer. One is a model of colorectal cancer. We discuss results pertaining to the time it takes to develop cancer and the location of the mutations. The other model is a general
Moran-type model. We discuss results pertaining to the rate of adaptation.

The need to simulate rare events occurs in many application areas,
including telecommunication, finance, insurance, computational physics
and chemistry. However, virtually any simulation problem involving
rare events will have a number of mathematical and computational
challenges. As it is well known, standard Monte Carlo sampling
techniques perform very poorly in that the relative errors under a
fixed computational effort grow rapidly as the event becomes more
rare. In this talk, I will discuss large deviations, rare events and
Monte Carlo methods for systems that have multiple scales and that are
stochastically perturbed by small noise. Depending on the type of
interaction of the fast scales with the strength of the noise we get
different behavior, both for the large deviations and for the
corresponding Monte Carlo methods. Using stochastic control arguments
we identify the large deviations principle for each regime of
interaction. Furthermore, we derive a control (equivalently a change
of measure) that allows to design asymptotically efficient importance
sampling schemes for the estimation of associated rare event
probabilities and expectations of functionals of interest. Standard
Monte Carlo methods perform poorly in these kind of problems in the
small noise limit. In the presence of multiple scales one faces
additional difficulties and straightforward adaptation of importance
sampling schemes for standard small noise diffusions will not produce
efficient schemes. We resolve this issue and demonstrate the
theoretical results by examples and simulation studies. Applications
of these results in chemistry problems and in mathematical finance
will also be discussed.

The closely placed phosphate charges along the charged biopolymer DNA backbone leads to strong electrostatic repulsion. However, when the DNA is immersed in an aqueous solution containing monovalent or divalent cations from added salts, the free energy of the system is lowered when counterions from the bulk condense on the backbone of the DNA. According to counterion condensation theory, each phosphate charge is reduced by the factor z theta, where theta is the number counterions associated per phosphate charge, z is the valence of counterions. Brownian Dynamics simulations also can be used to quantitatively describe condensation of monovalent and multivalent ions (from added salt) on the backbone of DNA.

The tumor suppressor gene p53 is responsible for maintaining the integrity of the human genome and plays a vital role in DNA repairing machinery. Loss of p53 tumor suppressor activity is a frequent defect in ~ 50% of human cancers. MDM2 controls the stability of p53 through ubiquitation to target the tumor suppressor protein for degradation by the proteasome. Inhibition the interactions between p53 and the E3 ubiquitin ligase MDM2/MDMX will reactivate the p53 pathway and selectively kill tumor cells. Extensive molecular dynamics simulations were used to study hydrocarbon linker stapled alpha-helical peptides which could be potential inhibitors of p53 peptide and MDM2.

Differential geometry and Lie theory have traditionally provided the mathematical framework for our most intuitive physical theory: classical mechanics. However, as is well-known, in the last century physicists developed newer theories which incorporate different kinds of symmetries, and bold concepts like the uncertainty principle have arisen that need to be addressed mathematically. Mathematical physicsists' response has been a constant search for methods of quantization and superization, thus allowing the integration of older techniques into these newer, broader theories. This talk will explain one part of this story in more detail. In particular we will describe super quantum group theory, an eclectic collection of theorems and conjectures whose development is very much still in progress, but one that promises a solution to some foundational questions in mathematical physics. The mathematical background needed is limited (I will provide the relevant definitions), the physical background needed is none (I will, however, assume that all members of the audience were born in the twentieth century); the main prerequisite for this talk is a curious mind which is willing to accommodate some occasional vague language.

In this talk we will introduce a variation of the classical Turan problem of determining the maximum number of edges in an $n$-vertex graph that does not contain a fixed forbidden graph. We will present some results and open problems. The talk is intended for a general audience and should be accessible for advanced undergraduates.

We performed molecular dynamics simulations to study the character of hydrophobic interaction between two nanoscale particles in water. For a systematic study of water density fluctuations induced by the hydrophobic interaction, we prepared a graphene plate and also other model plates made of “carbon” atoms that had different interaction strength with water. We calculated the interaction between two identical “carbon” plates immersed in water, and calculated the fluctuations in the number of water molecules in the confined space between two plates. The result showed that fluctuations in some cases are strongly enhanced compared to the fluctuations observed next to a single plate. If the character of water fluctuations in the confined space determines the character of hydrophobic interactions, then it is possible to conclude that the interaction between graphene plates in water is hydrophobic.
In another study, we investigated the effect of roughness on hydrophobic interaction (the rough hydrophobic surface was created by attaching non-polar headgroups to the graphene plates). Our study demonstrated that roughness enhances hydrophobic interactions. As a result of this enhancement, we observed a dewetting transition between two rough hydrophobic surfaces, which would not occur between the corresponding particles with smooth surfaces.

