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2006 Archive

Moduli spaces in Riemannian geometry

I will discuss various interesting moduli spaces which arise naturally in Riemannian Geometry, and some examples of how to compactify these moduli spaces. I will then discuss a partial compactification of the moduli space of anti-self-dual Riemannian metrics in dimension four.

Introduction and lecture survey

The goal is to read the following book: \vskip .1in

\noindent A first course in String Theory&quot;, by Barton Zwiebach. \vskip .1in

\noindent In the hands of Witten and other mathematical physicists, string theories (and more generally, the techniques of quantum field theory) have had a gigantic impact on geometry and topology over the last twenty years. Though the genuine physical significance of these
subjects is still debatable (many of the mathematically-interesting
models are what physicists would call toy models&quot;) and physicists'
methods often lack adequate mathematical foundations, they have a
strong internal consistency and lead to numerous surprising mathematical conjectures which could scarcely have been contemplated without this insight. \vskip .1in

\noindent We want in our seminar to try to understand the way physicists work. This is actually quite a hard task, because their goals, intuition and language are really different from ours, and require continual translation. (I would sum up the distinction by saying that mathematicians deal mostly in nouns, physicists in verbs.) Zwiebach's
book, for an MIT undergraduate physics course, seems to me unusually
readable. Its level of mathematical sophistication is not very high,
so we will probably be able to worry mostly about the actual
physics. Perhaps later on we can try to read some of the rather more
advanced book \vskip .1in

\noindent String theory&quot;, by Joseph Polchinski.

Symplectic reflection algebras and quantum Hamiltonian reduction

I will speak on a joint work with Etingof, Ginzburg, and Oblomkov,
in which we constructed the spherical subalgebras of symplectic
reflection algebras by quantum Hamiltonian reduction of differential
operators on representation spaces of affine Dynkin quivers.

Coupling nonpolar and polar solvation in implicit solvent models

Recent studies on the solvation of atomistic and nanoscale solutes indicate that a strong coupling exists between the hydrophobic, dispersion, and electrostatic contributions to the solvation free energy, a facet not considered in current implicit solvent models. We suggest a theoretical formalism, which accounts for coupling by minimizing the Gibbs free energy of the solvent with respect to a solvent volume exclusion function. The resulting differential equation is similar to the Laplace-Young equation for the geometrical description of capillary interfaces, but is extended to microscopic scales by explicitly considering curvature corrections as well as dispersion and electrostatic contributions. Unlike existing implicit solvent approaches, the solvent accessible surface is an output of our model. The presented formalism is illustrated on spherically or cylindrically symmetrical systems of neutral or charged solutes on different length scales. The results are in agreement with computer simulations and, most importantly, demonstrate that our method captures the strong sensitivity of solvent expulsion and dewetting to the particular form of the solvent-solute interactions.

On the chromatic number of the plane

In our talk, we will discuss a classical problem of combinatorial geometry going back to E. Nelson, P. Erd\&quot;os, and H. Hadwiger. The main question is to find the minimum number of colors needed to paint all the points in a space so that the distance between any two points of the same color would not belong to a fixed set of positive reals. We will present a survey of various results concerning the problem. In particular, we will exhibit some amazing connections between this problem and the Borsuk partition problem (which is to determine the minimum number of parts of smaller diameter into which an arbitrary bounded $n$ - dimensional set can be partitioned.)

Examples of piecewise smooth Lagrangian fibrations

Given an integral affine manifold one can easily produce a symplectic manifold fibered by Lagrangian tori. When the affine base has singularities of certain type, one should be able to obtain compact symplectic manifolds by means of gluing suitable singular fibres. In this lecture I will give models of Lagrangian fibrations which can be used to produce such compactifictions. These models resemble the singular behaviour expected to appear in generic&quot; special Lagrangian fibrations, in particular, our models are given by piecewise smooth maps. This type of non-smoothness can be encoded into semi-global symplectic invariants.

Spectral estimates and processes with almost periodic covariance function

Results on spectral estimates for stationary processes are discussed. Questions that arise on spectral estimation for the nonstationary class of processes with almost periodic covariance function are considered. Determination of lines of spectral support is of particular interest.

Contact geometry: An uneven plane field

Contact geometry has been around, in one form or another, for over two centuries, used primarily in the study of optics and differential equations. But it wasn't until the 1980s that the field of contact topology really came into its own. That is, in the last two decades, mathematicians have made great progress in answering questions about the global nature of spaces equipped with a contact structure. \vskip .1in

\noindent In my talk, I will explain what a contact structure is mostly by drawing pretty pictures. Then I'll explain a bit about doing knot theory in the presence of a contact structure. \vskip .1in

\noindent No knowledge of geometry or topology will be required to understand my talk. Scout's honor! If you know what a vector is, you'll be just fine. Also, I promise that within the first fifteen minutes of my talk, you'll understand the dumb joke I used for the title. And if that doesn't whet your appetite, I'm not sure what will (other than the cryptic hints Maia and Henning are dropping about the possible presence of food).

Searching for good $abc$-triples

An *$abc$-triple* is a triple of pairwise coprime positive integers $a$, $b$, $c$ with $a$ + $b$ = $c$. The *radical* $r$ of such a triple is the product of the distinct prime numbers dividing $abc$, and the *quality* $q$ is defined by $q = (log c)/log r$. For example, the triple given by $a = 5, b = 27, c = 32$ has $r = 30$ and $q = (log 32)/log 30 = 1.018975235$... The *$abc$-conjecture* asserts that for any real number $Q &gt; 1$, the number of $abc$-triples with quality greater than $Q$ is finite. It is known that there do exist infinitely many $abc$-triples with quality greater than $1$. The main subject of the lecture is an algorithm for listing, given a large integer $N$, all $abc$-triples with $c$ at most $N$ and quality greater than $1$. As a byproduct, the algorithm yields an upper bound for the number of such triples, as a function of $N$.

The exponential transform in higher dimensions

To any domain in the complex plane there can be associated a certain function of two complex variables, called the exponential transform. It first arose in operator theory in the 1970s, and has later been studied for its own sake by Mihai Putinar and myself. It turns out to have quite remarkable and useful properties. \vskip .1in

\noindent More recently we have tried to generalize the exponential transform to higher dimensions (several real variables). The object we come up with, a kind of renormalized Riesz potential at critical exponent,
turns out to have at least some interesting properties. I will give an overview of what we know so far.

Nonlinear knowledge in kernel approximation

Prior knowledge over arbitrary general sets is incorporated into
nonlinear kernel approximation problems in the form of linear
constraints in a linear program. The key tool in this incorporation is
a theorem of the alternative for convex functions that converts
nonlinear prior knowledge implications into linear inequalities
without the need to kernelize these implications. Effectiveness of the
proposed formulation is demonstrated on two synthetic examples and an
important lymph node metastasis prediction problem. All these problems
exhibit marked improvements upon the introduction of prior knowledge
over nonlinear kernel approximation approaches that do not utilize
such knowledge. (Joint work with my PhD student Edward (Ted) W. Wild)

Energy propagation for scalr radiation outside of a spherically symmetric black hole

Pieter Blue and myself
concerning the nature of non-spherically symmetric finite energy waves
outside of a Schwarzchild black hole. In particular, I will discuss our
proof of the so called &quot;Morawetz Estimate&quot; which is a kind of monotonicity
bound describing, in a unified way, the strength of wavefronts which form as
an arbitrary burst of radiation splits into two pieces; one piece escaping
to (asymptotically flat) infinity, and the other falling across the event
horizon into the black hole. The main technical obstacle one must overcome
in proving this kind of estimate is to capture (in a mathematical way) how
an arbitrary wave &quot;decides to split up&quot;. I will try to discuss this aspect
in detail, in particular how it relates to well established ideas in quantum
potential scattering.

Energy propagation for scalr radiation outside ofa spherically symmetric black hole

Pieter Blue and myself concerning the nature of non-spherically symmetric finite energy waves outside of a Schwarzchild black hole. In particular, I will discuss our proof of the so called &quot;Morawetz Estimate&quot; which is a kind of monotonicity bound describing, in a unified way, the strength of wavefronts which form as an arbitrary burst of radiation splits into two pieces; one piece escaping to (asymptotically flat) infinity, and the other falling across the event horizon into the black hole. The main technical obstacle one must overcome in proving this kind of estimate is to capture (in a mathematical way) how an arbitrary wave &quot;decides to split up&quot;. I will try to discuss this aspect in detail, in particular how it relates to well established ideas in quantum potential scattering.

Laplace's method for differential equations in the rough path theory

I will prove aymptotic results (large deviation and Laplace's method) for the laws of solutions of (formal stochastic) differential equations in the rough path sense. My result can be regarded as a &quot;rough path version&quot; of famous results for finite dimensional SDEs. However, formulated on a general Banach space, my results contain something new. Examples include: (1) solutions of SDEs on M-type 2 Banach spaces, and (2) heat processes (or heat kernel measures) on loop spaces.

A rough guide to Brownian motion

Mathematical Brownian motion was orginally devised as a model for the random bouncing of pollen grains in a liquid. But these days it's used to model everything from stock prices to atmospheric noise to drunken mathematicians.

In my talk, I'll show you how Brownian motion arises and talk about some of its many weird properties, as well as some applications. I'll discuss how they can help solve PDEs without hard analysis. I'll throw around funny-sounding names like Donsker, Wiener, and Doob. Time permitting, I'll also say a little about stochastic calculus, the art of differentiating the non-differentiable and making sense out of noise.

This talk should be accessible to all and no prior experience with probability is required.

Local Relative Trace Formula on GL(2)

By comparing the relative and Kuznetsov trace formulas in the global
setting, Jacquet developed a method for characterizing the image of the
base change map associating automorphic representations of $U(2, {\bf A}_E/{\bf A}_F)$ to automorphic representations of $GL(2,{\bf A}_E)$.
In this talk I will discuss recent work of mine defining, proving and
comparing local versions of the relative and Kuznetsov trace formulas on
GL(2). When evaluated with matching functions, the local Kuznetsov trace
formula and the local relative trace formula are equal and thus there is
an equality between their local distributions on the spectral sides.

Information theory and probability estimation: From Shannon to Shakespeare via Laplace, Good, Turing, Hardy, Ramanujan, and Fisher

Joint work with Prasad Santhanam, Krishna Viswanathan, and Junan Zhang \vskip .1in

\noindent Standard information-theoretic results show that data over small, typically binary, alphabets can be compressed to Shannon's entropy limit. Yet most practical sources, such as text, audio, or video, have essentially infinite support. Compressing such sources requires estimating probabilities of unlikely, even unseen, events, a problem considered by Laplace. Of existing estimators, an ingenious if cryptic one derived by Good and Turing while deciphering the Enigma code works best yet not optimally. Hardy and Ramanujan's celebrated results on the number of integer partitions yield an asymptotically optimal estimator that compresses arbitrary-alphabet data patterns to their entropy. The same approach generalizes Fisher's seminal work estimating the number of butterfly species and its extension authenticating a poem purportedly written by The Bard. The talk covers these topics and is self-contained.

Additive functionals on multiway search trees

We derive asymptotics of moments and limiting distributions, under the
random permutation model on $m$-ary search trees on $n$~keys, of
functionals that satisfy recurrence relations of a simple additive
form. Many important functionals including the space requirement,
internal path length, and the so-called shape functional fall under
this framework.

The limiting behavior of these functionals exhibit intriguing phase
changes. For suitably small'' input (or {\it toll}) sequences we
have asymptotic normality if $m \leq 26$ and typically periodic
behavior otherwise. For moderate'' toll sequences we have
convergence to non-normal distributions if $m \leq m_{0}$ (where
$m_{0} \geq 26$) and typically periodic behavior otherwise. For
large'' toll sequences we have convergence to non-normal
distributions for all values of~$m$.

Recent research greatly sharpens the understanding of the periodic
cases. For example, Chauvin and Pouyanne have shown that for $m \geq 27$ fixed, the space requirement equals
$$\mu(n+1) + 2 \Re[n^{\lambda_{2}} \Lambda] + \epsilon_{n} n^{\Re\lambda_{2}},$$
where $\mu \in \bf{R}$ and $\lambda_{2} \in \bf{C}$ are
certain constants, $\Lambda$ is a complex-valued random variable, and
$\epsilon_{n} \to 0$ a.s.\ and in~$L^{2}$ as $n \to \infty$. Using
the elementary but powerful contraction method,'' we identify the
distribution of~$\Lambda$.

As time and research progress permits, we will show how the
periodic-case result can be extended rather generally to other
additive functionals under the random permutation model, and beyond
those, to generalized P\'olya urn schemes and multi-type branching
processes.

(This is joint work with Jim Fill.)

Several complex variables are better than one

The theory of complex analysis in one variable is in some sense the calculus
student's dream come true. No annoying pathological cases, and things
generally just work right. However the zero sets of analytic functions in
one variable are downright boring (isolated points). In this talk I will
talk about the Hartogs Phenomenon which has a coolness factor at least double
that of the Maximum Principle. In particular it will tell us something about
how the zero sets of analytic functions behave when we have more than one
complex variable. You will also find out what the inhomogeneous d-bar
equation is, and how to exhibit solutions in certain cases, which should
really come in handy next time you are in a bar and need to show off
something more impressive than flipping ten beer coasters at once.

This talk should be accessible to all who have not slept through basic
calculus (or at least not slept through most of it).

Limits in algebraic geometry: subschemes vs. branchvarieties

Given a family of algebraic varieties indexed by $t\neq 0$, one can ask
for the limit variety'' at $t=0$. Grothendieck considered the case
that the varieties are all subvarieties of a fixed projective space,
and defined a limit {\it subscheme} which although very natural and
useful is typically not a variety. In particular its geometry can be
quite obscure.

I'll give an alternate definition of the limit, which is still a
variety but no longer sub''; we call it a {\bf branchvariety} of
projective space, a branched cover of a subvariety. I'll give loads
of examples.

Grothendieck defined a moduli space of subschemes, the Hilbert scheme.
I'll talk about the corresponding moduli stacks of branchvarieties,
and the ways in which they're better behaved than Hilbert schemes.

This work is joint with Valery Alexeev. {\bf The talk should be accessible
to anyone who's seen e.g. the normalization of an algebraic variety;
no scheme theory will be assumed.} (That's kind of the point.)

Introduction to SAGE and its components

This is the first annual mini-conference and workshop about the computer algebra system SAGE: http://modular.ucsd.edu/sage

SATURDAY:

10am William Stein: What is SAGE?

11am David Joyner: SAGE constructions

2pm Kristin Lauter: A Cryptographer's Laundry List

3pm Fernando Perez: IPython

4pm Steven Linton: GAP

5pm Sebastian Pauli: KASH

Applications and future directions

This is the first annual mini-conference and workshop about the computer algebra system SAGE:http://modular.ucsd.edu/sage

SUNDAY:

10am Wilson Cheung: SAGE on Solaris

10:30am Kyle Schalm: MPFR

1pm Steven Sivek: Online databases (e.g., Sloane's tables) via SAGE

1:30 George Havas: Proofs in finitely presented groups

2:30 Peter Mayr: Construction and the analysis of nearrings

3:30 Ifti Burhanuddin: Factoring integers and computing elliptic curve rational points

4:00 Jon Hanke: The 290 Theorem

5:00 David Kohel: SAGE Notions of Computing with Schemes

Cluster algebras and their quantizations

Cluster algebras, introduced jointly with S. Fomin, are a class of
axiomatically defined commutative rings equipped with a distinguished
set of generators (cluster variables) grouped into overlapping subsets
(clusters) of the same finite cardinality. The original motivation for
this theory was to create an algebraic framework for total positivity
and canonical bases in semisimple algebraic groups.  Since their
introduction, cluster algebras have found applications in a diverse variety of settings which include quiver representations, Teichmuller theory, Poisson geometry, discrete dynamical systems, tropical geometry, and algebraic combinatorics. We discuss the basics of the theory of cluster algebras and their quantum analogues constructed jointly with A. Berenstein.

