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#### 2008 Archive

##### Combinatorics and Power Series

In this talk I will explain why thinking about formal power series is useful, and what they can do for you. I will tell how to write down the coefficents of formal power series that take the form of exponential generating functions $E_g(E_{f_1}(x), \dots , E_{f_q}(x))$, using partitions. I will use those results with the added structure provided by thinking about my formal power series as Taylor series of functions to provide a formula for the derivatives of composite functions that look like $g(f_1(x), \dots , f_q(x))$. This talk will be accessible to all graduate students (first years included!) and no prior knowledge of combinatorics is required.

##### The n-category of cobordisms and TQFT

Since Atiyah and Segal described the notions of Topological Quantum Field Theory (TQFT) and Conformal Field Theory (CFT), mathematicians have been looking for a suitable n-category framework to describe cobordisms with corners and a more general TQFT involving such cobordisms as a functor from that higher category of cobordisms with corners to a higher category of vector spaces. The problem deals with such standard difficulties of higher category theory as weak vs. strict axioms, coherence, suitable diagrams, etc.: almost every higher category theorist has this problem in the back of her head, but nobody seems to have gotten through to a satisfactory solution. In the talk, I will describe the set up Mark Feshbach and I have found.

##### Towards a lifting of representations of finite reductive groups

The problem of understanding base change for p-adic groups forces one to consider a certain generalization of base change for finite groups. I will give an introduction to base change, and give a partial description of this generalization. (This is a preliminary report on joint work with Joshua Lansky.)

##### Cascade Processes in Social Networks

Social networks are often represented by directed graphs where the nodes are individuals and the edges indicate a form of social relationship. A simple way to model the diffusion of ideas, innovative behavior, or word-of-mouth effects on such a graph is to consider a stochastic process of infection'': each node becomes infected once an activation function of the set of its infected neighbors crosses a random threshold value. I will prove a conjecture of Kempe, Kleinberg, and Tardos which roughly states that if such a process is locally'' submodular then it must be globally'' submodular on average. The significance of this result is that it leads to a good algorithmic solution to the problem of maximizing the spread of influence in the network--a problem known in data mining as "viral marketing"'.
This is joint work with Elchanan Mossel.

##### Hurwitz spaces and string topology

String topology, originating in the work of Chas and
Sullivan in the late 90's, concerns itself with the algebraic and
topological properties of loop spaces of manifolds. Many interesting
connections to representation theory and symplectic geometry have
recently been established. Hurwitz spaces are moduli spaces of
branched covers of Riemann surfaces. In this talk we will propose a
generalization of this notion that serves as a bridge between the two
subjects, and allows for the construction of operations in string
topology governed by the moduli spaces of Riemann surfaces.

##### Symplectic geometry and invariants for low-dimensional topology

Over the past few years, ideas from symplectic geometry
have had a major impact on low-dimensional topology. Some of the most
impressive results stem from a set of invariants developed by Ozsvath
and Szabo. Though defined using symplectic geometry, they turn out to
be surprisingly powerful invariants of low-dimensional objects e.g.
knots, and three- and four-manifolds. In this talk, I will survey
these invariants and discuss how I have used them to prove results
related to knot theory, complex curves, surgery theory in dimension
three, and the theory of foliations and contact structures on
three-manifolds.

##### Curvature and continuity of optimal transport

In optimal transport theory, one wants to understand optimizing
phenomena occurring when transporting mass distributions in
Economics, Physics, Probability, Analysis, Geometry, and Biology. In
this talk, we will discuss continuity of optimal transport maps, in
view of a pseudo-Riemannian structure which we have formulated
recently. Curvature of this geometry plays an essential role for
continuity of optimal transportation. This natural geometric
framework provides new methods, elementary proofs and extensions of
some key ingredients in the regularity theory of Ma, Trudinger, Wang,
and Loeper. It also provides new examples, perspectives and research
directions.

This is joint work with Robert McCann (University of Toronto).

##### Mori cones of algebraic varieties

The Mori cone is a fundamental, often elusive, invariant
of an algebraic variety and is the central object of study in higher
dimensional algebraic geometry. In this talk I will explain Fulton's
conjecture, which predicts a very simple description of the Mori cone
of the moduli space of curves. I'll show how one can naturally obtain
upper and lower bounds for the Mori cone of a large class of
varieties. In the case of the moduli space of curves, the upper
bound is the cone described by Fulton's conjecture. In particular,
this gives a new possibitlity for the Mori cone and a new perspective
on Fulton's conjecture.

##### Image Inpainting

Image inpainting involves filling in part of an image (or video) using
information from the surrounding area. In this talk I will discuss the
connection of Navier-Stokes equations (NSE) in image inpainting. This
important connection suggests the possibility of other hybrid methods or
turbulence models in image inpainting. Recently, the three-dimensional
(3d) Navier-Stokes-Voight (NSV) equations, were suggested as a
regularizing model for the 3d NSE. We would like to investigate how we can
tune the relevant parameters of this model to optimize the end result in
image inpainting.

##### The u-invariant of fields

The u-invariant of a field is defined to be the maximal
dimension (number of variables) of a quadratic form which has no
nontrivial zeros. Although there are some expectations for what
u-invariants should be of most "naturally occuring" fields, these
invariants are unknown in a great number of situations. For example,
if $F$ is a nonreal number field, it is known that $u(F) = 4$, and it
is expected that the u-invariant of the rational function field $F(t)$
should be $8$. At this point, however, there is no known bound for
$u(F(t))$ (and no proof it is even finite).

Important progress on this type of problem was obtained by Parimala
and Suresh late last year, who showed that the u-invariant of a
rational function field $F(t)$ is $8$ when $F$ is $p$-adic ($p$ odd).
In this talk I will describe joint work with David Harbater and Julia
Hartmann in which we give an independent proof and a generalization of
this result using the method of field patching."

##### Moduli spaces of bundles and generalized theta functions

The Jacobian of any compact Riemann surface carries a natural
theta divisor, which can be defined as the zero locus of an explicit
function, the Riemann theta function. I will describe a generalization of
this idea, which starts by replacing the Jacobian with the moduli space of
bundles (sheaves) over a Riemann surface (or a higher dimensional base).
These moduli spaces also carry theta divisors, described via "generalized"
theta functions. In this talk, I will describe recent progress in the
study of generalized theta functions.

##### Resampling Methods in Change-Point Analysis

Change-point analysis deals with the question whether an observed
stochastic process follows one model or whether the underlying model
changes at least once during the observational period.
In change-point analysis critical values for testing procedures are
usually obtained by distributional asymptotics. However, convergence is
often rather slow. Moreover, these critical values do not sufficiently
reflect possible dependency. Using resampling methods one can often
obtain better approximations, especially for small sample sizes or
dependent data. Similarly one can obtain confidence intervals of the
change-point using resampling methods, which are often better than
asymptotic confidence intervals.

Recently sequential change-point analysis has become more and more
popular. In this setup, one gets the observations online', i.e.
sequentially one-by-one, after having observed a historic data set
without change. For each new observation one checks whether one can
still assume the null hypothesis. This is becoming more and more
important in such diverse fields as medicine, material science or finance.
In such a setting we can make use of the new incoming observations for
the bootstrap. From a practical point of view this is computationally
expensive, so one can think of alternatives which are cheaper but still
very good. From a theoretical point of view this means that we have new
critical values with each incoming observation, so the question is
whether this procedure remains consistent.

In this talk we focus on theoretically examining different bootstrap
procedures for change-point tests and comparing them in a simulation study.

##### Asymptotic behavior of general nonlinear diffusions

Asymptotic profiles for general nonlinear diffusions of the type $u_t =$ Laplacian $f(u)$ on the euclidean space will be characterized by scaling arguments and contractions in the Wasserstein euclidean distance.

##### The decomposition of global conformal invariants: On a conjecture of Deser and Schwimmer

Global conformal invariants are integrals of geometric scalars which remain invariant under conformal changes of the underlying metric. I will discuss (parts of) my recent proof of a conjecture of Deser and Schwimmer, which states that any such global invariant can be decomposed into standard building blocks" of three types. Time permiting, I will also present some applications and related open problems.

##### Algebraically Solvable Problems: Describing Polynomials as Equivalent to Explicit Solutions

We present a coefficient formula which provides some formation about the polynomial map $P|_{I_1\times\dotsb\times I_n}|$ when only incomplete information about the polynomial $P(X_1, \dotsc, X_n)$ is given. It is an integrative generalization and sharpening of several known results and has many applications, among these are:\vspace{-1ex} \begin{enumerate}\setlength{\itemsep}{-0.5ex}
\item
The fact that polynomials $P(X_1)\neq0$ in just one variable have at most $\deg(P)$ roots.
\item
Alon and Tarsi's Combinatorial Nullstellensatz.
\item
Chevalley and Warning's Theorem about the number of simultaneous zeros of systems of polynomials over finite fields.
\item
Ryser's Permanent Formula.
\item
Alon's Permanent Lemma.
\item
Alon and Tarsi's Theorem about orientations and colorings of graphs.
\item
Scheim's formula for the number of edge $$n$$-colorings of planar $$n$$-regular graphs.
\item
Alon, Friedland and Kalai's Theorem about regular subgraphs.
\item
Alon and F\"uredi's Theorem about cube covers.

##### Think global, act local

This talk will be a survey of local-to-global theorems in differential geometry. I will start out by giving some intuition about the curvature of Riemannian manifolds; this should be enough to get a feel for the results that I'll explain in the remainder of the talk. Then I'll begin to survey some results that link the local geometry (curvature) of the manifold to its global geometry. The idea is that sufficient knowledge about the local structure of a Riemannian manifold is sometimes enough to identify its global shape. Theorems of this kind are among the prettiest results in differential geometry and I will try to survey several of them, including the classification of spaces of constant sectional curvature, the Hadamard-Cartan theorem, and Klingenberg's sphere theorem. This talk should be very accessible, since many of the theorems can be understood intuitively. A bit like an index-free advertisement for differential geometry.

##### On a result of Waldspurger in higher rank

An important result of Waldspurger relates certain central $L$-values of automorphic forms on $GL(2)$ to period integrals over tori. Subsequently this result was reproved by Jacquet using the relative trace formula. We will explain some progress on extending Waldspurger's result to higher rank via a generalization of Jacquet's approach.

##### Brownian Motions Interacting Through Ranks and a Phase

Consider a particle in a finite dimensional Euclidean space
performing a Brownian motion with an instantaneous drift vector at every time point determined by the order in which the coordinates of its location can be arranged as a decreasing sequence. These processes appear naturally in a variety of areas from queueing theory, statistical physics, and economic modeling. One is generally interested in the spacings between the ordered coordinates under such a motion.

For finite $n$, the invariant distribution of the vector of spacings can be completely described and is a function of the drift. We show, as $n$ grows to infinity, a curious phenomenon occurs. We look at a transformation of the original process by exponentiating the location coordinates and dividing them by their total sum. Irrespective of the drifts, under the invariant distribution, only one of three things can happen to the transformed values: either they all go to zero, or the maximum grows to one while the rest go to zero, or they stabilize and converge in law to some member of a two parameter family of random point processes. This family known as the Poisson-Dirichlet's appears in genetics and renewal theory and is
well studied. The proof borrows ideas from Talagrand's analysis of Derrida's Random Energy Model of spin glasses. We also consider another alternative starting with a countable collection of Brownian motions. This countable model is related to the Harris model of elastic collisions and the discrete Ruzmaikina-Aizenmann model for competing particles.

This is based on separate joint works with Sourav Chatterjee and Jim Pitman.

##### Definable relations in the field of reals with a subgroup of the unit circle

In this talk (intended for general algebraic audience)
I will discuss the problem of characterization of definable relations in algebraic structures and give a survey of some classical results in this direction.
Also, I will present my joint work with Boris Zilber, where
we describe definable relations in the field of reals augmented by a binary predicate for a finite rank multiplicative group of complex numbers contained in the unit circle.

##### Invariant chiral differential operators

Given a finite-dimensional Lie algebra $g$ and a $g$-module $V$, the
ring $D(V)^g$ of invariant differential operators is a much-studied object
in classical invariant theory. It has a natural vertex algebra analogue.
First, $D(V)$ has a $VA$ analogue $S(V)$ known as a $\beta\gamma$-system or
algebra of chiral differential operators. The action of $g$ on $V$ induces an
action of the corresponding affine algebra on $S(V)$. The invariant space
$S(V)^{g[t]}$ is a commutant subalgebra of $S(V)$, and plays the role of
$D(V)^g$. In this talk, I'll describe $S(V)^{g[t]}$ in some basic but
nontrivial cases: when $g$ is abelian and the action is diagonalizable, and
when $g$ is one of the classical Lie algebras $sl(n), gl(n)$, or so$(n)$, and $V = C^n$. The answer is often a surprise: for example, when $g = C = V, S(V)^{g[t]}$ is the Zamolodchikov $W_3$ algebra with central charge $c=-2$.

##### A Linear, First Order PDE with No Solution

In 1956 Hans Lewy shocked the world. For decades, people had believed that just like ODEs, well-behaved PDEs should have a general existence theorem, and that it was only a matter of time before it was proven. However, Lewy constructed a very simple example to prove them all wrong. He was in as much shock and awe as they were.
We'll talk about this equation and the proof, which coincidentally uses a lot of CR geometry. The talk should be accessible to anyone who has ever taken a partial derivative.

##### A Spectral Radius Formula for the Fourier Transform on Locally Compact Motion Groups and Applications to Random Walks

% ----------------------------------------------------------------
The Beurling\,--\,Gelfand spectral radius formula for the Fourier transform on $\mathbb{R}$ (say) states that \linebreak $\lim_{n\to\infty}\lVert f^n\rVert^{1/n}_1= \sup_\chi\big\lvert\widehat{f}(\chi)\big\rvert$ for any $f\in L^1(\mathbb{R})$,
where $f^n:=f\ast\cdots\ast f$ denotes $n$-fold convolution of $f$ with itself,
$\widehat{f}$ is the Fourier transform of $f$, and the supremum extends over
all characters (i.e., the exponentials $e^{2\pi i\xi}$) of the group
$\mathbb{R}$. This extends to measures, relating
$\lim_{n\to\infty}\lVert\mu^n\rVert^{1/n}$ to the Gelfand transform
$\widehat{\mu}$ (which coincides with the Fourier transform of $\mu$ when
restricted to characters), where now $\lVert\ \rVert$ is the total variation
norm. We establish such a formula for measures in compact groups, and more
generally
semi-direct product of the form $A\times_\varphi K$ where $A$ is Abelian
and $K$ is compact; these include all Abelian groups, all compact groups,
and also the Euclidean motion groups (whence the name locally compact motion
groups). We then apply this formula to obtain some known results about norm
convergence of random walks on compact groups to the uniform distribution,
and some new results about random walks on general motion groups; in particular
we characterize mixing and ergodicity of random walks on such groups by means
of their Fourier transform.