On June 1st and June 8th, Ken Intrilligator and myself will run
an informal seminar on cluster algebras as entering into string
theory and some of my own work on mirror symmetry. Anyone interested
is welcome.

Kac-Moody groups generalize Lie groups but are typically infinite
dimensional. This defense will quickly introduce discrete and
topological Kac-Moody groups and outline a direct comparison between complex topological Kac-Moody groups and discrete Kac-Moody groups over the algebraic closure of the field with p elements. This result uses newly constructed homotopy decompositions for the "unipotent" factors of parabolic subgroups of a discrete Kac-Moody group in terms of unipotent algebraic groups. Additional applications will be given and the topics of infinite Coxeter groups, BN-pairs, and root group data systems will be visited.

A multiplicative cascade is a randomization of any measure on the unit interval, constructed from an iid collection of random variables indexed by the dyadic intervals. Given an arbitrary initial measure I will describe a method for constructing a continuous time, measure valued process whose value at each time is a cascade of the initial one. The process also has the Markov property, namely at any given time it is a cascade of the process at any earlier time. It has the further advantage of being a martingale and, under certain extra conditions, it is also continuous. I will discuss applications of this process to models of tree polymers and one dimensional random geometry.

Joint work with Ben Rifkind (University of Toronto).

Embedding problem for real-algebraic hypersurfaces dates back to
1978 when Webster proved that real-algebraic hypersurfaces is embeddable into a hyperquadric of possibly higher dimension. In a recent paper joint with Peter Ebenfelt, we showed that this is not true for the spheres case. We will exhibit an explicit example of a close, strictly pseudoconvex hypersurface and show that it is not locally holomorphically embeddable into a sphere of any dimension whatsoever by showing that the point at infinity is an obstruction for local embedding at all point.

By the work of Harer, the reduced homology of the complex of curves is a fundamental cohomological object associated to all torsion free finite index subgroups of the mapping class group. We call this homology group the Steinberg module of the mapping class group. It was previously known that the curve complex has the homotopy type of a bouquet of spheres. Here, we give the first explicit homologically nontrivial sphere in the curve complex and show that under the action of the mapping class group, the orbit of this homology class generates the reduced homology of the curve complex.

The main well developed numerical methods for Stochastic PDEs are
Stochastic Galerkin method and Stochastic Collocation method. The error
estimators of linear Poisson problem from those two methods corresponding
to numerical solutions, mean and second moment of numerical solution are
analyzed properly already. However, the analysis of other types of linear
and nonlinear models are still open. My talk will consider a stochastic
nonlinear Diffusion Reaction model, and analyze well-posedness of its weak
form in a new extended group of Banach spaces, additionally the
discretization of weak solution will be discussed.

In this talk I will present recent small data global well-posedness results for energy-critical nonlinear Schroedinger equations on specific compact manifolds, such as tori and spheres. Key ingredients are certain multilinear estimates of Strichartz type as a replacement for the classical dispersive estimates which fail in this setup.

In the recent FPSAC meeting in Nagoya, Michele D'Adderio
posed the problem of proving that a family of polynomials in N[q,t] that q,t-enumerate the convex polyominos in the nxm square are symmetric in q,t. In this talk we show how two beautiful bijections of Michele D'Adderio and Angela Hicks combine to yield the symmetry result as well as its connection to the theory of Macdonald Polynomials and Diagonal Harmonics.

We consider a switched (queueing) network in which there are constraints on which queues may be served simultaneously; such networks have been used to effectively model input-queued switches, wireless networks and more recently data-centers. The scheduling policy for such a network specifies which queues to serve at any point in time, based on the current state or past history of the system. As the main result, we shall discuss a new class of online scheduling policies that achieve optimal scaling for average queue-size for a class of switched networks including input-queued switches. Time permitting, we shall discuss various exciting open questions in the domain of stochastic networks.
This is based on joint work with Neil Walton (Univ of Amsterdam) and Yuan Zhong (MIT).

This refers to the following problem. Given a sequence of numbers, when are they the moments
of some measure (or distribution if you like to normalize to the case that is known as probability.)
That is, given a sequence $(a_n)^{\infty}_{n=0}$ when is there a measure such that $a_n = \int x^n d\mu{x}.$
To solve this problem, we introduce some basic modern analysis, specifically Hilbert space techniques, which
were invented early in the last century to solve problems like the Hamburger moment problem.

This lecture is an appetizer for my two books in OUP: Numerical Methods for Nonlinear Elliptic Differential Equations, A Synopsis, and Numerical Methods for Bifurcation and Center Manifolds in Nonlinear Elliptic and Parabolic Differential Equations, 2010 and 2011. We extend for the first time the linear discretization theory of Schaback, developed for meshfree methods, to nonlinear operator equations, relying heavily on methods of Boehmer, Vol I. There is no restriction to elliptic problems nor to symmetric numerical methods like Galerkin techniques. Trial spaces can be arbitrary, but have to approximate the solution well, and testing can be weak or strong. We present Galerkin techniques as an example. On the downside, stability is not easy to prove for special applications, and numerical methods have to be formulated as optimization problems. Results of this discretization theory cover error bounds and convergence rates. As an example we present the meshless method for some nonlinear elliptic problems of order 2. Some numerical examples are added for illustration.

