This talk presents a strategy for computational wave propagation that
consists in decomposing the solution wavefield onto a largely incomplete set of
eigenfunctions of the weighted Laplacian, with eigenvalues chosen randomly. The
recovery method is the ell-1 minimization of compressed sensing. For the
mathematician, we establish three possibly new estimates for the wave equation
that guarantee accuracy of the numerical method in one spatial dimension. For
the engineer, the compressive strategy offers a unique combination of
parallelism and memory savings that should be of particular relevance to
applications in reflection seismology. Joint work with Gabriel Peyre.

Answer: Kodaira put a compact Hodge manifold in
projective space.
We will spend an hour building up to the statement of the Kodaira
embedding theorem. Any graduate student should feel welcome as we will
build up the basic background material necessary to understand the
statement of the theorem. If time permits, we will present a *very*
brief outline of how a proof might
look and some applications. See you there.

\noindent In this talk, I will discuss a novel class of nonlinear dispersive equations, which describe the dynamical evolution of self-gravitating relativistic matter. In fact, the analysis of these model equations will give a mathematical vindication of Chandrasekhar's acclaimed physical theory of gravitational collapse. In particular, I will present results concerning the well-posedness of the initial-value problem, the singularity formation of solutions (blowup), as well as solitary wave solutions and their stability. Time permitting, I will also discuss some recent and ongoing work. \\

\noindent This is partly joint work with J\"urg Fr\""ohlich (ETH Z\""urich)

This term's topology seminar will be on Morse theory and its applications.
We'll look first at the basic idea of gradient flow on finite-dimensional manifolds, and how this gives us cell and handle decompositions. Then we'll look at the original applications, to geodesics on Riemannian manifolds and to Bott periodicity. Finally we'll look at more modern developments, including perhaps the Morse category of a manifold, Fukaya's approach to the cup product and Massey products, Floer homology, circle-valued Morse theory and combinatorial Morse theory.\\

\noindent As usual this is a learning seminar, where the volunteering participants give talks. At the first meeting I'll give an overview lecture and we'll try to arrange speakers for the rest of term.

In this talk, we present an analog of dual equivalence for
affine permutations. Exploiting the connection between affine
permutations and n-cores, this establishes the Schur positivity of the
strong Schur functions introduced by Lam, Lapointe, Morse and Shimozono,
which are generalizations of the k-Schur functions introduced by
Lapointe, Lascoux and Morse. Time permitting, we will show how this
approach may ultimately lead to an explicit connection between Macdonald
polynomials (and, more generally, LLT polynomials) and k-Schur
functions. \\

\noindent This is joint work with Sara Billey at the University of Washington.

We give a positive answer to Burstall's question of whether there exists an interesting theory of isothermic submanifolds of dimension $>2$ in $R^n$. We relate chains of such manifolds to solutions of a system of PDEs and describe their moduli space. We also describe Christoffel and Darboux/Ribaucour transforms of isothermic chains.

As seen in \textit{Die Hard with A Vengeance}, we investigate the classic puzzle of making 4 gallons using only a 3- and a 5-gallon jug. We'll look at various generalizations of this puzzle, when and how they are solvable, and how quickly one can arrive at a solution.

By invoking a result originally due to G. Shimura, we give a new proof
of a generalization of a theorem of Deuring concerning supersingular
elliptic curves, namely, a mass formula for superspecial abelian
varieties with PEL-structures in characteristic p. This mass formula
is then applied to estimate the dimension of Siegel modular cusp
spaces modulo p and count the number of irreducible components in the
supersingular region of the Siegel modular variety.

\noindent Many interesting examples of dynamical systems involve several well separated time scales. In many applications, for example in molecular dynamics simulations, one is only interested in the slow aspects of dynamics, or on the long-time behavior of the solutions. However, when the different scales are coupled, small or fast perturbations can build up to an observable effect that cannot be neglected. \\

\noindent In this talk I will discuss several types of models and address some of the analytic and computational difficulties common to many systems evolving on multiple time scales. We give a complete characterization of the slow aspects of the dynamics and devise efficient computational algorithms that take advantage of the scale separation. It is shown that the computational cost is practically independent of the spectral gap. Among the systems studied are highly oscillatory ODEs and a benchmark model of elastic spheres with disparate masses.

\noindent In my colloquium talk, I will explain how various fundamental structures from group representation theory appear naturally in the context of discrete harmonic analysis and can be applied to solve concrete problems from digital signal processing. I will begin by describing our solution to the problem of finding a canonical orthonormal basis of eigenfunctions of the discrete Fourier transform (DFT). Then I will explain how to generalize the construction to obtain a larger collection of functions that we call "The oscillator dictionary". Functions in the oscillator dictionary admit many interesting properties, in particular, I will explain two of these properties which arise in the context of problems of current interest in communication theory. This is joint work with Shamgar Gurevich (Berkeley) and Nir Sochen (Tel Aviv). \\

\noindent There is a sequel to my colloquium talk, which will be slightly more specialized and will take place during the algebraic geometry seminar. Here, my main objective is to introduce the geometric Weil representation which is an algebra-geometric ($ \ell $-adic Weil sheaf) counterpart of the Weil representation. Then, I will explain how the geometric Weil representation is used to prove to main result stated in my colloquium talk. In the course, I will explain Grothendieck's geometrization procedure by which sets are replaced by algebraic varieties and functions by sheaf theoretic objects.

This talk is a sequel to my colloquium talk, given earlier in the day.   My main objective is to introduce the geometric Weil representation which is an algebra-geometric
(l-adic Weil sheaf) counterpart of the Weil representation. Then, I will
explain how the geometric Weil representation is used to prove to main
result stated in my colloquium talk. In the course, I will explain
Grothendieck's geometrization procedure by which sets are replaced by
algebraic varieties and functions by sheaf theoretic objects.

My talk is composed of three parts. The first part is on high order
residual distribution (RD) schemes for steady state hyperbolic
conservation laws. High order RD schemes are conservative schemes that
overcome the restriction of mesh sizes in high order finite difference
schemes, and yet have comparable computational costs. It has a broad
range of applications from Navier-Stokes equations to semiconductor
simulations. I will present the design of the scheme, a Lax-Wendroff
type theorem and the numerical results. In the second part, I will
discuss the applications in systems biology. The modeling of the two
biological systems--cell polarization and multi-stage cell lineages,
and the computational aspect will be discussed. New efficient
numerical schemes for both time evolution and steady state
reaction-diffusion equations that arise in many biological models will
be presented in the third part.

Consider a parabolic subgroup of a reductive group G. By a theorem of Richardson (1974), the adjoint action of P on its nilpotent radical has an open dense orbit.
In general, there is an infinite family of orbits, so the description of the P-orbits is a ``wild'' problem.In type A there exists a translation of this problem into a question of representation-type of a category of representations of an algebra due to Hille and R\"ohrle (1999). In this talk I will describe their approach and explain how it can be extended to deal with parabolic subgroups of orthognal groups.

Data from real-world graphs often contains incomplete information, so we
only observe subgraphs of these graphs. It is therefore desirable to
understand how a typical subgraph relates to the underlying host graph.
We consider several interrelated problems on both random trees and
random subgraphs obtained by taking edges of the host graph
independently with probability $p$. In the second case, we study the
emergence of the giant component. We also use the spectral gap to
understand discrepancy and expansion properties of a random subgraph.
The Erd\H{o}s-R\'enyi random graph is the special case of this where the
host graph is the complete graph $K_n$. Additional applications include
taking a contact graph as the host graph, and viewing random subgraphs
as outbreaks of a disease.

t has been over two decades since M. Gromov initiated the study
of pseudo-holomorphic curves in symplectic manifolds. In the past decade
we have witnessed mathematical constructions of Gromov-Witten theory for
algebraic varieties, as well as many major advances in understanding their
properties. Recent works in string theory have motivated us to extend our
interests to Gromov-Witten theory for Deligne-Mumford stacks. Such a
theory has been constructed, but many of its properties remain to be
understood. In this talk I will explain the main ingredients of
Gromov-Witten theory of Deligne-Mumford stacks, and I will discuss some
recent progress regarding main questions in Gromov-Witten theory of
Deligne-Mumford stacks.

In 1980, I. Morrison proved that slope stability of a vector bundle of rank 2 over a compact Riemann surface implies Chow stability of the projectivization of the bundle with respect to certain polarizations. Using the notion of balanced metrics and recent work of Donaldson, Wang, and Phong-Sturm, we show that the statement holds for higher rank vector bundles over compact algebraic manifolds of arbitrary dimension that admit constant scalar curvature metric and have discrete automorphism group.

Ever been curious about continued fractions? This talk will cover the basics, and demonstrate how they are closely related to weighted square-and-domino tilings. In particular we'll do a neat bijection that "compresses" a board with periodic weights into a smaller period-1 board, and then show how this can be used to calculate periodic continued fractions. In addition, one can prove a number of Fibonacci/Lucas identities combinatorially using this model.

For many integer sequences $X=(x_n)_n$, it is
a natural question to describe the set $P_X$ of all
prime numbers $p$ that divide some non-zero term
of the sequence, and to quantify the `size' of $P_X$. \\

\noindent We focus on the case of linear recurrent sequences,
where we have fairly complete results for recurrences
of order 2 based on the Chebotarev density theorem,
and mostly open questions for higher order recurrences.

As part of a summer professional development institute, we investigated how a focus on the explanatory power of different proofs helped in-service teachers enhance their understanding of mathematical proof. I will present example problems and several solutions in order to illustrate different types of proof and what we mean by explanatory power. In addition, I will discuss the different types of teaching practices that the instructor of the institute used in order to facilitate changes in participants' understanding of proof. This talk is intended for undergrads, grads, and professors.

Recent work of G. Prasad and Rapinchuk has produced families of groups which are locally isomorphic but not globally isomorphic. In the case of unitary groups associated to division algebras with an involution of the second kind we describe the corresponding Langlands functoriality between their automorphic representations.

I will present results of molecular dynamics simulations of hemispherical hydrophobic pockets remaining in direct contact with water. The considered pockets of three different sizes represent simple models of nonpolar cavities often found in proteins' binding sites where they are important for hydrophobic interactions with ligands. A detailed analysis of solvent behaviour reveals significant density fluctuations inside the pockets resulting from cooperative movements of individual water molecules. \\

\noindent I will also consider a process of translocation of methane molecule from bulk solvent into the pockets and discuss the obtained potentials of mean force. Their analysis in the light of observed changes in water distribution around the interacting hydrophobic

If $Q_N(z)$ is a polynomial of degree $N$ and $P > 0$, then estimates for the size of the set where the logarithmic derivative $Q'(z)/Q(z)$ has modulus greater than P are given in terms of $P$ and $N$. These estimates are shown to be essentially the best possible. This is joint work with V. Ya. Eiderman.

In this talk, we introduce and study multivariate spacings. The spacings are
developed using the order statistics derived from data depth. Specifically,
the spacing between two consecutive order statistics is the region which
bridges the two order statistics, in the sense that the region contains all
the points whose depth values fall between the depth values of the two
consecutive order statistics. These multivariate spacings can be viewed as a
data-driven realization of the so-called ``statistically equivalent blocks".
These spacings assume a form of center-outward layers of ``shells" (``rings"
in the two-dimensional case), where the shapes of the shells follow closely
the underlying probabilistic geometry. The properties and applications of
these spacings are studied. In particular, the spacings are used to
construct tolerance regions. The construction of tolerance regions is
nonparametric and completely data driven, and the resulting tolerance region
reflects the true geometry of the underlying distribution. This is different
from most existing approaches which require that the shape of the tolerance
region be specified in advance. The proposed tolerance regions are shown to
meet the prescribed specifications, in terms of $\beta$-$content$ and
$\beta$-$expectation$. They are also asymptotically minimal under elliptical
distributions. Finally, we present a simulation and comparison study on the
proposed tolerance regions. \\

\noindent This is joint work with Prof. Regina Y. Liu from Rutgers University.

Hydrophobic interactions drive relatively apolar molecules to stick together
in an aqueous solution. Such interactions are crucial to the structure,
dynamics, and function of biological systems. The implicit (or continuum) solvent approach is an efficient
way to model such interactions. In this talk, I will first describe a class of variational
implicit-solvent models for solvation. Central in these models is a
free-energy functional of all possible solute-solvent interfaces, coupling both
non-polar and polar contributions. Minimization of this free-energy functional
determines equilibrium solute-solvent interfaces which conceptually replace
solvent accessible surfaces (SAS) or solvent excluded surfaces (SES). I will then
describe a level-set method for capturing equilibrium solute-solvent interfaces.
In our level-set method, a possible solute-solvent interface is represented by the zero
level set (i.e., the zero level surface) of a function and
is evolved to reduce the free energy of the system, eventually into� an equilibrium solute-solvent interface.
This� method is applied to the study of a large concave wall in water, together with a
small solute molecule. Our level-set calculations determine the solute-solvent interface locations and free energies very accurately
compared with molecular dynamics simulations that have been previously reported.
We also capture the bimodal behavior of the potential of mean force of the underlying hydrophobic interactions.
In addition, we find the curvature correction to the surface tension has a significant influence on the solute-solvent
interface profile in the concave region. All these demonstrate that our mean-field approach and numerical techniques
are capable of efficiently and accurately describing hydrophobic interactions with significant geometric influences.
This is joint work with Li-Tien Cheng, Piotr Setny, Joachim Dzubiella, Bo Li, and J. Andrew McCammon.

\indent The subject of pattern avoiding permutations has its roots in
computer science, namely in the problem of sorting a permutation
through a stack. A formula for the number of permutations of length
$n$ that can be sorted by passing it twice through a stack (where
the letters on the stack have to be in increasing order) was
conjectured by West, and later proved by Zeilberger. Goulden and
West found a bijection from such permutations to certain planar
maps, and later Cori, Jacquard and Schaeffer presented a bijection
from these planar maps to certain labeled plane trees, called
beta(1,0)-trees.