##### Some Image Processing Issues on the Mobile Platform

Image processing on the mobile platform is a crucial component of the multimedia function of a mobile device. Image processing on the mobile platform is constrained by low-power, low computational cost, and real-time requirement. In this presentation, I will talk about some problems and solutions for image processing on the mobile platform. Specifically, I will describe some real-time algorithms and results related to image up-scaling, contrast enhancement, and content protective wide-screen scaling. I will also talk about some challenges for applying image processing algorithms to video signals.

##### Scarf's Lemma and the Stable Paths Problem

We address a question in graphs called the stable paths problem, which
is an abstraction of a network routing problem concerning the Border
Gateway Protocol (BGP). The main tool we use is Scarf's Lemma, an
important result from game theory. This talk will describe Scarf's
Lemma and how it is related to other results more familiar to
combinatorialists, and then will explain its implications for the
stable paths problem.

##### Games we play Quantum Mechanically

We discuss the classical game theory and see how it can be generalized to
the quantum setting. I will try to cover quantum mechanics relevant to the
topic, nothing more than finite dimensional vector space. Then we will see
examples of some games and strategies, and then look at what can be said

##### A waiting time problem arising from the study of multi-stage carcinogenesis

We consider a model of a population of fixed size $N$ in which each individual gets replaced at rate one and each individual experiences a mutation at rate $u$. We calculate the asymptotic distribution of the time that it takes before there is an individual in the population with $m$ mutations. A variety of different behaviors are possible, depending on how $u$ changes with $N$. These results have applications to the problem of determining the waiting time for regulatory sequences to appear and to models of cancer development. This talk is partly based on joint work with Rick Durrett and Deena Schmidt.

##### Pentagons and Partitions

I aim to give you a glimpse of the beautiful subject of modular forms by way of unravelling the mystery of what pentagonal numbers could possibly have to do with the partition function.

##### Orthogonal Representations of Hopf Algebras

Let H be a Hopf algebra over an algebraically closed field of characteristic not 2.
Assume that the antipode of H has period 2, and let V be a finite-dimensional representation
of H. Then if V is self-dual, it must be either orthogonal'' or symplectic'' , in the
sense that it admits a non-degenerate H-invariant bilinear form which is either symmetric or
skew-symmetric.

We consider various Hopf algebras constructed from finite groups, and investigate when all
of their (fin dim) representations are orthogonal in the above sense.

##### Opers with irregular singularity and spectra of the shift of argument subalgebra

The universal enveloping algebra of any simple Lie algebra g contains a family of commutative subalgebras, called the quantum shift of argument subalgebras. Recently B. Feigin, L. Rybnikov and myself have proved that generically their action on finite-dimensional modules is diagonalizable and their joint spectra are in bijection with differential geometric objects on the projective line called "opers". They have regular singularity at one point, irregular singularity at another point and are monodromy free. Interestingly, they are associated not to G, but to the Langlands dual group of G. In addition, we have shown that the quantum shift of argument subalgebra corresponding to a regular nilpotent element of g has a cyclic vector in any irreducible finite-dimensional g-module. As a byproduct, we obtain the structure of a Gorenstein ring on any such module. I will talk about these results and explain the connection to the geometric Langlands correspondence.

##### \bf \huge On the category of bounded $(g,k)$-modules

This talk is based on my joint work with I. Penkov. Let g be a simple Lie
algebra, and $k$ be a reductive subalgebra in $g. A (g,k)-$module $M$ is
bounded if it is locally finite over $k$ and the multiplicities of all
irreducible finite-dimensional modules in $M$ are uniformly bounded.
(Two examples from classical representation theory are ladder modules
in Harish-Chandra theory and cuspidal modules in case when $k$ is a
Cartan subalgebra).

I will formulate several general results about bounded modules
involving primitive ideals theory and geometry (localization). Then I
concentrate on the example when $g=B_2$, and $k$ is the principal
$sl(2)-$subalgebra, where the complete classification of irreducible
simple bounded $(g,k)-$modules is done.

##### Infinite Dimensional Gelfand Pairs

The simplest infinite dimensional Gelfand pairs are the
ones of the form $(G,K) = \varinjlim (G_n,K_n)$ where
the $(G_n,K_n)$ are finite dimensional Gelfand pairs.
Here we take "Gelfand pair" to mean that the action of
$G$ on a suitable Hilbert space $L^2(G/K)$ is multiplicity
free, and we study several cases where that multiplicity
free property holds. The strongest results are for
cases where the $G_n$ are semidirect products $N_n\rtimes K_n$
with $N_n$ nilpotent. Then the $N_n$ are commutative or
$2$--step nilpotent. In many cases where the derived algebras
$[\mathfrak{n}_n,\mathfrak{n}_n]$ are of bounded dimension we construct
$G_n$--equivariant isometric maps
$\zeta_n : L^2(G_n/K_n) \to L^2(G_{n+1}/K_{n+1})$
and prove that the left regular representation of $G$ on the
Hilbert space $L^2(G/K) := \varinjlim L^2(G_n/K_n),\zeta_n$
is a multiplicity free direct integral of irreducible unitary
representations.

##### Tantalizers

A tensor power centralizer algebra (tantalizer)
is the algebra of commuting operators for a Lie group
or quantum group action on tensor space. The favourite examples
are the group algebra of the symmetric group and the Brauer algebra.
This talk will survey some recent work on tantalizers: giving definitions
and recent results for affine and graded
BMW algebras and some two boundary tantalizers.

##### Weyl algebras and W-algebras

We will discuss the connections between Weyl and finite W-algebras
and also their Gelfand-Tsetlin representations.

##### A Freudenthal Supermagic Square

Freudenthal Magic Square, which in characteristic 0 contains the simple exceptional
finite dimensional Lie algebras, is enlarged in several ways,
through the use of some simple alternative and Jordan superalgebras.
New simple modular Lie superalgebras are thus obtained.

##### Perfect crystals and Young walls

We will discuss the close connection between the theory of perfect crystals and combinatorics of Young walls. We will also discuss the construction of Young walls associated with adjoint perfect crystals.

##### On some of the mathematics in Garrett Lisi's E(8) Theory of Everything

A physicist, Garrett Lisi, has published a highly controversal, but fascinating, paper purporting to go beyond the standard model in that it unifies all 4 forces of nature by using as gauge group the exceptional Lie group $E(8)$. My talk, strictly mathematical, will be about an elaboration of the mathematics of $E(8)$ which Lisi relies on to construct his theory.

##### Group gradings on algebras of finitary linear transformations

In this joint work with Mikhail Zaicev we extend our results about
group gradings of matrix algebras to the algebras of finitary linear
transformations. Using the combinatorial techniques of functional
and Jordan algebras of finitary linear transformations.

##### \bf \huge $q-$Partition Algebras

The partition algebra is the centralizer of the symmetric group acting on tensor powers of its natural (permutation) module. It has a diagrammatic basis that generalizes Brauer's centralizer algebra for the orthogonal group. Many of these diagram centralizer algebras have q-generalizations: for example the q-symmetric group is the Iwahori-Hecke algebra and the q-Brauer algebra is the BMW algebra. We will introduce a candidate for a q-partition algebra --- constructed using Harish-Chandra restriction and induction on the finite general linear group over a field with q elements --- and we will illustrate some preliminary computations in this algebra.

##### Simple highest weight modules for non-twisted affine algebras

Mathieu used admissible highest weight modules of finite
dimensional Lie algebras to determine all torsion free modules.
There is a natural analogue of admissible modules for affine Lie
algebras. We use the work of Chari, Chari and Pressley and of Rao
to construct all admissible modules for non-twisted affine algebras.
This involves the construction of all simple highest weight modules
for $A_1^{(1)}$in $\tilde{O}$ having finite dimensional weight spaces. We believe
that this construction will actually produce all simple highest
weight modules in $\tilde{O}$ having finite dimensional weight spaces for
arbitrary non-twisted affine algebras. This is joint work with Frank Lemire

##### Demazure Crystals for quantum affine algebras

The crystals associated with integrable modules
of quantum affine algebras can be realized as subsets of
semi-infinite tensor products of certain finite crystals called
perfect crystals. In particular for the quantum affine algebra of
special linear type these crystals have a nicer presentation in
terms of colored Young diagrams. The crystals for Demazure
modules are finite subsets of the crystals for the associated
integrable modules. In this talk we will present some old and new
results on the Demazure crystals.

##### Quantum groups and pointed Hopf algebras

Each of the traditional finite quantum groups is associated to
a Lie algebra, and therefore has at its heart a Dynkin diagram.
A recent classification by Andruskiewitsch and Schneider shows
that the seemingly much larger class of finite dimensional
pointed Hopf algebras is not so different: Each such Hopf algebra
has at its heart a collection of linked Dynkin diagrams from
which its structure is largely determined.

In this talk, we will give an overview of the classification
and the current state of knowledge about these pointed Hopf
algebras, including a finite generation result in cohomology.

##### Flag Algebras

A substantial part of extremal combinatorics studies relations existing
between densities with which given (fixed size) combinatorial structures may
appear in unknown (and presumably very large) structures of the same type.
Using basic tools and concepts from algebra, analysis and measure theory, a
general framework is developed that allows to treat all problems of this
sort
in an uniform way and reveal mathematical structure that is common for most
known arguments in the area (the central objects in this theory are called
Flag Algebras whence comes the title.) In this talk I hope to give a general
impression, illustrated by simple examples, of how things look like in this
framework, and discuss its advantages and drawbacks compared to the
naive''
approach. I will also mention some concrete results obtained with the
help of
these methods.

##### Distribution of Segment Lengths in Genome Rearrangements

The study of gene orders for constructing phylogenetic trees was
introduced by Dobzhansky and Sturtevant in 1938. Different genomes
may have homologous genes arranged in different orders. In the early
1990s, Sankoff and colleagues modelled this as ordinary (unsigned)
permutations on a set of numbered genes 1,2,...,n, with biological
events such as inversions modelled as operations on the permutations.
Signed permutations may be used when the relative strands of the genes
are known, and circular permutations'' may be used used for circular
genomes. Pevzner, Bafna, and Hannenhalli further developed the model
into the breakpoint graph,'' which has been very successful in
analyzing sequences of inversions. We use combinatorial methods
(generating functions, commutative and noncommutative formal power
series, asymptotics, recursions, and enumeration formulas) to study
the distributions of the number and lengths of conserved segments of
genes between two or more unichromosomal genomes, including signed and
unsigned genomes, and linear and circular genomes. This generalizes
classical work on permutations from the 1940s-60s by Wolfowitz,
Kaplansky, Riordan, Abramson, and Moser, who studied decompositions of
permutations into strips of ascending or descending consecutive
numbers. In our setting, their work corresponds to comparison of two
unsigned genomes (known gene orders, unknown gene orientations).

##### Low-dimensional representations of finite simple groups and conjectures of Katz Kollar and Larsen

In the first part of the talk we will
survey recent results on representations of
finite (quasi)simple groups. In the second part
we will discuss proofs of some conjectures of
Katz, Kollar, and Larsen. This is joint work
with R. M. Guralnick.

##### Metric geometry and unfoldings of polyhedra

Most of us as children saw those paper or cardboard cutouts,
which we could call "foldouts", whose edges glue to form
(boundaries of) 3-dimensional convex polyhedra. Just how did
anyone figure out how to make them? Given a 3-dimensional
convex polyhedron, does there always exist a foldout in the
plane? What about higher dimensions? These questions have
surprising answers, depending on the precise meaning of
"foldout". One method is to treat boundaries of polyhedra like
Riemannian manifolds. Algorithmic concerns then raise
fundamental issues of computational complexity for the
combinatorics of geodesics on polyhedra. This talk is on joint
work with Igor Pak.

##### Science Education in the 21st Century: Using the tools of science to teach science

Guided by experimental tests of theory and practice, science
has advanced rapidly in the past 500 years. Guided primarily
by tradition and dogma, science education meanwhile has
remained largely medieval. Research on how people learn is
now revealing how many teachers badly misinterpret what
students are thinking and learning from traditional science
classes and exams. However, research is also providing insights
on how to do much better. The combination of this research
with modern information technology is setting the stage for a
new approach that can provide the relevant and effective
science education for all students that is needed for the 21st
century. I will discuss the failures of traditional educational
practices, even as used by “very good” teachers, and the
successes of some new practices and technology that
characterize this more effective approach, and how these results
are highly consistent with findings from cognitive science.

##### Spatiotemporal Analysis of Environmental Health Risk

Massive data sets with complex spatiotemporal structures are common in
environmental studies. In order to account for such spatiotemporal structures,
spatially and temporally correlated random effects are often incorporated into
generalized linear models for such data. The estimation of these models often
poses theoretical and computational challenges. We propose an orthodox best
linear unbiased predictor (BLUP) approach to these models. Our approach is
illustrated with application to Ohio lung cancer data where the annual lung
cancer deaths for 88 counties were obtained from 1968-1988. With estimated
spatial and temporal random effects, we will also discuss the identification of
high/low risk areas, spatial clustering as well as temporal trend.

##### On Innovation in Urban Mathematics Education

For his work on nurturing minority student high achievement in mathematics, Professor Treisman was named a MacArthur Fellow in 1992. In December 1999, he was named one of the outstanding leaders of higher education in the 20th century by the magazine Black Issues in Higher Education. In February 2006 he was named “2006 Scientist of the Year” by the Harvard Foundation of Harvard University for his outstanding contributions to mathematics. In all his work, he is an advocate for equity and excellence in education for all children.

##### Witten equation and quantum singularity theory

A long standing problem in Gromov-Witten theory is to compute
higher genus Gromov-Witten invariants of compact Calabi-Yau manifold such
as quintic 3-folds. The defining equation of these Calabi-Yau manifold
has a natural interpretation in Landau-Ginzburg/singularity theory. More
than 15 years ago, Witten proposed a PDE as a replacement of familiar
Cauchy-Riemann equation in the Laudau-Ginzburg/singularity setting.
Furthermore, he proposed two remarkable conjectures for his conjectural
theory for ADE-singularity. In the talk, we will present a moduli theory
of the solution spaces of Witten equation. As a consequence, we solve
Witten's conjectures for quantum theory of ADE-singularities. At the end
of the talk, we will sketch a plan to compute higher genus GW-invariants of
Calabi-Yau manifolds.