On average, I only get to do this once every 7 years

"Symmetric algebra" is a fancy way of saying "polynomial ring." The symmetric algebra of a k-vector space V is the enveloping commutative algebra of V in the category kVect, and can be realized as the polynomial ring generated by any basis of V.

Quantum symmetric algebras are analogues of polynomial rings in the category of modules over the quantized universal enveloping algebra of a semisimple Lie algebra. Familiar examples include quantum polynomial and matrix algebras as well as coordinate algebras of quantum Euclidean and symplectic vector spaces, but there are more exotic ones also. I will describe the general construction of quantum symmetric algebras and show that they satisfy a universal mapping property analogous to the one for ordinary symmetric algebras. This requires an appropriate notion of commutativity for algebras in Uq(g)-Mod.

I will try to illustrate the general theory with simple examples.

Quantum cellular automata (QCA) are models of quantum computation of particular interest from the point of view of quantum simulation. Quantum lattice gas automata (QLGA - equivalently partitioned quantum cellular automata) represent an interesting subclass of QCA. Prior work on QCA has investigated the relationship between these two classes of models. In the present paper we establish necessary and sufficient conditions for unbounded, finite Quantum Cellular Automata (QCA) (finitely many active cells in a quiescent background) to be Quantum Lattice Gas Automata. We define a local condition that classifies those QCA that are QLGA, and we show that there are QCA that are not QLGA.

Simulation of fermionic systems has been a topic of interest in quantum
simulation since Feynman's first papers on the topic. It has been known
for some time how to simulate fermionic systems and scalable proposals
for electronic structure calculations on quantum computers require some
solution to this problem. Current work makes use of the Jordan-Wigner
transformation to track phases arising from exchange anti-symmetry. For
a single term in a fermionic Hamiltonian on N modes the Jordan wigner
transformation requires an overhead of O(N) gates. In this talk I will
give an alternative to the Jordan Wigner transformation, originally
developed by Bravyi and Kitaev, which reduces this overhead to O(log N).
We give the details of this transformation for electronic structure
Hamiltonians and give the minimal basis model of the Hydrogen molecule
as an example.

The Jacobi ensemble is one of three ensembles of classical random matrix theory. It has a corresponding matrix form, so that a natural endeavor is to prove universality for the spectral properties of the matrix form. In joint work with L. Erdõs we provide the first such result. More interestingly, this matrix form has special relevance to other areas of mathematics because its eigenvalues describe the angles between random subspaces. We will consider random subspaces spanned by Euclidean and Fourier vectors and show how the Jacobi ensemble is related to discrete uncertainty principles.

The notion of convexity of vector spaces can be generalized to dimension-free sets of matrices. A natural question that then arises is how to identify a set's matrix convex hull. We introduce some techniques for getting a grasp on the matrix convex hull of semi-algebraic sets and look at one of these in particular: The bent TV screen. This is the set

$$ \{ (X, Y) : I - X^2 - Y^4 \succeq 0 \} $$

It is convex in the scalar case but is not matrix convex. In fact, there is no known "simple" formula for its matrix convex hull.

Polynomial invariants provide a tool to characterise quantum states
with respect to local unitary transformations. Unfortunately, the
situation becomes very complicated already for mixed states of three
qubits due to combinatorial explosion.

After an introduction to the mathematical background and general tools,
the talk will present preliminary results for mixed quantum states and
Hamiltonians for three-qubit systems.

The talk is based on joint work in progress with Robert Zeier.

Consider two random subspaces of a finite-dimensional vector space -- i.e. two random projection matrices P and Q. What is the dimension of their intersection? This (random) integer is almost surely equal to its minimal possible value, which corresponds to the subspaces being in general position. Many more delicate questions about the geometry of the configuration are encoded by the principle angles between the subspaces, which are determined by the eigenvalues of the operator-valued angle matrix PQP.
The situation is much more complicated in infinite-dimensions. Even the question of whether two random projections are likely to be in general position is difficult to make sense of, let alone answer. Nevertheless, understanding the operator-valued angle in an infinite-dimensional setting is of critical importance to the biggest open problem in free probability theory -- the so-called ``Unification Conjecture'' -- with ramifications for operator algebras, information theory, and random matrices.
In this talk, I will discuss recent and ongoing joint work with Benoit Collins, addressing the configuration of random subspaces in an infinite-dimensional context. Using a mixture of techniques from stochastic processes, PDEs, and complex analysis, we prove the general position claim and give a complete understanding of the associated geometry. This work proves an important special case of the Unification Conjecture, and has interesting implications for the original finite-dimensional setting as well.

