This is the organizational meeting for our topology learning semiar. The topic this quarter is formal group laws and their applications in topology. Nitu will give an introduction to the topic and we will distriubute the talks.

This lecture will talk about semidefinite programming (SDP) and
its applications in global polynomial optimization. Firstly, after
introducing SDP, we will how to represent k-elliptic curves by SDP.
Secondly, after an overview of the sum of squares (SOS) relaxation, which
can be reduced to SDP, we will present gradient SOS relaxation. While the
general SOS relaxation has a gap in finding the global minimum, the gradient
SOS relaxation can find the global minimum whenever a global minimizer
exists. Lastly, we will show how to exploit sparsity in SOS and its
applications in sensor network localization.

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
mathematics research in overcoming these obstacles. In
particular, I will discuss some of the mathematical models
used in studying fusion stability and refueling, how
solutions to those models may be approximated, and introduce
some model improvements to better simulate fusion processes.

I will begin by explaining the statement of the Homological
Mirror Symmetry conjecture for Fano toric varieties and outline how
Lefschetz fibrations have been used to prove the conjecture in some cases.
I will then show how Mikhalkin's flavour of tropical geometry can be used
to prove half of the homological mirror conjecture for all smooth
projective toric varieties (dropping the Fano condition!).

We introduce the concept of automatic structure. Informally, these
are structures that can be defined in terms of automata. By automata
we mean any of the following machines: finite automata, tree automata,
Buchi automata, and Rabin automata.

Nerode and the speaker initiated the systematic study of automatic
structures in 94. An important property of automatic structures is
that these structures are closed under the first order interpretations
and have effective semantics. In particular, the first order theory
of any automatic structure is decidable.

The theory of automatic structures has become an active research
area in the last decade with new and exciting results. In this talk
we survey recent results in the area and outline some of the interesting
proofs. The talk will provide many examples.

Some of the results of the talk are published in LICS 01-04 and
STACS04 conference proceedings. Results are joint with Nerode,
Rubin, Stephan, and Nies.

I will begin with the basics of the two subjects, with the goal of
explaining how each has been used as a tool in the other. The first half
will be devoted to foundational results, dating to the 1930s for potential
theory and the 1980s for complex dynamics. The second half will be
devoted to more recent developments.

Many mathematical objects come in continuous families, prompting the
desire to define a ``universal family'' that contains each such object
exactly once up to isomorphism. When this isn't possible (because the
family would be too bad to be worthwhile -- I'll talk about this
behavior),
we can try to come close, by including only ``stable'' objects.

Frequently the universal family is constructed by starting with a
bigger family that includes each object many times, then dividing
by a group action that implements the isomorphisms. There are two ways to
do this, one algebro-geometric (complex) and one symplecto-geometric
(real), and I'll give some idea of why they agree.

The main example will be the space of $N$ ordered points on the
Riemann sphere, modulo M\"obius transformations. These are unstable if two

K-theory is a valuable tool with applications in various areas of mathematics, including topology, geometry, algebraic geometry, and number theory. In this talk I will attempt to give a rough outline of what K-theory is about, in the algebraic as well as the geometric context. Given a ring R and a space X, I will define the groups K(R) and K(X), and explain the relation between the two. I also hope to outline some applications in various fields. This talk should be accessible and I hope interesting to students of both algebra and geometry alike.

Vanishing theorems for cohomology groups are one of the
essential tools of modern algebraic geometry, and have particularly
important applications in higher dimensional geometry. Under strong
positivity assumptions on line bundles, for example ampleness, there are
well-known "standard" vanishing theorems, like those of Kodaira, Nakano
and Kawamata-Viehweg. They have very useful partial analogues, called
Generic Vanishing Theorems - first discovered by Green and Lazarsfeld -
when the positivity hypotheses are weakened. I will describe all of the
above and their importance, and then explain that recent techniques based
on Fourier-Mukai functors and homological algebra can be used to widely
extend the context of generic vanishing, and relate it to standard
vanishing. As an application, I will explain how to generalize the results
of Green-Lazarsfeld to a version of Kodaira vanishing under weak
positivity hypotheses.

We describe various results on Galois groups of iterates of a given quadratic polynomial. Joint work with Rafe Jones (University of Wisconsin).

We consider sufficient conditions for regularity
of Leray-Hopf solutions of the Navier-Stokes equation. By a
result of Neustupa and Panel, a Leray-Hopf weak solution is
regular provided a single component of the velocity is
bounded. In this talk we will survey existing and present
new results on one component and one direction
regularity. We will also show global regularity for a class
of solutions of the Navier-Stokes equation in thin
domains. This is a joint work with M. Ziane.

One of the central problems in the study of von Neumann algebras
is to find computable invariants which can distinguish nonisomorphic
algebras. In the 1980s, Dan Voiculescu developed a noncommutative probability
theory in order to understand a particular class of such von Neumann algebras.
Specifically, he defined subsets of $R^n$ called microstate spaces which
model the behavior of a generating set of a given von Neumann algebra.
Since these spaces are subsets of $R^n$, classical analytic tools such as
volume can be applied to them.

I will discuss how the application of ideas from geometric measure theory
to microstate spaces has provided insight into the general problem of
invariants and answered some longstanding questions in von Neumann
algebras.

Consider the complete regular binary tree of depth M oriented from the root to the leaves. To each node we associate a random variable and those variables are assumed to be independent. Under the null hypothesis, these random variables have the standard normal distribution while under the alternative, there is a path from the root to a leaf along which the nodes have the normal distribution with mean A and variance 1, and the standard normal distribution away from the path. We show that, as M increases, the hypotheses become separable if, and only if, A is larger than the square root of 2 ln 2. We obtain corresponding results for other graphs and other distributions. The concept of predictability profile plays a crucial role in our analysis.