\indent We show that these labeled plane trees are in one-to-one
correspondence with permutations that avoid the generalized patterns
3-1-4-2 and 2-41-3. We do this by establishing a bijection between
the avoiders and the trees. This bijection translates 7 statistics
on the trees into statistics on the avoiders.

\noindent Moreover, extensive computations suggest that the
avoiders are structurally more closely connected to the
beta(1,0)-trees---and thus to the planar maps---than two-stack
sortable
permutations are.

In connection with this we give a nontrivial involution on the
beta(1,0)-trees, which specializes to an involution on unlabeled
rooted
plane trees, where it yields interesting results.

\indent Random matrix theory, a very young subject, studies the behaviour of the eigenvalues of matrices with random entries (with specified correlations). When all entries are independent (the simplest interesting assumption), a universal law emerges: essentially regardless of the laws of the entries, the eigenvalues become uniformly distributed in the unit disc as the matrix size increases. This {\em circular law} was first proved, with strong assumptions, in the 1980s; the current state of the art, due to Tao and Vu, with very weak assumptions, is less than a year old. It is the {\em universality} of the law that is of key interest. \\
\indent What if the entries are {\em not independent}? Of course, much more complex behaviour is possible in general. In the 1990s, ``$R$-diagonal'' matrix ensembles were introduced; they form a large class of non-normal random matrices with (typically) non-independent entries. In the last decade, they have found many uses in operator theory and free probability; most notably, they feature prominently in Haagerup's recent work towards proving the invariant subspace conjecture. \\
\indent In this lecture, I will discuss my recent joint work with Haagerup and Speicher, where we prove a universal law for the resolvent of any $R$-diagonal operator. The circular ensemble is an important special case. The rate of blow-up is, in fact universal among {\em all} $R$-diagonal operators, with a constant depending only on their fourth moment. The proof intertwines both complex analysis and combinatorics.\\

This talk will assume no knowledge of random matrix theory or free probability.

We study a Siegel-Weil identity between a theta series and
an Eisenstein series of genus three. In particular, this yields a
Siegel modular form of genus three whose Fourier coefficients count
the number of arithmetic embeddings of definite quaternion orders into
the Coxeter order of integral octonions.

I will talk about some of the main open problems
in the theory of Black Holes. I will talk in particular
on recent results concerning uniqueness and stability.

I will describe recent results of Ionescu-Klainerman
and Alexakis-Ionescu-Klainerman which remove the crucial
assumption of analyticity in the
well known result of Hawking, Carter and Robinson
concerning the uniqueness of the Kerr solution among
stationary solutions.

A {\em noncommutative projective surface} is a noetherian graded domain of Gelfand-Kirillov dimension 3; their classification is one of the most important areas of research in noncommutative algebraic geometry. We complete an important special case by classifying all noncommutative projective surfaces that are {\em birationally commutative}: to wit, they are graded subrings of a skew polynomial ring over a field. We show that birationally commutative projective surfaces fall into four families, parameterized by geometric data, and we obtain precise information on the possible forms of this data. This extends results of Rogalski and Stafford on rings generated in degree 1, although our proof techniques are significantly different.

For a reductive group G and its endoscopic group H over a p-adic field, the functorial transfer from H to G should be characterized in terms of the character identity. For automorphic induction for GL(n), this identity was established by Henniart and Herb, up to a constant. We discuss a relation of this constant to the Kottwitz-Shelstad transfer factor, in particular, to the epsilon factor normalization.

We will outline the proof of a quantitative version of the following Sard
type theorem. Given a Lipschitz function $f$ from the $k-$dimensional unit
cube into a general metric space, one can decomposed $f$ into a finite
number of Bi-Lipschitz functions $f|_{F_i}$ so that the $k-$Hausdorff
content of $f([0, 1]^k \smallsetminus \cup F_i$) is small. The case where
the metric space is $\mathbb{R}^d$ is a theorem of P. Jones (1988). This
positively answers problem 11.13 in ``Fractured Fractals and Broken
Dreams" by G. David and S. Semmes, or equivalently, question 9 from
``Thirty-three yes or no questions about mappings, measures, and metrics"
by J. Heinonen and S. Semmes.

\indent Bootstrap percolation is the following deterministic process on a graph
$G$. Given a set $A$ of initially `infected' vertices, and a threshold $r
\in \mathbb{N}$, new vertices are subsequently infected if they have at
least $r$ previously infected neighbours. The study of this model
originated in statistical physics, and the process is closely related to
the Ising model. The set $A$ is usually chosen randomly, each vertex being
infected independently with probability $p \in (0,1)$, and the main aim is
to determine the critical probability $p_c(G,r)$ at which percolation
(infection of the entire graph) becomes likely to occur.\\

I will give a survey of the area, focusing on the following recent result,
proved jointly with Bollobas and Morris:\\

The bootstrap process has been extensively studied on the $d$-dimensional
grid $[n]^d$, with $2 \le r \le d$, and it was proved by Cerf and Manzo
(building on work of Aizenman and Lebowitz, and Cerf and Cirillo) that
$$p_c\big( [n]^d,r \big) \; = \; \Theta\left( \frac{1}{\log_{r-1} n}
\right)^{d-r+1},$$ where $\log_{r-1}$ is the $(r-1)$-times iterated
logarithm. However, the exact threshold function was only known in the case
$d = r = 2$, where it was shown by Holroyd to be $(1 +
o(1))\frac{\pi^2}{18\log n}$. In this talk we show how to determine the
exact threshold for all fixed $d$ and $r$, concentrating on the crucial
case $d = r = 3$.

Data that we encounter in practice often have meny zero-valued
observations. Anaylizing such data without any consideration given
to how the zeros arose might lead to misleading results. In this talk,
we propose a new feature extraction method for very non-normal data.
Our method extends principle component analysis (PCA) in the same
manner as the generalized linear model extends the ordinary linear
regression model. As an example, we analyze multivariate species-size
data from a purse-seine fishery in the eastern Pacific Ocean.
The data contain many zero-valued observations for each variable
(combinations of species and size). Thus, as an error distribution we
use the Tweedie distribution which has a probability mass at zero and
apply Tweedie-generalized PCA (GPCA) method to the data.

I will present some results from a recently completed project that ties together several objects: restricted in a certain way permutations, $(2+2)$-free partially ordered sets, and a certain class of involutions (chord diagrams). Each of these structures can be encoded by a special sequence of numbers, called ascent sequences, thus providing bijections, preserving numerous statistics, between the objects.\\

\noindent In my talk, I will also discuss the generating function for these classes of objects, as well as a restriction on the ascent sequences that allows to settle a conjecture of Pudwell on permutations avoiding $3\bar{1}52\bar{4}$.\\

\noindent This is joint work with Mireille Bousquet-Melou (Bordeaux), Anders Claesson (Reykjavik University) and Mark Dukes (University of Iceland).

The kissing number $k(n)$ is the maximal number of equal nonoverlapping
spheres in $n$-dimensional space that can touch another sphere of the same
size. This problem in dimension three was the subject of a famous
discussion between Isaac Newton and David Gregory in 1694. In three
dimensions the problem was finally solved only in 1953 by Sch\"utte and

Let F be a local field of characteristic 0. We consider distributions on GL(n+1,F) which are invariant under the adjoint action of GL(n,F). We prove that such
distributions are invariant under transposition. This implies that an irreducible representation of GL(n+1,F), when restricted to GL(n,F) "decomposes" with multiplicity one.

Such property of a group and a subgroup is called strong Gelfand property. It is used in representation theory and automorphic forms. This property was introduced by Gelfand in the 50s for compact groups. However, for non-compact groups it is much more difficult to establish.

For our pair (GL(n+1,F),GL(n,F)) it was proven in 2007 in [AGRS] for non-Archimedean F, and in 2008 in [AG] and [SZ] for Archimedean F. In this lecture we will
present a new proof which is uniform for both cases. This proof is based on the above papers and an additional new tool. If time permits we will discuss similar
theorems that hold for orthogonal and unitary groups.

This talk will give an example of ideas from number theory
being deployed in the service of complex analytic geometry. We consider
the problem of the formal classification of flat meromorphic connections
on a complex manifold. We will first recall the answer in the
one-dimensional case (the Turrittin-Levelt theorem) and its relevance to
the asymptotic behavior of solutions of meromorphic differential
equations (the Stokes decomposition). We will then describe a
higher-dimensional analogue, whose proof is much subtler: it uses
analytic geometry not just over the complex numbers, but also over
certain complete nonarchimedean fields (e.g., formal power series). The
methods we use are ultimately inspired by Dwork's study of the p-adic
variation of zeta functions of algebraic varieties.

In 1951, Kenneth Arrow showed that under a reasonable definition of 'fair', there is no fair election system in a society of at least two individuals with at least three options. We'll make this statement into a precise theorem, prove it, and then discuss a generalization. While axioms and choice will both appear in this talk, the Axiom of Choice will not.

A tight coupling between cell structure, ionic fluxes and intracellular Ca2+ transients underlies the regulation of cardiac cell function. To investigate how a distribution of Ca2+ handling proteins may affect these coupled processes we developed a 3-D model of Ca2+-signaling, buffering and diffusion in rat ventricular myocytes. The model geometry was derived from the experiment. A diffusion modeling software using finite element tool kit (FEtK) libraries was implemented to solve the 4 coupled PDE systems. We concluded that the cardiac cell function is tightly regulated by the localization of Ca2+-handling proteins and strongly relays on the presence of mobile and stationary Ca2+ buffers and cell geometry.

If M is an arithmetic or geometric object, one can often attach to it a complex analytic function L(M,s). This is called the L-function of M and provides a powerful tool to study its various properties. We will consider the case when M= (F,g) where F is a Siegel modular form of genus two and g a classical modular form. In this setup we prove the following result: for s lying in a certain set of so called critical points, the corresponding values L(M,s) are algebraic numbers up to certain period integrals and behave nicely under automorphisms. This is predicted by an old conjecture of Deligne on motivic L-functions. The main tool used in our proof is an integral representation of the L-function involving the pullback of an Eisenstein series defined on a unitary group.

I will present a unified framework for a large-sample theory of
stationary and non-stationary processes. Topics in classical time
series analysis will be revisited and they include the estimation
of covariances, spectral densities and long-run variances. I will
also talk about high dimensional covariance matrices estimation
and inference of mean and quantiles of non-stationary processes.

In this talk we will explain how multiple Dirichlet series can be
employed to exploit analysis in several complex variables in order to
obtain arithmetic information. Then we will talk about their connections
to Weyl groups and Kac-Moody algebras.

Motivated by the relative trace formula of Jacquet and experience
on period integrals of automorphic forms, we take the first steps towards
formulating a ``relative'' Langlands program, i.e. a set of conjectures on
H-distinguished representations of a reductive group G (both locally and
globally), where H is a spherical subgroup of G. We prove several results in
this direction. Locally, the spectrum of H/G is described with the help of
the dual group associated to any spherical variety by Gaitsgory and Nadler.
Globally, period integrals are conjectured to be Euler products of explicit
local functionals, which we compute at unramified places and show that they
are equal to quotients of L-values. If time permits, I will also discuss an
approach which shows that different integral techniques for representing
L-functions (e.g. Tate integrals, Rankin-Selberg integrals, period
integrals) are, in fact, the same. This is in part joint work with Akshay
Venkatesh.

It is a classical problem in graph theory to look for those graphs that maximize the number of copies of a subgraph H and are F-free; the Turan problem being the most well known example of such problem. In this talk I will explain how the incidence graphs of projective planes of order $n$ are exactly those $n$ by $n$ bipartite graphs that are $C_4$-free and maximize the number of eight cycles. An analogous characterization of projective planes as $C_4$-free graphs that maximize the number of six cycles was previously known. I will also explain how a more general conjectural characterization of (the incidence graphs of) projective planes relates to some interesting geometric questions on projective planes. Finally I will mention some related open problems concerning so-called generalized polygons.

I will discuss the relationship between convergence of the Ricci flow on a Fano manifold and algebraic stability of the manifold with the anticanonical polarization. I will show that if the curvature remains bounded along the flow then stability implies convergence of the flow and so in particular existence of a Kahler-Einstein metric.

Detecting the Higgs boson is one of the highest priorities of the current
generation of particle physicists. While the Higgs may be a fundamental
particle, the interesting possibility exists that it is instead composed
of "quarks" of a new gauge theory at a higher energy scale, termed
Technicolor. In fact, the Higgs is expected to be discovered at the Large
Hadron Collider in the next two years, and if the particle's mass is on or
above a certain, not unreasonable scale, then a form of Technicolor may
become a prime candidate to explain its origin. I will describe this
puzzle and numerical methods which we are utilizing to explore it.

W. T. Gan and S. Takeda have defined the local packets of GSp(4) using theta correspondence. We shall discuss how to use the trace formula technique to derive character identities satisfied by these packets.

On Fano manifolds we study the supremum of the possible t such that there exists a metric in the first Chern class with Ricci curvature bounded below by t. For the projective plane blown up in one point we show that this supremum is 6/7.

Many signal-processing applications seek to approximate a signal as a superposition of only a few elementary atoms drawn from a large collection. This is known as sparse reconstruction. The theory of compressed sensing allows us to pose sparse reconstruction problems as structured convex optimization problems. I will discuss the role of duality in revealing some unexpected and useful properties of these problems, and will show how they lead to practical, large-scale algorithms. I will also describe some applications of the resulting algorithms.

\indent Consider a crystal formed of two types of atoms placed at the nodes of the integer lattice. The type of each atom is chosen at random, but the crystal is statistically shift-invariant. Consider next an electron hopping from atom to atom. This electron performs a random walk on the integer lattice with randomly chosen transition probabilities (since the configuration seen by the electron is different at each lattice site). This process is highly non-Markovian, due to the interaction between the walk and the environment.