The study of Cohen-Macaulay rings and modules has been a central topic in Commutative Algebra for many years. Among other things, they have played a major role in the investigations of the "Homological Conjectures", a set of problems on finite projective dimension, intersection theory, and related subjects. More recently, as a result of advances in the homological conjectures and developments in Arithmetic Geometry, a number of questions have come up about "almost" Cohen-Macaulay modules and algebras. In this talk I will give some background to these topics, discuss what almost Cohen-Macaulay algebras are and why they are interesting, and present various recent developments and open problems in this area.

In this talk I shall present an isoperimetric comparison
theorem for the Ricci flow on surfaces, inspired by Hamilton's
isoperimetric estimate. I will show how this can be used to prove the
Hamilton/Chow theorem, that the Ricci flow converges to a constant
curvature metric, thus for example providing a proof of the famous
uniformization theorem. This was joint work with Ben Andrews.

Answering a question of Paul Erd\H{o}s, Antal Balog and Endre
Szemerédi proved that a finite set $A \subset \mathbb{Z}$ with many
three term arithmetic progressions must have a long arithmetic
progression. We will
discuss the proof of this result which uses the Balog-Szemer\'{e}di-Gowers
Theorem, Freiman's Theorem, and Szemeredi's Theorem on arithmetic
progressions. No previous knowledge of additive number theory will be
assumed.

I will survey some recent progress on the classification of von
Neumann algebras arising from countable groups and their measure
preserving actions on probability spaces. This includes the finding of the
first classes of (superrigid) groups and actions that can be entirely
reconstructed from their von Neumann algebras.

Rapid advance of quantum technologies demands novel mathematical tools for engineering complex quantum systems. Characterization of the structural and dynamical properties of large-scale quantum devices, e.g., a quantum computer with 100 qubits, is among the current challenges. The major obstacle is the size of the Hilbert space and therefore the required experimental and computational resources that grow exponentially with the number of the system components. Recently, compressed sensing method has been applied for efficient characterization of quantum systems. Originally developed in classical signal processing, compressed sensing is a method to compress high-dimensional signals with a small number of measurements assuming that the signals live on a low-dimensional manifold, and then to
reliably reconstruct them. In this presentation, I talk about the compressed sensing theory for quantum inversion problems, its first experimental realization, and the new problems motivated by quantum applications.

[1] A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome,
M.P. Almeida, A. Fedrizzi and A. G. White,
”Efficient measurement of quantum dynamics via compressive sensing”,
Phys. Rev. Lett 106, 100401 (2011).

[2] A. Shabani, M.Mohseni, S. Lloyd, R. L. Kosut and H. Rabitz,
”Estimation of many-body quantum Hamiltonians via compressive sensing”,
Phys. Rev. A 84, 012107 (2011).

Practically every result that is presented in an elementary course
in number theory (i.e. Math 104 at UCSD) is used in the proof that this
joint work with D. Meyer works and gives an algorithm in the quantum
computing class analogous to P (polynomial).

Kirchhoff's Matrix Tree Theorem is one of the classical results in
spectral graph theory and it gives a formula for the number of
spanning trees of a graph in terms of the eigenvalues of its
Laplacian.
In 1973, Chvatal introduced the notion of graph toughness and made two
important conjectures: 1. Any graph with sufficiently large toughness
is Hamiltonian. 2. Any graph with sufficiently large toughness is
pancyclic.
The first conjecture is still open, but the second conjecture was
disproved by Alon who showed that there exist graphs with arbitrarily
large toughness and girth. The key to Alon's argument was determining
a close relation between the toughness of a regular graph and its
eigenvalues.
Independently and around the same time 1995, Brouwer found a slightly
better result relating the toughness of a regular graph to its
eigenvalues. In this talk, I will present some tight connections
between the eigenvalues of a connected regular graph and the maximum
number of edge-disjoint spanning trees in the graph that can be seen
as a spectral version of Nash-Williams/Tutte Theorem. I will show some
improvements of Brouwer's bound in certain ranges of toughness and
discuss another problem of Brouwer related to the toughness of graphs
attaining equality in the Hoffman ratio bound for the independence
number. This is joint work with my Ph.D. student, Wiseley Wong.

We discuss algebras over a field with the unusual property that all of their subalgebras are noetherian. We discuss some of the general results one can prove about such algebras. Some well-known algebras associated to elliptic curves turn out to have this property, and we discuss these examples in detail.