Joint work with Emmanuel Candes, Hannes Helgason and Ofer Zeitouni.

Vertex algebras arose out of conformal field theory, and were
first defined mathematically by Borcherds in 1986. Since then, they have
found applications in many areas of mathematics, including representation
theory, number theory, finite group theory, and geometry. Vertex algebras
are vector spaces (generally infinite-dimensional) which are equipped with
a family of bilinear products (indexed by the integers) which in general
are neither commutative nor associative. In many ways they behave like
ordinary associative algebras, and the usual categorical and formal
algebraic notions like homomorphisms, ideals, quotients, and modules over
vertex algebras are easy to define.

In this talk, I'll define vertex algebras, give some basic examples,
indicate how to do computations, and hopefully state some interesting open
problems.

How can you tell the dimension of a manifold? One answer lies in studying
the flow of heat on the manifold. {\em Heat flow} is a smoothing process on
Riemannian manifolds, whose long-term behaviour is intimately linked to
global geometry. However, the {\em short-time} smoothing behaviour is
universal: it depends only upon the dimension of the manifold,
and determines the dimension uniquely.

\medskip

In non-commutative geometry, the over-arching principal is to study a
non-commutative algebra, pretend it is an algebra of smooth functions
or differential operators on a {\em non-commutative manifold}, and import
analytic and algebraic tools from global analysis to discover geometric
facts about this manifold.

\medskip

While using heat flow is an excessively difficult way to determine the
dimension of a manifold, it yields one approach to define dimension for
non-commutative manifolds. In the context of {\em free probability} (one
branch of non-commutative geometry concentrating on analytic properties
of free groups), this leads, inexorably, to the somewhat comical-sounding
conclusion that {\em all free groups have dimension $6$}.

\medskip

In this talk, I will outline those aspects of free probability which relate to
heat kernel analysis, and make the connection between dimension and
heat flow clear. I will also discuss recent joint work with Roland Speicher,
showing that {\em all free semigroups have dimension $4$}.

I will discuss a recent proof of a weak upper bound on the
coarsening rate of the discrete-in-space version of an
ill-posed, nonlinear diffusion equation. The continuum
version of the equation violates parabolicity and lacks
a complete well-posedness theory. In particular,
numerical simulations indicate very sensitive
dependence on initial data. Nevertheless, models based
on its discrete-in-space version, which I will discuss,
are widely used in a number of applications, including
population dynamics (chemotactic movement of bacteria),
granular flow (formation of shear bands), and computer
vision (image denoising and segmentation). The bounds
have implications for all three applications. This is
joint work with Selim Esedoglu (U. of Michigan
Mathematics).

In this talk I will introduce the study of finite
dimensional division algebras and the Brauer group, and I will discuss
the fundamental problem of computing the index of a Brauer class. It
turns out that algebraic geometry can play an important role in this
problem. In particular I will describe the theory of twisted sheaves
and how it allows one to solve this problem in certain cases.

The study of cycles on homogeneous varietes has seen a great deal of
activity in the past few years. In particular, new results about
cycles on quadric hypersurfaces has resulted in fundamental
breakthroughs in the theory of quadratic forms.

The goal of this talk will be to answer a basic question about cycles
on homogeneous of more general types. In particular, we will address
the problem of calculating the group of zero dimensional cycles on
such varieties. In the case of quadrics, this was first done by Swan
in 1989 (and also independently by Karpenko). Similar computations
were made later for certain classes of other homogeneous varieties by
Merkurjev and Panin.

By using the geometry of Hilbert schemes of points on homogeneous
varieties, we will describe how to extend the previous results and to
compute the group of zero cycles for some homogeneous varieties of
each of the classical types.

A new proof of the hook formula
Abstract: The hook-length formula is a well known result expressing the
number of standard tableaux of shape $\lambda$ in terms of the lengths
of the hooks in the diagram of $\lambda$. Many proofs of this fact have
been given, of varying complexity. I'll give a new and simple proof
which uses only some power series and partial fractions expansions.
Other versions of the hook formula will also be discussed.

Suppose a domain has a defining function which is plurisubharmonic at each boundary point.
Does it have a plurisubharmonic defining function?
I will report on some recent progress on this question. This is joint work with Anne Katrin Herbig.

Exotic spheres, manifolds homeomorphic but <italic>not diffeomorphic</italic> to the standard sphere, had eluded mathematicians since the dawn of differential topology. In 1956, John Milnor found one in seven dimensions. His short paper on the result stunned the math world and won him the Field's medal. In this talk, we'll survey the ideas of smooth structures on manifolds and smooth vector bundles, and with a few smooth moves we'll construct Milnor's exotic sphere.

Note: This is meant to be a highly accessible talk; some familiarity with algebraic topology will enhance the experience, but is not strictly necessary!

Elliptic cohomology is a field at the intersection of number theory,
algebraic geometry and algebraic topology. Its definition is very
technical and highly homotopy theoretic. While its geometric definition
is still an open question, elliptic cohomology exhibits striking formal
similarities to string theory, and it is strongly expected that a
geometric interpretation will come from there.

To illustrate the interaction between the two fields, I will speak about
my work on orbifold genera and product formulas:
After a very informal introduction to elliptic cohomology, I will
discuss string theory on orbifolds and explain how a formula by
Dijkgraaf, Moore, Verlinde and Verlinde on the orbifold elliptic genus
of symmetric powers of a manifold motivated my work in elliptic
cohomology. I will proceed to explain why elliptic cohomology provides a
good framework for the study of orbifold genera.