##### Orthogonally Based Pivoting Transformation on Engineering Applications

A pivoting linear transformation based on the orthogonal concept is
presented. This transformation allows for the solving of many problems in
linear algebra, such as matrix computations and linear systems of equations,
and the updating of solutions from scratch. The goal of this talk is to
show how the study of this method has led us to better understandings
and easier and more efficient solutions of engineering problems.

##### Searching the quantum symmetry in topology

The famous quantization principal in physics asserts that the
quantum world posses more symmetry than the classical world. In the
talk,
we will apply the above quantization principal to topology. Using
several key examples, we will illustrate that such a quantum symmetry
phenomenon yields striking conjecture/theorem in all cases.

##### Gallai-Edmonds decomposition for non zero roots

Classical matching theory such as the Gallai-Edmonds structure theorem
is
concerned about the multiplicities of zero roots of matching polynomial
of
graphs. We prove an analogue of half of the Gallai-Edmonds structure
theorem (namely the so-called Stability Lemma) for non zero roots. We
conjecture that the other half (analogous to Gallai's Lemma) is also
true.
This is a joint work with William Chen.

##### Hilbert Schemes

What can the geometry of a moduli space tell us?

Suppose we look at the set of all subspaces in projective space.
We can turn this into an geometric object (the Hilbert scheme) whose
points correspond to subspaces of projective space.
But what does the geometry tell us? In other words, what do the connected
components represent? What about the tangent space? What if it has
singularities? What do these things tell us about the subspaces we're
parametrizing? What if we map a curve into this space: does that give us
anything meaningful? I will discuss these concepts through examples.

It turns out the Hilbert scheme is a very robust object, but it comes at a
price: which will be encapsulated in a theorem aptly called "Murphy's law
for Hilbert schemes".

##### The spectral gap of a random subgraph of a graph

The spectral gap of the normalized Laplacian of a graph is strongly related to many important graph properties, including the mixing rate of random walks as well as expansion and discrepancy properties. Here we consider a random subgraph $H$ of a given graph $G$ where each edge in $G$ is taken to be in $H$ independently with probability $p$, and derive bounds on the spectral hap of $H$ in terms of the spectral gap of $G$. This can be thought of as an extension of earlier work on the Erd\H{o}s-R\'enyi $G(n,p)$ mode, which effectively treats a special case where the underlying graph is the complete graph $K_n$.

We also survey some history and related problems.

##### Convergence for General Full Discretizations of Center Manifolds for Parabolic Differential Equations

For parabolic differential equations, space and time discretization methods, so called full discretizations, are necessary to determine the dynamics on center manifolds. Until now, only very little seems to be known about the convergence of these full discretizations. We show that, allowing stable, center, unstable manifolds, for the standard space discretization methods, the space discrete center manifolds converge to the original center manifolds in the following sense: The coefficients of the Taylor expansion of a discrete center manifold and its normal form converge to those of the original center manifold. Then standard or geometric time discretization methods can be applied to the discrete center manifold system of ordinary differential equations. We prove convergence for these full discretizations and give a short outline for the Hopf-bifurcation as example.

##### Formality for the homotopy Gerstenhaber algebra of Hochschild cochains

In my talk I will try to explain the result
of my joint paper arXiv:math/0605141 with D. Tamarkin
and B. Tsygan in as popular way as possible.
First, I will convince you that the notion of
homotopy Gerstenhaber algebra'' is not so crazy.
Second, I will recall algebraic structures
on Hochschild cochains and formulate the main
theorem. Finally, I will explain the idea of the
proof referring to the well known bar/cobar
constructions for associative algebras.
Well, if the bar/cobar constructions are not well known
then they will be well known after my talk.

##### Avoiding small subgraphs in Achlioptas processes

Consider the following random process. At each round, one is
presented with two random edges from the edge set of the complete
graph on $n$ vertices, and is asked to choose one of them. The
selected edges are collected into a graph, which thus grows at
the rate of one edge per round. This is known in the literature
as an Achlioptas process, and has been studied by many
researchers, mainly in the context of delaying or accelerating
the appearance of the giant component.

In our work, we investigate the classical small subgraph problem
for Achlioptas processes. That is, given a fixed graph $H$, we
study whether there is a deterministic online algorithm that
substantially delays or accelerates a typical appearance of $H$,
compared to its threshold of appearance in the random graph $G(n, M)$. It is easy to see that one cannot accelerate the appearance
of any fixed graph by more than a factor of 2, so we concentrate
on the task of avoiding $H$. We determine thresholds for the
avoidance of all cycles $C_t$, cliques $K_t$, and complete
bipartite graphs $K_{t,t}$.

Joint work with Michael Krivelevich and Benny Sudakov.

##### Applied Quaternions

We'll discuss some geometric properties of the quaternions, and show how quaternions are used in describing
rotations in computer graphics.

##### Jim Lin - UCSD

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

##### Noncommutative algebras over uncountable fields

fundamental importance in the theory of non-commutative rings. In this
talk I will discuss some of these questions, beginning from the classical problems of Kothe and Kurosh, and up to more recent developments and open questions in the field. In particular, I will discuss some special results for algebras over uncountable fields.

##### On the $\sigma_2$-scalar curvature and its application

In this talk, we establish an analytic foundation for a fully
non-linear equation $\frac{\sigma_2}{\sigma_1}=f$ on manifolds with
positive scalar curvature. This equation arises from conformal geometry.
As application, we prove that, if a compact 3-dimensional manifold $M$
admits a riemannian metric with positive scalar curvature and $\int \sigma_2\ge 0$, then topologically $M$ is a quotient of sphere.

##### Krylov Deferred Correction and Fast Elliptic Solvers for Time Dependent Partial Differential Equations

In this talk, we discuss a new class of numerical methods for the accurate and efficient integration of time dependent partial differential equations. Unlike traditional method of lines $(MoL)$, the new Krylov deferred correction $(KDC)$ accelerated method of lines transpose $(Mol^T)$ first discretizes the temporal direction using Gaussian type nodes and spectral integration, and the resulting coupled elliptic system is solved iteratively using Newton-Krylov techniques such as Newton-GMRES method, in which each function evaluation is simply one low order time stepping approximation of the error by solving a decoupled system using available fast elliptic equation solvers. Preliminary numerical experiments show that the KDC accelerated $MoL^T$ technique is unconditionally stable, can be spectrally accurate in both temporal and spatial directions, and allows optimal time step sizes in long-time simulations. Numerical experiments for parabolic type equations including the Schrodinger equation will be discussed.

##### Oded Yacobi - UCSD, Graduate Student

Using disjoint circles and a single straight line, can you fill up 3-space in such a way that each pair of circles is linked, and the line passes through the interior of each circle? We investigate a solution to this via an elementary construction of a Hopf fibration.

##### Convex families of proper mappings between balls

We will study the space of proper holomorphic mappings between balls in different dimensions. To every proper mapping we can associate a finite rank hermitian form. As the resulting space is be convex, it is natural to define a convex family of proper mappings. While there are many proper mappings in codimension one when there is no further assumption, we will prove there is no convex family in codimension one. Stronger results can be proven when one restricts the maps to the monomial category. This is joint work with John D'Angelo.

##### Binding Number, Cycles and Cliques

In 1973 Woodall defined the binding number of a graph and showed that
a binding number of $3/2$ implies that the graph has a hamiltonian
cycle. He conjectured that the same threshold implies that the graph
has a triangle, and moreover, cycles of all lengths. This was later
proven by Shi Ronghua. In 1974 Andr\'{a}sfai, Erd\H{o}s and S\'{o}s
showed that a $K_r$-free graph with large enough minimum degree is
$(r-1)$-colorable. Several authors investigated further results about
homomorphisms and colorings of dense graphs with low clique number. In
this talk we revisit the connections among binding number, cycles and
cliques in graphs. This includes joint work with Jeremy Lyle.

##### Stabbing simplices by points and affine spaces

B\'ar\'any showed that there is a constant $c_d>0$
such that if $S$ is any $n$-point set in $R^d$, then there exists
a point in $c_d$ fraction of simplices spanned by $S$.
We present a simple construction of a point set for which there is
no point contained in many simplices. The construction is optimal
for $d=2$ and gives the first non-trivial upper bounds on $c_d$ for
$d\geq 3$. We will also discuss generalizations to stabbing simplices
by affine spaces. Joint work with Ji\v{r}\'\i{} Matou\v{s}ek
and Gabriel Nivasch.

##### Optimal stopping and free boundary characterizations for some Brownian control problems

We study a singular stochastic control problem with state constraints in two dimensions. We show that the value function is continuously differentiable and its directional derivatives are the value functions of certain optimal stopping problems. Guided by the optimal stopping problem we then introduce the associated no-action region and the free boundary and show that, under appropriate conditions, an optimally controlled process is a Brownian motion in the no-action region with reflection at the free boundary. This proves a conjecture of Martins, Shreve and Soner (SIAM J. Control Optim., 34: 2133-2171, 1996) on the form of an optimal control for this class of singular control problems. An important issue in our analysis is that the running cost is Lipschitz but not differentiable. This lack of smoothness is one of the key obstacles in establishing regularity of the free boundary and of the value function. We show that the free boundary is Lipschitz and that the value function is C2 in the interior of the no-action region. We then use a verification argument applied to a suitable C2 approximation of the value function to establish optimality of the conjectured control. This is a joint work with Amarjit Budhiraja.

##### On the regularized Siegel-Weil formula for the second terms and non-vanishing of theta lifts from orthogonal groups

In this talk, we will discuss (a certain form of) the
Siegel-Weil formula for the second terms (the weak second term
identity). As an application, we show the following non-vanishing
result of global theta lifts from orthogonal groups: Let $\pi$ be a
cuspidal automorphic representation of an orthogonal group $O(V)$ with
$\dim V=2r-j$ even and $0\leq j\leq r-1$. Then the global theta lift
of $\pi$ to $Sp(2r)$ does not vanish ''up to disconnectedness" if the
(incomplete) $L$-function $L^S(s,\pi)$ does not vanish at
$s=1+\frac{j}{2}$. (This is a joint with W. Gan.)

##### Introduction to de Rham Cohomology

The Topology Seminar will be following Bott & Tu's Differential Forms in Algebraic Topology. This week we will define the de Rham Complex of a smooth manifold. We'll also look at a Mayer-Vietoris sequence and integration of a differential form.

##### Mathematical relativity and the black hole codes

The recent (surprising(?)) success of the numerical binary
black hole collision codes poses a number of challenges
for mathematical relativists. Following a quick overview
of numerical relativity and the binary black hole
problem, I will give my explanation as to why the codes
worked. I will then focus on what I see as the key
barriers that need be overcome to permit further
development, and what contribution mathematical
relativity can make to this exciting field. This talk
will be aimed at the non-expert.

##### Boundary Integral Methods for Electrostatics in Biomolecules Part I: An Overview

In this talk I will give an overview of the recent development of boundary integral
methods for solving Poisson-Boltzmann equations (PBE) for the electrostatic
potential in a solvation system of biomolecules. Electrostatic forces are crucial
in determining the structure and dynamics of biomolecules and their interactions
with solvents. As a mean-field approximation, the PBE has proved to be a very useful
model of such electrostatics. However, numerically solving the PBE in a very efficient
and accurate way is challenging.

I will first describe the method and present some of my computational results. I will
then compare different numerical methods for solving the PBE with complicated geometry.
In particular, I will present the newly developed fast multipole method for the boundary
integral discretization of the PBE. I will finally mentions some open questions.

The results presented in talk are mainly from the work done in McCammon's group at UCSD.

##### Branching random walks and diffusions on hyperbolic spaces: recurrence, transience and Hausdorff dimension of limiting sets

This talk focuses on asymptotic properties
of geometric branching processes on hyperbolic spaces
and manifolds. (In certain aspects, processes on hyperbolic
spaces are simpler than on Euclidean spaces.)
The first paper in this direction was
by Lalley and Sellke (1997) and dealt with a homogenous
branching diffusion on a hyperbolic (Lobachevsky) plane).
Afterwards, Karpelevich, Pechersky and Suhov (1998) extended
it to general homogeneous branching processes on
hyperbolic spaces of any dimension. Later on, Kelbert and Suhov
(2006, 2007) proceeded to include non-homogeneous branching
processes. One of the main questions here is to calculate
the Hausdorff dimension of the limiting set on the absolute.
I will not assume any preliminary knowledge of hyperbolic
geometry.

##### A Linear, First Order PDE with No Solution

In 1956 Hans Lewy shocked the world. For decades, people had believed that just like ODEs, well-behaved PDEs should have a general existence theorem, and that it was only a matter of time before it was proven. However, Lewy constructed a very simple example to prove them all wrong. He was in as much shock and awe as they were.

We'll talk about this equation and the proof, which coincidentally uses a lot or CR geometry. The talk should be accessible to anyone who has ever taken a partial derivative.

##### New 'Dimensions' in Genome Annotation

Traditional Genome annotation involves the enumeration of open
reading frames and their functional assignment. Currently, there are
on-going efforts to identify all the interactions between these
components. The resulting map of interactions effectively represents
a 2D annotation. It takes the form of a stoichiometric matrix, if
the interactions are described with chemical equations. The
formulation and properties of this matrix are detailed and how it can
be used as the basis for computing allowable phenotypic functions.
The issues associated with the packing of the bacterial genome and
the function of the interaction map in 3D will also be discussed.
Finally, we will go over the issue of genomes changing in space and
time (4D) through adaptive evolution and describe the full re-
sequencing of bacterial genomes to map all genetic changes that occur
during adaptation. All of these efforts represent mathematical
challenges.

##### Scalable Implicit Methods for Magnetic Fusion Modeling

Fusion energy holds the promise of a clean, sustainable and safe energy source for the future. While research in this field has been ongoing for the last half century, much work remains before it may prove a viable source of energy. In this talk, I discuss some of the scientific and engineering challenges remaining in fusion energy, and the role of applied mathematics and scientific computation in helping overcome these obstacles. I then introduce some of the mathematical models used in studying fusion stability and refueling, and how solutions to those models may be approximated. Of particular interest in such approximation techniques is the ability of the relevant numerical methods to scale up to resolutions necessary for accurately modeling the underlying physics. To that end, I discuss some of my recent work in the development of fully implicit solution approaches for magnetic fusion modeling, presenting both numerical results and theoretical analysis demonstrating the benefit of these approaches over traditional methods.