A corollary of the Erd\H{o}s-Stone theorem is that, for any $0 \leq
\alpha < 1$, graphs with density greater than $\alpha$ contain an
(arbitrarily) large subgraph of density at least $\alpha+c$ for some
fixed $c = c(\alpha)$, so long as the graph itself is sufficiently
large. This phenomenon is known as a jump at $\alpha$. Erd\H{o}s
conjectured that similar statements should hold for hypergraphs, and
multigraphs where each edge can appear with multiplicity at most $q$,
for $q \geq 2$ fixed. Brown, Erd\H{o}s, and Simonovits answered this
conjecture in the affirmative for $q=2$, that is for multigraphs where
each edge can appear at most twice. R\"{o}dl answered the question in

We will show that smooth complete solutions to the Ricci flow of
uniformly bounded curvature are analytic in time in the interior of
their interval of existence. The analyticity is a consequence of
classical Bernstein-Bando-Shi type estimates on the temporal and
spatial derivatives of the curvature tensor, and offers an alternative proof of the unique continuation of solutions to the Ricci flow. As a further application of these estimates, we will show that, under the above global hypotheses, about any interior space-time point (x0, t0), there exist local coordinates x on a neighborhood U of x0 in which the representation of the metric is real-analytic in both x and t on some cylinder over U.

Biological membranes are composed of (among other things) hundreds of different lipids, which are believed to segregate into fluid rafts, which may be relevant to processes like virus assembly. I'll talk about the spherical cow version of cells, synthetic membranes with three components (saturated and unsaturated lipids and cholesterol), which also segregate into two fluid phases. Membranes are also particularly interesting from a physical standpoint because they have both two- and three-dimensional hydrodynamic behavior ("quasi-2D"), with many strange features, such as diffusion coefficients of membrane rafts being effectively independent of their size. These quirks are characteristic of many interfacial fluids, and also appear in thin layers of liquid crystals and protein films at the air-water interface. I'll show some continuum stochastic simulations of membrane domains and phase separation, discuss new ways of measuring membrane viscosity, and suggest why some well-known dynamical scaling laws can change their exponents or even break down for phase separation in membranes. If there's time, I will also discuss how the dynamics of protein diffusion can be altered by coupling to the lipid membrane.

A sequence $(a_1,..., a_n)$ of positive integers is a parking function if its nondecreasing rearrangement $(b_1 \leq ... \leq b_n)$ satisfies $b_i < i + 1$ for all $i$. Parking functions were introduced by Konheim and Weiss to study a hashing problem in computer science, but have since received a great deal of attention in algebraic combinatorics. We will define two new objects attached to any (finite, real, irreducible) reflection group which generalize parking functions and deserve to be called parking spaces. We present a conjecture (proved in some cases) which asserts a deep relationship between these constructions. This is joint work with Drew Armstrong at the University of Miami and Vic Reiner at the University of Minnesota.

We will discuss some recent developments in several new directions of
spectral graph theory and mention a number of open problems.

This talk will be an exposition of joint work with Man Wai (Mandy)
Cheung on the effect of the Ricci flow on homogeneous metrics of
positive sectional curvature on flag varieties over the complex,
quaternions and octonians. The speaker’s 1972 paper shows that these
metrics exist only in the case of the variety of flags in the two
dimension projective space over these fields. Here are some of the
results:
All cases can flow from strictly positive curvature to some negative
sectional curvature.
All cases can flow from positive definite Ricci curvature to
indefinite Ricci curvature
The quaternionic and octonianic cases can flow from strictly positive
sectional curvature to indefinite Ricci curvature (in the case of the
quaternions this is a result of Boehm and Wilking).
In the complex case the flow keeps the metrics of strictly positive
curvature in the metrics with positive definite Ricci curvature.

I consider a Markov process inspired by a toy model of a limit order book. "Bid" and "ask" orders arrive in time; the prices are iid uniform on [0,1]. (I'll discuss some extensions.) When a match is possible (bid > ask), the highest bid and lowest ask leave the system. This process turns out to have surprising dynamics, with three limiting behaviours occurring with probability one. At low prices (< 0.21...), bids eventually never leave; at high prices (>0.78...), asks eventually never leave; and in between, the system "ought to" be positive recurrent. I will show how we can derive explicitly the limiting distribution of certain marginals for the middle prices; this makes it possible to extract the numerical values above from a 0-1 Law result.

Alternating direction methods are a commonplace tool for general mathematical programming and optimization. These methods have become particularly important in the field of variational image processing, which frequently requires the minimization of non-differentiable objectives. This paper considers accelerated (i.e., fast) variants of two common alternating direction methods: the Alternating Direction Method of Multipliers (ADMM) and the Alternating Minimization Algorithm (AMA). The proposed acceleration is of the form first proposed by Nesterov for gradient descent methods. In the case that the objective function is strongly convex, global convergence bounds are provided for both classical and accelerated variants of the methods. Numerical examples are presented to demonstrating the superior performance of the fast methods.