We will present a martingale approach to proving the invariance principle (i.e. Gaussian fluctuations from the mean) for (irreversible) Markov chains and show how this can be transferred to a result for the above process (called random walk in random environment).

This is joint work with Timo Seppalainen.

We will explore the process of solving a math problem using parallel
processing and MPI. As an example, we will solve a discrete Poisson
equation using Jacobi's method. Basic issues in developing, optimizing,
and deploying parallel algorithms on a cluster of CPU's will be discussed.

Glioma (brain tumor) invasion depends on its microenvironment. We will present two models in this talk. We first analyze the migration patterns of glioma cells from the main tumor, and show that the various patterns observed in experiments can be obtained by a model's simulations, by choosing appropriate values for some of the parameters (chemotaxis, haptotaxis, and adhesion) of the PDE model. For the second part of talk, we introduce a multiscale model in order to get more detailed informations on cell migration. The results of such an approach are compared to the experimental data as well.

*This is joint work with Avner Friedman (MBI), Sean Lawler, Michal O. Nowicki, E. Antonio Chiocca (Oncological Neurosurgery), Jed Johnson, John Lannutti (Lannutti lab) at the Ohio State University, and Hans Othmer (University of Minnesota).

We consider a very simple curvature condition:
Given constant $c$ and a dimension $n$ we say that a
manifold $(M,g)$ satisfies the condition (c,n) if the scalar
curvature is bounded below by c times the norm of the Weyl
curvature. We show that in each large even dimensions there is precisely one
constant $c^2=2(n-1)(n-2)$ such that this condition is invariant under
the Ricci flow.

The condition behaves very similar to scalar curvature under conformal
transformations
and we indicate how this can be utilized to get a large source of examples.
Finally we speculate what kind singularities should develop under the
Ricci flow.

I will discuss our solution (joint with A. Myasnikov) of Tarski's
problems about elementary theory of free groups,
new techniques and directions that resulted from this solution.

We present a sequential quadratic programming (SQP)
algorithm for nonlinear optimization. We give a
brief overview of SQP methods in general and then
describe an active-set method based on inertia control
for solving the convex quadratic subproblems. We also
discuss the motivation behind this algorithm as well as
its applications.

The objective of clustering, or unsupervised classification, is to partition a set of observations into different groups, or clusters, based on their similarities. Following Hartigan, a cluster is defined as a connected component of an upper level set of the underlying density. In this talk, we introduce a spectral clustering algorithm on estimated level sets, and we establish its strong consistency. We also discuss the estimation of the number of connected components of density level sets.

In 1961, Erdös asked whether or not there exist words of
arbitrary length over a fixed finite alphabet that avoid patterns of the
form $XX'$ where $X'$ is a permutation of $X$ (called "abelian squares"). This
problem has since been solved in the affirmative in a series of papers
from 1968 to 1992. A natural generalization of the problem is to study
"abelian k-th powers", i.e., words of the form $X_1X_2...X_k $where $X_i$ is
a permutation of $X_1$ for $2 \le i \le k$.
In this talk, I will discuss "crucial words" for abelian k-th powers,
i.e., finite words that avoid abelian k-th powers, but which cannot be
extended to the right by any letter of their own alphabets without
creating an abelian k-th power. More specifically, I will consider the
problem of determining the minimal length of a crucial word avoiding
abelian k-th powers. This problem has already been solved for abelian
squares by Evdokimov and Kitaev (2004). I will present a solution for
abelian cubes (the case k = 3) and state a conjectured solution for the
case of $k \ge 4.$

This is joint work with Amy Glen and Bjarni V. Halldórsson (Reykjavík
University).

Einstein's constraint equations govern the geometric properties of space-time in relation to matter and energy. Motivated by the preservation of the fulfillment of these constraints in a Hamiltonian formulation, the conformal thin sandwich method is a successful approach to determining the solution in a number of parameter classes.

We will start with a quick excursion into some of the highlights of the
history of Algebra. This leads to some present trends which connect
Algebra to several other areas of Mathematics from Algebraic Geometry and
Topology to pure and applied Analysis. These topics will be illustrated
through some concrete examples, such as quantum groups, braid groups,
wonderful models, toric arrangements, splines, equivariant $K$-theory and
the index theorem.

There are many methods one may use to solve partial
differential equations numerically. For large scale
problems, direct methods are not computationally
feasible and therefore iterative methods tend to be the
best option. Multigrid methods are a particularly
attractive strategy for certain classes of
problems. Roughly speaking, in a multigrid approach, a
problem is solved on a hierarchy of grids. The purpose
of this talk is to discuss the benefits of a multigrid
strategy and various ways it may be introduced in
optimization. Of particular interest is the so called
nonlinear multigrid scheme.

In this talk, we give a brief introduction to a natural
generalization of groups, called n-groups.\\

Just as discrete groups represent the homotopy types of acyclic spaces,
n-groups realize homotopy types of connected topological spaces $X$ such that
$\pi_i(X)=0$ for $i>n$. In this talk, we adopt the formalism of simplicial sets,
and define n-groups as simplicial sets satisfying certain a filling
condition (introduced by Duskin).\\

In the first part of the talk, we explain what a 2-group look like: this
material is contained in any textbook on simplicial sets. We indicate how
2-groups arise in topological quantum field theory.

The multitype contact process is a stochastic model including space in the form of local interactions and describing the evolution of two species competing on a connected graph. While it is conjectured for the multitype contact process on the two- dimensional regular lattice that, regardless of their birth and death rates, species cannot coexist at equilibrium, we prove that two species with opposite strategies (specialist versus generalist) coexist on a connected graph including two levels of interactions.

The space of diagonal harmonics has emerged as one of the key ingredients in a program initiated by Garsia and Haiman to give a representation-theoretical proof of some conjectures in the theory of Macdonald polynomials.

The study of this particular space has provided a remarkable display of connections between several areas, including representation theory, symmetric function theory, and combinatorics. Over two decades since the introduction of the diagonal harmonics, the bivariate Hilbert series of the diagonal harmonics has been the object of a variety of algebraic and combinatorial conjectures. In the following lecture, we will define the diagonal harmonics and explore some of the combinatorial objects related to this space. We assume only a basic understanding of undergraduate algebra and a passing appreciation for either free food or beautiful mathematical pictures.

Glaucoma is the second leading cause of blindness worldwide. Often the optic nerve head (ONH) glaucomatous damage and ONH changes occur prior to loss of visual function and are observable in vivo. Thus, digital image analysis is a promising choice for detecting the onset and/or progression of glaucoma. In this work, we present a new framework for detecting glaucomatous changes in the ONH using the method of proper orthogonal decomposition (POD)1. A baseline topograph subspace is constructed using POD for each eye to describe the ONH structure of the eye at a reference/baseline condition. The bases that form the baseline subspace capture the topograph measurement variability and any inherent structure variability of the ONH at baseline. Any glaucomatous changes in the ONH of an eye present during a follow-up exam are estimated by comparing the ONH topograph acquired from the follow-up exam with its baseline topograph subspace representation. Image correspondence measures of correlation, Euclidean distance, and image Euclidean distance (IMED) are used to quantify the ONH changes. An ONH topograph library built from the Louisiana State University experimental glaucoma study is used to demonstrate the performance.

This a joint work with Wanke Yin.
Let $M\subset \mathbb{C}^{n+1}$ ($n\ge 2$) be a real
analytic submanifold defined by an equation of the form:
$w=|z|^2+O(|z|^3)$, where we use $(z,w)\in {CC}^{n}\times CC$
for the coordinates of ${C}^{n+1}$. We first derive a pseudo-normal form
for $M$ near $0$. We then use it to prove that $(M,0)$ is holomorphically
equivalent to the quadric $(M_\infty: w=|z|^2,\ 0)$ if and only if it can
be formally transformed to $(M_\infty,0)$, using the rapid convergence
method. We also use it to give a necessary and sufficient condition
when $(M,0)$ can be formally flattened. Our main theorem generalizes a
classical result of Moser for the case of $n=1$.

All rings are associative with $1\not = 0$. A ring $A$ is decomposable if $A=A_{1}\times A_{2}$, otherwise $A$ is indecomposable.

We consider three types quivers of rings: Gabriel quiver, prime quiver and Pierce quiver.
Gabriel quiver and Pierce quiver are defined for semiperfect rings.
Let $A$ be an associative ring with the prime radical $Pr(A)$.
The factorring $\bar{A} = A/Pr(A)$
is called the diagonal of $A$. We say that a ring $A$ is a $FD$-ring if $\bar{A}$ is a finite direct product
of indecomposable rings. We define the prime quiver of $FD$-ring with $T$-nilpotent prime radical.

We discuss the properties of rings and its quivers, for example, a
right Noetherian semiperfect ring is semisimple Artinian if and only if
its Gabriel quiver is a disconnected union of vertices (without arrows).

Given a word over some alphabet, we can form a graph with the letters of
the alphabet as vertices, and with two vertices adjacent if those
letters occur alternatingly in the word. A motivation for studying the
class of graphs represented by words (in the described manner) comes
from algebra, but another application is in robot scheduling.\\

\noindent When considering a class of graphs, several immediate questions pop up:\\

\noindent - Which graphs belong (and which ones do not) to the class,\\
- How large do the words need to be to represent all such graphs, and\\
- Can we come up with alternative representations that in particular
make it easier to answer structural and algorithmic questions about
these graphs?\\

I will discuss recent answers to these questions. This is joint work
with Magnus M. Halldorsson (Reykjavik University) and Artem Pyatkin
(Sobolev Institute of Mathematics).

Systems characterized by expensive, nonconvex, noisy functions in moderate dimensions (n=2 to 24) necessitate the development of maximally efficient derivative-free optimization algorithms. Starting with the well-known Surrogate Management Framework (SMF), our lab has developed a new, highly efficient derivative-free optimization algorithm, which we dub LAttice-Based Derivative-free Optimization via Global Surrogates (LABDOGS). This algorithm combines a highly efficient, globally convergent surrogate-based Search algorithm with an efficient Poll which incorporates a minimum number of new function evaluations chosen from nearest-neighbor points. All function evaluations are coordinated with highly uniform noncartesian lattices derived from n-dimensional sphere packing theory. Representative numerical tests confirm significant improvements in convergence of lattice-based strategies as compared with otherwise identical codes coordinated using Cartesian grids.

The second topic of our talk focuses on incorporating approximate function evaluations into a surrogate-based optimization scheme. Assuming the accuracy of each function evaluation in a statistical setting diminishes towards zero in proportion with the reciprocal of the square root of the sample time, we have developed an algorithm for sampling the function only as accurately as warranted. The algorithm we have developed, dubbed $\alpha$-DOGS, maintains the globally convergent behavior of the LABDOGS Search while focusing the bulk of the computation time on regions of parameter space where the existing approximate function evaluations indicate that the true function minimum might lie.

In this talk, I discuss the reduction of the number of squares needed to express a sum of squares in the free *-algebra R. I will give examples of sums which are irreducible in this sense, and prove bounds on the minimal number of terms needed to express an arbitrary sum of squares of a given degree in a given number of variables

Singular value optimization problems arise in various applications in control theory. For instance the $H_{\infty}$ norm of the transfer function of a linear dynamical system, and the distance problems such as complex (or real) stability and controllability radii have singular value optimization characterizations. These problems are non-convex and non-smooth. The existing commonly employed algorithms for these problems are derivative-free, but do not exploit the Lipschitz nature of singular values in a systematic manner. Here we solve these problems largely depending on a Lipschitz optimization algorithm due to Piyavskii and Shubert, that never got attention in the context of optimization of eigenvalues or singular values. The Piyavskii-Shubert based algorithm outperforms the commonly employed algorithms for medium to large scale problems when a few digit accuracy is sought.

Recently Wang, Zheng, Boyd, and Ye proposed a further convex relaxation of the SDP relaxation for the sensor network localization problem, which they called edge-based SDP (ESDP). The ESDP is easier to solve than the SDP and, in simulation, yields solution about as accurate as the SDP relaxation. We show that, when the distance measurements are exact, we can determine which sensors are correctly positioned in the ESDP solution by checking if their individual traces are zero. On the other hand, we show that, when the distance measurements are inexact, this check is unreliable for both ESDP and SDP solutions. We then propose a robust version of ESDP relaxation for which small individual trace is a reliable check of sensor position accuracy. Moreover, the position error for such a sensor is in the order of the square root of its trace. Lastly, we propose a coordinate gradient descent method, using log-barrier penalty, to solve ESDP. This method is more efficient than interior-point method for solving SDP or ESDP and is implementable in a distributed manner. (This is joint work with Ting Kei Pong.)

There are at least $p^{2n^3/27+O(n^2)}$ groups of order $p^n$,
and in 2006 those of order $p^7$ were classified in over 600 pages of work.
Yet, with such a multitude of groups, a structure theory seems impossible.
One approach is to decompose the $p$-groups via central
and related products to reduce the study to indecomposable groups. Using rings
and Jordan algebras, a theorem is proved on the uniqueness of these decompositions,
asymptotic estimates are given which show there are roughly equal numbers of
decomposable and indecomposable groups, and the indecomposable groups are
categorized into classical families.

The efficient computation of complex flows on the sphere, governed by the 2D shallow-water equations, is of acute importance in the modeling and forecasting of weather phenomenon on the earth. Some of the most powerful supercomputer clusters every built have been fully dedicated to this problem. In the years to come, increased performance in such clusters will be derived in large part from massive parallelization, to tens of thousands and even hundreds of thousands of computational nodes in the cluster. To facilitate such scalability, switchless interconnect systems coordinating the communication within the cluster are absolutely essential, as such systems eliminate an otherwise significant bottleneck (that is, the switch) impeding the communication between the nodes.