##### Boundary Integral Methods for Electrostatics in Biomolecules

In the second part of the talk, I will focus on the mathematical aspects of integral
equation methods (IEM) and fast multipole methods (FMM). I will discuss why integral
equation methods are preferred for many problems as well as their limitations. Using
a simple example in one dimensional space, I will discuss the fundamental ideas of
the tree code and fast multipole methods, and how the ideas can be applied to more
general problems.

##### Totally nonnegative matrices, juggling patterns, and the affine flag manifold

Consider $b\times n$ real matrices such that every $b\times b$ minor has nonnegative determinant. Since there are polynomial relations between these minors, not every pattern of zero vs. strictly positive is achievable. In a widely circulated prepreprint, Alex Postnikov gave many ways to index the patterns that are.

I'll describe a new indexing, by bounded juggling patterns'', which will require a brief foray into the mathematics of juggling (with demonstrations). It turns out that many of the natural concepts from the matrix picture have been known to jugglers for 20 years.

Unbounded juggling patterns form a group, the affine Weyl group, and thereby index the Schubert varieties on the (infinite-dimensional) affine flag manifold. I'll explain how the complicated finite-dimensional geometry of Postnikov's stratification is induced from what is actually much more familiar infinite-dimensional geometry.

##### The Banach-Tarski paradox and amenability

We discuss how to cut up a pea into finitely many pieces that
can be rearranged to form a ball the size of the sun (but don't try this at home!), and how this leads to the notion of amenability. We give also a proof of Tarski's Theorem, a gem in mathematics.

##### The story of the Sparsest Cut problem

In the past decade methods from Riemannian geometry and Banach space theory have become a central tool in the design and analysis of approximation algorithms for a wide range of NP hard problems. In the reverse direction, problems and methods from theoretical computer science have recently led to
solutions of long standing problems in metric geometry. This talk will illustrate the connection between these fields through the example of the Sparsest Cut problem. This problem asks for a polynomial time algorithm which computes the Cheeger constant of a given finite graph. The Sparsest Cut problem is known to be NP hard, but it is of great interest to devise
efficient algorithms which compute the Cheeger constant up to a small multiplicative error. We will show how a simple linear programming formulation of this problem leads to a question on bi-Lipschitz embeddings of finite metric spaces into $L_1$, which has been solved by Bourgain in 1986. We will then proceed to study a quadratic variant of this approach which
leads to the best known approximation algorithm for the Sparsest Cut problem. The investigation of this semidefinite relaxation" leads to
delicate questions in metric geometry and isoperimetry, in which the
geometry of the Heisenberg group plays an unexpected role.

##### Inverse Problem in Galois Theory

In a couple of recent works with Khare and Larsen we have
developed a method to construct finite Lie groups of type $C_n, B_n$ and $G_2$ (and potentially more) as Galois groups over the field of rational numbers. The main tools are functorial lifts and self-dual automorphic forms on $GL(n)$ as a source of $l$-adic representations (Kottwitz, Clozel, Harris-Taylor). In this talk I will discuss how one can control the image of the $l$-adic representation by picking local components of automorphic representations: deeply embedded tame parameters and Jordan subgroups, as main tools.

##### Extremals and explicit values of conformal functionals.

The examples will be: [1] Spectral determinants of conformally covariant operators (e.g. conformal Laplace and Dirac operators), and [2] Total $Q$-curvature. Both of these
satisfy conformal invariance. We recently found a striking universality in the variational structure when the base manifold is the round sphere, in any such problem, by using
representation theory of the conformal group. As a corollary it gives us a neat proof of some of the extremal results in Kate Okikiolu's paper from Annals (2001), and of analogous results for many other examples. Lastly, I will mention the proof
of $B\% \ r-$ Schopka's conjecture on dimensional asymptotics of explicit determinants on spheres (i.e. the extremal values in the examples [1] above). Some of the work is joint with Bent Orsted.

##### On Constructing p-Adaptive Finite Element Methods

We formulate $p$-adaptive finite element methods based on the technique called gradient recovery. The answers to two open questions on the design of transit elements and the existence of approximate formulae for the error will be addressed.

##### Transition path and path ensemble optimization with gradient-augmented Harmonic Fourier Beads method

We present a simple method for solving a free boundary value
problem of locating either minimum free energy transition path
ensemble or minimum potential energy transition path between
two configurations on a corresponding energy surface. Our method, called gradient-augmented Harmonic Fourier Beads, employs the global Fourier representation of the path that is a curve interpolation of a discrete set of points on the surface - beads. To optimize the path curve, the method computes energy gradients for each bead from either convex optimization or molecular dynamics or Monte Carlo simulations subject to harmonic restraints. The path optimization is driven by primitive Steepest Descent procedure. Line integration of the Fourier transformed forces along the path curve provides complete and accurate structural and energetic information regarding all the intermediates and saddle points present. The utility of the HFB method is demonstrated by computing potentials of mean force for various transformations of diverse molecular systems.

##### On classification of discrete series representations of classical groups

here has been much recent progress on the classification of
discrete series representations of classical groups. In particular, we have the following two partial classifications: One, due to Jiang and Soudry, of generic representations in terms of Shahidi's L-invariants, and second, due to DeBacker and Reeder, depth zero representations can be organized into into $L$-packets, which are characterized in terms of character distributions. In particular, the two results are expressed in terms of two different languages. We show that the two classifications coincide where they overlap,
that is, for generic representations of depth 0. This result is relevant for the work on the Inverse Problem in Galois Theory.

##### Justification, Complexity, Self-Referentiality

Justification Logic is a relatively new field that studies provability,
knowledge, and belief via proofs, or justifications, explicitly present in
the language. Many justification logics have been developed that closely
resemble modal epistemic logics of knowledge and belief with one important
'there exists a proof of $F$,' justification logics operate with constructs $t :F$, where a justification term $t$ represents a blueprint of a Hilbert-style
proof of $F$. The machinery of explicit justifications can be used to analyze
well-known epistemic paradoxes such as Gettier's examples, to study
self-referential properties of modal logics, and to avoid Logical
Omniscience. This talk will focus on quantitative analysis of justification
logics. We will give an overview of what is known about their decidability
and complexity of the decision procedure. We will also analyze a realization
procedure that provides a bridge from a modal epistemic logic to its
justification counterpart. We will discuss the complexity of one such
realization procedure as well as provide its qualitative analysis that leads
to interesting corollaries about self-referentiality of modal logics.

##### Potential level-lowering for a genus two symplectic group

Given an algebraic automorphic representation of a genus two symplectic group over a totally real number field with an Iwahori-spherical component at a finite place $w$, we shall show - following Sorensen - that there is a congruent automorphic representation with a tamely ramified principal series component
at $w$. Thus after a base change to a finite solvable totally real extension, the level at $w$ can be lowered.

##### Numerical Relativity and a new frontier: \\ Connecting the seen with the unseen

Recent developments in the field of Numerical Relativity have not only provided key insights of binary black hole systems but also began influencing its future role. Undoubtedly one of the most important future drivers in the near future of the field will be its role as another element within the study of spectacular astrophysical phenomena involving strongly gravitation scenarios. Connecting (yet to be observed) gravitational waves with observations within the electromagnetic spectra will be one ultimate goal of this enterprise. This talk will summarise some interesting results on the binary black hole problem, connect with the data analysis efforts and illustrate the study of magnetized binary neutron star systems as a stepping stone towards more ambitious goals.

##### WANTED N! DERIVATIVES --> {\Huge\$}1000{\Huge\$}<-- REWARD

Let $S=\{ (a_1, b_1), (a_2,b_2),\dots, (a_n, b_n)\}$ be a set of lattice points in the first quadrant $\{(x,y): x \geq 0, y \geq 0\}$ and set
$\Delta_S (x,y) = \det \lVert x_i^{a_j} y_i^{b_j} \rVert_{i,j=1}^n$
Let $\mathbf{M}_S$ denote the linear span of all the partial derivatives of $\Delta (x,y)$. Computer data reveals that these vector spaces of polynomials intersect in the most remarkable ways. Since the eary 90's we have accumulated a variety of conjectures most of which are still open. In this talk we will give a glimpse of this amazing mathematical Kaleidoscope.

##### Level-raising congruences for algebraic automorphic representations Level-raising congruences for algebraic automorphic representations

Let $\pi$ be an algebraic automorphic representation of a reductive group $G$ over a totally real number field $F$. Assume $G$ is anisotropic at infinity, and $\pi$ is not congruent to an automorphic character. Suppose $w$ is a finite place of $F$ where the component of $\pi$ is unramified and congruent to the
trivial representation. Then there is an automorphic representation $\pi'$ of $G$ congruent to $\pi$, with the same central character and type at infinity, whose component at w is more ramified than that of $\pi$. Applications in rank one and two include showing that Saito-Kurokawa forms are congruent to generic ones, for the genus two symplectic group.

##### Automatic structures: at the interface of classical and feasible mathematics

Finite state automata are Turing machines with fixed finite bounds on resource use. Automata lend themselves well to real-time computations and efficient algorithms. Therefore, there is a long tradition of studying that part of mathematics which can be represented by automata. This talk will give a survey of this research. In particular, we discuss three major themes: how complicated can automatic structures be? can they be naturally described? how efficient are the associated algorithms? Examples include Thurston's automatic groups associated with 3-manifolds and automatic structures associated to model checking and program verification in computer science.

##### Bimodules, freeness and a new planar algebra

Haagerup and I introduced some 10 years ago a very simple
mechanism to construct classes of irreducible subfactors with exotic standard invariants. We will review this construction and describe the planar algebras arising from it (joint work with Das and Ghosh). Voiculescu's notion of freeness appears naturally in this context.

##### Pointwise estimates for nonlinear wave equations -- late time tails and distortion of traveling waves

For a class of semilinear wave equations we developed a
technique which provides pointwise estimates of the solutions. It offers
a qualitative as well as a quantitative asymptotic information on the
solutions. It also provides a link between the nonlinear terms present
in the equation and the asymptotic effects they generate.
After introducing the technique I will discuss some nonlinear phenomena
which persist even in the weak field regime but are absent in the linear
approximation: the nonlinear late time tails and the nonlinear
distortion of the traveling waves. These effects can be important for
the gravitational waves being the weak field solutions of the Einstein
equations.

##### Trust-region methods for large-scale unconstrained optimization

We consider methods for large-scale unconstrained optimization
based on finding an approximate solution of a quadratically
constrained trust-region subproblem. The solver is based on
sequential subspace minimization with a modified barrier
"accelerator" direction in the subspace basis.

##### The Hankel transform and the Kirilov model of the disctrete series of SL(2,R)

We present a new and simple proof for the Hankel inversion
formula. This formula was claimed by Hankel in 1869 and given many proofs
later. Our result generalizes similar results by Zemanian (1968) and
Duran (1990) to complex order Hankel transforms. The proof is based on
representation theoretic ideas and will allow us to completely describe
the Kirillov model for discrete series representations of $SL(2,R) and GL(2,R)$.

##### There is More to Examples than Meets the Eye

The talk will give an overview of the role of examples in mathematical thinking, learning, and teaching. It will elaborate on findings from a three-year study examining how secondary mathematics teachers select, construct, and use examples in their classrooms. The main purpose of the study was to characterize some aspects of teachers' use of instructional examples as well as their underlying considerations. In addition, the notion of example-space will be introduced, the kind of knowledge that is needed for judicious choice of examples will be discussed, and {\it a mathematics example-related teaching cycle} that accounts for ways in which teachers use examples in and for their classrooms will be presented.

Presented by UCSD and SDSU Mathematics and Science Education Joint Doctoral Program (MSED)

##### Coalescents and branching processes: how fast do they come down from infinity?

The main focus of this talk will be on the way coalescents come down from infinity, and in particular on the asymptotic behavior of the number of blocks at small times. We show that it is given by the number of families alive at small times in a branching process. This approach casts a new light on recent results of Berestycki et al. and Bertoin and Le Gall which gave this behavior in special cases and connects in a unified way several recent results of Bertoin and Le Gall, Birkner et al., and Berestycki et al. (Based on joint works with N. Berestycki and J. Schweinsberg, N. Berestycki and V. Limic)

##### The Burden of Proof

Teaching proof to unmotivated students can be quite a burden. I will discuss how math educators think about this difficult task. We will look at examples of student proofs, categorize some attempts at proof, and consider possible ways to help students improve their proving, formulating math education research questions in the process.

##### Mathematica 6 in Education and Research

This talk illustrates capabilities in Mathematica 6 that are directly applicable for use in teaching and research on campus. Topics of this technical talk include:

* 2D and 3D visualization

* Dynamic interactivity

* On-demand scientific data

* Example-driven course materials

* Symbolic interface construction

* Practical and theoretical applications

\noindent
Current users will benefit from seeing the many improvements and new features of Mathematica 6

\noindent
(http://www.wolfram.com/mathematica/newin6),
but prior knowledge of Mathematica is not required.

##### On several extensions of the Dyson constant term identity

Dyson's conjecture asserts that the constant term of certain
Laurent polynomial in the form of a product is the multinomial coefficients.

It was studied by many authors, including a proof by the Nobel prize
winner Wilson, an elegant recursive proof by Good, and a combinatorial
proof by Zeilberger. I will talk about an elementary approach by using
basic property of polynomials to several extensions of the Dyson constant
term identity.

##### Statistical Learning of Functions and Graphs in High Dimensions

A fundamental difficulty in statistical learning is the "curse of
dimensionality," where most learning problems become notoriously
difficult when the data are high dimensional. Even the simplest of
methods---the linear model---has proved to be interesting and
challenging to understand in the high dimensional setting, and has
attracted the recent attention of multiple communities, including
applied mathematics, signal processing, statistics, and machine
learning.

In this talk we present some recent work on several nonparametric
learning problems in the high dimensional setting. In particular, we
present theory and methods for estimating sparse regression functions,
additive models, and graphical models. For nonparametric regression,
we present a greedy algorithm based on thresholding derivatives that
achieves near optimal minimax rates of convergence. For additive
models, we present a functional version of methods based on L1
regularization for linear models. For graphical models, we present a
method for estimating the graph underlying an unknown graphical model
based only on observations.