Signal and image recovery from incomplete and inaccurate measurements

A common problem in applied science is to recover a signal or
image from a set of indirect measurements. For example, in
Magnetic Resonance Imaging (and many other imaging modes widely
used in medicine, astronomy, and other fields) we wish to reconstruct
an image from samples of its 2D Fourier spectrum. A natural
question arises: How many measurements do we need to recover
the object of interest? In this talk, we will discuss some recent
research that addresses this fundamental question, and present a
recovery framework that performs surprisingly well (both in theory
and in practice).

A special instance of this theory yields a novel sampling theorem:
Suppose that a finite N dimensional signal f has only B nonzero
Fourier coefficients at unknown frequencies. Then we can recover f
perfectly from a small number of samples (on the order of B log N)
in the time domain.

The recovery procedure is nonlinear, and consists of solving a
tractable convex program. Despite its nonlinearity, the recovery
procedure is exceptionally stable against measurement error.

The theory (and the recovery algorithm) can be extended to signals
and images that are sparse in a known representation, and &quot;sampled&quot;
using a specified set of measurement signals. We will show how our
ability to recover a sparse signal depends on the representation and
the measurements system obeying an uncertainty principle.

We will close by showing how these ideas can be applied to problems
in tomographic imaging and data compression.

Double Bruhat cells and cluster algebras

We will discuss geometric properties of double Bruhat cells and their
totally positive parts in complex semisimple groups. The ideas and
results to be discussed were influenced by the work of G.Lusztig and
obtained jointly with A.Berenstein, S.Fomin, B. and M.Shapiro, and
A.Vainshtein. These ideas provided the groundwork for the theory of
cluster algebras.

Representations of the symmetric group

I will begin with a few basic facts about the representation theory of finite groups and enumerate all the irreducible representations of the symmetric group. Then I’ll discuss the Frobenius characteristic map, which relates the symmetric group to symmetric polynomials. If time permits I will mention a beautiful connection with the representations of the general linear group, known as “Schur-Weyl Duality”.

An integrated study of discretization, adaptivity and iterative methods for solving partial differential equations

This talk will touch upon three general aspects of
numerical solution of partial differential equations: discretization
(for transforming a continuous problem to a discrete problem), grid
adaptation (for optimizing the discrete scheme) and iterative methods
(for solving the resulting discrete systems). A number of recent
results will be presented in all these different aspects and an
integrated application of these results will be illustrated through
an example from modeling of complex fluids.

The hardness of computing an eigenform

Computing the Fourier coefficients of modular forms is
an important problem in computational number theory.
In this talk we give some evidence that this task is likely
to be hard. In particular, we show that computing the
Fourier coefficients of any fixed Hecke eigenform is at
least as hard as factoring integers of the form pq (where
p and q are distinct primes). We will also discuss the
consequence of this result to the problem of computing a
basis of modular forms. This is joint work with Eric Bach.

Affine geometry and Engel-like identities characterizing finite solvable

PLEASE NOTE CHANGE OF TIME AND LOCATION

Brainstorm MathStorm

We will give a brief report on MathStorm activities in the past year or
two, and present a couple interesting projects in further details. We
will also outline future plans and welcome all interested graduate
students to participate and brainstorm MathStorm. Round-Table Pizzas will
be provided!

Asymptotics of Kahler-Einstein metrics and some

In this talk we will discuss a canonical asymptotic expansion up to infinite order of the K\&quot;{a}hler-Einstein metric on a compact complex manifold complement of a simple normal crossings divisor. This work is parallel to the asymptotics of Fefferman, Lee and Melrose on the strictly pseudoconvex domain in $\bf{C}^n$. At the end we will mention some possible applications related to complex geometry and algebraic geometry

Hierarchically structured branching-diffusing systems

The two-level superprocess is the diffusion limit of a two-level branching
Brownian motion, where particles are grouped into superparticles which
themselves duplicate or vanish according to a branching dynamic, in
addition to the motion and branching of the individual particles
themselves. We define three classes of initial states for two-level
superprocesses and describe the corresponding patterns of longtime
behavior, including two very different types of equilibria. Specifically,
we show that two of these classes of initial states lead to longtime
behavioral patterns in high dimensions that do not exist for ordinary,
single-level branching systems or superprocesses. (Joint work with
A.Greven.)

Symmetrize it, don't criticize it

Ask any professional mathematician what they most like to exploit, and the top two answers will most likely be graduate students and symmetry. This talk will feature one of the former speaking about the latter. Specifically, I will be talking about symmetric functions and trying to make the case that if you were stranded on a desert island and could bring only one mathematical object with you, the algebra of symmetric functions would make an excellent choice; one would almost certainly die of thirst/starvation before getting bored. After demonstrating some beautiful (and fundamental) algebraic and combinatorial properties of this algebra, I will show how they can be applied by sketching a proof of the hook-length formula and (if time permits) explaining the basics of Polya enumeration.

Computing Bernoulli numbers

I will describe a new very efficient algorithms for efficiently computing Bernoulli numbers, both the usual ones and the ones attached to Dirichlet characters. This is joint work with Kevin McGown.

p-local finite groups

A p-local finite group consists of a finite p-group S, together with a pair of categories that encode &quot;conjugacy relations&quot; among the subgroups of S, modelled on the fusion in Sylow p-subgroups of finite groups. To any p-local finite group, we associate a classifying space. These spaces share many of the same properties of p-completed classifying spaces of finite groups. In particular, a p-local finite group is characterized by its classifying space. This is a survey talk on the theory, from motivation to recent results.

Thermally-driven rare events and action minimization

Thermal or stochastic effects are prevalent in physical, chemical,
and biological systems. Particularly in small systems, noise can
overpower the deterministic dynamics and lead to rare events,''
events which would never be seen in the absence of noise. One
example is the thermally-driven switching of the magnetization in
small memory elements. Wentzell-Freidlin large deviation theory
is a mathematical tool for studying rare events. It estimates their
probability and also the most likely switching pathway,'' which
is the pathway in phase space by which rare events are most likely
to occur. We explain how large deviation theory and concepts from
stochastic resonance may be applied to analyze thermally-activated
magnetization reversal in the context of the spatially uniform
Landau-Lifschitz-Gilbert equations. The time-scales of the experiment
are critical. One surprising and physically relevant result is that
in multiple-pulse experiments, nonconvential short-time switching
pathways'' can dominate. The effect is dramatic: the usual pathway
(connected with the Arrhenius-law) underestimates the probability
of switching by an exponential factor.

An advantage of the method via large deviation theory is that it
generalizes to systems with spatial variation. To discuss the
complications and richness that emerge when spatial variation is
taken into account, we consider the (simpler) Allen-Cahn equation.
In this context, the rare event of interest is phase transformation
from $u= -1$ to $u= +1$, and the most likely switching pathway is a
pathway through function space. A natural reduced problem emerges
in the sharp-interface limit.'' We give a brief overview of some
results (rigorous in $d=1$, heuristic in $d&gt;1$).

The first part of the talk is joint work with Bob Kohn and
Eric Vanden--Eijnden. The second part includes work that is also
joint with Felix Otto and Yoshihiro Tonegawa.

Classifying descent and ascents modulo k

Given any fixed integer $k &gt; 1$, one can ask to find formulas for the number of permutations whose descent pairs satisfy certain conditions mod k. We will survey a number of results on these type of questions and we will show that number of remarkable identities emerge from this study.

Knotty by Nature (Who's down wid OPP?!)

Other People's Polynomials, that is. Vaughan Jones came up with his own in 1983 and revolutionized the centuries-old study of knots. Then he got the Fields Medal for it. In this talk, we'll discuss some knot theory basics, construct the Jones Polynomial and prove (!) its invariance, and explore some applications. Prerequisite: middle school algebra.

So, who's down with knot theo-ry? All the homies!

What makes the search problem computationally hard?

In this general talk I will discuss the complexity of NP-complete
combinatorial search problems for several computational models,
such as DPLL algorithms, Greedy algorithms,
Linear and Semidefinite relaxations. I will survey several lower bounds
for these models and discuss how the expansion of the underlying
graph affects the computational intractability of the search problem.

Class invariants: old and new

Complex multiplication provides a systematic method
to generate class fields using values of modular functions in the complex upper half plane. Using small' modular functions to reach this goal is an art that goes back to Weber, and for which the
modern machinery was provided by Shimura. In recent years, p-adic counterparts have been developed for the more traditional complex analytic algorithms. I'll discuss and compare old and new methods.

On finite element methods for fully nonlinear elliptic equations of second order

For the first time, we present for the general case of
fully nonlinear elliptic differential equations of
second order on $C^2$ domains in $R^n$, a
stability and convergence proof for a non standard
non conforming $C^1$ finite element method and the
variational crimes. Our proof is applicable to Davydov's $C^1$
finite elements on curved domains, available at the moment in
$R^2$ and probably for $R^3$ soon. The general case of elliptic
differential equations and systems of orders $2$ and $2m$ will
not be discussed in the lecture. The method applies to non
divergent quasilinear elliptic problems as well. Algorithms are
formulated to calculate the nonlinear system and to solve it by
a combination of continuation and discrete Newton methods. The
latter converges locally quadratically, essentially independent
of the actual grid size by the mesh independence principle. As
usual for curved domains we have to consider the necessary
quadrature and cubature approximations to avoid difficulties at
the boundary. Essential tools are the interplay between the weak
and strong form of the linearized operator and a new regularity
result for solutions of finite element equations.

The entropy condition for hyperbolic conservation laws

Since weak solutions of hyperbolic conservation laws may be nonunique,
typically an entropy condition is imposed in order to obtain uniqueness.
We discuss how the entropy condition implies regularity and structure of
solutions of scalar conservation laws. For the one-dimensional system of
isentropic Euler equations we show how the entropy condition gives
global existence of solutions with natural bounds.

Minimal surfaces in sub-Riemannian geometry and the Bernstein problem

In any sub-Riemannian group one can introduce a Levi-Civita connection adapted to the so-called horizontal subbundle. This can be used to introduce a horizontal connection on a codimension one submanifold and obtain, among other things, a notion of mean curvature. A smooth hypersurface is called H-minimal if its horizontal mean curvature vanishes everywhere. It is natural to pose an analogue of the classical Bernstein problem, starting with the basic prototype of the Heisenberg group. I will discuss a conjecture of Bernstein type in this setting, and show that, despite the fact that H-minimal surfaces are critical points of an appropriate area functional (the H-perimeter), a new phenomenon occurs: there exists smooth critical points of the H-perimeter which are not local minimizers. This is in striking discrepancy with the classical case. I will also discuss in detail how this pathological minimal surfaces suggest an attack to the Bernstein problem, and some interesting open questions.

MacMahon partition analysis and a remarkable diophantine system

Partition Analysis is a term coined by MacMahon for its algorithmic approach to the construction of explicit solutions of diophantine systems. This algorithm has recently been implemented in MAPLE and MATHEMATICA. After a brief introduction to the algorithm we show its power and limitations in its application to a remarkable diophantine system. Our task is to enumerate the number of ways of assigning weights to the vertices of the Hypercube so that all the maximal faces carry the same weight. Some solutions to this problem are given and its intimate connection to Representation Theory will be also be discussed.

The loop-erased random walk and the uniform spanning tree on the four-dimensional discrete torus

Let x and y be points chosen uniformly at random
from the four-dimensional discrete torus with side length n.
We show that the length of the loop-erased random walk from
$x$ to $y$ is of order $n^2 (log n)^{1/6},$ resolving a conjecture
of Benjamini and Kozma. We also show that the scaling limit
of the uniform spanning tree on the four-dimensional discrete
torus is the Brownian continuum random tree of Aldous. Our
proofs use the techniques developed by Peres and Revelle,
who studied the scaling limits of the uniform spanning tree
on a large class of finite graphs that includes the
d-dimensional discrete torus for $d &gt;= 5$, in combination with
results of Lawler concerning intersections of
four-dimensional random walks.

The Golod-Shafarevich Theorem

The Golod-Shafarevich theorem gives a sufficient condition for an associative algebra presented by generators and relators to be infinite dimensional. Immediately upon arrival in 1964, this result had important consequences in various fields of mathematics. Golod used the theorem to construct a finitely generated, infinite, torsion group--the first counterexample to the general Burnside problem. Shafarevich used it to construct the first infinite tower of class fields. The primary topics of this talk will be the theorem, the Burnside problem and Golod's example.

Non-abelian CFT: introduction to the Langlands program

This is an expository talk. I will describe how the local Langlands program gives a generalization of local class field theory to non-abelian extensions. If time permits, I'll describe the role played by non-abelian Lubin-Tate theory.

The Gaps Between the Zeros of Zeta(s)

Assuming the Riemann Hypothesis, one may ask questions about the gaps
between the ordinates of the nontrivial zeros of the Riemann zeta
function. In particular, are the zeros expected to be &quot;close&quot; to each
other often? I will present two analytic techniques that address this
question.

Don't eat lunch before the talk, since there will be free bagels!

Mutation patterns in populations with large family sizes

Suppose we take a sample of size $n$ from a population
and follow the ancestral lines backwards in time until
the most recent common ancestor. Under standard
assumptions, this process can be approximated by Kingman's
coalescent, in which two lineages merge at rate one.
Mutations that have occurred since the time of the most
recent common ancestor will lead to segregating sites,
which are positions in the DNA at which not all
individuals in the sample are the same. If we denote
by $M_k$ the number of mutations that affect k
individuals in the sample, then the sequence
$M_1, ... , M_{n-1}$, is called the site frequency
spectrum. We will explain how the site frequency spectrum
would be affected if some individuals have large numbers
of offspring, so that the coalescent process that
describes the genealogy of the population can have
multiple mergers of ancestral lines. This model may
be realistic for certain marine species. This is joint
work with Julien Berestycki and Nathanael Berestycki.

Stable constant mean curvature hypersurfaces and applications

First I will give some structure

theorems and nonexistences of complete noncompact weakly stable constant mean curvature hypersurfaces. Then I will explain how stablitiy appears in some geometric problems.

The scaling limit of the loop-erased random walk in three dimensions

We shall describe loop-erased random walk and
some of its properties and connections to other natural
probabilistic objects. We shall sketch a proof that it
has a scaling limit in three dimensions.

We will Riemann-Roch you

The Riemann-Roch theorem, proved in the middle of the nineteenth century, is one of the early highlights of complex geometry. The goal of the talk is to explain its statement, the somewhat enigmatic Riemann-Roch formula, and to show, by giving examples and applications, why it is so central to the theory of Riemann surfaces and to algebraic geometry.

We will begin by explaining what a Riemann surface (a.k.a. complex manifold of dimension 1) is and how Riemann came up with the idea. We then move on to see what kinds of functions can naturally be found on a Riemann surface. The Riemann-Roch formula makes a statement about the number of linearly independent functions on a (compact) Riemann surface with certain properties. While this sounds somewhat dry, the implications of the formula are quite exciting; we will see several applications to the classification of Riemann surfaces and will also use the theorem to relate Riemann surfaces to algebraic geometry. I will only assume some basic complex analysis and will try to explain
everything else.

Qubit complexity of continuous problems

For the foreseeable future the number of qubits will be a crucial computational
resource. We show how to lower bound the qubit complexity using the classical
query complexity.

We use this result to present a simple problem which cannot be solved on a
quantum computer in the standard quantum setting with deterministic queries
but can be solved on a classical computer using randomized queries (Monte Carlo). This suggests introducing a quantum setting with randomized queries.

We apply this setting to high dimensional integration and to path integration.
In particular, there is an exponential improvement in the qubit complexity of
path integration using the quantum setting with randomized queries.

We end by discussing future directions and where to learn more.