The present work introduces a new switchless interconnect topology for supercomputer clusters which are dedicated specifically for computing such flows on the sphere. This topology is based on a class of Fullerenes (i.e., Buckyballs) with octahedral symmetry. In this topology, each node has direct send/receive capabilities with three neighboring nodes, and the cluster is itself physically connected in a spherical configuration. This natural correspondence between the interconnect network and the discretized physical model itself tends to keep most communication local (that is, between neighbors) during the flow simulation, thereby minimizing the density of packets being passed across the cluster and increasing dramatically the overall computational speed. One of the most communication-intensive steps of the flow simulation is related to solving the Poisson equation on the sphere; it is shown that the present topology is particularly well suited to this problem, leveraging multigrid acceleration with Red/Black Gauss-Seidel smoothing.

No matter what discoveries are made at the Large Hadron Collider in Switzerland when it begins operating next year, its a sure thing that gauge fields (i.e., connections on vector bundles) will continue to play the central role in elementary particle theory that they have for the past 40 years.

The quantization of a pure gauge field amounts, informally, to the construction of a suitable measure on the configuration space of the gauge field, (i.e., the moduli space: connection forms modulo gauge transformations.) This is an infinite dimensional manifold which must be chosen large enough, in some distribution sense, to support this measure. In this talk I am going to show how one can hope to realize such nonlinear distribution spaces as spaces of geometric flows. Specifically, I will describe the state of the art for the Yang-Mills heat equation on a three manifold with boundary.

This term's plan is to read Jacob Lurie's new preprint:``On the classification of Topological Field Theories'', which is available on his MIT homepage. As usual, seminar participants will give the talks, and we'll try to parcel them out at the first meeting on April 7th. But everyone is welcome - we won't force you to speak if you don't want to! \\

\noindent In 1989 Atiyah (inspired by Segal and Witten) defined a TFT to be a monoidal functor from the category of (n+1)-dimensional cobordisms to the category of vector spaces. That is, it assigns a vector space to each closed n-manifold, and linear maps between these to each (n+1)-dimensional cobordism (that is, an (n+1)-dimensional manifold whose boundary is divided into "ingoing" and "outgoing" parts), satisfying natural composition laws. The idea comes from quantum field theory, in which each slab of spacetime between "past" and "future" spacelike hypersurfaces should define a unitary map between their corresponding Hilbert spaces of states. The difference is that in QFT, the metrics on such spacetime cobordisms matter, whereas in TFT the linear maps depend only on the underlying topology of the cobordisms. The general formalism of QFT suggests that one should be able to extend this algebraic structure into lower dimensions, assigning a category to each (n-1)-dimensional manifold, a 2-category to each (n-2)-dimensional manifold, and so on, ultimately assigning some kind of n-category to the point: this n-category ought to determine the whole TFT structure. Many attempts to formulate this sort of thing were made in the early 90s, but because of the lack of a solid definition of ``n-category'', made little progress. One can also extend into higher dimensions: k-parameter families of manifolds can be added into the picture, leading to theories in which the topology of diffeomorphism groups of manifolds enters naturally. A theory of this sort in 2 dimensions was worked out by Kevin Costello a few years ago under the name ``Topological Conformal Field Theory''. Lurie's new paper provides a complete formulation of TFTs incorporating all of the above features. He provides a solid definition of n-categories in the spirit of algebraic topology, and proves many foundational results about them. Then he shows how TFTs can be characterised using this language. In particular, he proves the remarkable ``Baez-Dolan cobordism hypothesis'', which states that the n-category of n-dimensional cobordisms is the free n-categ

Privacy-preserving support vector machine (SVM) classifiers are proposed for vertically
and horizontally partitioned data. Vertically partitioned data represent instances where
distinct entities hold different groups of input space features for the same individuals, but
are not willing to share their data or make it public. Horizontally partitioned data
represent instances where all entities hold the same features for different groups of
individuals and also are not willing to share their data or make it public. By using a
random kernel formulation we are able to construct a secure privacy-preserving kernel
classifier for both instances using all the data but without any entity revealing its
privately held data. Classification accuracy is better than an SVM classifier without
sharing data, and comparable to an SVM classifier where all the data is made public.

We introduce the area of extremal set theory via three
classical
results: the Erdos-Ko Rado theorem, Frankl's $r$-wise intersection
theorem, and the Kruskal-Katona shadow theorem. We then consider vector
space analogs of these problems. We prove a vector space analog of a
version of the Kruskal-Katona theorem due to Lov\'{a}sz. We apply this
result to extend Frank's theorem on $r$-wise intersecting families to
vector spaces. In particular, we obtain a short new proof of the
Erdos-Ko-Rado theorem for vector spaces.

We discuss metrics of positive curvature on 3-Sasakian 7-manifolds including one on a new diffeomorphism type.

For Markov random fields temporal mixing, the time it takes for the Glauber dynamics to approach its stationary distribution, is closely related to phase transitions in the spatial mixing properties of the measure such as uniqueness and the reconstruction problem. Such questions connect ideas from probability, statistical physics and theoretical computer science. I will survey some recent progress in understanding the mixing time of the Glauber dynamics as well as related results on spatial mixing.

In this talk, I plan to discuss how differential geometry can provide useful insights into the study of ordinary and partial differential equations. In particular, I will focus on the role of symplectic geometry in classical Lagrangian and Hamiltonian mechanics, as well as its generalization to the multisymplectic geometry of classical field theory. Finally, I will talk about how this perspective has paved the way for the development of ``geometric'' numerical integrators, which exactly preserve important structures, symmetries, and invariants.

Electronic structure calculations are commonly used to understand and predict the electronic, magnetic and optic properties of molecular systems and materials. They are also at the basis of ab initio molecular dynamics, the most reliable technique to investigate the atomic scale behavior of materials undergoing chemical reactions (oxidation, crack propagation, ...). In the first part of my talk, I will briefly review the foundations of the density functional theory for electronic structure calculations. In the second part, I will present some recent achievements in the field, as well as open problems. I will focus in particular on the mathematical modelling of defects in crystalline materials.

A novel Bloch band based level set method is proposed for computing
the semiclassical limit of Schrodinger equations in periodic media.
For the underlying equation subject to a highly oscillatory initial
data, a hybrid of the WKB approximation and homogenization leads to
the Bloch eigenvalue problem and an associated Hamilton-Jacobi system
for the phase in each Bloch band, with the Bloch eigenvalue be part
of the Hamiltonian. We formulate a level set description to capture
multi-valued solutions to the band WKB system, and then evaluate
total homogenized density over a sample set of bands. A superposition
of band densities is established over all bands and solution branches
when away from caustic points. The numerical approach splits the
solution process into several parts: i) initialize the level set
function from the band decomposition of the initial data; ii) solve
the Bloch eigenvalue problem to compute Bloch waves; iii) evolve the
band level set equation to compute multi-valued velocity and density
on each Bloch band; iv) evaluate the total position density over a
sample set of bands using Bloch waves and band densities obtained in
step ii) and iii), respectively. Numerical examples with different
number of bands are provided to demonstrate the good quality of the
method.

If a solution (M,g(t)) of Ricci flow develops a local singularity at a finite time T, there is a proper subset S of M on which the curvature becomes infinite as time approaches T. Existing approaches to Ricci-flow-with-surgery, due to Hamilton and Perelman, require one to modify the solution in a small neighborhood of S by gluing in a highly curved but nonetheless nonsingular solution. This must be done with careful regard to various surgery parameters in order to preserve critical a priori estimates. In case the local singularity is a rotationally-symmetric neckpinch (in any dimension $n>2$), we can now restart Ricci flow directly from the singular limit g(T), without performing an intervening surgery or making ad hoc choices. We show that any complete smooth forward evolution from g(T) is necessarily compact and has a unique asymptotic profile as it emerges from the singularity, which we describe. (This is joint work with Sigurd Angenent and Cristina Caputo.)

We study fronts of cells such as those invading a wound or in a growing tumor. First we look at a discrete stochastic model in which cells can move, proliferate, and experience cell-cell adhesion. We compare this with a coarse-grained, continuum description of this phenomenon by means of a generalized Cahn-Hilliard equation (GCH) with a proliferation term. 

There are two interesting regimes. For subcritical adhesion, there are propagating "pulled" fronts, similarly to those of Fisher-Kolmogorov equation. The problem of front velocity selection is examined, and our theoretical predictions are in a good agreement with a numerical solution of the GCH equation. For supercritical adhesion, there is a nontrivial transient behavior. The results of continuum and discrete models are in a good agreement with each other for the different regimes we analyzed.

Even though much progress has been made in mainstream experimental cancer research at the molecular level, traditional methodologies alone are sometimes insufficient to resolve important conceptual issues in cancer biology. In this talk, I will describe mathematical tools which help obtain new insights into the processes of cancer initiation, progression and treatment. The main idea is to study cancer as an evolutionary dynamical system on a selection-mutation network. I will discuss the following topics: Stem cells and tissue architecture; Stem cells and aging, and Drug resistance in CML.

A basic question in geometry and topology is to discover necessary (and perhaps sufficient) conditions for two manifolds to be locally or globally equivalent for some notion of equivalence. An example of a global topological invariant is the fundamental group of a topological space, because having isomorphic fundamental groups is a necessary condition for two spaces to be equivalent up to homotopy. We restrict ourselves to real hypersurfaces in $C^2$. I'll sketch Poincare's proof of the global inequivalence of the unit ball and polydisc, then outline a method due to Cartan, Chern, and Moser, about how to find a system of invariants for the local equivalence problem. Knowing a bit of differential geometry and complex analysis would be helpful, but isn't essential.

Numerical simulations of black hole binaries have made tremendous progress over the last years. The usefulness of such simulations is limited by their tremendous computational cost, which ultimately results from a separation of time-scales: Emission of gravitational radiation drives the evolution of the binary toward smaller separation and eventual merger. The time-scale for inspiral is far longer than the dynamical time-scale of each black hole. Therefore, the currently deployed explicit time-steppers are severely limited by Courant instabilities. Implicit time-stepping algorithms provide an obvious route around the Courant limit, thus offering a tremendous potential to speed up the simulations. However, the complexity of Einstein's equations make this a highly non-trivial endevour. This talk will first present a general overview of the status of Black Hole simulations, followed by a status report on the ongoing work aimed at implementing modern implict/explicit (IMEX) evolution schemes for Einstein's equations.

Let $A/F$ be an abelian variety
over a number field $F$ and let $P \in A(F)$
and $\Lambda \subset A(F)$ be a subgroup of the Mordell-Weil group.
For a prime $v$ of good reduction let
$r_v : A(F) \rightarrow A_v (k_v)$ be the reduction map.
During my talk I will show that the condition $r_v (P) \in
r_v (\Lambda)$ for almost all primes $v$ implies that
$P \in \Lambda + A(F)_{tor}$ for a wide class of abelian varieties.

Refreshments will be served at 3:30 P.M.\\

\noindent Richard Libby is responsible for counterparty risk oversight and control, related risk measurement and policy, operational and credit risk, economic and regulatory capital analysis, and model validation. \\

\noindent Prior to joining Barclays Global Investors in 1999, Richard was Vice President for Capital Markets Analytics at Bank of America with responsibility for credit derivatives and market risk systems and analytics. \\

\noindent Richard has a BA and MA in mathematics from the University of California, San Diego, and a PhD in mathematics from the University of California, Santa Cruz.

We convert the standard vector calculus description of Maxwell's Equations into the language of differential forms on Minkowski spacetime, which results in a very elegant reformulation (just two equations instead of four). We then show that this is actually invariant under Lorentz transformations, and describe what bothered Einstein so much that he had to formulate Special Relativity to fix things up.

The Fundamental Gap Conjecture due to S. T. Yau and M. van de Berg states that for a convex domain in $R^n$ with diameter $d$, the first two positive eigenvalues of the Dirichlet Laplacian satisfy \[\lambda_2 - \lambda_1 \geq \frac{3 \pi^2}{d^2}.\] $\lambda_2 - \lambda_1$ is known as the fundamental gap and has been studied by many authors. It is of natural interest to spectral geometers, and moreover, estimates for the fundamental gap have applications in analysis, statistical mechanics, quantum field theory, and numerical methods.

I will discuss joint work with Zhiqin Lu on the fundamental gap when the domain is a Euclidean triangle. Our first result is a compactness theorem for the gap function, which shows that the gap function is unbounded as a triangle collapses to a segment. I will outline our current work which indicates that the equilateral triangle is a strict local minimum for the gap function on triangular domains. Finally, I will discuss how these results combined with numerical methods may be used to prove the well known conjecture that among all triangular domains, the fundamental gap is minimized by the equilateral triangle.

Fluidization processes have many important applications in industry, in particular, in chemical, fossil, and petrochemical industries where good gas-solid mixing is required. Such mixing is commonly achieved through bubbles which are formed spontaneously and whose time-evolution appears to be governed by low-dimensional deterministic dynamics. A low-dimensional, computational agent-based bubble model is used to study the changes in the global bubble dynamics in response to changes in the frequency of the rising bubbles. A computationally-based bifurcation analysis shows that the collective bubble dynamics undergoes a series of transitions from equilibrium points to highly periodic orbits, chaotic attractors, and even intermittent behavior between periodic orbits and chaotic sets. Using ideas and methods from nonlinear dynamics and time-series analysis, long-term predictions for the purpose of developing control algorithms is possible through model fitting.