Lights Out is a single player game on graph G. The game starts with a coloring of the vertices of G with two colors, 0 and 1. At each step, one vertex is toggled which switches the color of that vertex and all of its neighbors. The game is won when all vertices have color 0. This game can be analyzed using linear algebra over a finite field, for example the number of solvable colorings of a graph is 2 to the rank of A + I, where A is the adjacency matrix of the graph, and I is the identity.
We consider the stochastic process arising from toggling a sequence of random vertices. We demonstrate how the process can be viewed as a random walk on an associated state graph. We then find the eigenvalues of the state graph, and use them to bound the rate of convergence and hitting times. We also provide bounds on the average number of steps until this random process reaches the all 0 coloring that are asymptotically tight for many families of graphs.

If two simple linear algebraic groups have the same F-isomorphism classes of maximal F-tori, are the two groups necessarily isomorphic? When F is a number field, it is an old question attributed to Shimura. We describe the recent solution to this question (which relies on the notion of weak commensurability introduced by Gopal Prasad and Andrei Rapinchuk) and its connection with the question "Can you hear the shape of a drum?" for arithmetic quotients of locally symmetric spaces.

This talk will cover some of the main problems in the field of nonlinear dispersive equations. I will discuss the stability, instability and blow-up for some simpler models such as the cubic Nonlinear Schr\"odinger equations

I will start with a basic introduction to planar algebras. I will
then discuss recent joint work with Scott Morrison (arXiv:1208.3637) and
recent joint work with Stephen Bigelow (arXiv:1208:1564). With
Morrison, we construct a new exotic subfactor planar algebra using
Bigelow's jellyfish algorithm. With Bigelow, we determine exactly
when a planar algebra has a presentation by generators and jellyfish
relations.

I will start with a basic introduction to planar algebras. I will
then discuss recent joint work with Scott Morrison (arXiv:1208.3637) and
recent joint work with Stephen Bigelow (arXiv:1208:1564). With
Morrison, we construct a new exotic subfactor planar algebra using
Bigelow's jellyfish algorithm. With Bigelow, we determine exactly
when a planar algebra has a presentation by generators and jellyfish
relations.

The Margulis Normal Subgroup Theorem states that any normal
subgroup of an irreducible lattice in a center-free higher-rank semisimple
Lie
group is of finite index. Stuck and Zimmer, expanding on Margulis'
approach, showed that any properly ergodic probability-preserving
ergodic action of such a lattice is essentially free.

I will present similar results: my work with Y. Shalom on normal
subgroups of lattices in products of simple locally compact groups and
normal subgroups of commensurators of lattices, and my work with J.
Peterson generalizing this result to stabilizers of ergodic
probability-preserving actions of such groups. As a consequence,
S-arithmetic lattices enjoy the same properties as the arithmetic
lattices (the Stuck-Zimmer result) as do lattices in certain product
groups. In particular, any nontrivial ergodic probability-preserving
action of $\mathrm{PSL}_{n}(\mathbb{Q})$, for $n \geq 3$, is
essentially free.

The key idea in the study of normal subgroups is considering
nonsingular actions which are the extreme opposite of
measure-preserving. Somewhat surprisingly, the key idea in
understanding stabilizers of probability-preserving actions also
involves studying such actions and the bulk of our work is directed
towards properties of these contractive actions.

In this talk I will show how understanding of the possible limiting measures of translations of a measure can help us to deal with certain counting problems. Then I talk about the limiting measures of translations of a horospherical measure. Finally I discuss how one can use this result to count the number rational points in a flag variety with respect to any line-bundle, reproving a result of Franke-Manin-Tschinkel (anti-canonial line-bundle) and Batyrev-Tschinkel (arbitrary line-bundle). (Joint with A. Mohammadi)

Must the Fourier series of an $L^2$ function converge pointwise almost everywhere? In the 1960's, Carleson answered this question in the affirmative, by studying a particular type of maximal singular integral operator, which has since become known as a Carleson operator. In the past 40 years, a number of important results have been proved for generalizations of the original Carleson operator. In this talk we will introduce the Carleson operator and survey several generalizations, and then describe new joint work with Po Lam Yung that introduces curved structure to the setting of Carleson operators.

Over the last hundred years, the circle method has become one of the
most important tools of analytic number theory. This talk (on joint work
with Roger Heath-Brown) will describe a new application of the circle
method to pairs of quadratic forms, via a novel two-dimensional analogue
of Kloosterman's version of the circle method. As a result, we prove
(under a mild geometric constraint) that any two quadratic forms with
integer coefficients, in 5 variables or more, simultaneously attain
prime values infinitely often.

I will begin with a friendly introduction to the deformation theory of
Galois representations and its role in modularity lifting, focusing on
the case of elliptic curves over Q. This will motivate the study of
local deformation rings and more specifically flat deformations. Next,
we will discuss Kisin’s resolution of the flat deformation ring at l = p
and describe conceptually the importance of local models of Shimura
varieties in analyzing its geometry. In the remaining time, we will
address the title of the talk; the additional structure we consider
could be a symplectic form, an orthogonal form, or more generally any
reductive subgroup G of $GL_N$. I will describe briefly the role that
recent advances in p-adic Hodge theory and local models of Shimura
varieties play in this situation.