Superspecial abelian varieties are positive characteristic
phenomenon. These are abelian varieties enjoying many special, even
super special l;-) properties, which I'll attempt to explain. After some
background about stratifications of moduli spaces and how, in that
context, the superspecial locus is the smallest, I will describe some
more recent work on superspecial abelian varieties. In particular, work
of M.-H. Nicole and A. Ghitza that makes use of the superspecial locus
to construct modular forms and mention very briefly some beautiful
graphs that can be constructed from the superspecial locus.

The relation between modular forms and Galois representations
is more popular than ever, thanks to recent progress on the
Fontaine-Mazur conjecture by Kisin, and on Serre's conjecture by Khare
and Wintenberger. After a short introduction to this circle of ideas, I
will discuss ongoing work on higher-dimensional generalizations, in the
context of Gross' philosophy of modular Galois representations.
I will attempt to make the talk as self-contained and accessible as
possible.

By a tiled order we mean a right Noetherian prime semiperfect and semidistributive ring with the nonzero Jacobson radical. For example, a serial tiled order $A$ is a Noetherian (but non-Artinian) serial indecomposable ring. A ring $A$ is decomposable if $A = A_1 \times A_2$, otherwise $A$ is indecomposable. Every serial tiled order $A$ is hereditary and Gorenstein, i.e., $inj.dim_A A_a = inj.dim_A {_A} A = 1$.

Let $R(A)$ be the Jacobson radical of a tiled order $A$. For any tiled order $A$ there exists a countable set of two sided ideals $I_1 \supset I_2 \supset \dots$, where $R^2 (A) \supset I_1, I_{k+1} \neq I_k$ and all quotient rings $A/I_k$ are Frobenius.

For any permutation $\sigma \in S_n$ there exists a Frobenius ring $B$ with the Nakayama permutaion $\sigma$. We consider the exponent matrices of tiled orders, in particular, Gorenstein matrices. We discuss the relations between exponent matrices and quivers of tiled orders, cyclic Gorenstein orders and doubly stochastic matrices, Gorentstein matrices and tiled orders of injective dimension one.

The Schur functions can be decomposed into "nonsymmetric Schur
functions" obtained through a certain specialization of Macdonald
polynomials. We explore several combinatorial properties of these
polynomials and a connection to crystal graphs.

How many times can a prime number be expressed as the sum of n squares? What is the asymptotic distribution of integer points on a family of ellipsoids $ax^2 + by^2 + cz^2 = n$ as $n$ tends to infinity?
I will explain how modular forms can be used to address these questions and others.

In this talk we will survey the development of the Green's
function for the Boltzmann equation. The talk will include
the motivation from the field of hyperbolic conservation
laws, the connection between the Boltzmann equation and the
hyperbolic conservation laws, and the particle-like and the
wave-like duality in the Boltzmann equation. With all these
components one can realize a clear layout of the Green's
function for the Boltzmann equation. Finally we will present
the application of the Green's function to an
initial-boundary value problem in the half space domain.

Variability in the number of simultaneous flows present can have a substantial impact on the perceived performance of packet networks such as the Internet. While the packet level behaviour of a given set of flows is by now well understood, less is known about the stochastic behaviour of the number of flows in progress on different routes through the network. In this talk we describe recent work on Brownian models of networks in heavy traffic. Joint work with Ruth Williams.

Algebraic K-theory is a highly complicated invariant of
algebraic varieties that encodes arithmetic, geometric and algebraic
information.
In this talk, I will try to make this distinction somewhat less vague and
explain how to isolate some of the algebraic and geometric information
K-theory provides about singularities, leading to proofs of various
longstanding conjectures.

This talk describes how a symbolic computer algebra
tool (NCAlgebra) that handles symbolic matrix
(noncommutative) products can be used to assist the
numerical solution of semidefinite programs where the
variables are matrices. The idea is to keep matrix
variables aggregated at all steps of a primal-dual
interior-point algorithm in which symbolic expressions
are automatically generated and used iteratively.

For a given elliptic curve $E$ over a finite field $F_q$, we let $N_k =
\#E(F_{q^k})$, where $F_{q^k}$ is a $k$th degree extension of the finite
field $F_q$. Because the Zeta Function for $E$ only depends on $q$ and
$N_1$, the sequence $\{N_k\}$ only depends on those numbers as well.
More specifically, we observe that these bivariate expressions for $N_k$
are in fact polynomials with integer coefficients, which alternate in sign
with respect to the power of $N_1$.

This motivated a search for a combinatorial interpretation of these
coefficients, and one such interpretation involves spanning trees of a
certain family of graphs. In this talk, I will describe this
combinatorial interpretation, as well as applications and directions for
future research. This will include determinantal formulas for $N_k$,
factorizations of $N_k$, and the definition of a new sequence of
polynomials, which we call elliptic cyclotomic polynomials.

One of the important features of elliptic curves which makes them the
focus of contemporary research is that they admit a group structure.
During the remainder of this talk I will describe chip-firing games, how
they provide a group structure on the set of spanning trees, and numerous
ways that these groups are analogous to those of elliptic curves.

The existence of K\"ahler-Einstein metrics on a compact K\""ahler manifold

Many mathematical objects come in continuous families, prompting the
desire to define a ``universal family'' that contains each such object
exactly once up to isomorphism. When this isn't possible (because the
family would be too bad to be worthwhile -- I'll talk about this behavior),
we can try to come close, by including only ``stable'' objects.