The Poisson-Boltzmann equation is widely used for modeling solvation effects. The computational cost of PB has largely restricted its applications to single-conformation calculations. The generalized Born model provides an approximation at substantially reduced cost. Currently the best GB methods reproduce PB results for electrostatic solvation energies with errors at $>$ 5 to 10 kcal/mol. When two proteins form a complex, the net electrostatic contributions to the binding free energy are typically of the order of 5 to 10 kcal/mol. Similarly, the net contributions of individual residues to protein folding free energy are $<$ 5 kcal/mol. Clearly in these applications the accuracy of current GB methods is insufficient. Here we present a simple scaling scheme that allows our GB method, $GBr^6$, to reproduce PB results for binding and folding free energies with high accuracy. From an ensemble of conformations sampled from molecular dynamics simulations, five were judiciously selected for PB calculations. These PB results were used for scaling $GBr^6$. Tests on protein binding and folding show that effects of point mutations calculated by scaled $GBr^6$ are accurate to within 0.5 kcal/mol or less. This method makes it possible to incorporate conformational sampling in electrostatic modeling without loss of accuracy.

The built environment is responsible for about 30\% of greenhouse gas emissions in the US. The design of green buildings that use significantly less energy, especially for cooling, requires mathematical models that can assist architects and designers to create new designs. I will discuss one aspect - the use of natural ventilation in buildings which are cooled by using the thermal energy they acquire either through solar heating or from gains within the building from people and equipment. This kind of analysis was used to optimize the design of the new San Diego Children's Museum, among others.

Finding a needle in a haystack was once a hard problem.
However, magnets made finding that needle much easier. In the modern
age, the vast amount of information is our haystack, and a particular
piece of information is our needle, and as the title suggests,
eigenvalues and eigenvectors are our magnets. In the last decades,
many researchers have found more and more ways to use eigenvalues and
eigenvectors as our magnets to find particular the pieces of
information we are looking for. Among these include the PageRank
algorithm and spectral bipartitioning. We will give the basic theory
behind these techniques and explore some examples.

It has been recognized in the last 5 years that specialized graphics hardware can also be used for general purpose computations. The architecture of these cards is such that SIMD computations are naturally a good fit for a certain class of applications. I will outline the programming model of modern graphics cards, sketch the history of the development of the supporting software stack and if there is interest I will outline how we have implemented lattice gauge theory algorithms leading to dramatic speedup of Monte Carlo simulations.

We continue the formulation of Maxwell's Equations in the language of differential forms. We describe how the Hodge star operator plays a role in relating the equations together, and also introduce the electromagnetic 4-potential, which unifies the classical electric scalar and magnetic vector potentials into one spacetime object. We then use both these tools to recast Maxwell's Equations as a wave equation, and investigate what it means for boundary value problems.

\indent It is never too early for grad students to begin thinking about choosing an area of specialty and choosing among the faculty who might supervise them. One of the most important choices a graduate student will make will be choosing a thesis advisor. However, it is a process that is unlike anything that students have encountered in their undergraduate education. For this reason, we felt like it would be useful for us to run a meeting where the actual process of finding an advisor is described by students who have only recently found thesis advisors.\\
\indent How did other students find a thesis advisor? What are the key factors to consider when choosing an advisor? What do professors look for before they accept a student as their thesis student? How does finding a thesis advisor lead to finding a thesis problem? We will discuss these questions.
We will have four graduate students-Jaime Lust, Allison Cuttler, Joey Reed and Ben Wilson describe their experiences finding a thesis advisor and what happens after a thesis advisor is found.\\
\indent We will also have one faculty, Jim Lin, describe what he looks for in a graduate student before he accepts him or her as a thesis student.\\

\indent All students, especially first, and second year students, are cordially invited to attend.

I will discuss the metric behavior of the Kahler-Ricci flow on Hirzebruch surfaces assuming that the initial metric is invariant under a maximal compact subgroup of the automorphism group. I will describe how, in the sense of Gromov-Hausdorff, the flow either shrinks to a point, collapses to $P^1$ or contracts an exceptional divisor. This confirms a conjecture of Feldman-Ilmanen-Knopf. This is a joint work with Jian Song.

A well known method of estimating the offset between two clocks in a data communication network involves exchanging timing messages between the clocks. Different distributions of the transmission delays in the two directions associated with the exchanged messages cause the estimator to be biased. Bootstrap bias-correction improves the estimator with respect to mean squared error. Studies on network traffic show that no single distribution adequately characterizes delays, and thus robustness of an estimator to different distribution assumptions is a critical property for an estimator to have. For common distribution assumptions for the transmission delays, the bias-corrected estimator has smaller mean squared error than the uncorrected estimator. Recent studies of Internet traffic show that delay distributions can be heavy-tailed. Evaluation of bootstrap bias corrected estimators in the context of heavy tailed network delays leads to some surprising results. Confidence interval procedures for clock offset and a brief discussion of estimating the difference in rates between two clocks will also be given.

Talk time runs from 11:15 AM until 12:15 PM.

Talk time runs from 4:45-5:45 PM.

A fertile area of recent research has found concrete polynomial time
lower bounds for solving hard computational problems on restricted
computational models. Among these problems are Satisfiability, Vertex
Cover, Hamilton Path, MOD6-SAT, and Majority-of-Majority-SAT, to name
a few. The proofs of such lower bounds all follow a certain
proof-by-contradiction strategy.

I will survey some of the results in this area, giving an overview of
the techniques involved. If there is time I will discuss an automated
search strategy for studying these proof techniques. In particular,
the search for better lower bounds can often be turned into the task
of solving a large series of linear programming instances.
Furthermore, the limits of these proof system(s) can be understood by
analyzing the space of possible linear programs

Computation of high frequency solutions to wave equations is important in many applications, and notoriously difficult in resolving wave oscillations. Gaussian beams are asymptotically valid high frequency solutions concentrated on a single curve through the physical domain, and superposition of Gaussian beams provides a powerful tool to generate more general high frequency solutions to PDEs. In this talk I will present a recovery theory of high frequency wave fields from phase space based measurements. The construction use essentially the idea of Gaussian beams, level set description in phase space as well as the geometric optics. Our main result asserts that the kth order phase space based Gaussian beam superposition converges to the original wave field in L2 at the rate of $\epsilon^{k/2-n/4}$ in dimension $n$. The damage done by caustics is accurately quantified. This work is in collaboration with James Ralston (UCLA).

We will have four panelists who have recently found jobs: Kristin Jehring, Assistant Professor, tenure track at St Mary's College, Indiana, Andy Niedermaier, Jane Street Capital, New York City, Mike Kinnally, Metron, a scientific consulting company in Reston, Virginia and Nate Eldredge, postdoc at Cornell University, Ithaca, New York.
They will describe their experiences applying for a job. Some of the questions they will answer are: How many applications should I send out? How do I prepare for an interview? What should I write in my cover letter and resume? What are important qualifications for a teaching job, postdoc job, tenure track research job and job in industry?
The discussion will be followed by a question and answer period.

The groups in question are modeled on an abstract Wiener space. Then a
group Brownian motion is defined, and its properties are studied in
connection with the geometry of this group. The main results include
quasi-invariance of the Gaussian (heat kernel) measure, log Sobolev
inequality (following a bound on the Ricci curvature), and the Taylor
isomorphism to the corresponding Fock space. The latter map is a
version of the Ito-Wiener expansion in the non-commutative setting.
This is a joint work with B. Driver.

Noncommutative noetherian rings will be studied
through rings of fractions. Examples will be presented and applications to baseball will be mentioned, if time permits.

I will discuss the problem of backwards-uniqueness or "unique-continuation" for the Ricci flow, and prove that two complete solutions $g(t)$, $\tilde{g}(t)$ to the Ricci flow on $[0, T]$ of uniformly bounded curvature that agree at $t=T$ must agree identically on $[0, T]$. A consequence is that the isometry group of a solution to the Ricci flow cannot expand in finite time.

In this talk, I will explain some similar results and interaction between locally symmetric spaces and moduli spaces of Riemann surfaces.

For example, let $M_g$ be the moduli space of Riemann of genus $g$, and $A_g$
be the moduli
space of principally polarized abelian varieties of dimension $g$, i.e.,
the quotient of the Siegel upper space by $Sp(2g, Z)$.
Then there is a Jacobian map $J: M_g \to A_g$, by associating to each Riemann
surface its Jacobian.

The celebrated Schottky problem is to characterize the image $J(M_g).$
Buser and Sarnak viewed $A_g$ as a complete metric space and showed that $J(M_g)$ lies in a very small neighborhood of the boundary of $A_g$ as $g$ goes to infinity. Motivated by this, Farb formulated the coarse Schottky problem: determine the image of $J(M_g)$ in the asymptotic cone (or tangent space at infinity) $C_\infty(A_g)$ of $A_g$, as defined by Gromov in large scale geometry.

In a joint work with Enrico Leuzinger, we showed that $J(M_g)$ is $c$-dense in $A_g$ for some constant $c=c(g)$ and hence its image in the asymptotic cone $C_\infty(A_g)$ is equal to the whole cone.

Another example is that the symmetric space $SL(n,R)/SO(n)$ admits several important equivariant cell decompositions with respect to the arithmetic group $SL(n, Z)$ and hence a cell decomposition of the locally symmetric space $SL(n, Z)/SL(n, R)/SO(n)$. One such decomposition comes from the Minkowski reduction of quadratic forms (or marked lattices). We generalize the Minkowski reduction to marked hyperbolic Riemann surfaces and obtain a solution to
a folklore problem: an intrinsic equivariant cell decomposition of the Teichmuller space $T_g$ with respect to the mapping class groups $Mod_g$,
which induces a cell decomposition of the moduli space $M_g$.

If time permits, I will also discuss other results on similarities between the two classes of spaces and groups.

A national cooperative of universities is developing a collection of video case materials about college math instruction. The project goal is to create a visually rich resource for helping novice instructors build teaching skills. The purpose of the presentation is to share some of the materials, review their development, discuss their potential uses, and gather comments to inform materials re-development. Attendance by all department members, from those very experienced in teaching college mathematics to those with a few days experience to those with intentions to teach in the future, is heartily encouraged. Video clips come from advanced as well as introductory undergraduate mathematics teaching and learning situations.

Hypoelliptic operators live in an interesting corner of the world of PDE, in which geometry plays a crucial role. Lie groups are a natural setting for the study of these operators, but even for simple examples such as the Heisenberg group, many questions remain open. I will give an overview and examples of what these objects are and how they behave, and discuss some recent results involving estimates for hypoelliptic heat kernels on
H-type groups, a class of Lie groups which generalize some of the properties of the Heisenberg group. All are welcome to attend.

I will discuss a construction of ancient solutions to Ricci flow on spheres and complex projective spaces which generalize Fateev's examples on three spheres. These examples supply counter-examples to some folklore conjectures on ancient solutions of Ricci flow on compact manifolds. This a joint work with Ioannis Bakas and Lei Ni.

Neural dynamics, from the dynamics of a single cell to the modeling of the activity of whole neural clusters, give rise to high dimensional dynamical systems. Unfortunately computers are too slow to solve them and math does not provide the theorems to make qualitative statements. This talk will be about models, what we can actually do and why the Fibonacci numbers keep popping up everywhere. Even when studying neural dynamics.

The category of exploded manifolds is an extension of the smooth
category with a good holomorphic curve theory and a `large scale'
related to tropical geometry. I will give examples to illustrate the
usefulness of working with exploded manifolds in order to study
holomorphic curves.

We develop a new approach to Gelfand-Zeitlin
theory for the symplectic group $Sp(n,\mathbb{C})$. Classical Gelfand-Zeitlin theory, concerning $GL(n,\mathbb{C})$, rests on the fact that branching from $GL(n,\mathbb{C})$ to $GL(n-1,\mathbb{C})$ is multiplicity-free. Since branching from $Sp(n,\mathbb{C})$ to $Sp(n-1,\mathbb{C})$ is not multiplicity-free, the theory cannot be
directly
applied to this case.

Let $L$ be the $n$-fold product of $SL(2,\mathbb{C})$. Our main theorem asserts that each multiplicity space that arises in the restriction of an irreducible representation of $Sp(n,\mathbb{C})$ to $Sp(n-1,\mathbb{C}$, has a unique irreducible $L$-action satisfying certain naturality conditions. We also given an explicit description of the $L$-module structure of each multiplicity space. As an application we obtain a Gelfand-Zeitlin type basis for the irreducible representations of $Sp(n,\mathbb{C})$.

n this joint work with Jingyi Chen and Weiyong He, we prove existence of long time smooth solutions to mean curvature flow of entire Lipschitz Lagrangian graphs. As an application, we obtain results on entire translating and self-expanding solutions to to the Lagrangian mean curvature flow.

Nonparametric and semiparametric mixture models are valuable
tools for solving many nasty problems when a population is
heterogeneous. While the maximum likelihood approach is straightforward,
its computation has long been known as being difficult, if not
intractable, due to the estimation of a distribution function defined on
an infinite-dimensional space. In this talk, I will describe some fast
algorithms that I recently developed for fitting these models; present
the results of their use in several applications, including the
over-dispersion problem, simultaneous hypothesis testing, the
Neyman-Scott problem and mixed effects models; and discuss some
implementation issues using R.

In this talk, we will discuss the problem of finding conformal metrics
with constant Q-curvature on a given compact four dimensional Riemannian
manifold (M,g) with boundary. This will be equivalent to solving a fourth
order nonlinear elliptic boundary value problem with boundary condition
given by a third-order pseudodifferential operator, and homogeneous Neumann
condition which has a variational structure. However when some conformally
invariant quantity associated to the problem is large, the Euler-Lagrange
functional associated is unbounded from below, implying that we have to
find critical points of saddle type. We will show how the search of saddle
points leads naturally to consider a new barycentric set of the manifold.

Brownian models provide tractable high-level descriptions
of networks in a variety of application areas. This talk will
review work in two areas: the modelling of multi-path routing
in the Internet, and the design of ramp metering policies for highway networks. \\

\noindent In both areas Brownian models are able to exploit the simplifications that arise in heavy traffic, and to make clear the main performance consequences of resource allocation policies.