An important open problem in quantum information concerns with the question whether entanglement between signal states can help in sending classical information over a quantum channel. Recently, Hastings proved that entanglement does help by finding a counter example for the long standing additivity conjecture that the minimum entropy output of a quantum channel is additive under taking tensor products. In this talk I will show that the minimum entropy output of a quantum channel is locally additive. Hastings' counter example for the global additivity conjecture makes this result somewhat surprising. In particular, it indicates that the non-additivity of the minimum entropy output is related to a global effect of quantum channels. I will end with a few related open problems.

We consider a classical continuum model which allows to describe essential electrokinetichenomena such as electro-phoresis and -osmosis. Applications and correspondingheory range from design of micro fluidic devices, energy storage devices,emiconductors to emulating communication in biological cells by synthetic nanopores.

Based on this classical formulation, we derive effective macroscopic equationshich describe binary symmetric electrolytes in porous media. Theeterogeneous materials naturally induce corrected transport parameters which weall "material tensors". A better understanding of the influence ofeterogeneous media on ionic transport is expected by the new formulation.he new equations provide also an essential computational advantage by reliablyeducing the degrees of freedom required to resolve the microstructure.

The presented results are gained by asymptotic multi-scale expansions.his formal procedure is then made rigorous by the derivation of error boundsetween the exact microscopic solution and the new upscaled macroscopic approximation.

A \emph{mixed graph} is a graph with directed edges, called
arcs, and undirected edges. A $k$-coloring of the vertices is
\emph{proper} if colors $1,2,\ldots,k$ are assigned to each vertex such
that vertices $u$ and $v$ have different colors if $uv$ is an edge and
the color of $u$ is less than or equal to (resp. strictly less than) the
color of $v$ if $uv$ is an arc. The \emph{weak (resp. strong) chromatic
polynomial} of a mixed graph is a counting function that counts the
number of proper $k$-colorings. This talk will discuss previous work on
reciprocity theorems for other types of chromatic polynomials, and our
reciprocity theorem for weak chromatic polynomials which uses partially
ordered sets and order polynomials. This is joint work with Matthias
Beck, Daniel Blado, Joseph Crawford, and Taina Jean-Louis.

The classical Koszul theory plays an important role in the representation theory of graded algebras. However, there are a lot of structures (algebras, categories, etc) having natural gradings with non-semisimple degree 0 parts, to which the classical theory cannot apply. Particular examples include polynomial rings over non-semisimple algebras, extension algebras of modules, etc. In this talk I'll introduce a generalized Koszul theory which does not demand the semisimple property. It preserves many classical results as Koszul duality and has a close relation to the classical one. Applications of this generalized theory to extension algebras of modules and modular skew group algebras will be described.

Logarithmic Sobolev inequalities (LSIs) show up in several areas of analysis; in particular, in probability. In this talk I will give some background and applications of LSIs. I will also discuss some recent work and show how LSIs can be used to give a new proof of the classical result that the empirical law of eigenvalues of a sequence of random matrices converges weakly to its mean in probability.

In non-commutative Iwasawa theory K-theoretic properties of Iwasawa algebras, i.e. completed group rings of e.g. compact p-adic Lie groups play a crucial role. Such groups arise naturally as Galois groups attached to p-adic representations as for example on the Tate module of abelian varieties. In this talk we address in particular the question for which such groups the invariant $SK_1$ vanishes. We reduce this vanishing to a linear algebra problem for Lie algebras over arbitrary rings, which we solve for Chevalley orders in split reductive Lie algebras. Also we shall try to indicate what arithmetic consequences the vanishing of $SK_1$ has.

Many materials problems require the accuracy of atomistic modeling in small regions, such as the neighborhood of a crack tip. However, these localized defects typically interact through long ranged elastic fields with a much larger region that cannot be computed atomistically. Many methods have recently been proposed to compute solutions to these multiscale problems by coupling atomistic models near a localized defect with continuum models where the deformation is nearly uniform. During the past several years, we have given a theoretical structure to the description and formulation of atomistic-to-continuum coupling that has clarified the relation between the various methods and sources of error. Our theoretical analysis and benchmark simulations have guided the development of optimally accurate and efficient coupling methods.

Motivated by the question whether (some) Diophantine equations are related to special values of $\zeta$- or $L$-functions we first describe the origin of classical Iwasawa theory. Then we give a survey on generalizations of these ideas to non-commutative Iwasawa theory, a topic which has been developed in recent years by several mathematicians, including the author.