The Conjecture of Birch and Swinnerton-Dyer for elliptic curves with complex multiplication by a nonmaximal order

Gross has refined the Birch--Swinnerton-Dyer Conjecture in the case of an elliptic curve with complex multiplication by a nonmaximal order.
Gross' Conjecture has been reformulated in the language of derived categories and determinants of perfect complexes. Burns and Flach have realized that this immediately leads to a refinement of Gross' Conjecture. The conjecture is now expressed as a statement concerning a generator of the image of a map of 1-dimensional modules. This conjecture is proved by a construction which shows it to follow from the Explicit Reciprocity Law and Rubin's Main Conjecture.

Models for commuting isometries

The von Neumann-Wold model for isometries on Hilbert space
has been important in many parts of mathematics from prediction theory
for stationary stochastic processes to the function theory of Hardy
spaces. Part of its power lies in its relative simplicity.

Various researchers have considered the structure of commuting n-tuples
of isometries including Berger-Coburn-Lebow in the mid seventies. In
this talk, I'll discuss some recent joint work with Bercovici and Foias
in which one attempts to obtain more detailed descriptions of n-tuples
in special cases. The results relate to canonical models and invariant
subspaces.

Cost thresholds for dynamic resource location

The traditional dynamic resource location problem attempts to
minimize the cost of servicing a number of sequential requests,
given foreknowledge of a limited number of requests. One artificial
constraint of this problem is the presumption that resource
relocation and remote servicing of requests have identical costs.
Parameterizing the ratio of relocation cost to service cost leads to
two extreme behaviors in terms of dynamic optimizability. The
threshold at which a specific graph transitions between these
behaviors reveals certain characteristics of the graph's
decomposability into cycles.

Cent($U(n)$), the cascade of orthogonal roots and the generalized exponents.

Let $\frak{b}$ be a Borel subalgebra of a
complex semisimple Lie algebra $\frak{g}$. Let
$\frak{h}\subset\frak{b}$ \ be a Cartan subalgebra and let
$\frak{n}$ be the nilradical of $\frak{b}$. Let
$\Delta_{+}\subset\frak{h}^{\ast}$ be the set of positive roots
corresponding to $\frak{b}$. Then there is a distinguished maximal
set $B\subset\Delta_{+}$ of strongly orthogonal roots called the
cascade of orthogonal roots. The center, Cent($U(\frak{n})$), of
the enveloping algebra of $\frak{n}$ is a module for
$H=\exp\frak{h}$.
\noindent{\bf Theorem.} Cent($U(\frak{n})$) {\it is a polynomial
ring in }$m$ {\it generators} $u_{1},...,u_{m}$ {\it where} $m=$
card$B$. {\it Furthermore, all} $H$-{\it weights in}
Cent($U(\frak{n})$) {\it are of multiplicity} $1$ {\it and the}
$u_{i}${\it can be chosen so that the weight vectors are all the
monomials} $u_{1}^{k_{1}}\cdots u_{m}^{k_{m}}$. {\it The} $u_{i}$
{\it are characterized up to scalar multiple as the weight vectors
which are also irreducible polynomials}.
We also have
\noindent{\bf Theorem.} {\it The set of weights in} Theorem 1 {\it
is exactly the set} $D_{cas}$ {\it of elements in the semigroup
generated by the linearly independent set} $B$ {\it which are also
dominant.}
The construction of the weight vectors can be given in terms of
matrix units for $U(\frak{g})$. Applications of the results are
given to the determination if minimal generalized exponents and
the proof that a Borel subgroup of $G$ has open coadjoint orbits
when $m=rank{\frak g}$.

A short proof of the Zeilberger-Bressoud q-Dyson theorem

Andrews' q-Dyson conjecture asserts that the constant term of a
certain Laurent polynomial is a product of simple factors. It was
proposed by Andrews in 1975 and proved by Zeilberger and Bressoud in
1985. We give the second proof of the theorem by using partial
fractions and iterated Laurent series. The underlying idea is that two
polynomials of degree n agreeing at n+1 points are identical. This is
a joint work with Ira Gessel.

Cooperative and non-cooperative resource sharing: delay prospective

From multi-description/multi-path routing to content distribution in
P2P networks to community networking, many forms of resource sharing
have, recently, been proposed to improve the network performance.
From the perspective of any one user and when ignoring the
interaction among users, all such schemes reduce to various forms of
providing parallelism. In this talk, we argue that focusing on
parallelism is by no means sufficient. When considering more users in
the system, these strategies provide forms of statistical
multiplexing advantage, while possibly increasing the network load
via increased redundancy, overhead increase, and even contention
inefficiency.

In this talk, we illustrate the issue of resource sharing in the
above context via a multi-queue multi-server problem. Even though
such model might not be perfectly realistic, it does capture some of
the above trade-offs. We use this model to provide analytical results
in a special case of homogeneous users and servers. Furthermore, we
prove the robustness of a certain locally optimal strategy to non-
cooperation in a Nash equilibrium/strategy context.

Corner percolation and the square root of 17

We consider a dependent bond percolation model on ${\bf Z}^2$,
introduced by Balint Toth, in which every edge is present
with probability 1/2, and each vertex has exactly two incident
edges, perpendicular to each other. We prove that all components
are finite cycles almost surely, but the expected diameter of
the cycle containing the origin is infinite. Moreover, we derive
the following critical exponents: the tail probability
$P($diameter of the cycle of the origin $&gt; n) \approx n^{-\gamma},$
and the expectation $E($length of a cycle conditioned on
having diameter $n) \approx n^\delta.$ We show that
$\gamma=(5-\sqrt{17})/4=0.219...$ and $\delta=(\sqrt{17}+1)/4=1.28...$
The relation $\gamma+\delta=3/2$ corresponds to the fact that the
scaling limit of the natural height function in the model is the
Additive Brownian Motion, whose level sets have Hausdorff dimension
3/2. The value of $\delta$ comes from the solution of a singular
sixth order ODE.

From alphas to zetas and two kinds of fields

When a mathematician is faced with a sequence of numbers that one wants to understand, one typically packages them together as a generating function.

For example, if one has an algebraic variety $V$ over a finite field $F_q$, a geometric object defined as the zero locus of a set of equations, one can consider the sequence of cardinalities $N_k$ over higher field extensions $F_{q^k}$. One particular generating function for the $\{N_k\}$, known as the Zeta Function of variety $V$, has lots of remarkable properties. These were conjectured by Weil in the 1940's and proven by Deligne in 1973, work which helped him earn a Fields Medal. In this talk I will give a snapshot of this work, for the case of curves, where the theory is already very rich.

On the mock theta functions

The mock theta functions were first stated by S.~Ramanujan in his last letter to G.~H.~Hardy in 1920. In a nutshell, these are functions that mimic the behavior of theta functions in some sense, but have asymptotic behaviors that are unlike those of ordinary theta functions. In particular, they are not modular. In this talk, we will provide an overview of the historical development of this subject and discuss works by G.~N.~Watson, G.~Andrews and most recently, K.~Ono.

Twisted K-homology and Lie groups

The twisted K-theory of a space was introduced by Donovan-Karoubi (1970)
for torsion twistings and by Rosenberg (1989) in general. Motivated by
applications in mathematical physics, the theory has attracted a lot of
interest in recent years. In this talk I will review some of the
foundations of twisted K-homology, and outline a new proof of the
Freed-Hopkins-Teleman theorem describing the twisted equivariant
K-homology of simple, simply connected compact Lie groups.

Eigenvalues and factors in graphs

A factor of a graph G is a spanning subgraph of G. A k-factor is a spanning k-regular subgraph. We describe some eigenvalue conditions that imply the existence of a 1-factor in a graph and discuss some open
problems.

Local polynomial fitting and smooth and non-smooth curve estimation

In regression analysis the interest is in exploring the relationship between a response
variable and some explanatory variable(s). Without knowledge about an appropriate
parametric relationship between the variables, one often relies on nonparametric methods.
Local polynomial fitting leads to estimators of the regression function and its derivatives
up to a certain order. We very briefly discuss the basic properties of these estimators
when the regression function is smooth. In particular we pay attention to the behaviour
of the estimators in boundary regions. This local modelling technique is applicable in a
variety of applications.
When the regression function is non-smooth, e.g. discontinuous, the estimates are
inconsistent in the non-smooth points. We briefly discuss some available nonparametric
methods. We discuss, among others, a nonparametric estimation method that searches
for compromising between the properties of jump-preserving and smoothing. The method
chooses, in each point, among three estimates: a local linear estimate using only data to
the left of the point, a local linear estimate based on only data to the right, and a local
linear estimate using data in a two-sided neighborhood around the point. The data-driven
choice among the three estimates is made by comparing, in some appropriate way, the
weighted residual mean squares of the three fits. This results into a consistent estimate.
We establish asymptotic properties of the estimator, and illustrate its performance via
simulations and examples.

Curves and surfaces: change-points, edge-detection and boundaries

Regression surfaces can also exhibit non-smooth or irregular features. To recover e.g.
a non-smooth behaviour one should avoid using a nonparametric technique that smooths
away this particular behaviour. We discuss how to estimate via local linear fitting a non-
smooth regression surface, in case of fixed or random design. The proposed procedure can
also be used for image denoising. Applications to surface estimation and image denoising
are shown.
Sometimes the interest is in estimating the location of the points/curves at which e.g.
a function is non-smooth, say discontinuous. This relates to the problem of change-point
estimation or edge detection (or boundary estimation). We discuss how one can use the
previously discussed techniques to some other problems, such as estimating non-smooth
densities.
These lectures are partly based on joint work with Alexandre Lambert and Peihua Qiu.

Introduction to the Seminar on the Madsen-Weiss proof of Mumford's conjecture

This is the introductory meeting for this quarter's topology learning seminar. The subject of the seminar will be introduced and the talks will be distributed.

The goal of the seminar is to study the Madsen-Weiss proof of Mumford's conjecture concerning the cohomology of the stable mapping class group. Knowing the group cohomology of this group means to know the cohomology of the stable moduli space of Riemann surfaces. While Mumford's conjecture deals with the geometry of the moduli space of surfaces, the methods of the proof given by Madsen and Weiss are mostly homotopy theoretic. The proof is a beautiful example for how the rather abstract machinery of homotopy theory can be used to address a concrete geometric question. Also, we think that the techniques used in the proof are interesting by themselves and thus worth studying.

The first couple of talks introduce the main players: moduli spaces of Riemann surfaces, mapping class groups, and Mumford's conjecture. Then we turn some fundamental concepts of homotopy theory. This is necessary in order to understand Ulrike Tillmann's work that identifies the computation of the cohomology of the stable mapping class group as a problem of stable homotopy theory. Based on her work, Madsen formulated a conjecture in the world of stable homotopy theory that implies Mumford's conjecture. Finally, the last third of the seminar will be concerned with giving/outlining the proof of Madsen's conjecture by Madsen and Weiss.

Induced-universal graphs for graphs with bounded maximum degree

Given a family $\digamma$ of graphs, a graph $U$ is universal for $\bf{F}$ if every graph in $\bf{F}$ is a subgraph of $U$. We say that $U$ is induced-universal for $\bf {F}$ if every graph in $\bf{F}$ is an {\it induced} subgraph of $U$. Recently Alon and Capalbo gave a construction for a universal graph for the family $\bf{F}$ of graphs on $n$ vertices with bounded degree which is within a constant factor of the best possible.

We give a construction for an induced-universal graph for the family of graphs on $n$ vertices with degree at most $r$, which has $Cn^{\lfloor (r+1)/2\rfloor}$ vertices and $Dn^{2\lfloor (r+1)/2\rfloor -1}$ edges, where $C$ and $D$ are constants depending only on $r$. This construction is nearly optimal when $r$ is even in that such an induced-universal graph must have at least $cn^{r/2}$ vertices for some $c$ depending only on $r$.

Our construction is explicit in that no probabilistic tools are needed to show that the graph exists or that a given graph is induced-universal. We extend our construction to multigraphs and directed graphs with bounded degree, and also graphs with bounded arboricity.

Classification of manifolds and the Poincar\'e conjecture

Manifolds arise naturally in areas of mathematics varying from topology to
geometry to analysis, and are important in applications to many fields
outside of math (e.g. physics). In this talk, I'll begin by explaining
what a manfiold is. Then I'll talk a bit about the classification of
manifolds in small dimensions. After that, I'll state the Poincar\'e
conjecture and explain how it is proved (for $n \geq 6$) using the
$h$-cobordism theorem. This talk should be accessible to anyone who can
see all the pretty pictures on the board.

The Liu-Yau quasi-local mass in spherical gravity

Liu and Yau recently introduced a new quasi-local mass in GR.
It is a function on a 2-surface in a 4-manifold. They showed that it was
positive. It is the maximum of the Brown-York energy over all 3-slicings
containing the given 2-surface. The Liu-Yau mass has unpleasant features, it
looks much more like an energy than a mass. In particular it is unboundedly
large on any solution of the Einstein equations, including Minkowski space!
In spherical symmetry, however, it has a natural physical interpretation.
Consider a regular spherical 3-slice filling the interior of the given
2-slice. Take the integral of the energy density of the interior. The
Liu-Yau mass of the boundary is the minimum of this total energy content
over all regular fillings. No other quasi-local mass gives such interior
information.

Just infinite algebras

Much is known about simple rings and these results are often extended to direct sums of simple rings or to a ring, R, whose factor ring R/I is simple for a suitable ideal I. An alternative approach to extending these results is to loosen the condition simplicity puts on ideals of a ring. Just infinite dimensional algebras are those which, in some sense, contain only few ideals.

I'll start by defining just infinite more precisely. My talk will summarize known results for and examples of just infinite rings. I'll also discuss the results of my dissertation- a few properties of just infinite algebras and an example of broadening theorems about simple rings to theorems about just infinite rings.

Fischer pairs and polynomial solutions for the Dirichlet problem

It is well-known that an ellipsoid $G$ has the following property (E) and
(E* resp.):
for any polynomial $f$ (entire function resp.), the
solution of the Dirichlet problem
for the restriction of $f$
to the boundary of the domain $G$, has an polynomial (entire resp.)
extension.

In this talk we give a positive answer to the following question of
D. Khavinson and H.S. Shapiro for a large class of domains:
the ellipsoids are the only domains satsifying (E) or (E*) in this class.
The results follow from a more general theorem in the context of so-called
Fischer pairs.

Another interesting consequence of our methods is the following result:
if a polyharmonic entire function of order $k$ vanishes on $k$
distinct ellipsoids then it vanishes
everywhere. This answers a question of W.K.
Hayman.

Absolute value equation solution via concave minimization

The NP-hard absolute value equation (AVE), Ax-$|x|$=b, where
A is an n-by-n real matrix and b is an n-by-1 real vector is solved by a succession of linear programs. The linear programs arise from a reformulation of the AVE as the minimization of a piecewise-linear concave function on a polyhedral set and solving the latter by
successive linearization. A simple MATLAB implementation
of the successive linearization algorithm solved 100 consecutively
generated 1000-dimensional random instances of the AVE with only
five violated equations out of a total of 100,000 equations.
Paper is available at: ftp://ftp.cs.wisc.edu/pub/dmi/tech-reports/06-02.pdf

Generating functions for permutations whose descent set contains a prescribed set

There are a number of papers in the literature that have proved generating functions for permutations and signed permutations according to the number of descents. We shall show how to generalize many of these type of results to get generating functions for permutations and signed permutations whose descent set contains a prespecified set S according to the number of descents. Our results will generalize previous results of Carlitz, Stanley, Foata, Han, and others.

Random matrix central limit theorems for non-intersecting random walks

This talk will focus on Gaussian Unitary Ensemble central limit
theorems for non-intersecting random walk with general increments. In
certain scaling regimes these walks exhibit Tracy$-$Widom and Sine kernel
fluctuations of random matrix theory. This is joint work with Jinho Baik.