Frequently the universal family is constructed by starting with a
bigger family that includes each object many times, then dividing
by a group action that implements the isomorphisms. There are two ways to
do this, one algebro-geometric (complex) and one symplecto-geometric
(real), and I'll give some idea of why they agree.

The main example will be the space of $N$ ordered points on the
Riemann sphere, modulo M\"obius transformations. These are unstable if

Many quantum games can be understood as protocols for the communication and processing of quantum information, and should be compared to classical games with communication. After introducing two of the standard quantum game protocols, I'll explain how this comparison works, and its consequences.

Waldspurger established a connection between the vanishing of
certain L-values and the vanishing of period integrals over tori. Subsequent
work of Gross, Zhang and others has, in certain cases, made this connection
more precise. I will describe a different approach, via the relative trace
formula, to obtain such refinements in general. This is joint work with
Kimball Martin.

One of the most important choices a graduate student will make in their graduate career 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 55 UCSD math 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 find a problem for a thesis?
The goal of this seminar is to share information about the process of finding a thesis advisor. We will have three graduate students-Manda Riehl, Jonathan Armel and Kristin Jehring describe their experiences finding a thesis advisor and what happens after a thesis advisor is found.
We will also have two faculty, Lance Small and Michael Holst describe what they look for in a graduate student before they accept him or her as their thesis student. Jim Lin will serve as moderator. All students, especially first, second and third year students, are cordially invited to attend.

The study of minimal surfaces is a central problem in geometry and analysis that dates back to the 1700's when the catenoid and helicoid were discovered. I will survey recent advances, focusing on joint work with Bill Minicozzi that describes the structure of a general embedded minimal surface in terms of the catenoid and helicoid. I hope to give an overview of how these results have played a role in the solution of some old problems.

We consider the conformal decomposition of Einstein's
constraint equations introduced by Lichnerowicz and York, on a
compact manifold with boundary. We show that there exists a solution
to the coupled Hamiltonian and momentum constraint equations when the
derivative of the mean extrinsic curvature is small enough, and
assuming that the Ricci scalar of the background metric is bounded,
though it can change sign on the manifold. The solutions are in
general not uniquely determined by the source functions and boundary
data. The proof technique is based on finding barriers for the
Hamiltonian constraint equation which are independent of the
solutions of the momentum constraint equation, and then using
standard fixed-point methods for increasing operators in Banach
spaces. This work generalizes a previous work by Isenberg and
Moncrief on closed manifolds.

A major advance in combinatorics occured in 1972,
when E. Szemeredi provided a complicated and ingeneous combinatorial
proof of the Erd\"os-Turan conjecture: every subset

Many quantum games can be understood as protocols for the communication and processing of quantum information, and should be compared to classical games with communication. After introducing two of the standard quantum game protocols, I'll explain how this comparison works, and its consequences.

\noindent Suppose we view the three-dimensional sphere as: $S^3 = \{(z,w) \subset \mathbb{C}^2|\ |z|^2 + |w|^2 = 1\}. $ If we are given a complex curve $V_f = \{(z,w)|0 = f(z,w) \in \mathbb{C} [z,w]\},$ we can then examine the intersection $K = V_f \cap S^3.$
In the transverse case, this intersection $K$ will be a link i.e. an embedded
one-manifold in the three-sphere. This talk will be interested in the question:
Question: Which links can arise from complex curves in the above manner?
I will discuss the history of this problem, focusing first on the case where
$f(z,w)$ has an isolated singularity at the origin where the question is completely
answered. I’ll then discuss how a powerful set of knot invariants
defined by Ozsvath and Szabo and independently by Rasmussen using the
theory of pseudo-holomorphic curves can provide information on the above
question. More precisely, Ozsvath and Szabo and Rasmussen defined a numerical
invariant of knots, denoted $\tau(K)$, which we show provides an obstruction
to knots arising in the above manner. More surprisingly, suppose
we focus on knots whose exteriors, $S^3 - K$, admit the structure of a fiber
bundle over the circle, the so-called $fibered$ knots. In this case we show
that $\tau(K)$ detects exactly when a fibered knot arises as the intersection of
the three-sphere with a complex curve satisfying a certain genus constraint.
Our proof relies on connections between Ozsvath-Szabo theory and certain
geometric structures on three-manifolds called contact structures.

A set of permutations on $n$ points is intersecting if, for
any two of its elements, there is some point which is sent to the same
point by both of them. How large can such a set be? Similarly, a set
of partial permutations (meaning injections defined on some $r$ points
of the $n$-set, for some fixed $r$) is intersecting if, for any two of its
elements, there is some point on which they are both defined and is
sent to the same point by both of them. Again, how large can such a
set be?

We shall survey and discuss some results on these problems. We will
also mention some fascinating conjectures in this area.

This talk includes joint work with Peter Cameron and Imre Leader.

New probability models are proposed for the analysis of marked point
processes. These models deal with the type of data that arrive
or are observed in possibly unequal time intervals such as
financial transactions, earthquakes among others. The models
treat both the time between event arrivals and the observed marks
as stochastic processes. We adopt a class of bivariate
distributions to form the bivariate mixture transition
distribution(BMTD). In these models the
conditional bivariate distribution of the next
observation given the past is a mixture of conditional
distributions given each one of the last p observations or a
selection of past p events. The identifiability of the model is
investigated, and EM algorithm is developed to obtain estimates
of the model parameters. Simulation and real data examples are
used to demonstrate the utility of these models.