We develop potential theoretical methods in the construction of
measure-valued branching processes. We complete results on the construction, regularity and other properties of the superprocess associated with a given right process and a branching mechanism.

The Witten-Reshetikhin-Turaev Topological Quantum Field Theory in particular provides us with the so-called quantum representations of mapping class groups. The geometric construction of these involves geometric quantization of moduli spaces, which produced a holomorphic vector bundle over Teichm\"uller space. This bundle supports a projectively flat connection constructed by algebraic geometric techniques by Hitchin. We will present a a Toeplitz operator approximation formula for the parallel transport of the Hitchin connection. We will discuss applications of this

In this talk, we will discuss the positive singular $\sigma_k$-Yamabe
problem on $S^n\setminus \Lambda$ where $\Lambda$ is a finite set of
points of the standard sphere $S^n$, and $k$ a positive integer
verifying $0\leq 2k<n$. Geometrically, the problem is to find a complete
metric on $S^n\setminus\Lambda$ which is conformal to the standard metric
and has constant positive $\sigma_k$-curvature. Analytically, it is
equivalent to finding a positive solution to a singular fully-nonlinear
equation. Using asymptotic analysis combined with Fredholm theory and
contraction mapping principle, we will show how to use the disposition of
the points of $\Lambda$ to get some existence results.

In addition to frequent single-nucleotide mutations, mammalian and many other genomes undergo rare and dramatic changes called genome rearrangements. These include inversions, fissions, fusions, and translocations. Although analysis of genome rearrangements was pioneered by Dobzhansky and Sturtevant in 1938, we still know very little about the rearrangement events that produced the existing varieties of genomic architectures. Recovery of mammalian rearrangement history is a difficult combinatorial problem that I will cover in this talk. Our data sets have included sequenced genomes (human, mouse, rat, and others), as well as radiation hybrid maps of additional mammals. \\

Coauthors:
Pavel Pevzner, UCSD, Department of Computer Science and Engineering
Guillaume Bourque, Genome Institute of Singapore

We investigate pairs of Euclidean TI-domains which are isospectral but not congruent. For Riemannian manifolds there are several examples which are isospectral but not isometric, see e.g. J. Milnor [2]. The method we use dates back to T. Sunada [3] considering the problem as a geometric analogue of a method in number theory which uses Dedekind zeta functions. Counter examples to M. Kac’s conjecture so-far
can all be constructed by a certain tiling method (“transplantability”) using special linear operator groups which act 2-transitively on certain associated modules. These can be represented by colored graphs, which yield information on the fixpoint structure of the groups. It is shown that if any such operator group acts 2-transitively onthe associated module, no new counter examples can occur.\\

\footnotesize
\noindent [1] M. KAC. Can one hear the shape of the drum?, Amer. Math. Monthly 73 (4, part 2) (1966), 1–23. \\
\noindent [2] J. MILNOR. Eigenvalues of the Laplace operators on certain manifolds, Proc. Nat. Acad. Sci. USA 51 (1964), 542. \\
\noindent [3] T. SUNADA. Riemannian Coverings and Isospectral Manifolds, Ann. Math. 121 (1980), 169–186.

We consider a system of particles which perform branching Brownian
motion with negative drift and are killed upon reaching zero, in the near-critical regime where the total population stays roughly constant
with approximately N particles. We show that the characteristic time
scale for the evolution of this population is of order $(\log N)^3$,
in the sense that when time is measured in these units, the scaled
number of particles converges to a version of Neveu's continuous-state
branching process. Furthermore, the genealogy of the particles is then
governed by a coalescent process known as the Bolthausen-Sznitman
coalescent. This validates the non-rigorous predictions by Brunet,
Derrida, Muller, and Munier for a closely related model. This is
joint work with Julien Berestycki and Nathanael Berestycki.

In this talk, we introduce a new family of combinatorial objects called Kepler walls. Roughly speaking, a Kepler wall is a wall built of bricks in which no two bricks are adjacent, and each brick below the top row is supported by a brick in the row above. Despite their unlikely definition, Kepler walls of unrestricted width are counted by binomial coefficients, as we will see by means of a constructive bijection. We will also see connections to other interesting and well-understood sequences, such as the Catalan and Fibonacci numbers.

\footnotesize This term's seminar will be on ``Khovanov homology and categorification''.\\

If one wants to show that some quantity takes only non-negative integral values, one of the best ways to do so is to show that it is ``secretly'' the dimension of some vector space. ``Categorification'' is the philosophy that one should look for interesting examples of this kind of thing throughout mathematics, hoping to find that for example: \\

\begin{enumerate} \item Non-negative integers are secretly dimensions of vector spaces

\item Integers are secretly virtual dimensions of formal differences of vector spaces (or superdimensions of supervector spaces)

\item Integer Laurent polynomials are secretly graded dimensions of Z-graded (super)vector spaces;

\item Abelian groups are secretly Grothendieck groups of additive categories \end{enumerate}

The Euler characteristic, for example, is an integer-valued invariant with wonderful properties and applications. We can ``categorify'' it by viewing it as the dimension (in the second sense above) of a more powerful vector-space valued invariant, homology. Why is homology more powerful? Because it is \textit{functorial}, capturing information about maps between spaces which the Euler characteristic can't. It's this appearance of functoriality that gives rise to the name ``categorification''.\\

In 1999 Mikhail Khovanov showed that the Jones polynomial for knots in 3-space can be categorified (in the third sense above). He showed how to associate to any knot a bunch of homology groups which turn out to be strictly stronger, as topological invariants, than the Jones polynomial; moreover, they are functorial with respect to surface cobordisms in 4-space between knots! The invention of Khovanov homology has not only had beautiful applications in topology (Rasmussen's proof of Milnor's conjectures about the unknotting numbers of torus knots) but also inspired a lot of work by algebraists which might ultimately explain what quantum groups ``really are''. \\

Our seminar will work through the most important papers about Khovanov homology and knot theory, beginning with those of Dror Bar-Natan, and if there's enough time we'll look at some of the more algebraic work too. \\

The seminar meets Tuesdays in 7218 from 10.30-12. \\

I will give the first talk next Tuesday, and after that we'll try to arrange a schedule of speakers for the rest of the term. Everyone is welcome to attend and/or speak, though

We consider a quadratic programming method designed for use in a
sequential quadratic programming (SQP) method for large-scale
nonlinearly constrained optimization.\\

\noindent Because the efficiency of SQP methods is determined by how the
quadratic subproblem is formulated and solved, we propose an
active-set method based on inertia control that prevents
singularity in the associated KKT systems. The method is able to
utilize black-box linear algebra software, thereby exploiting
recent advances in computer hardware. Moreover, the method makes
no assumptions on the convexity of the quadratic problems making
it particularly useful in SQP methods using exact second
derivatives. \\

\noindent In addition, the method can be applied to a regularized quadratic
subproblem involving an augmented Lagrangian objective function,
eliminating the need for a full-rank assumption on the constraint
matrix.

Geometric mechanics involves the use of differential geometry and symmetry techniques to study mechanical systems. In particular, it deals with global invariants of the motion, and how they can be used to describe and understand the qualitative properties of complicated dynamical systems, without necessarily explicitly solving the equations of motion. This approach parallels the development of geometric numerical methods in numerical analysis, wherein numerical algorithms for the solution of differential equations are constructed so as to exactly conserve the invariants of motion of the continuous dynamical system.

This talk will provide a gentle introduction to the role of geometric methods in understanding nonlinear dynamical systems, and why it is important to develop numerical methods that have good global properties, as opposed to just good local behavior.

For over two decades we have been proving
identities involving plethystic operators
(vertex operators for some people) by
manipulations which politely could
be called ``heuristic''. But deep down I
felt them to be quite ``fishy". But referees accepted
them and we felt nevertheless confident
since we always got the right answer,
as amply confirmed by computer experimentations.
But suddenly this summer an example popped up
where our manipulations yielded a patently
false answer. Panic? Yes ... for a while.
In this talk we will present how in the end
all of this finally, and belatedly
could be made completely rigorous.

We use a simple probability method to transform the celebrated Monge-Kantorovich problem in a bounded region of Euclidean plane into a Dirichlet boundary problem associated to a quasi-linear elliptic equation with 0-order term missing in its diffusion coefficients. Thus, we are also able to give a probability approach to the famous Monge-Ampere equation, which is known to be associated to the above problem.

Variational, level set and PDE based methods and their applications in digital
image processing have been well developed and studied for the past twenty years.
These methods were soon applied to medical image processing problems.
However, the study for biological shapes, e.g. surfaces of brains or other human
organs, are still in its early stage. The bulk of this talk explores some
applications of variational, level set and PDE based methods in biological shape
processing and analysis.\\

There will be three topics in this talk. The first one is on 3D brain
aneurysm capturing using level set based method, which is inspired by the technique 
of illusory contours in image analysis. The second one is on multiscale
representations(MSR) of 3D shapes, which is wavelet flavored but level set and PDE based. 
The third one is on Bregman iteration as a fast solver for L1-minimizations and its application 
to image processing problems in DNA sequencing.

T he talk is going to be on progress made in approximation theory of Lie
group-valued periodic functions (loops) by so-called polynomial loops.
This
is a relatively unexplored topic within the larger area of nonlinearly
constrained approximation, which includes the study of H\"{o}lder

We will discuss complex tori, and explain the role that a special class of functions, the theta functions, play in their study. I will also outline connections between theta functions and other special functions.

For over two decades we have been proving
identities involving plethystic operators
(vertex operators for some people) by
manipulations which politely could
be called ``heuristic''. But deep down I
felt them to be quite ``fishy''. But referees accepted
them and we felt nevertheless confident
since we always got the right answer,
as amply confirmed by computer experimentations.
But suddenly this summer an example popped up
where our manipulations yielded a patently
false answer. Panic? Yes ... for a while.
In this talk we will present how in the end
all of this finally, and belatedly
could be made completely rigorous. \\

This talk will be a continuation of the talk from last week.

Categorification is big business in representation theory these days,
and much of the inspiration for categorification comes from geometric
representation theory. We'll try to explain some of the geometric
inspiration for sl(2) categorification. As an application, we
describe an interesting equivalence of categories between the derived
categories of coherent sheaves on the cotangent bundle of dual
Grassmanians. \\

Joint with Sabin Cautis and Joel Kamnitzer.

The Peter-Weyl theorem is one of the results that made me decide to study representation theory. In a few words it tells you how to describe the space $L^2(G)$ in terms of the representation theory of a compact group $G$.

The idea of this talk is to informally develop enough theory to state and understand this theorem and some of its consequences, and in this way motivate the study of Lie groups and their representations. No previews knowledge of the subject is assumed.

Single-molecule biophysical tools permit measurements of the mechanical response of individual biomolecules to external load, revealing details that are typically lost when studied by ensemble methods. Kramers theory of diffusive barrier crossing in one dimension has been used to derive analytical solutions for the observables in such experiments, in particular, for the force dependent lifetimes. We propose a minimalist model that captures the effects of multidimensionality of the free energy landscape on the kinetics of a single-molecule system under constant applied force. The model predicts a rich spectrum of scenarios for the response of the system to the applied force. Among the scenarios is the conventional decrease in the lifetime with the force, as well as a remarkable rollover in the lifetime with a seemingly counterintuitive increase of the lifetime at low force followed by a decrease in the lifetime at higher forces. Realizations of each of the predicted scenario are discussed in various biological contexts. Our model demonstrates that the rollover in the lifetime does not necessarily imply a discrete switch between two coexisting pathways on the free energy landscape, and that the rollover can also be realized for a dynamics as simple as that on a single pathway with a single bound state. Our model leads to an analytical solution that reproduces the entire spectrum of scenarios, including the rollover, in the force-dependent lifetime, in terms of the microscopic parameters of the system.

A planimeter is a device that can measure the area of a
region by tracing its perimeter. We will see how the polar planimeter
is an elegant practical example of Green's theorem. We will use
Green's theorem to elucidate various features of the polar planimeter,
such as the neutral circle and what a compensating polar planimeter
compensates for. I will show off several examples of planimeters,
including polar, rolling and radial planimeters.

We introduce a new basis for quasisymmetric functions, called
"quasisymmetric Schur functions", and provide a combinatorial rule for
the multiplication of a quasisymmetric Schur function by a Schur
function. We extend this approach to develop similar multiplication
rules for Demazure characters and for Demazure atoms. This is joint
work with Jim Haglund, Kurt Luoto, and Steph van Willigenburg.

I will talk about a main theorem in Tian's JDG paper ``On a set
of polarized Kaehler metrics on algebraic manifolds''. A crucial
ingredient is to use Hormander's estimates to construct so called peak
sections (which has energy concentration at a isolated point and
prescribed polynomial growth order around that point).

We shall investigate $\gamma(R)$, the zero-divisor graph of a commutative ring $R$ (where $0 \ne 1$). Specifically, if $R$ is the given ring, let $\gamma(R) = (V,E)$ with $V = Z(R)$ and $E=\{ \{x,y\} | xy=0\}$. We will show elementary properties of $\gamma(R)$ and restrict our attention to graphs with finitely many vertices.