Many directions to a black hole

Black holes are intimately tied to the strongest gravitational fields
which are in turn responsible for espectacular phenomena both observed
and conjectured. It is in their surrounding where Einstein's theory display
many of its profund implications and present us with a challenge to
understand them. This talk will discuss a few examples &--&in 4 and larger
dimensions&--& where black holes play a key role and the efforts to unravel
their physical consequences.

Graphs &amp; probability a perfect matching

Erd\H{o}s in 1947 provided a bound on the so-called diagonal Ramsey numbers with a startling
and elegant proof drawn from probability. In 1960 Erd\H{o}s and Renyi published a landmark paper
introducing the notion of a random graph and exploring the rather quirky nature of these objects.
Slowly at first, and now with increasing speed, the probabilistic method in combinatorics has grown from these beginnings into a large and vibrant field with deep connections to many fundamental questions in mathematics and computer science.

In this talk, we will give a taste of probabilistic methods, using the subject of random graphs as a foil.
We'll give a whirlwind tour of the properties of random graphs, their applications, and a look at some very large graphs.
This talk should be accessible to anyone with a basic understanding of undergraduate probability, and of interest to all.

Phase field modeling and simulation of bio-membranes

Lipid membranes are interesting subjects of biological
studies. In this talk, we report some recent works done by
the Penn State group on the phase field modeling and simulations
of the vesicle membrane deformation under elastic bending
energy and the interaction with background fluid flows.
Connections are made to the well$-$known Willmore problem in
differential geometry and the Gamma convergence of nonlinear
functionals in the calculus of variation. We also discuss
how to effectively retrieve topological information within
the phase field framework which may have broad applications

The signature is an E-infinity ring mapThe signature can be regarded as a map from oriented topological bordism

The signature can be regarded as a map from oriented topological bordism
to symmetric L theory. We investigate its multiplicative properties and
show that there is simplicial model for the signature which is a map of
E symmetric ring spectra. The construction generalizes to a large class
of theories which are bordism-like in a suitable sense. The work is
joint with James McClure.

Some nice homotopy theory for Madsen-Weiss

This talk will build on Nitu$^\prime$'s talk to give some of the homotopy theory used in the Madsen-Weiss paper about the cohomology of the stable mapping class group $\Gamma_{\infty, 1+1}$. I will tell you about Quillen's plus construction and about operads as a recognition principle for infinite loop spaces, as well as at least hint at how they will be used in Tillmann's proof that ${\bf Z}\times{B}\Gamma_{\infty,1+1}^+$ is an infinite loop space.

Almost minimal CR submanifolds

In this talk we investigate what kind of CR manifolds have the following
local uniqueness property: If two holomorphic functions have the same
modulus on the manifold they are equal (up to a constant) in the whole
space. Minimal generic submanifolds have this property.

We will investigate the nonminimal case and we will give a sufficient
geometric property which we will call almost minimality. This condition is
related to the manifold being contained in a possibly singular real analytic
Levi-flat hypersurface.

Fragmentation of the continuous random tree

We consider the height process of a Levy process with no negative
jumps, and its associated continuous tree representation. Using
Levy snake tools developed by Duquesne and Le Gall, we construct a fragmentation process (at node and at height), which in the stable case corresponds to the self-similar fragmentation (at node and at height) described by Miermont. For the general fragmentation process we compute a family of dislocation measures. We compute also the asymptotic for the number of small fragments at time t. In the case of the fragmentation at node, this limit is increasing in t and discontinuous. In the alpha-stable case the fragmentation is self-similar and the results are close (but still different) to those obtained by Bertoin for general self-similar fragmentations under a strong additional assumtion which is not fulfilled here.

EureQa! an introduction to the p-adics

Whether your usual number system is broken down, or simply in need of an Eul-change, you will find plenty of absolute value in this talk. I'll remove that annoying ring from your definition of prime and roll-over your infinite odometer. Once your number system is complete, we'll travel local-to-global, and visit Hensel $&amp;$ Gertrud in a magical 20Dream-land where a series converges if and only if the the terms tend to zero.

This talk should be much more accessible than its abstract. I encourage you to attend, if only for the grapes.

Traces of CM values of modular functions

The values of the famous j-invariant at quadratic
irrationalities in the upper half plane are known as singular moduli
and are of particular interest in number theory.
Recently, Zagier realized the generating series of the traces of the
singular moduli as a classical meromorphic modular form of weight 3/2.
In this talk, we will discuss this and related results and give a
generalization to modular functions on Riemann surfaces of arbitrary
genus. Furthermore, we realize a certain generating series of arithmetic
intersection numbers and Faltings heights as the derivative of Zagier's
Eisenstein series of weight 3/2. This recovers a result of Kudla,
Rapoport and Yang.

This is joint work with Jan Bruinier.

Amenable algebras

We will begin with some motivation for the study of amenability
for affine associative algebras, partially based on the better established
theory of amenable groups. Following a paper of G. Elek, we will give three equivalent definitions of amenability for algebras and prove some of the basic algebraic properties of amenable algebras.

Tableaux combinatorics for the asymmetric exclusion process

The partially asymmetric exclusion process (PASEP) is an important model
from statistical mechanics which describes a system of interacting
particles hopping left and right on a one-dimensional lattice of n sites.
Particles may enter the system from the left with probability alpha, and
exit from the right with probability beta. The model is partially
asymmetric in the sense that the probability of hopping left is q times
the probability of hopping right. In this talk, we will describe a
surprising connection between the PASEP model and the combinatorics of
certain 0-1 tableaux called permutation tableaux. Namely, we prove that in
the long time limit, the probability that the PASEP is in a particular
configuration tau is a generating function for permutation tableaux of
shape lambda(tau), enumerated according to three statistics.

The tableaux in question come from total positivity on the Grassmannian
(via work of Postnikov).

Realizations of the associahedron and cyclohedron

Abstract: We describe different realizations with integer coordinates for
the associahedron (i.e., the Stasheff polytope) and for the cyclohedron
(i.e., the Bott-Taubes polytope), and compare them to the permutahedra of
type A and B, respectively.

The coordinates are integral and obtained by an algorithm which uses an oriented Coxeter graph of type A or B respectively as the only input and which
specializes to a procedure presented by J.-L. Loday for a certain
orientation of the Coxeter graph of An.

The described realizations have cambrian fans of type A and B as normal
fans. This settles a conjecture of N. Reading for cambrian fans of these
types.

This is a joint work with Carsten Lange

Boosting and Brownian motion

The computational task that lies in the core of many machine learning problems is the minimization of a cost function called the training error. This problem is frequently solved by local search algorithms such as gradient descent. The training error can usually be expressed as a sum over many terms, each corresponding to the loss of the model on a single training example. We show that the iteratively minimizing a cost function of this form by local search is closely related to the following game: Imagine you are a shepherd in charge of a large herd of sheep and your goal is to concentrate the sheep in a particular small area by nightfall. Your influence on the sheep movements is represented by vectors which define the direction in which you want each sheep to move. The lengths of the vectors correspond to the fraction of your ''energy'' that you spend on moving the particular sheep, and these lengths sum to one. The sheep then have to respond by moving in a way that has a slight correlation with the influence direction on average. We characterize the min/max solution to this game and show that by taking the appropriate small-step/continuous-time limit, this solution can be characterized by a stochastic differential equation. By solving this differential equation we re-derive some known boosting as well as design some new ones with desirable properties.

Recurrent extensions of self-similar Markov processes

Let $X=(X_t)_{t\ge 0}$ be a self-similar Markov process with values in the
non-negative half line, such that the state $0$ is a trap. We present a
necessary and sufficient condition for the existence of a self-similar
recurrent extension of $X$ that leaves $0$ continuously. This condition is
expressed in terms of the L\'evy process associated with $X$ by the Lamperti transformation.

Classification of hyperbolicity and stability preservers

Classification of hyperbolicity and stability
preservers
Julius Borcea
Department of Mathematics, Stockholm University, SE-106 91, Stockholm, Sweden
julius@math.su.se
A linear operator T on C[z] is called hyperbolicity-preserving or an HPO for short
if T(P) is hyperbolic whenever P 2 C[z] is hyperbolic, i.e., it has all real zeros.
One of the main challenges in the theory of univariate complex polynomials is to
describe the monoid AHP of all HPOs. This outstanding open problem goes back
to P´olya-Schur’s well-known characterization of multiplier sequences of the first
kind, that is, HPOs which are diagonal in the standard monomial basis of C[z].
P´olya-Schur’s 1914 result generated a vast literature on this subject and related
topics at the interface between analysis, operator theory and algebra but so far only
partial results under rather restrictive conditions have been obtained. In this talk
I will report on the progress towards a complete solution of both this problem and
its analog for (Hurwitz) stable polynomials as well as their multivariate versions
made in an ongoing series of papers jointly with Petter Br¨and´en and Boris Shapiro.
The concepts of hyperbolicity and stability have natural multivariate extensions:
a polynomial f 2 C[z1, . . . , zn] is stable if f(z1, . . . , zn) 6= 0 whenever
=(zj) > 0, 1 j n. A stable polynomial with real coefficients is called real
stable. Hence a univariate real stable polynomial is hyperbolic in the above sense.
We generalize the notion of multiplier sequences to multivariate polynomials and
give a complete characterization of higher-dimensional multiplier sequences. We
then classify all operators in the Weyl algebra An of differential operators that
preserve stability and show that real stability preservers in n variables are generated
by real stable polynomials in 2n variables via the symbol map. One of
the key ingredients in the proofs is a natural duality theorem for the Fischer-Fock
spa

The Feynman path integral isn't fine, man

The Feynman path integral interprets quantum amplitudes as sums over
all possible histories which are "weighted'' by the classical
action. In interpreting that statement, I'll review some classical
and quantum mechanics, and then convince you why the Feynman path
integral is reasonable. I'll then convince you that everything I
just said is mathematical nonsense. I'll also try to indicate some
nice math that has come out of attempting to make sense of this over
the past fifty years. I'll try to make the lecture as
self-contained as possible, but familiarity with basic concepts of
classical and quantum mechanics would be helpful. \textbf{Warning}:
This talk contains lots of hand waving.

Lifting of cusp forms from \tilde{SL}_2 to GSpin(1,4)

We construct liftings of cuspidal automorphic forms from the metaplectic group $\tilde{SL}_2$ to GSpin(1,4) using Maass Converse Theorem. In order to prove the non-vanishing of the lift, we derive Waldspurger's formula for Fourier coefficients of half integer weight Maass forms. We analyze the automorphic representation of the adelic spin group obtained from the lift and show that it is CAP to the Saito-Kurokawa lift from $\tilde{SL}_2$ to GSp_4.

Linear Equations and Lie Quadratics

Consider an ordinary differential equation

dx/dt = x(t)B
for a function x(t) taking values in the space of 3x3 matrices, where B is a skew-symmetric 3x3 matrix affinely dependent on t.

The theory of Lie quadratics and Riemannian cubics will be briefly reviewed, then applied to reveal some mathematical structure associated with this elementary linear ODE with (slightly) variable coefficients.

Traveling wave solutions of the Camassa-Holm equation

We classify all weak traveling wave solutions of the Camassa-Holm equation. It is proved that, in addition to smooth solutions, there are a multitude of traveling waves with singularities: peakons, cuspons, stumpons, and composite waves. The approach is also shown to apply to other related nonlinear wave equations.

Random Matrices and Infinitesimal Rotations

A theme in studying random matrices from the classical groups is that
Haar-distributed matrices are close to Gaussian matrices in many ways; one
way is that linear functions of random orthogonal matrices are
asymptotically normally distributed (as the dimension approaches
infinity). I will discuss an infinitesimal version of Stein's method,
introduced by Stein in 1995, which can be used to obtain a rate of
convergence in this and related theorems.

Jim Lin, Moderator - UCSD

We will have a discussion where panelists (Brian Miceli, grad student, Holly Hauschild, Warschawski Assistant Professor, Dan Rogalski, Assistant Professor) will share their personal experiences applying for jobs in academia. We will try to address the following questions.
What should you do before applying for a job? How many applications should I send out? How should I prepare for an interview? What if my spouse is also on the job market?
There will be a question and answer period following the discussion.

On the convergence of multilevel methods for elliptic problems

In this talk we present results related to the convergence of multilevel/multigrid methods, applied to symmetric positive definite linear systems. The presentation is based on the subspace correction framework. The abstract theory is applied to examples coming from finite element discretizations of scalar elliptic partial differential equations and we show how some of the known estimates of the convergence rate of the multigrid method can be obtained in a straightforward fashion. We also discuss some relations between a multiplicative Schwarz method convergence and the condition number of the corresponding additive Schwarz preconditioner.

Nontrivial Sha for curves of genus 2 arising from K3 surfaces

When doing a $2$-descent on the Jacobian $J$ of a curve of genus $2$, one wishes to determine whether or not certain twists of $J$ have
rational points. As $J$ and its twists are unwieldy, we consider the
quotient $K$ of a twist by the involution induced by multiplication by
$-1$ on $J$. We construct an explicit curve $C$ and corresponding twist
for which there is a Brauer-Manin obstruction to the existence of
rational points on $K$. This yields infinitely many twists of $C$ with
nontrivial Tate-Shafarevich group. This is joint work with Adam Logan
at Waterloo.

The Orbit structure of a Group Constructed from the Classical Analogue of the Gelfand-Zeit

Let $M(n)$ be the algebra (viewed as both a Lie and an associative algebra) of $n\times n$ matrices over $\mathbb{C}$. Let $P(n)$ denote the algebra of polynomials on $M(n)$. The associative commutator on the universal enveloping algebra induces a Poisson structure on $P(n)$. Let $J(n)$ be the commutative Poisson subalgebra of $P(n)$ generated by the invariants $P(m)^{Gl(m)} \text{ for } m=1,\cdots , n$. $J(n)$ gives rise to a commutative Lie algebra of vector fields on $M(n)$; $V=\{ \xi_{f}| f\in J(n)\}$. These fields integrate to an action of a commutative, simply connected complex analytic group $A\simeq\mathbb{C} ^{\frac{(n-1)n}{2}}$ on $M(n)$. Note that the dimension of this group is exactly half the dimension of the generic coadjoint orbits. Moreover, on the most generic orbits of $A$, the commutative Lie algebra $V$ is an algebra of symplectic vector fields of exactly half the dimension of the generic coadjoint orbits. We will discuss the orbit structure of the action of $A$ on $M(n)$. We will give a description of the work of Kostant-Wallach in the most generic case in such a form that can be used to establish a formalism for dealing with less generic orbits.

Southern California Number Theory Day

Invited Speakers: Florian Pop (Univ. of Pennsylvania), Niranjan Ramachandran (Univ. of Maryland, College Park), Karl Rubin (Univ. of California, Irvine), Daqing Wan (Univ. of California, Irvine)

Towards general Schur positivity and higher saturation

We prove Schur positivity conjectures due to
Fomin-Fulton-Li-Poon and Okounkov and give several related results.
We show that Schur polynomials satisfy an interesting
Schur-log-concavity property. We discuss a "higher Klyachko
problem" about inequalities between Littlewood-Richardson
coefficients and formulate a general Schur positivity conjecture
involving certain mysterious convex polytopes. The talk is
based on a joint work with Thomas Lam and Pavlo Pylyavskyy.

Random networks for communications

The theory of random graphs is a useful mathematical tool to describe many real world systems. Recently, the mathematical and engineering communities have shown a renewed interest in the geometric version of these models. The nodes are geometrically distributed at random, and pairs of nodes are connected by edges, whose presence depends on the random positions of the nodes in the plane. One emerging application is in the field of wireless communications, where radio transmitting stations communicate by radiating electromagnetic waves.
In this talk, first I review several models of random networks for communication that are directly related to continuum percolation. Then, I show some recent results on connectivity of dependent percolation models of interference limited networks. Finally, I introduce the Gupta-Kumar concept of throughput scaling, and argue how this can be obtained as a natural consequence of the RSW theorem in percolation theory.