The talk is based on work with Larry Wasserman, Pradeep Ravikumar, Han
Liu, and Martin Wainwright.

##### On the work of Xiao-Song Lin; from classical to quantum topology

In 1954, John Milnor introduced the notion of link homotopy and
his invariants of links which he used to classify 3 component links up to
homotopy. In 1987 the speaker and XS Lin acheived the classification,
for any number of components, essentially by refining the Milnor invariants.

The Habegger-Lin classification scheme was extended to other equivalence
relations in Lin's thesis and to more general concordance-type relations
satisfying a list of 6 axioms. Axioms 1-4 are local, axiom 5 says that
any string link (or 'pure tangle' as in pure braid) has an inverse, while
axiom 6 says the equivalence relation on links is generated by isotopy and
after 'closure').

In the early 90's Birman and Lin studied the work of Vassiliev on links
and described in simple terms the Vassiliev filtration. Bar-Natan adopted
their description as a definition of 'finite type' invariants and
eventually all this was tied back to the perturbative Chern Simons quantum
invariants via the Kontsevich Integral.

Early on, Lin suggested the Milnor invariants were of finite type, but
this is strictly true only of the string link invariants because Milnor's
invariants are only 'partially' defined, i.e. their indeterminacy depends
on the lower order invariants. The speaker and G. Masbaum actually gave
in 1997 a formula computing the Milnor string link invariants from the
Kontsevich Integral. The tree-like Feynman diagrams correspond to the
Milnor invariants.

The nagging problem that Vassiliev invariants of links are universally
defined, but Milnor invariants, which ultimately gave the link-homotopy
classification, are only partially defined, suggests that finite-type
invariants of links are deficient. It turns out that axiom 6 of the
aforementioned classification sheme is not satisfied so that Vassiliev
(finite type) invariants of links can and ought to be refined, as shown in
a recent preprint by the speaker and JB Meilhan.

See http://www.math.sciences.univ-nantes.fr/~habegger/) for the
aforementioned works. (For those interested in the non-perturbative CS
theory, one can also find at this address the work of BHMV on Topological
Quantum Field Theory derived from the Kauffman bracket, e.g. the Jones'
Polynomial.)

##### Subriemannian Geometry: When You Can't Get There From Here

A more accurate subtitle for this talk would have been "When You Can't Get There From Here By Going
Thataway, You Got To Go Somewhere Else First,", but that was too long to fit. A subriemannian manifold is a sort of
space in which certain directions of travel are illegal. This can describe lots of problems involving systems with
too many degrees of freedom. I'll talk about several examples, including rolling balls, falling cats, Carthaginian
queens, and drunks with planimeters (if you don't know what a planimeter is, go ask John Eggers), and also about
Chow's Theorem, which says that maybe you can get there from here after all. If time permits, there might be some
applications to PDEs and some open problems mentioned. This talk will be accessible to anyone who has heard of a
smooth manifold.

##### A fast multi-prime approach to compute the Hilbert class polynomial

The computation of the Hilbert class polynomial has
applications ranging from explicit class field theory to
cryptography. Several new algorithms to compute it have
been developed during the last 5 years, each having its pro's
and cons. In this talk we will present a significant speed
up of the Chinese remainder theorem approach'. We will give
a detailed run time analysis of the new algorithm, using
tools from both analytic number theory and arithmetic
geometry. The resulting run time is almost optimal: one of
the bottlenecks is writing down the answer.

• ##### 05/08/08

We will have three panelists who have recently found jobs: Steve Butler, post doc, UCLA, Maia Averett, Mills College, Oakland California, and Dave Clark Assistant Professor, Randolf Macon College, Ashland, Virginia They will describe their experiences applying for an academic job. Some of the questions they will answer are: How many applications should I send out? How do I prepare for an interview? What should I write in my cover letter and resume? What are important qualifications for a teaching job, postdoc job, tenure track research job? If you are soon to be on the job market, this is a terrific opportunity to find out the optimal way to apply for jobs.
The discussion will be followed by a question and answer period.

##### Applications of Ramanujan graphs in Cryptography.

This talk will explain a new construction of secure cryptographic hash functions from Ramanujan graphs. First we will explain cryptographic hash functions and the importance of the collision-resistance property. After a brief overview of expander graphs, we will give a construction of provable collision resistant hash functions from expander graphs in which finding cycles is hard.

As an example, we give a family of optimal expander graphs for provable collision resistant hash function constructions: the family of Ramanujan graphs constructed by Pizer. Pizer described a family of Ramanujan graphs, where the nodes of the graph are isomorphism classes of supersingular elliptic curves over $F_p^2$, and the edges are n-isogenies, n a prime different from $p$. When the hash function is constructed from one of Pizer's Ramanujan graphs, then collision resistance follows from hardness of computing isogenies between supersingular elliptic curves.

Joint work with Denis Charles and Eyal Goren

##### Protein Folding, Topological Frustration and Biological Function

In order to function most proteins have to fold to a unique three
dimensional shape. Thus, the same protein sequence helps in both
biological function and protein folding. I will give a brief introduction
to protein folding, the priniciple of minimal frustration and the funneled
energy landscapes. I will then discuss the interplay of folding and function
through the structure of the protein in the context of the beta-trefoil
fold family of proteins.

##### Sum-product estimates for sets of numbers and reals

An old conjecture of Erd\H os and Szemer\'edi states that if $A$
is a finite set of
integers then the sum-set or the product-set should be large. The sum-set of
$A$ is defined
as $A+A=\{a+b | a,b \in A\}$ and the product set is $A\cdot A=\{ab | a,b \in A\}.$
Erd\H os and Szemer\'edi conjectured that the sum-set or the product set is
in the size of $A,$ i.e. $\max (|A+A|,|A\cdot A|)\geq c|A|^{2-\delta}$ for
any positive $\delta$.
I proved earlier that $\max (|A+A|,|A\cdot A|)\geq c|A|^{14/11}/\log{|A|},$
for any finite set of complex numbers, $A.$
In this talk we improve the bound further for sets of real numbers.

##### Brownian Motion

Brownian motion is a stochastic process named after botanist Robert Brown,
who is credited with having discovered it in 1827 after observing the
erratic behavior of pollen grains. It has since become one of the most
important ideas in probability theory and has wide variety of applications
everywhere from physics to finance.

The talk will begin with a brief discussion of the necessary probability
background, including the notions of expectation, martingales, and random
walks. I will then define Brownian motion and discuss some of its more
interesting properties. If time permits, I will also discuss how Brownian
motion is used to define stochastic differential equations.

##### Vriational implicit-solvent modeling and the level-set computation of biomolecular structures and interactions \\ Part I: An overview

Understanding biomolecules---their structures, dynamics, and
interactions with solvent---is essential to revealing mechanisms
and functions of biological systems. While atomistic simulations
that treat both solvent and solute molecules explicitly are usually
more accurate, implicit or continuum solvent models for biomolecules
are far more efficient. With an implicit solvent, the free energy
and structure of an underlying solvation system is described
through the solute particles and the interface that separates the
solutes and solvent.

Dzubiella, Swanson and McCammon [Phys. Rev. Lett.104, 527 (2006)
and J. Chem. Phys. 124, 084905 (2006)] developed a class of
variational implicit-solvent models. Central in these models is a
free-energy functional of all admissible solute-solvent interfaces,
coupling both nonpolar and polar contributions of an underlying
system. An energy-minimizing interface then defines an equilibrium
solute-solvent interface. Cheng et al. [J. Chem. Phys. 127, 084503
(2007)] developed a robust level-set method for numerically
capturing such interfaces.

In this talk, I will give an overview of the recent development
of variational implicit-solvent approach for solvation systems.
I will point out how various kinds of mathematical concepts and
techniques from differential geometry and partial differential
equations can be applied to this approach.

Joint work with Jianwei Che, Li-Tien Cheng, Joachim Dzubiella,
J. Andy McCammon, and Yang Xie.

##### Regular Graphs for Irregular Times

We will introduce regular graphs, and explore the eigenvalues associated to these graphs. We will then see what the eigenvalues of a graph tell us about its expansion. And, why this is actually interesting. Finally we will discuss irregular graphs and what statements we may make about their eigenvalues.

##### Variational implicit-solvent modeling and the level-set computation of biomolecular structures and interactions \\ Part II: The level-set method

Understanding biomolecules---their structures, dynamics, and interactions
with solvent---is essential to revealing mechanisms and functions of
biological systems. While atomistic simulations that treat both solvent
and solute molecules explicitly are usually more accurate, implicit or
continuum solvent models for biomolecules are far more efficient. With an
implicit solvent, the free energy and structure of an underlying solvation
system is described through the solute particles and the interface that
separates the solutes and solvent.

Dzubiella, Swanson and McCammon [Phys. Rev. Lett.104, 527 (2006) and J. Chem.
Phys. 124, 084905 (2006)] developed a class of variational implicit-solvent
models. Central in these models is a free-energy functional of all admissible
solute-solvent interfaces, coupling both nonpolar and polar contributions
of an underlying system. An energy-minimizing interface then defines an
equilibrium solute-solvent interface. Cheng et al. [J. Chem. Phys. 127,
084503 (2007)] developed a robust level-set method for numerically capturing
such interfaces.

In this talk, I will begin with a brief introduction of the level-set method.
I will then give details of the application of this method to the implicit-solvent
computation of nonpolar molecules. Finally, I will present some new results
on the coupling of the level-set method with molecular mechanics for
implicit-solvent modeling of molecules.

This is joint work with Jianwei Che, Joachim Dzubiella, Bo Li, J. Andy McCammon,
and Yang Xie.

##### Invitation to MathStorm

MathStorm is a mathematics consulting group started by
graduate students at UCSD several years ago. It provides an
opportunity for graduate students to help or collaborate
with students and professors from other UCSD departments,
as well as with people from outside the university. I'll
describe some typical problems participants might work on
and discuss why this could be a useful or profitable thing

##### Arithmetic partial differential equations

We develop an arithmetic analogue of linear partial differential
equations in two independent space-time'' variables. The
spatial derivative is a Fermat quotient operator, while the time
derivative is a usual derivation. This allows us to flow''
integers or, more generally, points on algebraic groups with
coordinates in rings with arithmetic flavor. In particular, we
show that elliptic curves have certain canonical arithmetic
flows'' on them that are arithmetic analogues of the convection,
heat, and wave equations. The same is true for the additive and
the multiplicative group and also for modular curves.

##### A Finite Element Based Level Set Method and Simulations of Stefan Problems

The level set method is a numerical technique to track moving interfaces. I will discuss my research on a finite element based level set method and present some simulations of crytal growth and dendritic solidification.

##### Distribution of Segment Lengths in Genome Rearrangements

The study of gene orders for constructing phylogenetic trees was
introduced by Dobzhansky and Sturtevant in 1938. Different genomes
may have homologous genes arranged in different orders. In the early
1990s, Sankoff and colleagues modelled this as ordinary (unsigned)
permutations on a set of numbered genes $1,2,\ldots,n$, with
biological events such as inversions modelled as operations on the
permutations. Signed permutations are used to indicate the relative
strands of the genes, and circular permutations may be used for
circular genomes. We use combinatorial methods (generating functions,
asymptotics, and enumeration formulas) to study the distributions of
the number and lengths of conserved segments of genes between multiple
genomes, including signed and unsigned genomes, and circular and
linear genomes. This generalizes classical work from the 1940s--60s
by Wolfowitz, Kaplansky, Riordan, Abramson, and Moser, who studied
decompositions of permutations into strips of ascending or descending
consecutive numbers. In our setting, their work corresponds to
comparison of two unsigned genomes (known gene orders, unknown gene
orientations).

##### Optimal stochastic impulse control with delayed reaction

We study impulse control problems of jump diffusions with delayed
reaction. This means that there is a delay $\delta>0$ between the
time when a decision for intervention is taken and the time when the
intervention is actually carried out.
We show that under certain conditions this problem can be transformed
into a sequence of iterated no-delay optimal stopping problems and
there is an explicit relation between the solutions of these two
problems.
The results are illustrated by an example where the problem is to
find the optimal times to increase the production capacity of a firm,
assuming that there are transaction costs with each new order and the
increase takes place $\delta$ time units after the (irreversible)
order has been placed.

The presentation is based on joint work with Agn\es Sulem: Optimal
stochastic control with delayed reaction", Applied Mathematics and
Optimization (to appear)

##### \bf \Huge An introduction to Malliavin calculus for L$\acute{e}$vy processes and applications to finance

The purpose of this lecture is to give a non-technical, yet rigorous introduction to Malliavin calculus for L$\acute{e}$vy processes and its applications to finance. The lecture consists of two parts:

Part 1 deals with the Brownian motion case. We first use the Wiener-It\^{o} chaos expansion theorem to define the Mallavin derivative in this context and then study some of its fundamental properties, including the chain rule and the duality property (integration by parts). Then we apply it to finance. Examples of applications are

(i) the hedging formula in complete markets provided by the Clark-Ocone theorem,

(ii) parameter sensitivity results", e.g. a numerically tractable
computation of the delta-hedge" and othergreeks" in finance.

Part 2 deals with the general L$\acute{e}$vy process case. To some extent a similar presentation of the Malliavin derivative can be given here as in Part 1, but there are also basic differences, for example regarding the chain rule. Examples of applications to finance are

(i) optimal hedging in incomplete markets (based on the Clark-Ocone formula L$\acute{e}$vy processes),

(ii) optimal consumption and portfolio with partial information in a market driven by L$\acute{e}$vy processes.

The presentation is mainly based on the forthcoming book
G. Di Nunno, B. $\O$ksendal and F. Proske:
Malliavin Calculus for L$\acute{e}$vy Processes and Applications to Finance".
Springer 2008/2009 (to appear).

##### Self-organized criticality via stochastic partial differential equations

Models of self-organized criticality which can be described by stochastic partial differential equations with noncoercive mono- tone diffusivity function and multiplicative Wiener forcing term (as e.g. the Bak/Tang/Wiesenfeld- and Zhang-models) are analyzed. Existence and uniqueness of nonnegative strong solutions are proved. Previously numerically predicted transition to the critical state in 1-D is confirmed by a rigorous proof that this indeed happens in finite time with high probability.