\footnotesize

Articular cartilage is a resilient soft tissue that supports load joints at the knee, shoulder and hip. Cartilage is primarily comprised of interstitial water (roughly 80\% by volume) and extracellular matrix (ECM). Cells called chondrocytes are dispersed through ECM and maintain and regenerate the tissue. Chondrocytes are surrounded by a narrow layer called pericellular matrix (PCM), which is believe to be important for modulating the biomechanical environment of chondrocyte. In this study, computational models will be presented to analyze the multiscale micromechanical environment of chondrocytes.\\

Firstly, we will discuss transient finite element method (FEM) to model linear biphasic mechanics of a single cell within cartilage layer under cyclic loading. The FEM model was employed to analyze the effects of frequency on mechanical variables in cellular environment under macroscopic loading at 1\% strain and in the frequency range 0.01 0.1 Hz. In this frequency range, intracellular axial strains exhibited up to a ten-fold increase in magnitude relative to 1\% applied strain. The dynamics of strain amplification exhibited a two-scale response that was highly dependent on ratios of typical time scales in the model, such as the loading period, gel diffusion times for the cell, the PCM and the ECM. In conjunction with strain amplification, solid stress in the surrounding ECM was reduced by up to 35\%. We propose here that the computational model developed in this study has potential application in correlating mechanical variables in the cellular microenvironment to biosynthetic responses induced by cyclic loading of native cartilage or engineered cell-gel constructs.\\

Secondly, we will discuss the formulation, implementation and application of multiscale axisymmetric boundary element method (BEM) for simulating in situ deformation of chondrocyte and the PCM in states of mechanical equilibrium. The BEM was employed to conduct a multiscale continuum model to determine linear elastic properties of the PCM in situ. Taken together with previous experimental and theoretical studies of cell-matrix interactions in cartilage, these findings suggest an important role for the PCM in modulating the mechanical environment of the chondrocyte. \\

This is joint work with Mansoor Haider (NCSU), and our experimental colleague, Farshid Guilak (Duke).

In addition to the Deligne-Mumford compactification for the moduli space of genus four curves, there are a number of additional compactifications that arise naturally. In this talk I will discuss joint work with Radu Laza where we compare some of these spaces. The description we obtain is similar to that for genus three curves (work of Hyeon-Lee), as well as to some previous results we have for the moduli space of cubic threefolds.

Exponential integrators are methods for the solution of ordinary differential equations which use the matrix exponential in some form. As the solution to linear equations is given by the exponential, these methods are well suited for stiff ordinary differential equations where the stiffness is concentrated in the linear part. Such equations arise when semi-discretizing semi-linear differential equations. The biggest challenge for exponential integrators is that we need to compute the exponential of a matrix. If the matrix is not small, as is the case when solving partial differential equations, then an iterative method needs to be used. Methods based on Krylov subspaces are a natural candidate. I will describe the efforts of Will Wright (La Trobe University, Melbourne) and myself to implement such a procedure and comment on our results.

Partial differential equations (PDEs) can be used to model numerous
physical processes, from steady-state heat distribution to the
formation of black holes. When the solution changes over time,
special techniques and considerations must be taken for their accurate
solution. In this talk, I will briefly introduce the concepts,
discuss a few of the concerns, and show some numerical results from
simple model problems.

We describe several corollaries of the Littlewood-Richardson refinements, including a method for multiplying two Schur functions with different numbers of variables and expanding the result as a sum of key polynomials. We use interactions between Schur functions and quasisymmetric Schur functions to prove a conjecture of Bergeron and Reutenauer. We show that their conjectured basis is indeed a basis for the quotient ring of quasisymmetric functions by symmetric functions, which also provides a combinatorial proof of Garsia and Wallach's results about the freeness and dimension of QSym/Sym. This is joint work with Aaron Lauve.

Neuroscientists have become increasingly interested in exploring
dynamic
relationships among brain regions. Such a relationship, when directed from
one region toward another, is denoted by ``effective connectivity.'' An fMRI
experimental paradigm which is
well-suited for examination of effective connectivity is the slow
event-related design.
This design presents stimuli at sufficient temporal spacing for determining
within-trial
trajectories of BOLD activation. However, while several analytic methods for
determining
effective connectivity in fMRI studies have been devised, few are adapted to
the
characteristics of event-related designs, which include non-stationary BOLD
responses and nesting of responses within trials and subjects.
We propose a model tailored for exploring effective connectivity
of multiple brain regions in event-related fMRI designs - a semi-parametric
adaptation of vector autoregressive (VAR) models, termed "stimulus-locked
VAR"
(SloVAR). Connectivity coefficients vary as a function of time
relative to stimulus onset, are regularized via basis expansions, and vary
randomly across subjects. SloVAR obtains flexible, data-driven estimates of
effective
connectivity and hence is useful for building connectivity models when prior
information
on dynamic regional relationships is sparse. Indices derived from the
coefficient estimates can also be used to relate effective connectivity
estimates
to behavioral or clinical measures. We demonstrate the SloVAR model
on a sample of clinically depressed and normal controls, showing that
early but not late cortico-amygdala connectivity appears crucial to
emotional control and
early but not late cortico-cortico connectivity predicts depression severity
in the depressed group, relationships that would have been missed in a more
traditional VAR analysis.

Reto Muller investigated a new geometric flow which consists
of a coupled system of the Ricci flow on a closed manifold $M$ with the harmonic map flow of a map $\phi$ from $M$ to some closed target closed manifold $N$, given by $\frac{\partial}{\partial t} g = - 2 Ric + 2 \alpha \nabla \phi \bigotimes \nabla \phi, \frac{\partial}{\partial t}\phi = \tau_{g}\phi $, where $\alpha$ is a positive coupling constant. This new flow shares many good properties with the Ricci flow.

We will define useful definitions of growth on an algebra. In particular, we will consider Gelfand-Kirillov (GK) dimension. After stating some nice properties of GK dimension of algebras, we will sketch a combinatorial proof of the Bergman Gap Theorem.

Chemotaxis is characterized by the directional cell movement following external chemical gradients. It plays a crucial role in a variety of biological processes including neuronal development, wound healing and cancer metastasis. Ultimately, the accuracy of gradient sensing is limited by the fluctuations of signaling components, e.g. the stochastic receptor occupancy on cell surface. We use concepts and techniques from interrelated disciplines (statistics, information theory, and statistical physics) to model the stochastic information processing in eukaryotic chemotaxis. Specifically, we address the following issues:

\begin{enumerate} \item What are the physical limits of the gradient estimation? \\ \item How much information can be reliably gained by a chemotaxing cell? \\ \item How to optimize the chemotactic performance? \\ \end{enumerate}

Through answering those questions, we expect to derive extra insights for general biological signaling systems.

A multivariate real polynomial is non-negative if its value is at least zero for all points in $\mathbb{R}^n$. Obvious examples of non-negative polynomials are squares and sums of squares. What is the relationship between non-negative polynomials and sums of squares? I will review the history of this question, beginning with Hilbert's groundbreaking paper and Hilbert's 17th problem. I will discuss why this question is still relevant today, for computational reasons, among others. I will then discuss my own research which looks at this problem from the point of view of convex geometry. I will show how to prove that there exist non-negative polynomials that are not sums of squares via ``naive" dimension counting. I will discuss the quantitative relationship between non-negative polynomials and sums of squares and also show that there exist convex polynomials that are not sums of squares.

Knots are embeddings of circles in the three dimensional sphere. We will discuss an equivalence relation called knot concordance and the group of equivalence classes under connect sum

We will discuss the structure of the knot concordance group, finite order concordance classes and open problems in the area.

We study how to solve sum of squares (SOS) and Lasserre's
relaxations for large scale
polynomial optimization. When interior-point type methods are used,
typically only small
or moderately large problems could be solved. This paper proposes the
regularization
type methods which would solve significantly larger problems. We first
describe these
methods for general conic semidefinite optimization, and then apply
them to solve large
scale polynomial optimization. Their efficiency is demonstrated by
extensive numerical
computations. In particular, a general dense quartic polynomial
optimization with 100
variables would be solved on a regular computer, which is almost
impossible by applying
prior existing SOS solvers.

Venn diagrams are tools used to represent relationships among
sets. They are easy to understand but can be difficult to draw if they
involve more than three sets. The quest for a method to construct symmetric
Venn diagrams has led to some interesting theorems about partially ordered
sets. We describe several of these theorems, their relationship to Venn
diagrams, and a conjecture that unifies this research.

There has been much recent progress in the study of Ricci solitons on nilpotent and solvable Lie groups. In this talk, I will define the Ricci Yang-Mills flow which is related to the Ricci flow. I will also define Ricci Yang-Mills solitons, which are generalized fixed points of the Ricci Yang-Mills flow. These metrics are related to Ricci solitons; however, they are defined on principal G-bundles and are designed to detect more of the bundle structure. On nilpotent Lie groups, one can say precisely in what sense Ricci Yang-Mills solitons are related to Ricci solitons. I will provide examples of 2-step nilpotent Lie groups that admit Ricci Yang-Mills solitons but that do not admit Ricci solitons. This is joint work with Mike Jablonski.

The Wiener Chaos is a natural orthogonal decomposition of the $L^2$ space of a Brownian motion, naturally associated to stochastic integration theory; the orders of chaos are given by the range of multiple Wiener-Ito integrals.

In 2006, Nualart and collaborators proved a remarkable central limit theorem in the context of the chaos. If $X_k$ is a sequence of $n$th Wiener-Ito integrals (in the $n$th chaos), then necessary and sufficient conditions that $X_k$ converge weakly to a normal law are that its (second and) fourth moments converge -- all other moments are controlled by these.

In this lecture, I will discuss recent joint work with Roland Speicher in which we prove an analogous theorem for the empirical eigenvalue laws of high-dimensional random matrices.

The concept of average is highly useful (and much maligned) concept in all of mathematics and in life. However, few people stop to think about what an average really \emph{is}. As it turns out, it is a very important theoretical concept in mathematics, and it isn't just something that helps one lie with statistics. It is really the heart of measure and integration theory. In this talk we'll learn how measure theory and integration unifies various different kinds of averages, and one big result: Jensen's inequality, and its applications to relating more exotic means to one another.

\footnotesize Single-molecule force spectroscopy experiments involve imposition of controlled forces at the single molecule level and observing the corresponding mechanical behavior of the molecule. The molecular resistance to deformation can be utilized for studying transition pathways of molecules in terms of energy, time scales and even number of transition states. These have found applications in a wide variety of problems, for instance, to understand folding-unfolding dynamics of biomolecules, ligand-receptor binding, transcription of DNA by RNA polymerase, motion of molecular motors to name a few.
Existing analyses of force measurements rely heavily on theoretical models for reliable extraction of kinetic and energetic properties. Despite significant advances, there remain gaps in fully exploiting the experiments and their analyses. Specifically, the effect of pulling device stiffness or compliance has not been comprehensively captured. Hence, the best models for extracting molecular parameters can only be applied to measurements obtained from soft pulling devices (e.g., optical tweezers) and result in well-documented discrepancies when applied to stiff devices (e.g., AFM). This restriction makes pulling speed the only control parameter in the experiments, making reliable extraction of molecular properties problematic and prone to error. \\

Here, we present a one-dimensional analytical model derived from physical principles for extracting the intrinsic rates and activation free energies from rupture force measurements that is applicable to the entire range of pulling speeds and device stiffnesses. The model therefore is not restricted to the analyses of force measurements performed with soft pulling devices only. Further, the model allows better design of experiments that specifically exploits device stiffness as a control parameter in addition to pulling speed for a more reliable estimation of energetic and kinetic parameters. The model also helps explain previous discrepancies noted in rupture forces measured with devices of different effective stiffnesses and provides a framework for modeling other stiffness-related issues in single-molecule force spectroscopy.

\footnotesize Single-molecule force spectroscopy experiments involve imposition of controlled forces at the single molecule level and observing the corresponding mechanical behavior of the molecule. The molecular resistance to deformation can be utilized for studying transition pathways of molecules in terms of energy, time scales and even number of transition states. These have found applications in a wide variety of problems, for instance, to understand folding-unfolding dynamics of biomolecules, ligand-receptor binding, transcription of DNA by RNA polymerase, motion of molecular motors to name a few.
Existing analyses of force measurements rely heavily on theoretical models for reliable extraction of kinetic and energetic properties. Despite significant advances, there remain gaps in fully exploiting the experiments and their analyses. Specifically, the effect of pulling device stiffness or compliance has not been comprehensively captured. Hence, the best models for extracting molecular parameters can only be applied to measurements obtained from soft pulling devices (e.g., optical tweezers) and result in well-documented discrepancies when applied to stiff devices (e.g., AFM). This restriction makes pulling speed the only control parameter in the experiments, making reliable extraction of molecular properties problematic and prone to error. \\

Here, we present a one-dimensional analytical model derived from physical principles for extracting the intrinsic rates and activation free energies from rupture force measurements that is applicable to the entire range of pulling speeds and device stiffnesses. The model therefore is not restricted to the analyses of force measurements performed with soft pulling devices only. Further, the model allows better design of experiments that specifically exploits device stiffness as a control parameter in addition to pulling speed for a more reliable estimation of energetic and kinetic parameters. The model also helps explain previous discrepancies noted in rupture forces measured with devices of different effective stiffnesses and provides a framework for modeling other stiffness-related issues in single-molecule force spectroscopy.

We are interested in the behaviour of Ricci-flat Kahler metrics on a compact Calabi-Yau manifold, with Kahler classes approaching the boundary of the Kahler cone. The case when the volume approaches zero is especially interesting since the corresponding complex Monge-Ampere equation degenerates in the limit. If the Calabi-Yau manifold is the total space of a holomorphic fibration, the Ricci-flat metrics collapse to a metric the base, which `remembers' the fibration structure.

We briefly survey line search algorithms for unconstrained
optimization.
Next, we consider the search direction and line search strategies used
in
several algorithms that implement a quasi-Newton method for simple
bounds,
including algorithm L-BFGS-B. In this context, we discuss two
currently-used line search algorithms and introduce a new method meant
to
combine the best properties of two different strategies. We present a
modified L-BFGS-B method using the new line search and demonstrate its
significant performance gains by numerical tests using the CUTEr test
set.

The asymptotic expansions to the Boltzmann equations provide a clue of the
connection from kinetic theory to fluid mechanics.
The Hilbert expansion turns out to be useful to verify compressible fluid
limits. As its applications, we rigorously establish the compressible Euler and
acoustic limits from the Boltzmann equation and the Euler-Poisson limit from
the Vlasov-Poisson-Boltzmann system. Moreover, we prove a global-in-time
convergence for a repulsive Euler-Poisson flow for irrotational monatomic gas.