Index juggling and the Gauss-Bonnet theorem

What gets a lot of geometers and topologists to play well together, besides bringing toroidal pastries to a gathering, is to talk about local-to-global results. A central result of this kind for surfaces is the Gauss-Bonnet Theorem, which relates the Gaussian curvature of a surface (a local thing) to its Euler characteristic (a global thing). Instead of going into a thoroughly modern definition of Euler characteristic, it is possible to instead relate the total curvature to another closely related concept for which we can draw lots of pretty pictures: singular points of vector fields and their indices.

In this talk, I'll introduce vector fields on surfaces, and the concept of index, via pretty pictures and decorated oranges. Then I'll talk about Gaussian curvature, as intuitively as I can (which will involve some, ahem... hand-waving, and I'll try keep another kind of index juggling to a minimum!). Next up is the theorem and its proof via juggling indices. Finally, I'll give a little peek under the hood by mentioning triangulations of surfaces and how it relates to all this (with more hand-waving).

Asymptotics for extremal moments and monodromy of complex singularities

We present a finite algorithm for the computation of
any moment of the solution
(= the characteristic function of $\{p < 1\},$ with $p$ a polynomial,
assumed to have isolated (complex) critical points)
of the truncated extremal $n$-dimensional $L$-moment problem,
linearly in terms of a finite set of generating moments, in the context of dynamic
(i.e. time-dependent) moments. We find that a system of such generators is provided by the moments
corresponding to a basis for ${\bf R}[x_1,...,x_n]/ I_{\nabla_p},$ where $I_{\nabla_p}$ is the gradient ideal
of $p.$
From this, based on a well-known algebraic formalism for asymptotics of the Fourier transform,
we obtain computations for the coefficients of the asymptotic expansions for the moments
in terms of the generators and the monodromy of $p.$

Communities from seed sets

Expanding a seed set into a larger community is an important task
for various algorithms that run on the world wide web graph, including
topic-sensitive search ranking, community finding, and topic
distillation. We present an algorithm that expands a seed set using
random walks and maximum flow computations. Our algorithm finds a
community with small conductance by adding nodes related to the
seed, while examining only a small neighborhood of the entire graph.

Class Numbers of Cyclotomic Fields

Even since Kummer's 19th century attempts to prove Fermat's Last Theorem, cyclotomic fields have been at the center of attention. Indeed, in the second half of the 20th century the subject enjoyed a strong revival in the hands of Iwasawa.

Despite tremendous progress, many questions remain. Vandiver's conjecture is perhaps the most outstanding one. In this talk we discuss the problem of calculating class numbers of cyclotomic fields. This is a computational problems that, even using the fastest computers, has only been solved in a handful of cases. We present an experimental approach to determining these class numbers.

Blowup analysis for the Yamabe equation in high dimensions.

In this talk we will address the problem of analyzing the
blow-up behavior of a sequence of solutions to the Yamabe equation in high dimensions. We will show how sharp pointwise estimates can be used to prove that the Weyl tensor vanishes up to order $[\frac{n-6}{2}]$ at a blowup point. This gives a proof of the compactness of the set of solutions to the Yamabe problem, a result conjectured by R. Schoen around 15 years ago. This is joint work with Marcus Khuri and Richard Schoen.

Lie Groups and their Lie algebras: A linear problem is easier than a non-linear one !!

In this talk, we will discuss some of the fundamental theorems of Lie theory. We will discuss how to associate to a Lie group (i.e. a smooth manifold that is also a group) a linear space, known as its Lie algebra. We will then discuss how we can use Lie algebras to classify different Lie groups. We will discuss the Fundamental Theorem of Sophus Lie which tells us exactly when two Lie groups are locally isomorphic, as well as the equivalence of the categories of real simply connected Lie groups and finite dimensional real Lie algebras. If time permits, we will discuss the study of real compact groups and their classification by certain types of complex Lie algebras. We will also discuss algebraic groups (groups that are algebraic varieties) and algebraic Lie algebras.

New results for the Schroedinger map equation

By direct iteration we establish local well-posedness for the subcritical SM in all dimensions $\geqq$ 2 and global well-posedness in all dimensions $\geqq$ 3 assuming smallness of the initial data.

Efficent Computation of Epitaxial Growth

To begin, I will describe a fast Monte-Carlo algorithm for simulating epitaxial surface growth, based on the continuous-time Monte-Carlo algorithm of Bortz, Kalos and Lebowitz. When simulating realistic growth regimes, much computational time is consumed by the relatively fast dynamics of the adatoms. To solve this problem, we allow adatoms to take larger steps, determined by the local geometry, effectively reducing the number of transitions required. We achieve nearly a factor of ten speed up,for growth at moderate temperatures and moderate to low deposition rates.

In the second part of the talk, I will discuss the computation of stained heteroepitaxial growth. Elastic effects are incorporated using a ball and spring type model. An efficient method based on combining Fourier and multigrid formulations is presented. The equations for the elastic displacement of atoms in the film are extended to a rectangular region by the use of fictitious atoms and a connectivity matrix, allowing the application of standard multigrid ideas. Except for the top layer, the atoms in the substrate are completely removed and replaced by equivalent forces which can be efficiently evaluated using a fast Fourier transform. This formulation has been implemented in both two and three dimensions.

Finally we introduce various approximations in the implementation of KMC to improve the computation speed. Numerical results show that layer-by-layer growth is unstable if the misfit is large enough resulting in the formation of three dimensional islands.

Affine insertion and Schubert calculus on the affine Grassmannian

By Bott's work on loop groups on simple Lie groups
and a result of Quillen, the homology of the affine Grassmannian Gr
of SL(k+1) is a ring that is naturally embedded in the ring of symmetric functions.
Since Gr is a Kac-Moody G/P it is natural to consider the Schubert bases
of its homology and cohomology. The Schubert structure constants of the homology of Gr
are particularly interesting, as they include the genus zero Gromov-Witten invariants for
Grassmannians, or equivalently the structure constants for the fusion tensor product for
the Weiss-Zumino-Witten conformal field theory model.

In the summer of 2004 I conjectured that the Schubert basis of homology
of Gr is given by the k-Schur functions of Lascoux, Lapointe, and Morse. This was
recently proved by Thomas Lam. Lapointe and Morse also had a dual basis called the dual
k-Schur functions or affine Schur functions; these are the Schubert basis of cohomology
of Gr.

The dual k-Schur functions may be realized as generating functions for
k-tableaux or weak tableaux, which are defined using the weak order on the affine
symmetric group.

We introduce strong tableaux, whose definition is based on the strong Bruhat order on the
affine symmetric group. The generating functions of strong tableaux are symmetric
functions,
which we prove are equal to the k-Schur functions.

We give an algorithm called affine insertion, which maps certain
biwords to pairs of tableaux, one strong and one weak. This bijection
proves an analogue of the Cauchy identity coming from the pairing between the homology
and cohomology of Gr.

As k goes to infinity, weak and strong tableaux both converge to semistandard tableaux
and our bijection converges to the usual
row insertion Robinson-Schensted-Knuth correspondence.

As corollaries we obtain Pieri rules for the Schubert calculus of
both the homology and cohomology of the affine Grassmannian.
Both of these are new.

This is joint work with Thomas Lam, Luc La

The support vector machine as a statistical learning tool

The main objective of statistical learning is the characterization of an unknown
dependency between observations (measurements) on observational
units and certain properties of these observational units. All the measurements
are assumed to be observable. The dependent properties are only
available for a subset (i.e. the learning set) of observation units. The Support
Vector Machine (SVM), a machine learning concept, has recently attracted
increasing attention in the statistics community. One of the reasons is, that it
can handle the situation of far more variables than observational units as it is
now common in bioinformatics applications. The general problem of estimating
unknown dependencies and to use them for prediction occurs in many
other applications such as environmental sciences, quantitative economics,
and finance.
The SVM (V. N. Vapnik: The nature of statistical learning theory.
Springer, NY, 2000) is a classification method specially suitable for overlapping
classes. It produces nonlinear boundaries by constructing a linear
boundary in a transformed version of the feature space. We discuss the SVM
in the context of statistics under unspecified stochastic assumptions. For
instance complexity control can be achieved in a similar way as in penalized
(regularized) binary regression. Apart from the classical (machine learning)
algorithms statistical estimation concepts are considered.

The Nullstellensatz in a noncommutative setting

We will concentrate on the connection of Hilbert's Nullstellensatz and the Jacobson radical. The talk will focus on PI-algebras and follow a paper of Amitsur.

Convergence and Optimality of Adaptive Mixed Finite Element Methods

In this talk we shall analyze the convergence and optimality of adaptive mixed finite element methods for second order elliptic partial differential equations. The main difficulty for the mixed finite element method is the lack of minimization principle and thus the failure of orthogonality. A quasi-orthogonality property is proved using the fact that the error is orthogonal to the divergence free subspace, while the part of the error containing divergence can be bounded by data oscillation.

Weighted Delaunay triangulations and discrete Laplacians.

We will investigate the use of weighted
triangulations as a discrete analogue of Riemannian geometry. We will then introduce discrete Laplacian operators, which are particularly weighted Laplacians on the 1-skeleton of a metric (Euclidean) triangulation in the sense of Laplacians on graphs. We will investigate some of the properties of these Laplacians, including an interesting optimality result for weighted Delaunay triangulations originally proven by Rippa for (unweighted) Delaunay triangulations.

On the second Neumann eigenvalue

I will review results and open problems regarding simplicity of the lowest nonzero Neumann Laplacian eigenvalue on domains, and properties of the corresponding eigenfunctions related to the hot spots conjecture. I'll present a new simplicity result for a family of smooth planar domains, that is based on analysis of mirror couplings. Joint work with K. Burdzy.

Long-Lived Private Information in a Continuous Time Economy

Portfolio Choice, Optimal Consumption, and Utility Gain
Abstract: We study the consumption-investment problem of an agent with a constant relative risk aversion (CRRA) preference function, who possesses private information about the future prospects of a stock. We examine the value of the information to the agent by comparing the utility equivalent with and without the information of the agent. The value of private information to the agent depends linearly on the wealth of agents and decreases with both the propensity to intermediate consumption and risk aversion. Agents with low coefficients of relative risk aversion value private information more highly. Consistent with the empirical literature, the optimal portfolio holdings of informed agents are correlated with expected returns on the risky asset. Highly risk averse informed agents consume a greater fraction of their wealth when they are informed than when they are uninformed, but the opposite is true of agents with low degrees of risk aversion.

Just-Infinite Algebras and an Extension of a Theorem of Herstein

Much is known about simple rings and these results are often
extended to direct sums of simple rings. An alternative approach to extending these results is to loosen the condition simplicity puts on ideals of a ring. Just infinite dimensional algebras are those which, in some sense, contain only few ideals.

My talk will begin with examples and a more precise definition of just infinite, after which I will prove the principal results of my dissertation. The first is a theorem that all just infinite algebras are prime;
the second result is an example of broadening theorems about simple rings to theorems about just infinite rings.

Ultraproducts and applications to ring theory

We will define the notions of ultrafilters and ultraproducts, and describe some of their basic properties. We will also discuss a few applications to ring theory.

Random linear extensions of grids

A grid poset -- or "grid" for short -- is a product of chains. We ask,
what does a random linear extension of a grid look like? In particular,
we show that the average "jump number," i.e., the number of times that
two consecutive elements in a linear extension are incomparable in the
poset, is close to its maximum possible value. The techniques employed
rely on entropy arguments. We mention several interesting questions

A little algebra goes a long way

Applying simple algebraic ideas in combinatorics often leads to new and beautiful theorems. I will present some of these results.

Demazure structure inside Kirillov-Reshetikhin crystals

The conjecturally perfect Kirillov-Reshetikhin (KR)
crystals are known to be isomorphic as classical crystals to
certain Demazure subcrystals of crystal graphs of irreducible
highest weight modules over affine algebras.
Under some assumptions we show that the classical
isomorphism from the Demazure crystal to the KR crystal,
sends zero arrows to zero arrows. This implies that the affine
crystal structure on these KR crystals is unique.

Enumerating (multiplex) juggling sequences

We consider the problem of enumerating periodic $\sigma$-juggling sequences of length $n$ for multiplex juggling, where $\sigma$ is the initial state (or $landing ~ schedule$) of the balls. We first show that this problem is equivalent to choosing 1's in a specified matrix to guarantee certain column and row sums, and then using this matrix, derive a recursion. This work is a generalization of earlier work of Fan Chung and Ron Graham. \\
This is joint work with Ron Graham.

Collinear Points in Permutations and the Finite Plane Kakeya Problem

The talk will cover some work I’ve done recently on a very interesting
question (the size of the largest finite plane Besicovitch set) that connects
finite geometry, additive combinatorics, and old-fashioned counting. There
are a number of nice problems that are wide open, more or less unexamined,
and quite accessible.

Graph computing and drawing

I will survey some useful tools, both internal and external, for computing with graphs. In particular, I will discuss some practical strategies for drawing graphs appropriate for papers and slides, and will offer some demonstrations. I will also survey the topic of graph drawing in general, perhaps with some connection to ideas in computational geometry if time permits.

Global well-posedness in Sobolev space implies global existence for weighted $L^2$ initial data for $L^2$ -critical NLSThe $L^2$-critical defocusing NLS

The $L^2$-critical defocusing NLS initial value problem on $\Bbb{R}^d$ is known to be locally well-posed for initial data in $L^2$. Hamiltonian conservation and the pseudoconformal transformation show that global well-posedness holds for initial data $u_0$ in Sobolev $H^1$ and in the weighted space $(1+|x|) u_0 \in L^2$. For the $d=2$ problem, it is known that global well-posedess also holds for data in $H^s$ and also for data in the weighted space $(1+|x|)^{\sigma} u_0 \in L ^2$ for certain $s$, $\sigma < 1$. The talk will presents a new result: If global well-posedness holds in $H^s$ then global well-posedness and scattering holds in the weighted space with $\sigma = s$.

Eigenvalues and eigenvectors of irregular graphs

The eigenvalues of regular graphs have been well studied. They have
strong connections with the expansion constant (Alon-Milman, Tanner), diameter
(Chung), chromatic and independence number (Hoffman) of a graph.
In this talk, I will discuss the eigenvalues of irregular graphs. One of the first
results of spectral graph theory due to Collatz and Sinogowitz (1957) states
that the spectral radius of a graph is between the average degree and the
maximum degree of a graph with equality iff the graph is regular. When the
graph is irregular, I will show how can we improve these inequalities. I will
conclude with a list of open problems.
This is based on joint work with David Gregory (Queen’s University at
Kingston, Canada) and Vlado Nikiforov (University of Memphis, USA).

The rank of random graphs

The Erdos Renyi random graph G(n,p) is generated by independently
and randomly choosing whether or not to draw an edge between each pair
of points, with p being the probability the edge is included. We consider the
adjacency matrix of this graph, or equivalently a random symmetric matrix
where each entry above the diagonal is independently chosen to be either 1
(with probability p), or 0 (with probability 1-p). We are primarily interested
in the following two questions:
1. Is this matrix likely to be nonsingular?
2. If the matrix is singular, how close will it be to full rank?
I will discuss answers to both of these questions in the range 0.5 ln
n/n<p<1/2, along with conjectured answers for smaller p.
This talk is based on joint work with Professor Van Vu

Minimal polynomials and radii of elements in finite-dimensional power-associative algebras

We begin by revisiting the definition and some of the properties of the minimal polynomial of an element of a finite-dimensional power-associative algebra $\cal A$ over an arbitrary field $\mathbb{F}$. Our main observation is that $p_a$, the minimal polynomial of $a\in\cal A$, may depend not only on
$a$, but also on the underlying algebra. Restricting attention to the case where $\mathbb{F}$ is either $\mathbb{R}$ or $\mathbb{C}$, we proceed to define $r(a)$, the {\it radius} of an element $a$ in $\cal A$, to be the largest root in absolute value of the minimal polynomial of $a$. As it is, $r$ possesses some of the familiar properties of the classical spectral radius. In particular, $r$ is a continuous function on $\cal A$. In the third part of the talk we discuss stability of subnorms acting on subsets of finite-dimensional power-associative algebras. Following a brief survey, we enhance our understanding of the subject with the help of our findings about the radius $r$.
Our main new result states that if $\cal S$, a subset of an
algebra $\cal A$, satisfies certain assumptions, and $f$ is a
continuous subnorm on $\cal S$, then $f$ is stable on $\cal S$ if and only if $f$ majorizes the radius \nolinebreak$r$.