##### Geometric mechanics of charged ribbons, or orientation - dependent nonlocal interactions along charged filaments

We derive equations of motion for the dynamical folding of
biological molecules (such as DNA), that are modeled as continuous
filamentary distributions of interacting rigid charge conformations.
The equations of motion for the dynamics of such a system are
nonlocal when the screened Coulomb interactions, or Lennard-Jones
potentials between pairs of charges are included. These nonlocal
dynamical equations are derived using Euler-Poincar'e variational
formulations, extending earlier work for exact geometric rods. In the
absence of nonlocal interactions, the equations reduce to the
Kirchhoff theory of elastic rods. An elegant change of variables
separates the dynamics geometrically into "horizontal" and "vertical"
components.

This is joint work with Francois Gay-Balmaz(EPFL), David Ellis (Imperial),
Darryl D. Holm (Imperial), and Tudor Ratiu (EPFL).

##### Towards Quantum Algorithms using the Curvelet Transform

The curvelet transform is a directional wavelet transform, introduced by Candes and Donoho (2002). I will present some preliminary results on curvelet-based quantum algorithms. First, the quantum curvelet transform can be implemented efficiently, for a simple class of Haar curvelets," and possibly for other curvelets as well.

Next, consider the following example. Given a state that is a uniform superposition over a ball in $\mathbb{R}^n$, we compute the (continuous) curvelet transform. We then measure the state, and observe a scale $a$, direction $\theta$ and location $b$. With significant probability, $a$ is small (corresponding to a fine-scale element), and $b$ and $\theta$ determine a line that passes close to the center of the ball. This suggests an interesting quantum algorithm for finding the center of a radial function. However, there remain some technical obstacles in carrying this result over to the discrete setting, and in designing a suitable mother curvelet that can be implemented efficiently.

##### A New Combinatorial Approach to q-Series Identities

In this talk, we present a new combinatorial context for q-series
identities. By simply rewriting many partition theoretic identities,
we can make them more amenable to tilings of an infinitely long
board. The advantage of this approach is that beautiful yet complex
bijections relating two different collections of partitions are
replaced by simple constructions involving the same collection of
tilings. Classical results of Euler, Sylvester, Lebesgue, Rogers,
and Ramanujan will be discussed.

##### An Introduction to the Dynamical Systems of Chemical Reaction Networks

In a chemical reaction network, the concentrations of chemical species
evolve in time, governed by the differential equations of mass-action
kinetics. This talk provides an introduction to the algebraic study
of chemical reaction network theory. Chemical reactions can be
represented by directed edges of a graph. A basic question is whether
such a network has a steady state. The locus of all steady states is
defined by the steady state ideal. We introduce the space of toric
dynamical systems of a digraph representing a chemical network. The
nicest chemical reaction networks are the toric dynamical systems:
their steady state loci and moduli spaces are toric varieties. In
chemistry, they are the systems whose steady states are a special
kind, called complex balancing steady states. We make the connection
also to the deficiency theory of M. Feinberg. No prior knowledge of
chemical reaction network theory or toric geometry will be assumed

This is joint work with Gheorge Craciun, Alicia Dickenstein, and Bernd
Sturmfels.

##### An Introduction to Mathematical Population Genetics and Coalescent Theory

This talk will provide a brief introduction to Mathematical Population Genetics, focusing on Coalescent Theory. We will introduce the classical Wright-Fisher population model and explain why, in this model, the genealogy of the population can be described by Kingman's coalescent, in which each pair of ancestral lineages merges into a single lineage at rate one. We will then derive some consequences of this model. At the end of the talk, we will discuss some recent research related to models of populations with large family sizes.

##### Hat-guessing games and their applications. \footnote{There are no applications.}

Ten prisoners are locked together in a room and told that tomorrow
they'll all be placed in a line, each prisoner facing the back of the
man in front of him. The guards will then put red and blue hats on the
prisoners, so that each man can see the hats in front of him but not
his own hat. Then, from the back of the line to the front, each
prisoner is forced to guess the color of his own hat. Each one who
guesses correctly is set free, while the rest go back to prison --
and, critically, each of the prisoners can hear the guesses of the
people behind him. Is there a good strategy for the prisoners? \\

\noindent We'll examine this problem, a hat-guessing game that's really an
error-correcting code in disguise, and finally a hat-guessing game
that will make you wonder whether the axiom of choice is such a great
idea. The ideas in this talk are simple, and I'll spend most of the
time treating these games as riddles or puzzles rather than talking
about the deeper theories hidden in their solutions.

##### Video Cases Project and Teacher Development

A discussion of the Video Cases Project as a means to examine teaching and to develop a continuing program of teacher development.

##### Developing Video Cases of College Math Instruction

A national cooperative of universities is developing a collection of videos about college math instruction as a resource for helping novice instructors to build their teaching skills. Currently midway through the 3-year project, initial draft video and text materials are ready for review. The purpose of the presentation is to share some of these materials, discuss their potential uses, and gather comments on how to shape and re-develop them. Attendance by all department members, from those very experienced in teaching mathematics to those with a few days experience to those with intentions to teach in the future, is heartily encouraged. Video clips come from advanced as well as introductory undergraduate mathematics teaching and learning situations.

##### Basic Category Theory

Talk time runs from 10:30 until 12:00.

##### Bi-Quadratic Optimization and Semidefinite Programming Relaxations

This talk discusses the so-called bi-quadratic optimization over
unit spheres. We show that this problem is NP-hard and there is no
polynomial time algorithm returning apositive relative approximation
bound.
After that, we present various approximation methods based on
semidefinite
programming (SDP) relaxations. Our theoretical results are: For
general
bi-quadratic forms, we develop a $1/\max{\{m,n\}}$-approximation algorithm;
for
bi-quadratic forms tha are square-free, we get a relative
approximation
bound $1/mn$. When $\min{\{m,n\}}$ is a constant, we present two polynomial
time
approximation schemes (PTASs) which are based on sum of squares (SOS)
relaxation hierarchy and grid sampling of the standard simplex. For
practical computational purposes, we propose the first order SOS
relaxation, a convex quadratic SDP relaxation and a simple minimum
eigenvalue method, and give their quality analyses. Some illustrative
numerical examples are also given.

##### The Art of Subtracting Infinity

We begin this lecture by reviewing the concept of the trace of a
matrix. Then we move on to study the vibrations of a guitar string
of length $\pi$. The noise that the string produces contains a base
tone and higher overtones. In the idealized mathematical model, the
list of frequencies of all these tones are
$$1,\ 2^2,\ 3^2,\ 4^2,\ \dots,$$
and so the wavelengths are
$$1,\ \frac1{2^2},\ \frac1{3^2},\dots\dots$$
We explain why the {\it total wavelength},
$1+\frac1{2^2}+\frac1{3^2}+\dots$ can be computed by integrating the
{\it Green's function} $x(1-x/\pi)$ along the guitar string, giving
an answer of $\pi^2/6$.

The trouble starts when we try to replace the guitar string by a
sphere and carry out the same process. The resulting formula
$$\sum_{n=1}^\infty \frac{2n+1}{n(n+1)}\ =\ \int_{S^2} G(x,x)\,dS(x)$$
simply boils down to $\infty=\infty$. However, by subtracting
infinity very carefully from each side, we obtain an interesting
formula which works for any surface, not just a sphere. Although
this mathematics may never be used to help us design musical
instruments, it is related to the vortex theory of fluids, and
Einstein's relativity.

##### New Permutation Statistics: Alternating Descents and Alternating Major Index

Last year, Denis Chebikin introduced the alternating descent of a
permutation. For a permutation $\sigma = \sigma_1 \ldots \sigma_n$ in
the symmetric group $S_n$, let $AltDes(\sigma) = \{2i+1: \sigma_{2i+1} <\sigma_{2i+2}\} \cup \{2i: \sigma_{2i} > \sigma_{2i+1}\}$. Then we can define $altdes(\sigma) =|AltDes(\sigma)|$ and $altmaj(\sigma) =\sum_{i \in AltDes(\sigma)} i$. We shall show how to derive
the generating function for the joint distribution of $altdes$ and
$altmaj$ over the symmetric group. Slight variations of our technique
also allow us to find similar generating functions for the groups $B_n$
and $D_n$.

##### Image Inpainting

Abstract: Image inpainting involves filling in part of an image (or video) using information from the surrounding area. In this talk I will discuss the connection of Navier-Stokes equations (NSE) in image inpainting. This important connection suggests the possibility of other hybrid methods or turbulence models in image inpainting. Recently, the three-dimensional (3d) Navier-Stokes-Voight (NSV) equations, were suggested as a regularizing model for the 3d NSE. We would like to investigate how we can tune the relevant parameters of this model to optimize the end result in image inpainting.

##### Minimization: One Dimension at a Time

This talk will cover one of the more popular methods for unconstrained
minimization and explore techniques to improve upon it.  In particular,
I will cover the BFGS method, a type of quasi-Newton method, where one
seeks the minimizer by using a sequence of approximate quadratic models.
The most important technique to improve upon this method is developed
from the following idea, which is the heart of the talk: can we minimize
a function over a k-dimensional subspace in such a way as to make
the answer turns out to be 'yes', which in turn motivates similar
techniques for the nonlinear case.  Some benefits are: we operate with
significantly smaller matrices, have a smaller memory footprint, and can
reinitialize the curvature at each iteration at little to no cost, which
dramatically improves convergence when the function is ill-conditioned
near the solution.  Some knowledge of linear algebra will be helpful but
not necessary - I will aim to make the talk as accessible as possible.

##### A generalization of variable-splitting theorems and Segre variety mapping

First of all, I will explain our variable-splitting
theorems (J. Korean Math. Soc. 37 (2000), no. 2, 245--251
and Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2 (2003), no. 3,
433--448).
Then I will formulate a generalization of our theorems as
a parameter version.
Since I believe that Segre variety mapping is a useful tool
to attack the problem, I will give you an another proof of
Kim-Zaitsev-A.H. theorem, which appears in above second
mentioned paper.

##### Benchmarking Derivative-Free Optimization Algorithms

As optimization problems become increasingly complex,
the availability and computability of derivatives
becomes problematic. As a result, it is important to
use so-called direct search methods, which are
optimization algorithms that do not use derivative
information. We present a small collection of test
problems that is designed to facilitate the
benchmarking of different direct search algorithms.
All implementation has been done in Matlab. This is
joint work with Jorge More of Argonne National Lab.

##### Applications of the Combinatorial Nullstellensatz

In this talk I will discuss a combinatorial
specialization of Hilbert's celebrated nullstellensatz.
This version has numerous applications in various
branches of mathematics, such as combinatorics
and graph theory, number theory, and geometry. We will
discuss some striking and elegant proofs in each
of these areas, together with some open problems which
might yield to an approach using the nullstellensatz.

##### Jumping Sequences

\indent Given the cost $w(i,j)$ of jumping from integer $j$ down
to integer $i$ one can ask what is the minimal total sum cost of
jumping from $n$ to $1$, and what can be said about the jumping
pattern that achieves this minimal cost? The optimal jumping patterns
can be encoded into a sequence $a(1),a(2),a(3),\ldots$ so that for any
$n$ the optimal thing is to jump from $n$ to $a(n)$ to $a(a(n))$ and
so on until $a(a(\cdots(a(n))\cdots))=1$, this sequence is called the
jumping sequence for the given weight function.

These sequences arise from a problem of secure network broadcasting.
In this talk we will give some basic results about some sequences
associated with various weights, including showing that for the weight
function $w(i,j)=(i+j)/i^2$ that the only values which appear
arbitrarily deep in jumping patterns are the Pell numbers.

(Joint work with Ron Graham and Nan Zang)

##### Spatiotemporal Statistics: When Truth is Bigger Than Proof

Spatiotemporal statistics (STS) in a modern knowledge synthesis context
accounts for important issues of real-world system modelling and prediction,
including core scientific theories, site-specific information, datasets with
varying physical meanings, critical reasoning modes, and ~Shuman
agent-natural system~T interactions.  The underlying methodological postulate
is that everyday truth is bigger than technical proof, i.e. knowing how to
prove something and effectively process the relevant numerical data do not
necessarily imply that one understands the deeper meaning of what one has
proved and that possesses an adequate interpretation of the data that one has
processed. These considerations are particularly significant in the case of
interdisciplinary real-world systems that vary in a composite space-time
domain under conditions of multi-sourced uncertainty. Accordingly, modern
STS seeks to develop a rigorous mathematical framework that is able to
integrate ~Struth~T with ~Sproof~T in the living experience sense considered
above.

##### On the law of series in ergodic theory

In a discrete time ergodic aperiodic dynamical system, one can look at the recurrence times for small measure sets. As the measure of these sets decrease to zero, within several regular enough setups, one has proved that the normalized hitting time distributions or return time distributions tend to the exponential law. Such limiting distribution is called an asymptotic distribution. We have proved with T. Downarowicz that in a positive entropy system any asymptotic along cylinder sets (whence for symbolic dynamical systems) must be subexponential. This turns out to be interpreted as a rule saying that clusters (grouped repetitions of visits to a set) are whatever at least as large as they can be in the independent process case, where their appearances are ruled by the exponential distribution. We also present some material saying that typically a symbolic dynamical system should present along a sequence of lengths of cylinders of upper density one, absolute clustering.

##### Concentrate Harder: Concentration inequalities and their use in probabilistic combinatorics.

\indent When dealing with random variables, we are often concerned with their
expected value - what we expect them to be. But often this isn't enough;
we don't just want to know the expectation, but we are concerned with
how close to the expectation our random variable is likely to be. Here
is where concentration inequalities come into play: we wish to bound the
probability that a random variable differs greatly from it's mean. The
use of the inequalities of Markov, Chebyshev, Chernoff, Azuma and
beyond have become key fixtures in arguments in probabilistic
combinatorics.

In this talk we attempt to survey these inequalities, and provide some
beautiful (and hopefully illuminating) examples of their use from the
literature (and, perhaps, some examples from my 'day job'.)

Plus, I will attempt to spell Tchebycheff a different way every time I
write it.

##### $1$-Motives, Special Values of $L$-functions, Quillen $K$--theory and Étale Cohomology

We will discuss our recent proof (joint work with C. Greither) of a conjecture linking $\ell$--adic realizations of $1$-motives and special values of equivariant $L$--functions in characteristic $p$, refining earlier results of Deligne and Tate. As a consequence, we give proofs (in the characteristic $p$ setting) of various central classical conjectures on special values of $L$--functions, namely those due to Coates-Sinnott, Brumer-Stark, and Gross. If time permits,
we will indicate how this theory can be extended to characteristic $0$.