Last time I described the (na\"ive) algebraic geometry way to define

For K a C.M. abelian extension of a totally real base-field k with
Galois group G, Solomon has recently constructed for each prime p a Z p[G] ideal of Q p[G] related to values of Dirichlet L-functions at s=1 and conjectured that this ideal is contained within Z p[G]. Jones has subsequently shown that for each odd p the Equivariant Tamagawa Number Conjecture (or ETNC) implies that Solomon's ideal should actually be contained within the Fitting ideal of the class-group of O K. I shall explain how to define analogous ideals related to values of Dirichlet L-functions at integers r strictly greater than 1 and provide a sketch of the techniques used to show that the ETNC relates these `higher Solomon ideals' to the Fitting ideals of certain natural cohomology groups (and thus, when the Quillen-Lichtenbaum conjecture is valid, to Fitting ideals of Quillen K-groups of O K). In particular, for certain choices of K/k and r these results are
unconditional as the relevant cases of the ETNC and Quillen-Lichtenbaum conjecture are known to be valid.

We analyze stochastic volatility effects in the context of the bond market. The short rate model is of Vasicek type and the focus of our analysis is the effect of multiple scale variations in the volatility of this model. Using a singular perturbation approach we can identify a parsimonious representation of multiscale stochastic volatility effects. The results are illustrated with numerical simulations. We also present a framework for model calibration and look at applications to bond option pricing.

Evaluating certain Siegel modular functions at CM points on the moduli space of principally polarized abelian surfaces give algebraic
numbers which we call class invariants. The construction of class
invariants is motivated by explicit class field theory, specifically,
the construction of units with possible applications to Stark conjectures.
Class invariants can also be viewed as invariants of the binary sextic
defining a genus 2 curve whose Jacobian corresponds to the CM point
on the moduli space. The explicit construction of genus two curves
with CM is motivated by cryptographic applications.

When evaluating certain Siegel modular functions at CM points,
the coefficients of the minimal polynomials have striking factorizations.
In joint work with Eyal Goren, we studied primes that appear in the
factorization of the denominators, and proved a bound on such primes
closely related to the discriminant of the CM field. In more recent work,
we study the primes appearing simultaneously in the numerators of
CM values of certain Siegel modular functions in dimension 2.
This work generalizes the work of Gross and Zagier for the modular
j-function and is related to a conjecture of Bruinier and Yang on
intersection numbers.

All groups under our consideration are finitely generated. Asymptotic cones (AC) of groups were
introduced by M.Gromov in 1981. He used them for the description of groups with polynomial
growth. AC of groups are homogeneous geodesic, metric spaces. There exists a group having
non-homeomorphic cones. All AC of G are R-trees iff the group G is hyperbolic. In a recent joint
paper with D.Osin and M.Sapir, we called a group G lacunary hyperbolic (LH) if at least one AC
of G is an R-tree. We characterize LH groups as direct limits of hyperbolic groups satisfying
certain restrictions on the hyperbolicity constants and injectivity radii. We show that the class
of LH groups is very large. Many group-theoretical couner-examples (E.G., some Tarski
monsters) are LH groups. Among new examples, we construct a group having an AC with
a non-trivial countable fundamental group. This solves Gromov's problem of 1993.

I'm going to explain the theorem of Hopkins and Miller (and partly Goerss) that gives a version of Lubin-Tate deformation theory in the context of algebraic topology. More concretely, the theorem says that there is a functor $(k,\Gamma) \mapsto E_{(k,\Gamma)}$ from formal groups laws over perfect fields of characteristic $p>0$ to a very nice category of commutative ring spectra, namely $E_{\infty}$-ring spectra. It has the property that the formal group law of the cohomology theory associated with the ring spectrum $E_{(k,\Gamma)}$ is the universal deformation of $(k,\Gamma)$. By functoriality, we obtain an action of the (extended) Morava stabilizer group on the spectrum $E_{(\mathbb{F}_{p^n},H_n)}$, where $H_n$ denotes the Honda formal group law of height $n$. One application is the construction of the higher real $K$-theories $EO_n$ as the homotopy fixed point spectra obtained from the action of finite index subgroups of the Morava stabilizer subgroup.

Markov models on trees arise naturally in many fields, notably in molecular
biology - as models of evolution; in statistical physics - as models of
spin systems; and in networking - as models of broadcasting. In this talk,
I will discuss various inference problems motivated especially by
applications in statistical phylogenetics, i.e. the reconstruction of
evolutionary histories of organisms from their molecular sequences. In
particular, I will consider the "root reconstruction" problem: how
accurately can one guess the value at the root of the tree, given the state
at the leaves? I will focus on recent work establishing new conditions for
the impossibility of such reconstruction. I will also discuss the related
"phylogenetic reconstruction" problem: given enough samples at the leaves,
can one reconstruct the tree that generated this data and, if so, how
efficiently? I will present a recent result on a sharp transition in the
number of samples required to recover the tree topology, using a connection
to the root reconstruction problem above. Time permitting, I will describe
briefly connections to computational learning theory and network tomography
as well. This is joint work with S. Bhamidi, C. Borgs, J. Chayes, C.
Daskalakis, E. Mossel, and R. Rajagopal.

In this talk, I will talk about an incredibly wonderful theorem of Charles
Fefferman about biholomorphisms of strictly pseudoconvex domains. I will
also talk about a certain kernel named after a famous movie actress. You
will find out what is a biholomorphic mapping, strictly pseudoconvex domain,
kernel, etc... I might also tell you some historical background, where I
will make up the bits that I don't actually know.

I will present some recent results on the Serrin-type conditional regularity of the Navier-Stokes equations. Basically, if one component of the weak solution of the Navier-Stokes equation belongs to a Serrin type space of regularity then the weak solution is regular and is unique. The second part of the talk is devoted to the primitive equations of the ocean with the Dirichlet boundary condition for which we prove the global regularity. This is a joint work with I. Kukavica.