The chromatic number of random graphs

The chromatic number $\chi(G)$ of a graph $G$ is one the most important graph invariants. It is known, due to Bollob\'as as well as Luczak, that for the random graph $G(n,p)$, $\chi(G(n,p)) \sim d/2\ln d$ where $d = np$ is the expected average degree. Similarly, Frieze and Luczak showed that for a random $d$-regular graph that $\chi(G_{n,d}) \sim d/2\ln d$. What can we say in the case where, instead of a $d$-regular graph, we instead look at a random graph with a given degree sequence ${\bf d} = (d_1,\dots,d_n)$ with average degree $d$? In this talk, I look at a recent preprint of Frieze, Krivelevich, and Smyth who show under a condition on ${\bf d}$ that $\chi(G_{n,{\bf d}}) = \Theta(d/\ln d)$. I also look some at the problem of translating this result to a random graph with a given expected degree sequence.

Harvey Friedman - Department of Mathematics, Ohio State University

We give a general overview of progress and expectations in three directions.

1. What kind of mathematical results can we prove only by going beyond the
usual axioms for mathematics (ZFC)? (Some systematic investigations in the
integers). Buzzwords: templates, Boolean Relation Theory, Order Calculus.

2. Is there a relationship between ordinary commonsense naive thinking and
abstract mathematics? (They are unexpectedly close, and in some sense even
equivalent). Buzzwords: interpretation, Concept Calculus.

3. In what sense and to what extent can we achieve certainty, at least in
mathematics? (With the use of existing computer technology, we can, in
various senses). Buzzwords: proof assistants, certificates.

Boundaries of Levi-flat hypersurfaces

In this joint work with Pierre Dolbeault and Giuseppe Tomassini
we consider the problem of characterizing compact real submanifolds of $C^n$ that bound Levi flat hypersurfaces. The problem is well studied in $C^2$ but surprisingly little is known in higher dimension. In this talk I will, in particular, explain the fundamental difference between $n=2$ and higher dimension
showing why the known methods in $C^2$ do not apply.

High-order surface relaxation vs. the Ehrlich-Schwoebel effect in thin-film growth

The surface of an epitaxially growing thin film often exhibits a
mound-like structure with its characteristic lateral size increasing in time. In this talk, we consider two competing mechanisms for such a coarsening process: (1) surface relaxation described by high-order gradients of the surface profile; and (2) the Ehrlich-Schwoebel (ES) effect which is the upper-lower terrace asymmetry in the adatom attachment and detachment to and from atomic steps. We present a theory based on a class of continuum models that are mathematically
gradient-flows of some effective free-energy functionals describing these mechanisms. This theory consists of two parts: (1) variational properties of the energies, such as ground states'' and their large-system-size asymptotics, showing the unboundedness of surface slope and revealing the relation between some of the models;
(2) rigorous bounds for the scaling law of the roughness, the rate of increase of surface slope, and the rate of energy dissipation, all of which characterize the coarsening process. Predictions on scaling laws made by our theory agree well with experiments.

Stochastic modelling methods for gene finding

In this talk, the problem of finding genes from the genome (DNA
sequence) is formulated as a problem in stochastic modelling and
classification. No prior knowledge of biology is assumed and the talk
will be completely self-contained in this respect. A new classification
algorithm, called Mixed Memory Markov Model (4M) algorithm, is presented,
and its significance (probability of generating an incorrect
classification) is analyzed using sound statistical principles. It is
also shown that, on nearly 70 bacterial genomes, the 4M algorithm performs
as well or better than the currently most popular algorithm, known as
Glimmer-2. (But the emphasis of the talk is on the statistical aspects.)

Chip-firing games

A chip-firing game is a game played by $n$ players where each player is alotted a certain set of neighbors and a certain number of tokens, here called
chips. (In math-speak this would be a graph $G=(V,E)$ with $n$ vertices $v \in V$ with nonnegative integer weights, and the edges $e\in E$ determine
which vertices are adjacent.) During the course of each round, starting with player $1$, every player whose chip-stack is greater than his/her number of
neighbors must give a chip to each of them. If a player's chip-stack is too small, play skips over to the next player. Once everyone has played,
player $1$ goes again if possible, and play continues this way.
In this talk, we will discuss several mathematical questions that come up from this model:
\begin{enumerate}
\item For what initial configurations will play eventually stop?
\item Does the order of the players matter?
\item Which initial configurations will appear again and again?
\end{enumerate}
We then consider a new game, called the dollar game as defined by N. Biggs, where an additional player, the bank, is added. The bank takes chips
from the players as above, but can only give all of his/her neighbors chips if play would stop otherwise. However the bank player
is allowed to go into debt and thus can give his/her neighbors chips whenever play would stop. We will discuss how this new game gives
rise to a group structure, and applications. If there is time we will discuss the Matrix-Tree Theorem which lies at the nexus between Algebraic
Combinatorics and Graph Theory.

\vspace{1em}

All are welcome. Even if you do not know what a graph is, all will be explained.

Tricks with spectral sequences

Spectral sequences present a fairly complicated algebraic gadget with which to study some interesting geometric questions. In this talk, the goal is to show how certain methods that depend on the structure of a spectral sequence lead to satisfying results. Examples will abound.

Automorphisms of the semigroup of endomorphisms of free algebras of homogeneous varieties.

Let $F=F(x_1,..x_n)$ be a free algebra of a homogeneous variety of linear algebras freely generated by $X={x_1,...x_n}$ , $End F$ be the semigroup of endomorphisms of $F$ and Aut End $F$ be the group of automorphisms of the semigroup $End F$.
We investigate the structure of the group Aut End $F$ for the variety of Lie algebras, the variety of $m$-nilpotent associative algebras, the variety of commutative algebras over so-called $R_1MF-$domains. These domains contain, in particular, Bezout domain, unique factorization domains. As a consequence, a complete description of the group of automorphisms of the full matrix semigroup of nxn matrices over $R_1MF-$domains !
is obtained.

Stability reversal in non-homogeneous static fluids in high-dimensional spaces

Fluid models have been used as toy-models of event horizons in general relativity and its generalizations to more than four spacetime dimensions. We study here a fluid model for the Gregory-Laflamme instability in black strings. With consider a Newtonian, incompressible, static, axially-symmetric fluid with surface tension, in n dimensions plus one periodic dimension. The fluid configurations are those that minimize the
fluid surface area for fixed volume. Homogeneous fluid configurations are known to be stationary solutions of this functional, and they are stable in a dynamical sense above a critical value of the fluid volume. Below that value Plateau-Rayleigh instabilities occur. We show in this
article that at this critical value of the volume there is a pitchfork bifurcation point. We prove that there are infinitely many other pitchfork bifurcation points at smaller values of the fluid volume. Each bifurcation solution represents a non-homogeneous static fluid configuration and its
stability depends on the space dimension. By stability we mean in the sense of minimum of the above functional. We show that the non-homogeneous configurations are all unstable if n less or equal 10, and they all become stable if n bigger or equal 11. This stability inversion for high space dimensions could be of interest in gravitational theories in more than four dimensions and in string theory.

Chiral equivariant cohomology

I will discuss a new cohomology theory that extends $H$. Cartan's cohomology theory of $G^*$ algebras. The latter is an algebraic abstraction of the topological equivariant cohomology theory for $G-$spaces, where $G$ is a compact Lie group. Cartan's theory, discovered in the 50s and further developed by others in the 90s, gave a de Rham model for the topological equivariant cohomology, the same way ordinary de Rham theory does for singular cohomology in a geometric setting. The chiral equivariant cohomology takes values in a vertex algebra and includes Cartan's cohomology as a subalgebra. I will give a brief introduction to vertex algebras, and then discuss the construction of the new cohomology and some of the basic results and examples. This is a
joint work with Bong Lian and Bailin Song.

Counting Descents with Prescribed Tops and Bottoms

Given sets $X$ and $Y$ of positive integers and a permutation
$\sigma = \sigma_1 \sigma_2 \cdots \sigma_n$, an $X,Y$-\emph{descent}
of $\sigma$ is a descent pair $\sigma_i > \sigma_{i+1}$ whose top''
$\sigma_i$ is in $X$ and whose bottom'' $\sigma_{i+1}$ is in $Y$. We give two formulas for the number $P_{n,s}^{X,Y}$ of $\sigma \in S_n$ with
$s$ $X,Y$-descents. $P_{n,s}^{X,Y}$ is also shown to be a hit number of a certain Ferrers board. This work generalizes results of Kitaev and Remmel on counting descent pairs whose top (or bottom) is equal to 0 mod $k$.
(This is joint work with Jeff Remmel.)

Ergodic properties of crystallization processes

We are interested in a birth-and-growth process where germs are born according to a Poisson point process with invariant under translation (in space) intensity measure. The germs can be born in free space and then start growing until occupying the available space. In this general framework, the crystallization process can be characterized by the random field giving for a point in the space state the first time this point is reached by a crystal. We prove under general conditions on speed growth and geometrical shape of free crystals that this random field is mixing
in the sense of ergodic theory and obtain estimates for the absolute regularity coefficient.

Fractal strings, hearing the sound of a drum, and the Riemann hypothesis

What do compact curves of infinite length, continuous, nowhere
differentiable functions, and uncountable sets of measure zero have in
common? It's not that they don't exist. The Koch Curve, the Weierstrass
Function, and the Cantor Middle Thirds Set are all fractal sets and
respective examples of each of the above. What do they have to do with
Mark Kac's famous question: Can one hear the shape of a drum?"
This talk is about how this question, Spectral Geometry, and
Fractal sets come together in the work of Michel Lapidus and others to
formulate a statement equivalent to the Riemann hypothesis. The talk
will go roughly as follows: \\

1. Pictures of fractals and a definition of non-integral, fractal, dimension.

2. A statement of the Riemann hypothesis.

3. One can hear the shape of a fractal string" if and only if the
Riemann hypothesis holds. \\

The talk will involve mostly proofs by pictures and should be
completely accessible.

Heat Eisenstein series and applications

I will define the heat Eisenstein series and sketch some of their theory, as developed by Jorgenson and Lang in the 90s and early 00s. Then, I will describe their role in an ongoing project to generalize theta relations and spectral zeta functions to arithmetic quotients of real Lie groups in higher rank. Finally, I will state some of my results on exact fundamental domains and explain how they are used or are expected to be used in this program.

Pseudo-random graphs: properties and applications

An $(n,d,\lambda)$-graph is a d-regular graph on $n$ vertices so that the absolute value of each eigenvalue of its adjacency matrix, besides the largest one, is at most $\lambda$. I will survey some of the remarkable pseudo-random properties of such graphs in which $\lambda$ is much smaller than $d$, describe various constructions, and present several applications of these graphs in the solution of problems in Extremal Combinatorics, Geometry and and Complexity.

Lie algebras with one defining relation

In the talk we consider Lie algebras with one defining relation.
Starting with an analog of Freiheitssatz (Shirshov's theorem), we give an example of a Lie algebra over a field of prime characteristic with cohomological dimension one which is not a free Lie algebra (this gives a counterexample to a hypothesis that the analog of Stallings-Swan theorem takes place for Lie algebras; the problem in the case of field of zero
characteristic is still open, we formulate some related conjectures).
For a finitely generated free Lie algebra L we construct an example of two elements $u$ and $v$ of $L$ such that the factor algebras $L/(u)$ and $L/(v)$ are
isomorphic, where $(u)$ and $(v)$ are ideals of L generated by $u$ and $v$, respectively, but there is no automorphism $f$ of $L$ such that $f(u)=v$.
We consider also the situation where a Lie algebra with one defining relation is a free Lie algebra.

High throughput docking -- How can mathematics help?

In this presentation, I will briefly introduce the concept
of
high throughput docking (HTD) and how it is used in pharmaceutical and
biotech industries. A successful HTD requires not only accurate
description of molecular interactions, but also efficient numerical
algorithms. A couple of computational methods that we have developed
for
docking will be discussed. They demonstrate how this practical problem
can greatly benefit from relatively simple mathematical methods. I
will
also discuss some challenges ahead for further improving HTD and
general
high throughput virtual screening.

Stochastic modeling in single molecule biophysics

Recent advances in nano-technology allow scientists for the first time to follow a biochemical process on a single molecule basis. These advances also raise many challenging data-analysis problems and call for a sophisticated statistical modeling and inference effort.
First, by zooming in on single molecules, recent single-molecule experiments revealed that many classical models derived from oversimplified assumptions are no longer valid. Second, the stochastic nature of the experimental data and the presence of latent processes much complicate the inference. In this talk we will use the modeling of subdiffusion phenomenon in enzymatic reaction to illustrate the statistics and probability challenges in single-molecule biophysics.

This talk is based on joint work with Sunney Xie.

\bigskip
\bf
\it
Jointly sponsored by the Department of Mathematics and the Division
of Biostatistics and Bioinformatics

The geometry of schemes (and other tall tales!)

This talk will motivate the use of schemes in understanding varieties. Schemes are notorious for their lack of transparency. In order to motivate the practicality of schemes, I will discuss the basics of varieties: specifically coordinate rings (the ring you get from a variety) and the various properties that they have.

After that, schemes will be (roughly) defined and I will come to the heart of the talk---examples of what happens when we relax various conditions on coordinate rings. Very strange phenomena come up: we can get extremely local information, we can get points on varieties with tangent vectors, we can describe some strange varieties over the reals, etc. If time permits, I will talk about schemes as they pertain to number theory and give an example or two.

This talk is meant to introduce schemes in a geometric sense. Most importantly, I want to brush aside the technical baggage that comes with high-end commutative algebra, sheaves, etc. in order to get at the heart of the geometry. Those with a penchant for pictures will be none too displeased.

Universality questions in random matrix theory

This talk is addressed to a wide audience.

One of the main features of Random Matrix Theory (RMT) is that it provides
accurate models for correlated quantities that describe various complex
systems arising in a broad variety of problems in physics, pure and
applied mathematics, and in other branches of knowledge.
We will start by presenting various examples of such quantities.

A mathematical reason for a wide applicability of RMT is that there should
be certain versions of the Central Limit Theorem but now for certain
classes of correlated random variables.
In particular, the limiting behavior of the eigenvalues of a large random
matrix should be independent of the details of the distribution of the
matrix elements. This loose statement is known as the Universality Conjecture.

We then introduce the three classical types of invariant random matrix
ensembles. For the technically simplest case of the so-called unitary
ensembles, we explain how all the statistical quantities of interest can
be expressed in terms of orthogonal polynomials on the real line.

The Universality Conjecture in the bulk of the spectrum for the unitary
ensembles was established by Deift-Kriecherbauer-McLaughlin-Venakides-Zhou
in 1999. The other two classes of invariant ensembles, orthogonal and
symplectic, are more difficult to analyze. In the end of the talk, we will
describe our recent work with Deift where we prove the Universality
Conjecture both in the bulk and at the edge of the spectrum for the
remaining two classes of invariant ensembles in great generality.