For finitely generated groups, growth of the elements of the group, and the series (or generating functions) associated to the growth function, have been widely studied. Recently researches have begun to study the growth of conjugacy classes in these groups. Disconcertingly, the conjugacy growth series had been found by Rivin not to be rational for free groups with respect to a free basis. In this talk I will introduce the notion of geodesic conjugacy growth functions and series, and discuss the effects of various group constructions on rationality of both the geodesic conjugacy and (spherical) conjugacy languages whose growth is measured by these functions. In particular, we show that rationality of the geodesic conjugacy growth series, as well as on regularity of the geodesic and spherical conjugacy growth series is preserved by both direct and free products. This is joint work with Laura Ciobanu.

The classical Hasse invariant is defined via the determinant of the
Hasse-Witt matrix. It allows cutting out the so-called ordinary locus
within the special fiber of a modular curve: this is the affine locus
where the Hasse invariant is invertible. For more general Shimura
varieties, the ordinary locus may be empty, and the Hasse invariant is
then trivial. On the other hand, there exist for all Shimura varieties
of PEL-type so-called generalized Hasse-Witt invariants which are
vector-valued, but they are typically not robust enough to carry over
the usual applications of the classical Hasse invariant. In this talk,
we specialize to the scalar-valued cases that are most similar to the
classical invariant (joint work with W. Goldring).

I will discuss some common mathematical structures arising in information-processing networks in computer science, statistics and machine learning, and quantum information and many-body systems. These seemingly disparate fields are connected by variations on the graphical modeling language of tensor networks, or more generally monoidal categories with various additional properties.

Tools from algebraic geometry, representation theory, and category theory have recently been applied to problems arising from such networks. Basic questions about each type of information-processing system (such as what probability distributions or quantum states can be represented, or what word problems can be solved efficiently) quickly become interesting problems in shared algebraic geometry, representation theory, and category theory. The result has been new insights into problems ranging from recognizing images to classifying quantum phases of matter and interesting challenges in pure mathematics.

The spectra of Erdos-Renyi random graphs have been long studied. We consider random graphs of which each edge is determined by an independent random indicator variable with the expected value not all equal in general. We prove that the eigenvalues of the adjacent matrix and the normalized Laplacian matrix of such random graphs can be approximated by those of the `expectation graph ’.

Shimura varieties are generalizations of modular curves which
are at the heart of the programs of both R. Langlands and S. Kudla. We
will focus on the geometric infrastructure built using Barsotti-Tate
groups, and we will present some arithmetic applications related to the
above programs or their p-adic variants.

We apply the variational implicit solvent model (VISM), numerically minimized by the level-set method, to study hydration effects in the high-affinity binding of the B2 bicyclo[2.2.2]octane derivative to the CB[7] molecule. For the unbounded host molecule, we find two equilibrium dielectric interfaces with two different types of initial wraps. The host cavity shows capillary evaporation when the initial guess is a loose wrapping; while the host is completed hydrated when a tight initial guess is prescribed. In good agreement with MD simulation results, the hydrated case is more favorable due to solvent-solute electrostatic interaction advantage. For the guest binding we find reasonable agreement to experimental binding affinities. Dielectric interfaces of the system during the binding process are given by level-set simulations. Individual free energy contributions show that water-mediated hydrophobic interactions based on decreasing water unfavorable concave surface upon binding and electrostatic interactions are two major driving forces for the binding process. The findings are in line with recent computer simulations and experiment data. With further refinement the VISM could be a promising tool for an efficient calculation of molecular binding affinities.

There are two basic theorems on arithmetic properties of probability measures on Euclidean space: the Levy decomposition of infinitely divisible probability measures as convolutions of Poisson and Gaussian measures, and the Khintchine factorization of arbitrary probability measures in terms of indecomposable measures and measures without indecomposable factors. Both theorems have been generalized by K. R. Parthasarathy to measures on an Abelian locally compact group. Within this framework the role of Gaussian factors will be discussed. Moreover, characterizations of Gaussian measures (in the sense of Cramer and Bernstein) will be presented whose validity depends on the structure of the underlying group.

We discuss recent work by Gross and Siebert defining logarithmic Gromov-Witten invariants.

A $G$-strand is a map $\mathbb{R}\times\mathbb{R}\to G$ into a Lie group $G$ that follows from Hamilton's principle for a certain class of $G$-invariant Lagrangians. $G$-strands on finite-dimensional groups satisfy $1+1$ space-time evolutionary equations. A large class of these equations have Lax-pair representations that show they admit soliton solutions. For example, the $SO(3)$-strand equations may be regarded physically as integrable dynamics for solitons on a continuous spin chain. Various other examples will be discussed, including collisions of solutions with singular support (e.g., peakons) on ${\rm Diff}(\mathbb{R})$-strands, in which ${\rm Diff}(\mathbb{R})$ is the group of diffeomorphisms of the real line $\mathbb{R}$, for which the group product is composition of smooth invertible functions.