The quaternions form an interesting and useful number system which is a (noncommutative!) extension of the complex numbers. We define the quaternions and give some of the famous history surrounding Hamilton's discovery of them. We describe some applications of quaternions to geometry and algebra.

As an example, consider the following problem. Given positive
integers $a_1,…,a_d$ that are relatively prime, let S be the set of
integers that can be written as a nonnegative integer combination of
these $a_i$. We can think of the $a_i$ as denominations of postage stamps
and S as the postal rates that can be paid exactly using these
denominations. What can we say about the structure of this set, S? What
is the largest integer not in S (called the Frobenius number)? How many
positive integers are not in S?

We attack these problems using the generating function $f_S(x)$, defined
to be the sum, over all elements s of S, of the monomials $x^s$. We will
build up the general theory of computing generating functions – for
this and other problems – and then use these generating functions to
answer questions we’re interested in. We will approach these problems
from an algorithmic perspective: what can we do in polynomial time?

In 1990, Jones received a Fields Medal, in part, for his work on knots and knot invariants. In particular, he developed what is now known as the Jones polynomial which can serve to distinguish two knots from one another. In this talk, we introduce the Jones polynomial and its basic properties and how it can be helpful to scouts who need to know their knots.

on-specific effects are ubiquitous in nature and have relevance in colloidal
science, electrochemistry, and geological and biological physics. The molecular
origin and the coarse-grained modeling of these effects are still widely unexplored.
In this talk we attempt to give more molecular insight into the individual
correlations in aqueous electrolyte systems which give rise to the ion-specific
behavior in bulk (e.g., the osmotic pressure) or in confinement (e.g., between
colloidal or biological surfaces). Particularly, we present a nonlocal
Poisson-Boltzmann theory, based on classical density functional theory,
which captures and rationalizes ion-specific excluded-volume correlations
(the 'size effect') in dense electrolytes and may help understanding the
restabilization of proteins, clays, and colloids at high salt concentrations.
The importance of electrostatic correlations at low dielectric constants is
briefly discussed.

lectrical Impedance Tomography (EIT) is a medical imaging
technique which attempts to
find conductivity inside the human body.
Mathematically speaking, EIT is an inverse
problem. In inverse problems, experimental data is
used to approximate some property
(or control) of the system of interest. For EIT,
this experimental data is electric potential
on the body's surface. One big concern with EIT is
that it is a highly ill-posed problem.
In our context, this means that the conductivity is
highly dependent on experimental
noise.
In this talk I will describe the mathematical
model used for the forward problem of EIT.
The inverse problem will then be described as a
constrained least squares problem.

We consider the model problem of detecting whether or not in a given sensor network, there is a cluster of sensors which exhibit an unusual behavior. Formally, suppose we are given a set of nodes and attach a random variable to each node which represent the measurement that a particular sensor transmits. Under the normal circumstances, the variables have a standard normal distribution. Under abnormal circumstances, there is a cluster (subset of nodes) where the variables now have a positive mean. The cluster is unknown but restricted to belong to a class of interest, for example discrete squares.\\

We also address surveillance settings where each sensor in the network transmits information over time. The resulting model is similar, now with a time series is attached to each node. We consider some well-known examples of growth models, including cellular automata used to model epidemics.\\

In both settings, we study best possible detection rates under which no test works. We do so for a variety of cluster classes. In all the situations we consider, we show that the scan statistic, by far the most popular method in practice, is near-optimal.\\

Joint work with Emmanuel Candes (Stanford) and Arnaud Durand (Universit$\mathrm{\acute{e}}$ Paris XI)

I will talk about the scattering operators on conformally compact Einstein manifolds based on the work of Graham and Zworski. A conformally compact Einstein manifold comes with a conformal manifold as its conformal infinity. I will show scattering operators, as spectral property of the bulk space, in many ways are related to global conformal property of the infinity. I will in particular talk about a recent joint work with Colin Guillarmou on the relation of the location of real scattering poles and the Yamabe constant of the conformal infinity.

An alternate form for the binomial tail is presented, which leads to a variety of bounds for the central tail. A few can be weakened into the corresponding Chernoff and Slud bounds, which not only demonstrates the quality of the presented bounds, but also provides alternate proofs for the classical bounds.

Two types of dynamical system are differential equations and Markov chains, representing continuous deterministic systems and discrete random systems respectively. For a Markov chain in which the jumps are ``small and frequent,'' the individual random jumps can average out to a ``drift'' as in a first-order differential equation. We'll explore a couple of general results of this type and do a couple examples, largely following a paper by Darling and Norris.

One of the most important choices a graduate student will make will be choosing a thesis advisor. It is never too early for students to begin thinking about choosing an area of specialty and choosing among the faculty who might supervise them.
How did other students find a thesis advisor? What are the key factors to consider when choosing an advisor? What do professors look for before they accept a student as their thesis student? How does finding a thesis advisor lead to finding a thesis problem? We will discuss these questions.
We will have four graduate students-Raul Gomez, Mike Scullard, Michael Ferry and Kevin McGown describe their experiences finding a thesis advisor and what happens after a thesis advisor is found.
We will also have one faculty, Jim Lin, describe what he looks for in a graduate student before he accepts him or her as a thesis student.
All students, especially first, second and third year students, are cordially invited to attend.

Traditionally weak stationarity of a random field $\{X(t) : t\in \mathbf{T}\}$ over an index space $\mathbf{T}$ is defined with respect to a translation operation in $\mathbf{T}$. But this classical notion of stationarity does not extend to related random fields, as for example to the field of averages of $\{X(t): t\in \mathbf{T}\}$. In order to equip this latter field with a stationarity property one introduces a generalized translation in $\mathbf{T}$ which arises from a generalized convolution structure in the space $M^b(\mathbf{T})$ of bounded measures on $\mathbf{T}$. There are two fundamental constructions providing such (hypergroup) convolution structures on the index spaces $\mathbf{Z}_+$ and $\mathbf{R}_+$, in terms of polynomial sequences and families of special functions, respectively.\\

In the present talk emphasis will be put on polynomially stationary random fields $\{X(n): n\in\mathbf{Z}_+\}$ which were studied for the first time by R.~Lasser and M.~Leitner about 20 years ago. In the meantime the theory has developed interesting applications such as regularization, moving averages and prediction.\\

For square-integrable radial random fields over graphs, J.P.~Arnaud has coined a notion of stationarity which yields spectral and Karhunen type representations.
These fields are related to polynomially stationary random fields over $\mathbf{Z}_+$, where the underlying polynomial sequence generates the Cartier-Dunau convolution structure in $M^b(\mathbf{Z}_+)$. An analogous approach related to special function stationarity of random fields over $\mathbf{R}_+$ seems promising, but requires further progress.

This is an overview talk for students on problems in the field of non-linear wave equations. We'll
first introduce several models from classical field theory,
and then discuss some open problems and current techniques
for approaching them. The focus of this and future
seminars will be on asymptotic
stability problems and decay estimates.

\small This talk will focus on the following two problems.

\begin{enumerate}\item Cone beam computed tomography (CBCT) reconstruction.

CBCT has been extensively studied for many
years. It is desirable to reconstruct the CBCT
image with as few x-ray projections as possible in
order to reduce radiation dose. In this talk, we
present our recent work on an iterative CBCT
reconstruction algorithm. We consider a cost
function consisting of a data fidelity term and a
total-variance regularization term. A
forward-backward splitting algorithm is used to
minimize the cost function efficiently. We test
our reconstruction algorithm in a digital patient
phantom and the reconstruction can be achieved
with 30 CBCT projections. Our algorithm can also
be applied in 4D CBCT reconstruction problem. A
proposed temporal regulation algorithm for 4DCBCT
reconstruction will also be discussed.

\item Treatment plan optimization.

When beam of radiation passes through a patient,
they may kill both cancerous and normal cells, so
the goal of the treatment is to kill the tumor (by
delivering the prescribed dose to it), while
sparing the organs-at-risk (by minimizing the dose
to it). We define our objective function as a
penalty-based one-side quadratic function based on
the dose received by each voxel. Overdosing
penalty is given to all voxles, while underdosing
penalty is only given to tumor voxels. The
decision variables can be intensity of each beam
bixel (IMRT Fluence Map Optimization), intensity
and shape of each beam aperture (IMRT Direct
Aperture Optimization), or aperture shape and
intensity in each beam angle (VMAT Optimization),
depending on various radiation techniques and/or
models.
\end{enumerate}
The reconstruction process and one of treatment
plan optimization models have been implemented on
Nvidia CUDA platform on GPU and a high computing
efficiency has been achieved.

A metric space is a set together with a notion of distance. An example would be 3-space with our usual definition of distance, but there are lots of examples which could be quite abstract. Suppose we're given two such spaces: how far apart are they? Does this even make sense? Is there a well-defined notion of the distance between abstract metric spaces? Can a sequence of abstract metric spaces converge? We will discuss these questions in relation to some recent research on curvature flows and geometry.

A tropical curve is a metric graph with possibly unbounded edges, and
tropical rational functions are continuous piecewise linear functions
with
integer slopes. We define the complete linear system $|D|$ of a divisor $D$
on
a tropical curve analogously to the classical counterpart. Due to work
of
Baker and Norine, there is a rank function $r(D)$ on such linear systems,
as well a canonical divisor $K$. Completely analogous to the classical
case, this rank function satisfies Riemann-Roch and analogues of
Riemann-Hurwitz.

After an introduction to these tropical analogues, this talk will
describe
joint work with Josephine Yu and Christian Haase investigating the
structure of $|D|$ as a cell complex. We show that linear systems are
quotients of tropical modules, finitely generated by vertices of the
cell
complex. Using a finite set of generators, $|D|$ defines a map from the
tropical curve to a tropical projective space, and the image can be
extended to a parameterized tropical curve of degree equal to $\mathrm{deg}(D)$.
The
tropical convex hull of the image realizes the linear system $|D|$ as an
embedded polyhedral complex.

We consider hyper-Kaehler manifolds of complex dimension 4 which are fibrations. It is known that the fibers are abelian varieties and the base is $\mathbb P^2$. We assume that the general fiber is isomorphic to a product of two elliptic curves. Our result is that such a hyper-Kaehler manifold is deformation equivalent to a Hilbert scheme of two points on a K3 surface.

In this talk, we study how to use gradient/derivative recovery techniques to formulate error estimate and error indicator for p-adaptive FEMs, where elements are allowed to have variable degrees. The study also suggests an approach to implement a fully automatic hp-adaptive FEM. In this approach, the
decision on whether to refine a given element into two child elements (h-refinement) or increase its degree (p-refinement) is made heuristically purely on information from error estimate. Several numerical results will be presented to show the efficiency of the methods.

In this talk we'll focus on various dispersive estimates for the wave equation on Minkowski Space.

The title of this talk is the same as that of an influential article published in 1960 by the physicist and mathematician Eugene Wigner. Wigner's thesis is that mathematics is obviously an effective tool in the sciences (especially physics), but it is unreasonably so: much more successful at describing natural laws than could reasonably be expected. He concludes, ``The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.'' I will survey the evidence that led him to this conclusion, examine some later attempts to explain this miracle, and hint at my own viewpoint. I hope to hear yours as well.

The occupied set grows by adding points x which have at least theta already occupied points in their neighborhoods. Such ``threshold growth'' models are interesting in many contexts. For growth on an integer lattice, explicitly computable approximations can be developed when the neighborhood range is large. Other cases will also be briefly addressed.

Solvation is ubiquitous in experiments. In this talk, an accurate classical density functional theory (DFT) is presented for predicting the microscopic structure and thermodynamic properties of an arbitrary molecule solvated in a molecular solvent. The novel free-energy functional is constructed in terms of solvent density which depends on position and orientation of solvent molecule. The key input is the inhomogeneous position and orientation dependent solvent direct correlation function, and this direct correlation function is calculated by the “homogeneous reference fluid approximation”, namely in terms of the direct correlation function of the pure solvent system (the c-function).

Towards precise prediction, we propose the following strategy: we first perform MD simulations of the pure solvent system, and then sample over many solvent configurations so as to compute the position and angle-dependent two-body distribution functions (the h-function). Subsequently applying the so-called molecular Ornstein-Zernike relation, we obtain the corresponding direct correlation function, which serves as input for the free energy functional. In the presence of a given molecular solute, which provides the external potential, this functional can be minimized with respect to water density , using a 3D Cartesian grid for position and Gauss-Legendre angular grid for orientations, to obtain, at the minimum, the absolute solvation free-energy of the solute and the equilibrium solvent density profile around it.

In comparison with direct MD simulation results, the DFT provides accurate representations of both microscopic structure and thermodynamic properties for a wide variety of solutes dissolved in molecular solvents including acetonitrile, water etc.. Unlike molecular simulations, DFT provides direct information on the free energy from which all thermodynamic properties can be derived.

What do a square-wheeled bicycle, a 17th-century French painting, and the Indiana legislature all have in common? They appear among the many bright stars on the mathematical horizon, or perhaps, more correctly in the Math[ematical] Horizons. Math Horizons, the undergraduate magazine started by the MAA in 1994, publishes articles to introduce students to the world of mathematics outside the classroom. Some of mathematics’ best expositors have written for MH over the years; here are some of the highlights from the first ten years of Horizons.

The coamoeba of a complex polynomial $f$ is defined to be the
image of the hypersurface defined by $f$ under the mapping $\text{Arg}$
that
sends each coordinate $z_k$ to its argument $\arg z_k$. We shall discuss the
connection between coamoebas and the multidimensional Mellin transforms
of rational functions.

In this talk we'll focus on various dispersive estimates for the wave equation on Minkowski Space with a potential.