##### Trust-search Methods for Unconstrained Optimization

Recent research on interior methods has
re-emphasized the role of sequential
unconstrained optimization for the
solution of nonlinear programming
problems. We focus on the numerical
linear algebra associated with a class of
trust-search'' methods that combine the
best features of line-search and
trust-region methods for unconstrained
optimization.

##### Winning Elections with Point-Set Topology

It is well known that to win a US Presidential election it is not sufficient win the popular vote. Rather, because of the electoral college, the geographical distribution of votes is crucial. Data from past elections shows that a useful statistic describing the distribution of votes is its "dimension". I will explain the topological definition for the "dimension" of a subset of Euclidean space, estimate it for the Kerry and Bush votes in the 2004 US Presidential election, and recall the consequences for the outcome of that election.

##### Mixing of mean-field Glauber dynamics

I will describe the three phases for the mixing time (the time to equilibriate) of the Glauber dynamics for the Ising model on the complete graph on $n$ vertices. At high temperature, the time required to mix is order $n(\log n)$, and there is a cut-off, meaning that in a window of order $n$, the distance to equilibrium drops from near one to near zero. At critical temperature, the dynamics mix in order $n^(3/2)$ steps. At low temperature, the mixing is exponentially slow, but if the dynamics are restricted to one of the two modes of the stationary distribution, then it mixes in order $n(\log n)$ steps. Joint work with M. Luczak and Y. Peres.

##### An esoteric subject noted for its difficulty and irrelevance

We'll think about finite sets and isomorphism, leading to discussion of coherence.

##### A Virtual Screening Study of the Acetylcholine Binding Protein using a Computational Relaxed-Complex Approach

The nicotinic acetylcholine receptor (nAChR) is a member of the
ligand-gated ion channel family and is implicated in many neurological
events. Yet, the receptor is difficult to target without high-resolution
structures. In contrast, the structure of the acetylcholine binding
protein (AChBP) has been solved to high resolution, and it serves as a
surrogate structure of the extra-cellular domain in nAChR. Here we conduct
a virtual screening study of the AChBP using the relaxed complex method,
which involves a combination of molecular dynamics simulations (to achieve
receptor structures) and ligand docking. The library screened through
comes from the National Cancer Institute, and its ligands show great
potential for binding AChBP in various manners. These ligands mimic the
known binders of AChBP; a significant subset docks well against all
species of the protein and some distinguish between the various
structures. These novel ligands could serve as potential pharmaceuticals
in the AChBP/nAChR systems.

##### Jim Lin - UCSD

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

##### Robust Multilevel Preconditioners for PDEs with Jump Coefficients

Making multigrid/DD methods converge (nearly) uniformly for
elliptic equations with strongly discontinuous coefficients
was an open problem. Recently, we proved that the multilevel
and DD preconditioners lead to a nearly uniform convergent
preconditioned conjugate gradient methods. In this talk, I
will present the theoretical and numerical justification of
these results. As an application of these elliptic solvers, I
will also present the auxiliary space preconditioners
(Hiptmair and Xu 2007) for H(curl) and H(div) systems, which
convert solving H(curl) or H(div) systems into solving several
Poisson equations. Another way to interpret these
preconditioners is to cast the H(curl) and H(div) systems into
a compatible discretization framework. Using this framework,
I will derive the algorithm for solving H(div) systems, and
use it to solve the mixed formulation of Poisson equation by
the augmented Lagrange method.

##### An Introduction to Biostatistics

In this seminar we aim to give a first introduction to the field of biostatistics, starting with what statistics is about. In a non-technical fashion, we will elaborate on three areas of biostatistics: clinical trials, survival analysis and computational biology (bioinformatics). Time permitting, we will also briefly describe the biostatistics graduate programs around the country, and the excellent career opportunities with a biostatistics degree.

##### The Langlands-Shahidi Method for the Split Classical Groups over Function Fields and Functoriality

The Langlands-Shahidi method over a global function field will be presented for the split classical groups. We will highlight the difficulties that arise in
positive characteristic with local-global arguments. Important applications to functoriality for generic representations of the split classical groups to an
appropriate GL(N) are then obtained when the Langlands-Shahidi method is used together with a converse of Cogdell and Piatetski-Shapiro. Finally, we will mention how
to go from the weak functorial lift to the strong lift. This has an important application to the Ramanujan conjecture, due to the fact that it is a theorem of Lafforgue for GL(N).

##### Auxiliary AMG Preconditioner for Maxwell Equations with Jump Coefficients

In this talk, I will explain auxiliary space preconditioning technique. Under this framework, we only need to construct certain stable decomposition of the finite element spaces to construct efficient preconditioners. Based on some regular decomposition results on H(curl), we construct a robust and efficient (AMG) preconditioner for Maxwell's equation with jump coefficients.

##### Spatially Adaptive Stochastic Numerical Methods  for Intrinsic Fluctuations in Reaction-Diffusion System

We shall discuss reaction diffusion systems arising in the study of biological systems where concentration fluctuations   play an important role.  This includes the study of cell signaling, cell polarization, and directed cell motility through detection  of shallow concentration gradients.  To study these systems  we shall formulate stochastic partial differential equations to  model intrinsic fluctuations in the concentration fields associated with these processes.  A number of challenges arise in  approximating the solutions of such SPDE's. Further issues  arise when considering non-periodic domains with complex geometries which arise naturally in modeling  biological systems.  We shall also discuss an approach to obtain  spatially adaptive numerical discretization of SPDE's, which specifically address how to discretize stochastic terms at coarse-refined  interfaces.  In the spirit of the seminar we shall try to make the talk accessible to a wide audience with ample time for  informal discussion.

##### Quantum Shannon Decompositions from Cartan Involutions

A number of quantum circuit decompositions and factorization algorithms
have been published in the past few years, the most successful of which
have all exploited the structure of the Lie algebra of the special unitary
group. The work presented here extends upon the best known universal quantum
circuit by making explicit the basis of the circuit's design in the Cartan
Decomposition and providing an explicit factoring algorithm which exploits
this Lie algebraic structure.

##### Mathematical and Numerical Study of Maxwell's Equations in Negative Index Material

Since 2000, there has been a growing interest in the
study of negative index metamaterials across many disciplines.
In this talk, I'll first derive the Maxwell's
equations resulting from negative index metamaterials.
Then I'll discuss some time-domain mixed
finite elements developed for solving these equations,
followed by succinct error estimates.
Finally, some numerical results and open issues will be presented.

##### Braided Monoidal Categories (continued) \\ Tannaka Reconstruction Theorems

\noindent 1. Ben Wilson, Braided monoidal categories (continued)

\noindent 2. Justin Roberts, Tannaka reconstruction theorems

##### Undergraduates' Comprehension of Mathematical Arguments

(Talk runs from 10:00-11:30 A.M.) \\

In reform-oriented mathematics classrooms, students are expected to learn from hearing the arguments of their classmates. In advanced mathematics classrooms, mathematics major are expected to learn from observing their professors present proofs and reading proofs in their textbooks. Both modes of instruction rest on the specific assumption that students are able to learn from the arguments of others. Unfortunately, research suggests that most students are not convinced and do not comprehend the arguments that they observe. in this talk, I will present two studies that investigate the reasoning processes that mathematics majors use when they read mathematical arguments. The data from these studies will be used for three purposes: (1) to provide data on what competences mathematics majors have and lack when reading arguments, (2) to explain why mathematics majors have difficulty with comprehending and evaluating mathematical arguments, and (3) to delineate strategies that successful students use to learn when reading mathematical text.

##### Implementing p-Version of Adaptive Finite Element Methods

The talk focuses on computational aspect of a new version of
Adaptive Finite Element Methods which allows elements of different
degrees.
We shall discuss p-refinement, p-unrefinement algorithms, basis
functions
for transition elements and problems arise in using this method with
a
domain decomposition solver on a parallel computer. Some demos of the
method will be presented.

##### Enumerative Geometry

Enumerative geometry is a branch of geometry devoted
to counting geometric objects. For example, one could ask:
How many lines are there passing through two points? (Easy, that's one
line.) Or one could ask: given five lines in the plane, how many conic sections are there tangent to all five lines?
(Harder, but the answer is still one.) Given a surface defined by
a cubic equation (say $x^3+y^3+z^3=1$), how many straight lines
are contained in the surface? (This was determined in the mid-19th
century, and the answer is 27.) Even harder, given a three-dimensional
object defined by an equation like $x^5+y^5+z^5+w^5=1$, how
many plane conic sections are contained in this object? (Much harder,
the answer is 609,250.) I will give some examples and techniques,
and explain the history of how the field of enumerative geometry

##### The Chromatic Number of a Random $d$-regular Graph

\indent A random $d$-regular graph is simply a graph, drawn uniformly at
random, from the probability space of all $d$-regular graphs on $n$
vertices. Researchers are interested in the properties of this graph that hold
with probability approaching 1 as the number of vertices grows large.

In this talk we discuss some of the main results in this area and the
tools that are used for proving them. We also describe a new result

This is joint work with Xavier P\'{e}rez and Nick Wormald.

##### Non-Reconstruction for Colorings on Trees

Consider $k$-colorings of the complete tree of depth $\ell$ and
branching factor $\Delta$. If we fix the coloring of the leaves, as
$\ell$ tends to $\infty$, for what range of $k$ is the root uniformly
distributed over all $k$ colors? This corresponds to the threshold
for uniqueness of the infinite-volume Gibbs measure. It is
straightforward to show the existence of colorings of the leaves
which freeze'' the entire tree when $k \le \Delta+1$. What happens
for a {\em typical} coloring of the leaves? When the leaves have a
non-vanishing influence on the root in expectation, over random
colorings of the leaves, reconstruction is said to hold.
Non-reconstruction is equivalent to extremality of the free-boundary
Gibbs measure. When $k<\Delta/\ln{\Delta}$, it is straightforward to
show that reconstruction is possible and hence the measure is not
extremal.

We prove that for $C>1$ and $k =C\Delta/\ln{\Delta}$, that the Gibbs
measure is extremal in a strong sense: with high probability over the
colorings of the leaves the influence at the root decays
exponentially fast with the depth of the tree. Closely related
results were also proven by Sly. The above strong form of extremality
implies that a local Markov chain that updates constant sized blocks
has inverse linear entropy constant and hence $O(N\log N)$ mixing
time where $N$ is the number of vertices of the tree.

This is joint work with Juan Vera, Eric Vigoda and Dror Weitz.

##### Basics of Mirror Symmetry

String theory suggests that the universe works in a 10 dimensional
construct: 4-dimensional spacetime and a 6-dimensional compact object called
a Calabi-Yau space. Physics suggests that it might be possible for two
different Calabi-Yaus to give identical physics. These are called mirror
pairs.

What is the geometric relation between these two objects? One incredible
suggestion is that counting curves in one Calabi-Yau is tied to the
coefficients of an integral on the mirror. Up to this point geometers had
enough trouble counting degree 2 or 3 curves in a Calabi-Yau and it usually
involved esoteric machinery. Now it could be calculated by just an integral
on a different space and looking at the coefficients.

I will outline some of conjectures and implications then try to do an
example on a particular three-fold.

##### Wee Teck Gan - UCSD

The Gross-Prasad conjecture says that the
period integral of a cuspidal representation of SO(n+1) x SO(n) over the diagonal SO(n) is nonzero if and only if the central critical value of a certain L-function does not vanish. There is a refinement of the conjecture, due to Ichino-Ikeda, which gives a formula for the central L-value in terms of the period integral. I will explain this refined conjecture, and discuss some joint work with Ichino in the case n=4. The case n=2 is a classic theorem of Waldspurger and the case n=3 is a recent theorem of Ichino.

##### Exceptional Vector Bundles Associated to Degenerations of Surfaces

If a smooth complex surface Y degenerates to a singular surface X
with a ordinary double point $x^2+y^2+z^2=0$, then the specialisation map
$H_2(Y,Z) \rightarrow H_2(X,Z)$ has nontrivial kernel, generated by the so called
vanishing cycle: a 2-sphere in $Y$ which collapses to a point in $X$. However,
in the theory of families of surfaces, more complicated degenerations
naturally occur which have no vanishing cycles. In some of these cases we
construct an exceptional vector bundle on the smooth fibre $Y$ which (roughly
speaking) is trivialised on $X$, and so is analogous to a vanishing cycle.
Here an exceptional vector bundle is a (holomorphic) complex vector bundle
which is rigid (cannot be deformed) and satisfies some additional
properties. They have previously been used to describe derived categories of
algebraic varieties.

##### Simultaneous confidence bands in time series

I will talk about statistical inference of trends in mean
non-stationary models, and mean regression and conditional variance
(or volatility) functions in nonlinear stochastic regression models.
Simultaneous confidence bands are constructed and the coverage
probabilities are shown to be asymptotically correct. The
Simultaneous confidence bands are useful for model specification
problems in nonlinear time series. The results are applied to
environmental and financial time series.

##### Cohomological Invariants of Line Bundle-Valued Forms

Over an algebraic variety, vector bundles with a symmetric
bilinear form, taking values in a possibly non-trivial line bundle,
are of increasing interest in arithmetic geometry. These are the
natural objects on which to define generalized trace maps. However,
until recently these forms have not enjoyed a theory of cohomological
invariants analogous to the Hasse-Witt invariants (when the line
bundle is trivial). I will give some arithmetic situations where line
bundle-valued forms arise, and I will survey the construction of new
invariants for these forms in "mod 4" etale cohomology.

##### The (Unreasonable) Effectiveness of Local Weak Convergence Methodology in Probability

The idea of local neighborhoods of probabilistic discrete
structures (such as random graphs) to the local neighborhood of limiting
infinite objects has been known for a long time in the probability community and has proved to be remarkably effective in proving convergence results in many different situations.\\

Here we shall give a wide range of examples of the above methodology. In
particular

\begin{enumerate}

\item We shall show how the above methodology can be used to tackle problem of flows through random networks, where we have a random network with nodes communicating via least cost paths to other nodes. We shall show in some models how the above methodology allows us to prove the convergence of the empirical distribution of edge flows exhibiting how macroscopic order emerges from microscopic rules.

\item We shall show how for a wide variety of random trees (uniform random
trees, preferential attachment trees from a wide variety of attachment
schemes etc), how the above methodology shows the convergence of the
spectral distribution of the adjacency matrix to a limiting non random
distribution function.

\item Time permitting we shall also show how how one can deduce convergence of the maximal matching for various families of random trees and what this
means about the spectral mass at 0.

\end{enumerate}

Joint work with David Aldous, Steve Evans and Arnab Sen.