Non-homogeneous Calderon-Zygmund operators, capacities with signed kernels, and Geometric Measure Theory: some solved and unsolved problems

There was a huge progress in understanding the geometric nature of removable singularities for some subtle classes of analytic and harmonic functions. This progress culminated in the solution of several problems of Vitushkin from the 50's. A 100 year old problem of Painlevé asking for geometric" description of removable sets for bounded analytic functions was solved by Xavier Tolsa. The progress was based on essentially several ingredients including: 1. very non-trivial connection between Geometric Measure Theory and Calderon-Zygmund operators and 2. a theory of Calderon-Zygmund operators in bad environment". The talk will present this story, which unites the methods of Complex Analysis, Harmonic Analysis, Geometric Measure Theory and Probability. Some unsolved problems will be listed.

Principal ideals of the exterior algebra

Let $F$ be a field and $n>0$ a positive integer.
Let $A = A_0 + A_1 + ...+ A_n$ be the exterior algebra of
dimension $2^n$ over $F$ with its natural grading.

Then a homogenous element w in $A_s$ generates a homogeneous
principal ideal $wA$. What is the maximum value of
$\dim_F(wA \cap A_r)$ for given $s,r,n$ as $w$ varies in $A_s$?

We state a conjecture. The (most interesting?) case
$(s,r,n) = (3,6,9)$ (where the max is 84) is directly
related to the exceptional Lie algebra $E_8$.

(By definition, A is generated by elements $e_1,...,e_n$
that satisfy $(1) e_i^2 =0$ and $(2) e_ie_j +e_je_i=0$ for
$1<=i,j<= n$. The generators $e_i$ all lie in $A_1$.
By definition, $A_s$ is the $F-span$ of all
$s-fold$ products $e_{j_1} ... e_{j_s}.)$

Classical dynamic density functional theory: Correlated Brownian motion in colloidal systems

Classical density functional theory (DFT) has been proven to be a
powerful mathematical tool to describe the equilibrium structure and
phase behavior of correlated many-body systems (e.g., dense colloidal
or
biological fluids) in bulk or under the action of external potentials.
Using the equilibrium functional a dynamic DFT can be constructed
which
accurately reproduces the strongly inhomogeneous steady-state or even
time-dependent structure of systems in non-equilibrium. This talk aims
at a pedagogical introduction exemplified by simple colloidal systems
out of equilibrium, such as colloidal sedimentation,
diffusion-controlled reactions, and driven polymer solutions.

Is infinity plus one prime? (Or: Factorization in the omnific integers)

In the 1970s, John Horton Conway introduced the {\sl surreal numbers}, a
number system that contains not only the real numbers, but contains many
infinite numbers as well. Among the elements of this field, one finds $\omega$
(the first infinite ordinal), $\omega + 1$ (infinity plus one''), $\omega - 1$
(`infinity minus one''), $\frac{1}{\omega}$ (an infinitesimal) and as well as
numbers like $\frac{\omega^{\omega}}{\sqrt{\omega^2 - \pi}} + \omega^{\frac{1}{\omega}}$. Within the surreal numbers, there is also a
natural generalization of the integers, called the {\sl omnific integers},
denoted $\mathbb{O}\mathbb{Z}$. The first three numbers listed above are
omnific integers, while the last two are not. With this new notion of integer,
one can revisit many of the classical questions of number theory. Most
immediately, what are the prime numbers? Can every omnific integer be factored
uniquely into omnnific prime numbers? In this talk, after introducing the
surreal numbers, we'll explore some questions about primes in the land of
$\mathbb{O}\mathbb{Z}$.

Combinatorial paella

Q:
What do ring homomorphisms, homogeneous and elementary symmetric functions, brick tabloids, and involutions have in common?

A:
If you mash them all together and stir them around, you can get some nice results for generating functions for various permutation statistics.

In this talk, we'll define all the stuff we'll need to use. Then we will prove some famous results, as well as mention some not-so-famous results. It will be an example of how combinatorial objects can take a problem out of its current context (e.g. the ring of symmetric functions, or representation theory, or any number of other areas) to prove theorems in an accessible way.

Shintani zeta-functions and Gross-Stark units for totally real fields

Let $F$ be a totally real number field and let
$p$ be a finite prime of $F$, such that $p$ splits
completely in the finite abelian extension $H$ of $F$.
Stark has proposed a conjecture stating the existence
of a $p-unit$ in $F$ with absolute values at the places
above $p$ specified in terms of the values at zero of
the partial zeta functions associated to $H/F$. Gross
proposed a refinement of Stark's conjecture which
gives a conjectural formula for the image of Stark's
unit in $F_p^\times/ \widehat E$, where $F_p$ denotes the completion
of $F$ at $p$ and $\widehat E$ denotes the topological closure of the
group of totally positive units of $F$. We propose a
further refinement of Gross' conjecture by proposing a
conjectural formula for the exact value of Stark's
unit in $F_p^\times$.

Which primes are the sum of two rational cubes

In the 19th century, Sylvester conjectured that every prime
which is congruent to 4, 7, or 8 modulo 9 can be expressed as the sum
of two rational cubes. In the 1950s, Selmer restated the conjecture
as part of a more general study of rational points on elliptic curves.
Later, the conjecture was seen to be a very special case of the conjecture of Birch and Swinnerton-Dyer. However, an unconditional proof remained elusive until Elkies announced a proof in the early 1990s for the case where $p$ is 4 or 7 modulo 9. We will describe how special values of modular forms allow one to attack Sylvester's conjecture and other similar problems concerning elliptic curves. We will conclude with current research, joint with John Voight, which uses these methods
to study the remaining open case, where $p$ is 8 modulo 9.

High order numerical methods for elliptic interface problems

Elliptic or parabolic equations with discontinuous coefficients are
seen
in many disciplines such as
electromagnetic, fluid dynamics and biophysics. For the accurate
numerical
solutions of these equations,
it is important to enforce the known jumps in the solution and/or its
gradient on the internal interfaces.
Failing to do this would result in solutions of low accuracy and
convergence of low order, or even divergence
of the computation. In this talk I will briefly review the elliptic
interface problems and their numerical
algorithms. I will then introduce a novel matched interface and
boundary
(MIB) method and its applications
to the solution of the electrostatic potential of macromolecules.

Regular subgraphs of random graphs

The $k$-core of a graph is the largest subgraph of minimum degree at least
$k$. In this talk, we will be discussing the appearance of $k$-regular
subgraphs of random graphs in the $Erdos-R \acute e nyi$ model of random graphs. Pittel, Spencer and Wormald
determined very precisely when the $k$-core of a random graph appears.
The $k$-core of a graph is revealed by performing a greedy
vertex-deletion algorithm, whereas no simple algorithm exists for
finding $k$-regular subgraphs when $k > 2$. I will present a
combination of algebraic, probabilistic and structural techniques
which pin down the point of appearance of a $k$-regular subgraph in
a random graph with edge-probability $p$ to a window for $p$ of
width roughly $2/n$. To introduce these techniques, a more
general discussion of degree constrained subgraphs will be given,
including a proof of a recent conjecture on generalized factors of
graphs. Recently published empirical evidence using statistical
physical methods supports our conjecture of a sharp threshold for
the appearance of $k$-regular subgraphs of random graphs, while this
conjecture, nevertheless, remains open.

Autonomous stochastic resonance produces epidemic oscillations of fluctuating size

It is known from simulations that stochastic epidemic models such as SIR, SIRS produce sustained oscillations, whereas the corresponding deterministic models produce damped oscillations. We use multiscale analysis to identify a parameter range in which stochastic SIR is approximated by a stochastic combination of sinusoids, where the coefficient processes run on a slow time scale. Ornstein-Uhlenbeck excursions form an envelope of the epidemic oscillations.

To modularity and beyond

In the past decade or so, several big conjectures in number theory have been proven, starting with Fermat's Last Theorem and including the recent proof of the so-called Sato-Tate conjecture.
All these advances involve proving that something is modular. I will try to explain what some of these conjectures/theorems say and what it means for something to be modular.

Linear algebra in tropical semirings

Tropical algebras are semirings with operations $a\oplus b=\max(a,b)$ (or $a\oplus b=\min(a,b)$) and $a\otimes b=a+b$. Tropical
algebra is usually approached as a subtopic of algebraic geometry. But
what if we take traditional linear algebraic problems and ask them
using tropical operations instead? We'll find that these problems are
frequently solvable, although not always in the same ways as they're
solved in conventional algebras. These problems have surprising
relevance to industrial systems control and operations research
problems.

Continued Fractions, Special Values of the Double Sine Function, and Stark Units over Real Quadratic Fields

In this talk we will present new results showing that every existent Stark unit over a real quadratic field may be expressed as a product of double sine function values in a compact and canonical form using a continued fraction approach due to Zagier and Hayes.

Negative-definite functions on the dual of a hypergroup

Hypergroups are locally compact spaces for which the space of bounded measures
can be made a Banach algebra by introducing an axiomatically determined
convolution. Prominent constructions of such convolutions over $\mathbb {Z}_+$ and $\mathbb {R}_+$ are
performed via polynomials or special functions respectively.
In order to establish canonical representations of convolution semigroups of
probability measures on a commutative hypergroup $K$ (in the sense of Schoenberg
correspondence and Levy-Khintchine decomposition) one needs to study
negative-definite functions on the dual $K$ $\hat{}$ of $K$ which in general is not a hypergroup.
With appropriate definition of negative-definiteness on $K$ $\hat{}$ some harmonic
analysis can be developed, and for large classes of (Euclidean) hypergroups
the structure of negative-definite functions and of the corresponding
convolution semigroups can be illuminated.

Linear cellular automata, sofic groups and stable finiteness of group rings

In this talk I'll first recall the notion of a "cellular automaton over a group" (also presenting celebrated
Conway's "Game of Life") and then consider the linear version
of the so-called "Garden of Eden Theorem" and Gottshalk's
"Surjunctivity problem" which arise from symbolic dynamics.
After recalling the notion of "amenability" for groups, I'll discuss the
Gromov-Weiss notion of a "sofic group".
Finally, I'll discuss Kaplansky's conjectures on the non-existence of
idempotents and on stable finiteness of group rings and relate these
problems to the theory of linear cellular automata.

Error estimation and adaptive computation for elliptic problems with randomly perturbed coefficients

We develop and analyze an efficient numerical method for computing
the response of the solution of an elliptic problem with randomly
perturbed coefficients. We use a variational analysis based on the
operator to deal with the perturbations in data. To deal with
perturbations in the diffusion coefficient, we construct a piecewise
constant approximation to the random perturbation then use domain
decomposition to decompose the problem into sub-problems on which
the diffusion coefficient is constant. To compute local solutions of
the sub-problems, we use the infinite series for the inverse of a
perturbation of an invertible matrix to devise a fast way to compute
the effects of variation in the parameter. Finally, we derive a
posteriori error estimates that take into account all the sources
of error and derive a new adaptive algorithm that provides a
quantitative way to distribute computational resources between all
of the sources.

Fragmentation and coalescence of conditioned Galton-Watson forests

Given a Galton-Watson forest of $k$ trees with some offspring
distribution conditioned to have n total vertices, suppose that an edge is
deleted uniformly at random from this forest. For what offspring
distributions is it true that the resulting forest is distributed like a
Galton-Watson forest of $k+1$ trees with the same offspring distribution
conditioned to have n vertices?

This question is related to recent work on fragmentation of partitions of
the set $[n]= {1,2,...,n}$. After reviewing the
known results about partitions, we show how to use generating functions to
pass from partitions to forests and determine the offspring distributions
for which this fragmentation model works. We will also discuss certain
interesting combinatorial interpretations of some of the offspring
distributions. Finally, we describe how to go in the reverse direction, and
determine how trees in a conditioned Galton-Watson forest should coalesce in
order to preserve the conditioned structure.

This is joint work with N. Berestycki and J. Pitman.

The projective plane: Or, how to make ends meet

We discuss the problem of how to find the number of simultaneous solutions to two polynomial equations in two variables. This amounts to calculating the number of intersection points of two curves in the plane. We discuss why this question has a much nicer answer if one is wily enough to replace the usual $(x,y)$-plane with something else.

What is a non-commutative space?

An important aspect of quantum mechanics is that it introduces non-commutative phenomena into physics: Heisenberg's uncertainty principle reflects the fact of mathematical life that matrices don't necessarily commute. Over the last 80 years the study of non-commutative structures has become an increasingly popular activity in many branches of mathematics. The two examples I will talk about are non-commutative topology and non-commutative measure theory. My main goal will be to explain what people mean when they use these words and why the terminology is justified. I hope this will be interesting for geometers and analysts alike. The main ingredients in non-commutative geometry are non-commutative rings, so if you're an algebraist, maybe you'll find something to like as well. Time permitting I'll describe some examples and theorems relating to the classification of non-commutative measure spaces (a.k.a. von Neumann algebras).

L-functions for multi-quadratic extensions of number fields at s=-1 and annihilation of the tame kernel

Brumer's conjecture uses abelian L-functions at $s=0$ to produce group-ring elements which should annihilate class groups of number fields.
We consider the analogous conjecture at $s=-1$, and give evidence in the setting of a number field obtained as the composite of several relative quadratic extensions of the base field.

Amoebae and algae of real algebraic curves

Amoebae and algae are two natural objects that can be associated to a
projective algebraic variety $V$. The amoeba is the logarithmic
projection of $V$ to ${\bf R}^n$ while the alga is the complementary
projection to the argument torus $(S^1)^n$.

It turns out that that the topology of a plane real algebraic curve
is intrinsically related to the geometry of its amoeba and alga.
The talk will survey known instances of this relation.

Random Tomography, Structural Biology, Shape Diffusions

Single particle electron microscopy is a powerful method that biophysicists employ to learn
about the structure of biological macromolecules. In contrast to the more traditional crystallographic
methods, this method images "unconstrained" particles, thus posing a variety of statistical problems. We
formulate and study such a problem, one that is essentially of a random tomographic nature. Although
unidentifiable (ill-posed), this problem can be seen to be amenable to a statistical solution, once strict
parametric assumptions are imposed.It can also be seen to present challenges from a data analysis point of
view (e.g. uncertainty estimation and presentation). In addition, motivated by the "physics" involved in
the data-collection process, we define and explore a new type of diffusions in D.G. Kendall's shape space.
These diffusions arise as the stochastic evolution of the orbits (under a group action) of a projected
motion.

Harmonic analysis of finite lamplighter random walks

Recently, a lot of papers have been devoted to the analysis of lamplighter random walks, in particular when the underlying graph is the infinite path $\mathbb{Z}$. In the present talk, we develop a spectral analysis for lamplighter random walks on finite graphs. In the general case, we use the $C_2$-symmetry to reduce the spectral computations to a series of eigenvalue problems on the underlying graph.
In the case the graph has a transitive isometry group $G$, we also describe the spectral analysis in terms of the representation theory of the wreath product $C_2\wr G$. We apply our theory to the lamplighter random walks on the complete graph and on the discrete circle. These examples were already studied by Haggstrom and Jonasson by probabilistic methods.

Biomedical image processing and geometric modeling

This talk includes three parts. In Part I, I will briefly
introduce a number of imaging techniques for 3D structure
reconstructions at different scales (including tissue, cellular,
subnanometer, and atomic levels). In Part II, a couple of image
processing algorithms will be presented, in order to extract features
of interest for geometric modeling. Part III is mesh generation, an
essential step for finite element method. I will introduce two
approaches for mesh generation: Delaunay triangulation and
Octree-based method.

Representable graphs

A graph $G=(V,E)$ is representable if there exists a word $W$ over
the alphabet $V$ such that letters $x$ and $y$ alternate in $W$ if
and only if $(x,y) \in E$ for each $x \neq y$. If $W$ is $k$-uniform
(each letter of $W$ occurs exactly $k$ times in it) then $G$ is
called $k$-representable.

The notion of representable graphs appears in study, by Seif and
later by Kitaev and Seif, of the Perkins semigroup which has played
central role in semigroup theory since 1960. Thus, the objects in
question being on border between combinatorics on words and graph
theory, come from needs in algebra.