In this talk we first review the critical threshold phenomena
for Euler-Poisson equations, which arise in the semiclassical
approximation of Schrodinger-Poisson equations and plasma
dynamics. We then present a phased space-based level set
method for the computation of multi-valued velocity and
electric fields of one-dimensional Euler-Poisson equations.
This method uses an implicit Eulerian formulation in an
extended space, which incorporates both velocity and electric
fields into the configuration space. Multi-valued velocity
and electric fields are captured through common zeros of two
level set functions, which solve a linear homogeneous
transport equation in the field space. The superposition
principle for multi-valued solutions is established.

Recently, using Gelfand-Zeitlin and the space of Hessenberg matrices, Wallach and I found natural Darboux coordinates (as a classical mechanical solution of the Gelfand-Zeitlin question) on $\frak g$ for the case where $\frak g$ is the space of all matrices. Now, at least locally, I do the same for any reductive $\frak g$ using a beautiful result of A. A. Tarasov on Fomenko-Miscenko theory and old results of mine on a generalization of the Hessenberg matrices.

To prove the Erd\H os-Ko-Rado theorem about extremal collections of
subsets of $1,\ldots,n$, they invented the {\em shifting} technique,
which preserves the number of subsets in a collection but simplifies
(in some senses) the collection. After a finite number of shifts,
one's collection becomes invariant under shifting, and easily studied.

Given a finite set of $n$ vectors in a $k$-dimensional vector space,
the collection of subsets that form bases of the vector space
satisfies some combinatorial properties. Abstracting them, Whitney
defined {\em matroids}. The matroids that are invariant under shifting
have been classified, and correspond to partitions inside a
$k \times (n-k)$ rectangle. The shift of a matroid usually is not a matroid.

I'll present a new version of the Littlewood-Richardson rule,
that starts with a certain matroid, and alternately shifts it
(breaking matroidness) and decomposes as a union of maximal submatroids.
The leaves of the tree so constructed are labeled with fully
shifted matroids, hence partitions. To actually carry out
such a calculation in practice requires some new algorithms.

Unlike all other known Littlewood-Richardson rules, this matroid
shifting rule has an easy generalization to multiplication of Schubert
(not just Schur) polynomials, where it is still a conjecture.

This work is joint with Ravi Vakil.

Let $N$ denote a manifold equipped with a finite Borel measure
$\gamma$. A vector field $Z$ on $N$ is said to
be admissible with respect
to $\gamma$
if $Z$ admits an integration by parts formula.
The measure $\gamma$ is
said to be quasi-invariant under $Z$
if the class of null sets of $\gamma$
is preserved by the flow generated
by $Z$. In this talk we study the law $\gamma$ of an elliptic
diffusion process with values in a closed compact manifold.
We construct a
class of admissible vector fields for $\gamma$, show that $\gamma$
is quasi-invariant under these vector fields,
and give a formula for the
associated family of Radon-Nikodym
derivatives $d\gamma_s\over d\gamma$.

In this talk, we use biological locomotion on small scales as an
inspiration (and an excuse) to solve a number of modeling
problems in small-scale fluid mechanics. We first solve for the
swimming kinematics of elastic swimmers, devices which exploit
flow-induced deformation of elastic filaments for
propulsion. More generally, we then show how soft surfaces can
be exploited for propulsion without inertia. Finally, we
describe how the viscoelastic nature of the surrounding fluid
can affect the kinematics and energetics of simple swimmers.

I will try to discuss Abstract Algebra from its emergence to
the present day and its place among other mathematical areas.

A number of recent results, including special discretization
schemes, adaptive methods and multilevel iterative methods for
the resulting algebraic systems, will be presented in this talk
for various partial differential equations (PDEs). With a
careful and combined use of qualitative properties of PDEs, the
underlying functional spaces and their discretizations, many
different kinds of equations will be treated with similar
techniques. After an introduction to some practically efficient
methods such as the algebraic multigrid method for the Poisson
equations, it will be shown how more complicated systems such as
linear elasticity equations, electro- magnetic equations, porous
media, Stokes equations and more general Newtonian/non-Newtonian
models can be reduced to the solution of a sequence of Poisson
equation and its simple variants. The efficiency of these
algorithms will be illustrated by theoretical analysis, numerical
examples and engineering applications.

Abstract: Brumer's conjecture states that Stickelberger elements
combining values of L-functions at s=0 for an abelian extension of
number fields E/F should annihilate the ideal class group of E when it
is considered as module over the appropriate group ring.
In some cases, an ideal obtained from these
Stickelberger elements has been shown to equal
a Fitting ideal connected with the ideal class group.
We consider the analog of this at s=-1, in which the class group
is replaced by the tame kernel, which we will define.
For a field extension of degree 2, we show that there is an exact equality
between the Fitting ideal of the tame kernel and the most natural
higher Stickelberger ideal; the 2-part of this equality is conditional on
the Birch-Tate conjecture.

Over the last two decades long memory time series have become an
established modeling tool in many areas of science and technology,
including geosciences, medical sciences, telecommunication networks
and to some extend financial econometrics. It has however been
recently realized that practically all statistical procedures
intended to detect and estimate long memory give spurious results if
a time series without long memory is perturbed by nonstationarities,
like trends or breaks (change-points). For example, if a mean of a
short memory time series changes, a test for the presence of long
memory will incorrectly indicate that the time series has long
memory. Similarly, a test for the presence of change point, will
incorectly show that that a change point is present if the time
series is stationary with long memory. A growing body of research
which has accumulated over the last decade is concerned with finding
and illustrating cases of such spurious inference, without
addressing the issue how to choose between the two modeling
approaches. In this talk we will discuss two new statistical tests
aimed at distinguishing between the two approaches and apply them to
a financial and a hydrological time series. The talk will focus on
the ideas rather than technicalities and will be broadly accessible.