##### A glimpse of chaos. Fractals, the Mandelbrot set, and complex dynamics.

Everybody has seen them: Computer generated images of fractals have become
very popular during the 80s and can be found for example on math books
written for a broad audience. But what is the actual mathematics behind
these pictures? This will be a more or less formal introduction to
iteration of holomorphic functions especially polynomial dynamics.
Pictures and a movie included.

##### Asymptotics in Homotopy Theory

At the heart of stable homotopy theory are the homotopy groups of spheres. These describe how to build spaces by attaching disks and are closely related to the study of framed manifolds. In this talk, I will discuss classical and geometric methods to compute these groups, and I will describe current work that governs the large-scale behavior.

##### Moduli of Curves and Hypergraphs

The Grothendieck-Knudsen moduli space $\bar M_{0,n}$ of stable, $n$-pointed
rational curves has a natural stratification given by topological type. It
is a natural question whether the boundary generates the effective cone of
divisors, or whether the one-dimensional strata generate the effective
cone of curves (the Fulton Conjecture). We construct many (non-boundary)
divisors that are generators of the effective cone, as well as rigid
curves intersecting the interior. The main technique is to identify
$M_{0,n}$ with the Brill-Noether locus of a reducible curve given by a
hypergraph. This is joint work with Jenia Tevelev.

##### Modern Total Positivity

Total positivity is the study of real matrices all of whose minors are
positive. I will talk about some old problems in total positivity,
and discuss some relations of total positivity with combinatorics,
representation theory, and algebraic geometry.

##### Lie group and homogeneous variational integrators and their applications to geometric optimal control theory

\indent The geometric approach to mechanics serves as the theoretical
underpinning of innovative control methodologies in geometric control
theory. These techniques allow the attitude of satellites to be
controlled using changes in its shape, as opposed to chemical
propulsion, and are the basis for understanding the ability of a
falling cat to always land on its feet, even when released in an
inverted orientation.

We will discuss the application of geometric structure-preserving
numerical schemes to the optimal control of mechanical systems. In
particular, we consider Lie group variational integrators, which are
based on a discretization of Hamilton's principle that preserves the
Lie group structure of the configuration space. In contrast to
traditional Lie group integrators, issues of equivariance and order-
of-accuracy are independent of the choice of retraction in the
variational formulation. The importance of simultaneously preserving
the symplectic and Lie group properties is also demonstrated.

Recent extensions to homogeneous spaces yield intrinsic methods for
Hamiltonian flows on the sphere, and have potential applications to
the simulation of geometrically exact rods, structures and
mechanisms. Extensions to Hamiltonian PDEs and uncertainty
propagation on Lie groups using noncommutative harmonic analysis
techniques will also be discussed.

We will place recent work in the context of progress towards a
coherent theory of computational geometric mechanics and
computational geometric control theory, which is concerned with
developing a self-consistent discrete theory of differential
geometry, mechanics, and control.

This research is partially supported by NSF grants DMS-0714223,
DMS-0726263, and DMS-0747659.

##### Infinitesimal Arrays of Group-Valued Random Variables

The talk is concerned with the limiting behavior of row sums of infinitesimal arrays of independent random variables taking values in a locally compact Abelian group. By a theorem of K.R. Parthasarathy, any possible limit of such row sums is weakly infinite divisible and as such a convolution product of an idempotent measure, a Dirac measure, a Gaussian measure and a generalized Poisson measure. Following the classical lines sufficient conditions in terms of the characteristics of the above factors are established in order to obtain convergence of the row sums. Specialization to symmetric arrays and to the torus group illustrates the slightly technical results.

##### Quantum Mckay correspondence

Let $G$ be a finite subgroup of SU(2) or SO(3). McKay correspondence describes the classical geometry of G-Hilb (the Hilbert scheme of $G$-clusters in $C^2$ or $C^3$) in terms of the representation theory of $G$. We determine the Gromov-Witten invariants of $G$-Hilb and describe it in terms of a root system canonically associated to $G$. This in particular describes the quantum cohomology of $G$-Hilb in terms of the associated root system.

##### From Molecular Dynamics to Biomechanics: Multiscale Methods for strongly Non-linear Problems

The mathematical modeling of complex problems in, e.g., mechanics,
often
gives rise to heterogeneous and strongly non-linear models, whose
numerical
treatment is far from trivial. For example, the numerical simulation
of
(bio-)mechanical problems based on realistic geometries and material
models puts high demands on the efficiency and reliability of the
simulation
methods, the handling of the geometries, and the design of the
numerical
software. In this talk, we discuss different non-linear multiscale
approaches
for the efficient simulation of constrained and non-linear
minimization
problems and their efficient and problem-open implementation. Within
our
multiscale approach, different models (or energy functionals) on
different
scales are used concurrently in order to resolve scale-dependent
non-linear
effects. Only a proper synchronization of the scale dependent models
on the
different scales will lead to an increase in convergence speed and
robustness.
Thus, particular emphasis has to be put on the transfer between the
different
scales as well as on the convergence properties of the non-linear
multilevel
iteration process. Examples from (bio-)mechanics including large
deformations, strongly
non-linear materials, frictional contact problems, and coupled
multiscale
simulations will illustrate the efficiency and robustness of our
approach.

##### Hilbert's 17th Problem and Global Optimization

This lecture will give a brief introduction of Hilbert's 17th problem: is every nonnegative polynomial a summation of squares of rational functions? After Artin's affirmative solution, there is a generalization to Positivstellensatz on the solvability of semialgebraic systems. The Positivstellensatz can be applied to solve global optimization problems of polynomial functions. The basic tool is semidefinite programming (SDP), a very nice convex optimization model.

##### Sign Patterns and Determinant Expansions for Chemical Reaction Networks

A signed matrix is actually a class of matrices consisting of all matrices whose (i,j)-th entry has the same sign. There is a classical theory associating a graph to such a class and analyzing the sign of determinants of matrices in the class. We extend this to give more refined determinant results. It has been recently observed that substantial numbers of chemical reaction networks (these play a prominent role in systems biology) have dynamics dx/dt = f(x) with the Jacobian of f(x) having a sign pattern or something similar. Our results will be applied to this. \\

Joint work with Bill Helton, Raul Gomez.

##### Independent dominating sets in graphs of girth 5

An early result in probabilistic combinatorics, due independently to
Arnautov, Payan and Lovasz is that if $G$ is a graph on $n$ vertices
with minimum degree $\delta$ then $G$ contains a dominating set of size
of at most $n(1+ \log(\delta+1))/(\delta+1)$. Here, under the further
assumptions that $G$ is of girth at least 5 and is $d$-regular we show
that, effectively, such a dominating set can be taken to be independent.
That is, we show that every $n$-vertex $d$-regular graph of girth 5
contains an independent dominating set of size $(n \log d)/d + O(n/d)$ as $d \to \infty$. We further give examples showing that the
girth requirement is necessary and that regularity is necessary
in the sense that if $G$ has minimum degree $\delta$, the smallest
independent set may be much larger than $(n \log \delta)/\delta$.
This is joint work with A. Harutyunyan, and J. Verstraete.

##### Response to the final report of the National Mathematics Advisory Panel

In response to a Presidential Executive Order, the National Mathematics Advisory Panel issued a controversial report in March 2008. The Report is significant because of its historical genesis, its subject matter--learning--and the strong position that the Report takes on the primacy of quantitative methods in education research. I briefly introduce the Report, and examine the argumentative grammar used to justify the Report's position on scientific evidence'' in mathematics education research. As appropriate, I will weave in points made by commentaries on the Report from leaders in the field. These commentaries will appear in a special December 2008 issue of the \textit{Educational Researcher} journal, for which I served as guest editor.

##### A High-Order Boundary Integral Method for Neumann Boundary Value Problems on Lipschitz Domains

We present a high-order Nystr\"om method for Neumann boundary

##### The Prime Numbers

I will discuss various properties of
the set of prime numbers whose proofs illustrate
the interaction between number theory and other
branches of mathematics, such as complex analysis,
algebra, geometry, topology etc.

##### Dispersive Properties of the Surface Water-Wave Problem

I will speak on the dispersive character of waves on the interface between vacuum and water under the influence of gravity and surface tension. I will begin by giving a preciese account of the formulation of the surface water-wave problem and discusion of its distinct features. They include the dispersion relation, its severe nonlinearity, traveling waves and the Hamiltonian structure. I will describe the recent work of Hans Christianson, Gigliola Staffilani and myself on the local smoothing effect of 1/4 derivative for the fully nonlinear problem under surface tension with some detail of the proof. If time permits, I will explore some oen questions regarding long-time behavior and stability.

##### Singular special Lagrangian $n$-folds

We discuss recent progress on understanding singular special Lagrangian $n$-folds. Our focus will be on joint work with N. Kapouleas using gluing methods to construct a wide variety of special Lagrangian cones in every dimension three and greater.

##### Helicoid-Like Minimal Disks

Colding and Minicozzi have shown that if an embedded minimal disk in $B_R\subset\mathbb{R}^3$ has large curvature then in a smaller ball, on a scale still proportional to $R$, the disk looks roughly like a piece of a helicoid. In this talk, we will see that near points whose curvature is relatively large the description can be made more precise. That is, in a neighborhood of such a point (on a scale $s$ proportional to the inverse of the curvature of the point) the surface is bi-Lipschitz to a piece of a helicoid. Moreover, the Lipschitz constant goes to 1 as $Rs$ goes to $\infty$ . This follows from Meeks and Rosenberg's result on the uniqueness of the helicoid of which, time permitting, we will discuss a new proof. Joint work with C. Breiner.

##### Beta Ensembles, Random Schroedinger, and Diffusion

The beta ensembles of random matrix theory are natural generalizations of
the Gaussian Orthogonal, Unitary, and Symplectic Ensembles, these classical cases corresponding to beta = 1, 2, or 4. We prove that the largest eigenvalues in the general ensembles have limit laws described by the low lying spectrum of certain random Schroedinger operators, providing a new characterization of the celebrated Tracy-Widom laws. As a corollary, a second characterization is available via the explosion probability of an
associated one-dimensional diffusion. A complementary picture is developed
for beta versions of random sample-covariance matrices. (Based on work with J. Ramirez and B. Virag.)

##### How to Win \$1,000,000 I will discuss complexity theory, the area of study that leads to the$\mathrm{P =? \ NP}$problem. By the end we'll formulate all sorts of conjectures that are believed to be true, but no one has any idea how to solve. These include the enigmatic$\mathrm{P = ?\ NP}$. If you could solve this problem, you would win the admiration of every complexity theorist along with the \$1 million Clay Math Millennium prize. Isn't
that worth an hour of your time?

##### The Topology of DNA-Protein Interactions

\indent The central axis of the famous DNA double helix is often topologically constrained or even circular. The topology of this axis can influence which proteins interact with the underlying DNA. Subsequently, in all cells there are proteins whose primary function is to change the DNA axis topology -- for example converting a torus link into an unknot. Additionally, there are several protein families that change the axis topology as a by-product of their interaction with DNA.

This talk will describe typical DNA conformations, and the families of proteins that change these conformations. I'll present a few examples illustrating how Dehn surgery methods have been useful in understanding certain DNA-protein interactions, and discuss the most common topological techniques used to attack these problems.

##### The $s$-invariant of the Cappell-Shaneson Spheres

The smooth 4-dimensional Poincare conjecture is something of an orphan. No significant progress has been made in a while, and no one is even really sure whether it's true or false. Some plausible counterexamples have been known for over 20 years, and I'll tell you about a particular family of these, the Cappell-Shaneson spheres, which we've recently been thinking about again. The obvious approach to a counterexample is to find an invariant which distinguishes it from the standard 4-sphere; sadly no such invariants are known. We're taking a different approach by extracting a local' problem, involving the slice genus of certain knots and links. Rasmussen's $s$-invariant, related to the Khovanov homology of a link, gives bounds on the slice genus, and thence a potential obstruction. Unfortunately, the links are huge, and calculating the s-invariant is hard. Nevertheless we've made some progress (a potientially dangerous shortcut, a new algorithm, and a new method of extracting the s-invariant), and even have an answer in one case. (With Michael Freedman, Robert Gompf and Kevin Walker)

##### Computing Equilibria of Continuous Games

There has been much recent interest in effective methods to compute Nash or
correlated equilibria for finite games. In this talk we present an overview
of some of our recent results for the computation of equilibria in games
where the players have an infinite number of pure strategies. In particular,
we discuss games where the payoff functions are a polynomial expression of
the actions of the players. In the zero-sum case, we show that the value of
the game, and the corresponding optimal mixed strategies, can be computed by
solving a single semidefinite programming problem, thus providing a natural
generalization of the well-known LP characterization of finite games. We
also discuss some further extensions to the general nonzero sum case, for
both Nash and correlated equilibria. Much of the material is joint work with
Asu Ozdaglar and Noah Stein.

##### An Exotic Sphere with Positive Sectional Curvature

I'll discuss joint work with Peter Petersen that shows that the Gromoll-Meyer exotic 7-sphere admits positive sectional curvature. I'll discuss the history of the problem and give a coarse outline of our solution.

##### The Steenrod Problem of Realizing Polynomial Cohomology Rings

In 1960 Norman Steenrod asked which graded polynomial rings occur as the cohomology ring of a space. Progress on this question has been made throughout the last fifty years, and the final step in its solution was recently given by K. Andersen and myself. My talk will be a survey of this problem and its solution.

##### TBA

Talk time runs until 10:30.

##### Approximating Maxcut in Infinite Graphs

Talk time runs until 11:00.

##### Knot Invariants via Algebraic Geometry

We explain how to construct knot and tangle invariants (such as Khovanov homology) by studying holomorphic vector bundles on certain compact, complex manifolds. Topologically these complex manifolds are just products of the same projective space $P^1$. Conjecturially, if one used Grassmannians $\mathrm{Gr}(k,n)$ instead of projective spaces this would give a series of new knot invariants analogous to Khovanov homology. \\

\noindent (Joint work with Joel Kamnitzer.)

##### An Invitation to Algebraic Statistics

Algebraic statistics is concerned with the development of techniques in
algebraic geometry, commutative algebra and combinatorics, to address
problems in statistics and its applications. This lecture gives an
elementary and self-contained introduction to this subject. Particular
emphasis will be placed on connections to optimization. Along the way,
the speaker will advertise his new book with Mathias Drton
and Seth Sullivant, as well as an upcoming MSRI workshop.