In my talk, I will discuss some properties of representable graphs.
In particular, I will give examples of non-representable graphs and
will discuss certain wide classes of graphs that are 2- and
3-representable. I will conclude with a number of open problems in
this direction of research.

This is a joint work with Artem Pyatkin.

Unimodality and real zeros of P-Eulerian polynomials

The $P-$Eulerian polynomial $W_P(x)$ of a finite naturally
labelled poset $P$ is the generating function for linear extensions of $P$ by
number of descents. We will discuss some recent work related to the
question of whether the coefficients of $W_P(x)$ are unimodal and
whether $W_P(x)$ has real zeros. In particular we will explain the
beautiful work of Branden on sign-graded posets.

Large deviations for parabolic Anderson and other random media models

Many random media models involve random variables attached to sites in a state space
such as $Z^d$ or $R^d$. In these models, the sums of values of these random variables
(or integrals) along paths through the state space are are of great interest. In many cases,
the supremum or infimum of these sums are physically relevant. Examples of this are
first passage percolation and the parabolic Anderson model.
The growth of these extrema are generally linear in the path length and satisfy a law of
large numbers. In this talk we examine deviations above the mean and below the mean for
a variety of models and show they are generally very unbalanced.

A primer in analytic number theory

This talk is an introduction to analytic number theory via two very classical results. We will start with a discussion of the Riemann zeta function, including a quick proof of the functional equation and some discussion of the location of the zeros. Then we will go through Dirichlet's proof that there are infinitely many primes congruent to a modulo $q$ when $(a, q) = 1$.

To keep the talk at a survey level, some results will be stated without proof. (Translation: I'm going to avoid long, techincal calculations as much as possible.)

Nonvanishing of the Central Value of Certain L-functions

Based on the recent progress in the Langlands functoriality
for classical groups, we establish the relation between the central
value of the Rankin-Selberg L-functions for GL(n) x GL(m) and certain
periods of automorphic forms on classical groups. Such a relation is
a natural generalization of Gross-Prasad conjecture made about 10 years ago. The lecture is based on the joint work with Ginzburg and Rallis.

On the Langlands functoriality for classical groups

The Langlands functoriality conjecture is one of the basic problems in the theory of automorphic forms and representations. In the past four decades, there was essentially no progress made until a few years ago when the Langlands functoriality was established for generic cuspidal automorphic representations from classical groups
to general linear groups in a series of papers by Cogdell$-$Kim$-$Piatetski$-$Shapiro$-$Shahidi, Ginzburg$-$Rallis$-$Soudry,
and Jiang-Soudry. Many important applications were found afterwards. The problem of establishing the theory for general cuspidal automorphic representations remains a big open problem. In this talk, I will report on some recent progress on this problem, based on my work and my work with Soudry

Measuring degree of controllability of a linear dynamical system

A linear time-invariant dynamical system is
controllable if its trajectory can be adjusted
to pass through any pair of points by the proper selection of an
input.
Controllability can
be equivalently characterized as a rank
problem and therefore cannot be verified
reliably numerically in finite precision.
To measure the degree of controllability of a system
the <em>distance to uncontrollability</em> is introduced as the
spectral or
Frobenius norm of the
smallest perturbation yielding an uncontrollable system.
For a first order system we present a polynomial time
algorithm to find the nearest uncontrollable system
that improves the computational costs of the previous techniques.
The algorithm locates the global
minimum of a singular value optimization problem
equivalent to the distance to uncontrollability.
In the second part for higher-order systems we derive a singular-value
characterization and exploit
this characterization for the computation of the higher-order distance
to
+uncontrollability to low
precision.

Lifting of characters over metaplectic covers of ${\tensy SL(n)}$ over a ${\tensy p}$-adic field

We generalize the approach of Adams by obtaining a character fornula relating the irreducible characters of $PGL(n)$ with those of certain covers of $SL(n)$. We then study the lifting of functions between covers of $SL(n)$ and $PGL(n)$. We use orbital integrals to obtain a formula for the lifting of characters as a dual to the formula relating the characters. This is based on the approach of Kazhdan and Flicker.

The cyclic Sieving phenomenon: A midwest tradition

In work in the mid-90's, John Stembridge noted that several combinatorial statistic generating functions carried enumerative information when the variable was set to $-1$. He dubbed this the "$q=-1$ phenomenon. In joint work with Vic Reiner and Dennis Stanton, we have noted that Stembridge's phenomenon generalizes to when $q$ is an appropriate root of unity. We called this the "cyclic sieving phenomenon." I will describe several instances of this phenomenon and some open problems.

Existence and uniqueness of the measure of maximal entropy for the Teichmuller flow on the moduli space of abelian differentials \\ \small [joint work with B.M. Gurevich]

The moduli space of abelian differentials carries a natural Lebesgue measure class, and, by the Theorem of H.Masur and W.Veech, the
Teichmueller flow on the moduli space of abelian differentials
preserves a finite ergodic measure in the Lebesgue measure
class. The entropy of the flow with respect to the absolutely
continuous measure has been computed by Veech in 1986.

The main result of this talk, obtained by B.M. Gurevich and the
speaker, is that the absolutely continuous measure is the
unique measure of maximal entropy for the Teichmueller flow.

The first step of the proof is an observation that the absolutely
continuous measure has the Margulis property of uniform expansion
on unstable leaves. After that, the argument proceeds in Veech's
space of zippered rectangles. The flow is represented as a
symbolic flow over a countable topological Bernoulli chain
and with a Hoelder roof function depending only on the future.
Following the method of Gurevich, the flow is then approximated
by a sequence of flows whose suspension functions depend on only
one coordinate in the sequence space. For these, conditions for
existence and uniqueness of the measure of maximal entropy are
known by theorems of Gurevich and Savchenko. Since the roof function
of our initial flow is Hoelder, the approximation is rapid enough
and yields maximality of entropy for the smooth measure as well
as the uniqueness of the measure of maximal entropy.

Reformulation of the resistive MHD system for ensuring discrete preservation of constraints

We investigate the system of partial differential equations used in
resistive magnetohydrodynamic modeling of fusion plasmas. This system
couples the Euler and Maxwell equations for evolution of a charged
fluid
in an electromagnetic field, hence the magnetic field in the resulting
PDE system must evolve on a divergence-free constraint manifold. As
traditional numerical solution approaches often violate these
constraints, we investigate a reformulation of the resistive MHD
system
to allow for accurate evolution of the continuum-level equations,
while
simultaneously ensuring that the solution satisfies the solenoidal
constraint.

Convergence of excursion measures of one-dimensional diffusion processes with an exit boundary

Convergence of excursion measures is discussed by means of time change of the Brownian excursion measure. As an application an invariance principle of meanders of positive recurrent diffusion process is obtained. (Based on a joint work with P. J. Fitzsimmons.)

String topology and moduli spaces of Riemann surfaces

String topology (defined by Chas and Sullivan) is the study of
the topology of the space of loops (or strings) in a manifold. Chas and
Sullivan's work, as well as recent work by Cohen, Jones, Godin, and others
focuses on defining various algebraic operations on the space of loops (or
its homology). One can phrase many of these constructions in the language
of field theories used by physicists (though our approach will be purely
mathematical). I'll give an introduction to these sort of field theories
from the point of view of algebraic topology, and explain how various
flavors of string topology fit into this framework.

Moduli of metaplectic bundles on curves and Theta-sheaves

We give a geometric analog of the Weil representation of the metaplectic
group, placing it in the framework of the geometric Langlands program.
For a smooth projective curve $X$ we introduce an algebraic stack
Bun$\backsim$G of metaplectic bundles on $X$. We give a
tannakian description of the Langlands dual to the metaplectic group.
Namely, we introduce a geometric version Sph of the (nonramified)
Hecke algebra of the metaplectic group and describe it as a tensor
category. The tensor category Sph acts on the derived category
D(Bun$\backsim$G) by Hecke operators.
Further, we construct a perverse sheaf on Bun$\backsim$G corresponding
to the Weil representation and show that it is a Hecke eigensheaf with
respect to Sph.

Langlands correspondence for loop groups

The classical Langlands correspondence relates representations
of a reductive algebraic group over a local non-archimedian field $F$ and
representations of the Galois group of $F$. If we replace $F$ by the field
$C((t))$ of complex Laurent power series, then the corresponding group
becomes the (formal) loop group. It is natural to ask: is there an
analogue of the Langlands correspondence in this case? It turns out that
the answer is affirmative, and there is an interesting theory which may be
viewed as both "geometrization" and "categorification" of the classical
theory. I will explain the general set-up for this new theory and give
some examples using representations of affine Kac-Moody algebras.

On the class of the differential-functional equations

The central topic of the talk is the class of Markov processes associated with the random walk on the group of the affine
transformations of the real line. The corresponding generators L have a form of the functional-differential operators with
the linearly transformed argument. These or similar operators were studied by Poincare, Birkhoff, Kato and other classics from the pure analytical point of view. We will present the complete analysis of the bounded L-harmonic functions (i.e. Martin
boundary of our Markov processes).

Cryptography and expander graphs

This talk will explain one of the basic building blocks of cryptography, cryptographic hash functions, and
relate them to another beautiful mathematical object: expander graphs.

The Symmetry Group of Differential Equations

While he was studying partial differential equations, Sophus Lie came up with the idea of trying to solve them by using their symmetry group. His idea was to apply Galois Theory to differential equations instead of polynomials. Lie's key observation was that these symmetry groups are locally determined by their Lie algebras.

Normally Lie groups of differential equations are only locally defined , i.e. they are only defined in a neighborhood of the identity element. However if we enlarge the manifold where the group is acting we can find a globally defined group action whose restriction to the original manifold is the original action.

In this talk we will calculate the symmetry group of the line equation $y''=0$ and see that, despite the simplicity of this equation, the symmetry group is not globally defined! However, the action can be enlarged to a well defined action on $RP^2$. We will do the same with Maxwell's equations obtaining, in this way, a conformal model of the universe where the symmetry group of Maxwell's equations is well defined.

On the non-vanishing of global theta lifts from even orthogonal groups

We will discuss the non-vanishing problem of global theta
lifts from even orthogonal groups to symplectic groups, especially
focusing on the first occurrence conjecture of the global theta lifts.

Surfaces in finite covers of 3-manifolds: the Virtual Haken Conjecture

As with many areas of topology and geometry, a starting point in the study of 3-manifolds is to try to understand codimension one objects in them, namely embedded surfaces. A particularly useful class of surfaces are the "incompressible" ones which are topologically essential; a 3-manifold containing such a surface is called a Haken manifold. There are many 3-manifolds which are not Haken, but if we ask about immersed, rather than embedded, surfaces the situation becomes much more mysterious. A closely related question is this: Suppose M is a 3-manifold with infinite fundamental group, does M have a finite cover which is Haken? The Virtual Haken Conjecture posits that the answer to this question is yes.

This talk will survey some recent results in this area, focusing on my work with (variously) William Thurston, Dylan Thurston, and Frank Calegari. From the point of view of Thurston's Geometrization Conjecture, this is really a question about hyperbolic 3-manifolds, that is, lattices in PSL(2, C). This opens the door to a rich array of tools that might seem quite surprising in light of the purely topological description of the problem above. Indeed, some unlikely-sounding terms that I will probably mention in my talk are "the Classification of Finite Simple Groups" and "the Langlands Conjecture", as well as such topological oddities as "random 3-manifolds"!

Density formula of the occupation time of one-dimensional diffusion processes

For the density of the occupation time of general one-dimensional diffusion processes, continuity and its asymptotic behaviors at extremal points are studied. \\
(Based on a joint work with S. Watanabe and Y. Yano.)

On the periods of automorphic forms on special orthogonal groups and the Gross-Prasad conjecture

A period of an automorphic form on a reductive group $G$ over anumber fieldis defined by its integral over a subgroup $H$ of $G$. Suchperiods are often related to special values of automorphic $L$-functions.In this talk, we present a conjecture in the case of special orthogonalgroups, which can be regarded as a refinement of the global Gross-Prasadconjecture about the restriction of automorphic representations of $SO(n+1$) to $SO(n)$. If time permits, we also discuss a relation of ourconjecture to Arthur's conjecture on the multiplicity of representationsin the space of automorphic forms. This is a joint work with Tamotsu Ikeda.

Freeman Dyson's challenge for the future: The mock $\tensy \theta$ functions

Ramanujan's legacy to mathematics is well documented with connections to some of the deepest subjects in modern number theory: Deligne's proof of the Weil Conjectures, the proof of Fermat's Last Theorem, the birth of probabilistic number theory, the introduction of the "circle method" among others. Although most of Ramanujan's mathematics is now well understood, one baffling enigma remained. In his last letter to Hardy (written on his death bed), Ramanujan gave 17 examples of functions he referred to as "mock $\theta$ functions". Over the next fifty years, ad hoc works by many number theorists (such as Andrews, Atkin, Cohen, Dyson, Selberg, Swinnerton-Dyer, Watson...) clearly pointed to the importance of these strange functions. Their work motivated Freeman Dyson to proclaim:

"Mock $\theta$-functions give us tantalizing hints of a grand synthesis still to be discovered. Somehow it should be possible to build them into a coherent group-theoretical structure ... This remains a challenge for the future." -- Freeman Dyson, 1987

Over the last year, Kathrin Bringmann and I have written a series of three papers on this enigma. Extending recent work of Zwegers, we solve Dyson's challenge in terms of harmonic Maass forms. We fully developed the arithmetic (i.e. p-adic and Galois theoretic) and analytic properties of all such forms, and we have applied these results to solve open problems in additive number theory.

Random walks, geometry and comparative genomics

We will see how a problem in genome rearrangement leads to describe a new
kind of phase transition for random walks on graphs. This phase transition
is related to the well-known Erdos-Renyi double jump phenomenon for random
graphs. I will particularly try to describe the effect that the scale of
mutations may have on the analysis of the problem, and will outline two
possible approaches: one borrowing ideas from hyperbolic geometry and the
other based on cheating and using branching random walks.

Hydrodynamic limits of spatially structured coalescents

We are motivated by a question arising in population genetics, and
try to describe the effect of migratory fluxes and spatial
structure on the genealogy of a population. This leads to the
study of systems of particles performing simple random walk on a
given graph, and where particles coalescence according to a
certain mechanism (typically, Kingman's coalescent) when they are
on the same site. We obtain various asymptotic results for this
process, at both small and large time scales, which are of
intrerest to population genetics. We will also discuss some
related conjectures.

A congruence for S-units of a totally real field

Let $p$ be a prime number, $K$ a $CM$ field containing a primitive $p^(n)th$ root of unity and k the maximal real subfield of $K$. We show that
a special case of a conjecture of Solomon gives rise to a (conjectural)
congruence mod $p^n$, relating certain $S$-units of $k$ to the principal
semi-local units of $K$, via the use of local Hilbert symbols.
We will discuss the progress made in proving this conjecture and (briefly) some computational verifications.

Covering hypergraphs with cuts

Covering a graph with subgraphs of certain type has a long history in graph theory and it goes back to Boole, Ore and M.Hall. The problem of covering a graph by cliques or bicliques has been studied by many researchers including Erdos, Goodman, Posa, Chung, Graham, Pollak, Alon etc. Few of the results obtained for graphs extend to hypergraphs. In this talk, I will present some of these results and show some new results
regarding the covers of an $r$-uniform hypergraph by $k$-cuts such that the total size of the cuts (the sum of the number of edges of
all cuts) is minimum. This is joint work with Andr$\rm\acute e$ K$\rm\ddot u$ndgen
(Cal.State San Marcos).

Toric and hypertoric combinatorics

I will discuss three classes examples in which the geometry of an
algebraic variety illuminates a certain combinatorial object. The first
example will involve the classical relationship between toric varieties and
polytopes. The next two examples will deal with a relatively new class of
spaces called hypertoric varieties. These may be thought of as
quaternionifications of toric varieties, and they interact richly with the
combinatorics of matroids, or of finite collections of hyperplanes in a
vector space.