In this talk we will discuss questions about finite dimensional central simple p-algebras. In particular we will
discuss the first construction of a noncyclic p-algebra due to
Amitsur and Saltman. From there we will talk about the structure
of these algebras under prime to p extensions and the conditions
under which they are indecomposable.

200 prisoners, scheduled for execution, are given one chance for
survival. Their 200 names are put in a row of 200 boxes, one name per
box. Each prisoner will enter this room, one at a time, and open 100
boxes with the goal of finding his or her own name. If every
prisoner does this successfully, all will go free. If any one
fails to find his or her own name, all will be executed. The
boxes will be closed after each prisoner and once a prisoner has
entered the room, any attempt at communication with the others
will be punished by execution. However, the prisoners are
allowed to strategize beforehand. In fact, a strategy exists which
gives, roughly, a $\frac{1}{3}$ probability of survival. Can you find
it? No Googling!

We'll consider this problem and the fate of other unfortunate
hypothetical prisoners, mostly as an excuse to discuss some
combinatorics of permutation enumeration.

Solutions of incompressible Navier-Stokes equations can
exhibit a wide spectrum of different types of behavior.
In various regimes,
the equations contain as special limiting cases
for example the classical
heat equation, the non-linear Schroedinger
equation, various other dispersive equations
with strange dispersion relations,
various non-trivial finite-dimensional dynamical systems,
some classical geometric semilinear elliptic equations, etc.
In addition, when thinking about
realistic fluid flows and applications,
ideas from statistical mechanics enter the picture. In
the lecture I will explain
(a limited number of) some PDE aspects of these equations.

This term's topology learning seminar will be on Ozsvath-Szabo homology.

About 7 years ago, Ozsvath and Szabo invented this construction (which they call "Heegaard Floer homology") in an attempt to give a different definition of Seiberg-Witten theory. Their theory has been incredibly successful in applications to low-dimensional topology.

In brief, they show how to associate a family of homology groups to a 3--manifold by choosing a Heegaard splitting and computing a suitable Lagrangian intersection Floer homology. Some of the most important features of the construction are:

1. 4--dimensional cobordisms induce maps between homology groups; the invariants of closed 4-manifolds are conjecturally equal to the Seiberg-Witten invariants.

2. There is a version of the homology for knots in $S^3$, which leads to an exact formula (not just a bound!) for the genus of knots. Consequently their homology distinguishes the unknot, and can be used to prove many old conjectures about surgery on knots.

3. The theory gives rise to powerful invariants of contact structures on 3--manifolds and can distinguish tight from overtwisted.

4. The homology for knots, unlike all earlier gauge-theoretic invariants, can actually be calculated by purely combinatorial means. There is a strong hope that this will eventually lead to a complete combinatorial calculation of the Ozsvath-Szabo/Seiberg-Witten/Donaldson invariants of 4-manifolds.

The first meeting will be Tuesday 10th April, in room 7218, at 10.30am.
I will give an introductory talk and then we will arrange the schedule of speakers for the rest of the term. Anyone is welcome to attend - attendance does not necessarily lead to being volunteered for a talk!

Sinc--interpolation is an infinitely smooth interpolation on the whole real
line based on a series of shifted and dilated sinus--cardinalis functions used as
Lagrange basis. It often converges very rapidly, so for example for functions
analytic in an open strip containing the real line and which decay fast enough at
infinity. This decay does not need to be very rapid, however, as in Runge's function
$1/(1+x^2)$. Then one must truncate the series, and this truncation error is much
larger than the discretisation error (it decreases algebraically while the latter
does it exponentially).

\vskip0.5em
In our talk we will give a formula for the error commited when merely using
function values from a finite interval symmetric about the origin.
The main part of the formula is a polynomial in the distance between the nodes
whose coefficients contain derivatives of the function at the extremities.

The concept of relative measure was fairly popular among Polish
mathematician of 1930 in Lvov. Steinhaus and Urbanik were working
on introducing a relative measure and relative measurability into
the area of random variables.
Recent work on signal processing and time series has led to
re-discovery of the 'old-school' theorems and application to
data generated by signals or time-series. Some fundamental work
was done by Garnder and continuation of this work was done
by Leskow and Napolitano.

A short informal introduction to nonstochastic approach
to time series inference via relative measurability will
be presented. Applications to signal forecasting will be
presented.

Let G be a group acting``nicely" on a space, $X$, with some
structure(topological, differentiable, algebraic, combinatorial.). A
basic problem is to find an effective way of determining if two points
$x,y$ are in the same orbit (or at least in an appropriate closure of an
orbit). In this lecture I will look at methods that can be used in
concert with computers to
approach such problems through the determination of "enough" computable
invariant functions. There will be several examples including measures of
quantum entanglement.

There will be a 10 minute organizational meeting for the schedule of talks this quarter, after which I will discuss a recent proof of the local Langlands conjecture for GSp(4). Joint with with Shuichiro Takeda.

A q-query Locally Decodable Code (LDC) is an error-correcting
code that encodes an n-bit message x as a codeword $C(x)$, such that one can
probabilistically recover any bit $x_i$ of the message by querying only $q$
bits
of the codeword $C(x)$, even after some constant fraction of codeword bits
has
been corrupted. The goal of LDC related research is to minimize the length
of
such codes.

A q-server private information retrieval (PIR) scheme is a cryptographic
protocol that allows a user to retrieve the $i-th$ bit of an $n-bit$ string $x$
replicated between $q$ servers while each server individually learns no
information about $i$. The goal of PIR related research is to minimize the
communication complexity of such schemes.

We present a novel algebraic approach to LDCs and PIRs and obtain vast
improvements upon the earlier work. Specifically, given any Mersenne prime
$p=2^t - 1$, we design three query LDCs of length $Exp(n^{1/t})$, for every
$n$. Based
on the largest known Mersenne prime, this translates to a length of less
than $Exp(n^{10^{-7}})$, compared to $Exp(n^{1/2})$ in the previous
constructions.
We also design 3-server PIR schemes with communication complexity of
$O(n^{10^{-7}})$ to access an n-bit database, compared to the previous best
scheme with complexity $O(n^{1/5.25})$.

It has often been conjectured that there are infinitely many Mersenne primes. Under this conjecture, our constructions yield three query locally
decodable codes of subexponential length and three server private
information
retrieval schemes with subpolynomial communication complexity.

In this talk, I am going to discuss some recent results concerning the derivation, from many body quantum mechanics, of a cubic nonlinear Schroedinger equation, known as the Gross-Pitaevskii equation, for the dynamics of Bose-Einstein condensation. This is a joint work with L. Erdos and H.-T. Yau.

Recently, Wang, Chakrabarti, Wang and Faloutsos have shown that the spectral radius of a graph plays an important role in modeling virus propagation in networks. This led Van Dam and Kooij to consider the following problem: which connected graph on n nodes and diameter D has minimal spectral radius ? Van Dam and Kooij answered this question for $D=n-1,n-2,n-3,n/2,2,1$ and provided a conjecture for the case $D=n-e$, when e is fixed. In this talk, I give an overview of their work and I will outline a proof of their conjecture for $e=4$ and
possible extensions for $e>4$.

This is joint work in progress with Edwin Van Dam (University of Tilburg,
The Netherlands).

This will be an expository talk. I'll begin by introducing the concepts of reduite (reduced function) and balayage (swept measure) in classical potential theory and their interpretations in terms of Brownian motion. I'll then discuss the extension of these ideas to Markov processes as in Hunt's fundamental memoir. After introducing h-transforms I'll be able to define the interior reduite and discuss some of its properties following Walsh. If time permits I'll give some indications of recent work in this area by Fitzsimmons and myself.

As any good TA knows, adding fractions can be tricky. But being able to work with fractions can enrich your mathematical life in many ways. Unfortunately, if you happen to be a ring, this luxury is not inherent. However, you could still hope to embed into some larger ring containing inverses--even to the extent that you become a field or division ring.

We will look first at the theory of localization for commutative rings where everything is fairly straightforward. Then, we will see what happens when we try to generalize to noncommutative rings and look at some examples of good and bad cases.

And the quote... Thoreau.

We will discuss a recent proof of the local Langlands conjecture for GSp(4). This is joint work with Shuichiro Takeda.

Prior knowledge over arbitrary general sets is incorporated into
nonlinear support vector machine approximation and classification problems as linear constraints of a linear program. The key tool in this incorporation is a theorem of the alternative for convex functions that converts nonlinear prior knowledge implications into linear inequalities
without the need to kernelize these implications. Effectiveness of the
proposed formulation is demonstrated on synthetic examples and on
important breast cancer prognosis problems. All these problems
exhibit marked improvements upon the introduction of prior knowledge
over nonlinear kernel approaches that do not utilize
such knowledge.

We seek for the Hilbert series of the ring of invariant polynomials
in the $2n+n^2$ variables $\{u_i,v_j,x_{i,j}\}_{i,j=1}^n$ under the
action of $GL_n[C]$ by right multiplication on the row vector
$u=(u_1,u_2,\ldots ,u_n)$, left multiplication on the column vector
$v=(v_1,v_2,\ldots ,v_n)$ and by conjugation on the matrix $\|x_{i,j}\|_{i,j=1}^n$.
We reduce the computation of this Hilbert series to the evaluation of
the constant term of a certain rational function. Remarkably, the
final result hinges on the explicit evaluation of certain
Kostka-Foulkes polynomials.

The goal is to explain what a stack is and why people think they are interesting objects to study. I'll start by saying what people (who want to classify a certain class of mathematical objects, say triangles or elliptic curves or...) expect from a moduli space and why it can easily happen that such a moduli `space' doesn't exist. This is the situation in which stacks get their chance. They are some kind of a generalization of a the notion of a `space', where the meaning of space depends on what you are interested in (for example, in the context of algebraic geometry one might say that stacks are a generalization of the notion of a scheme). I'll then use an example to motivate the axioms appearing in the definition of a stack. Depending on how much time I have I will say what an algebraic stack is and maybe how Deligne and Mumford employed the notion to prove something interesting.

On a Hilbert modular surface over $\mathbb Z$, there are two
families of arithmetic cycles. One family consists of the
Hirzebruch-Zagier divisors
$\mathcal T_m$ of codimension $1$, indexed by positive integers $m$, and
another consists of the CM cycles $CM(K)$ of codimension 2, indexed by
quartic CM number fields $K$. When $K$ is not biquadratic, $\mathcal T_m$
and $CM(K)$ intersect properly, and a natural
question is, what is the intersection number? In this talk, we present a
conjectural formula for the intersection number of
Bruinier and myself. We give two partial results in this talk. If time
permits, I will also briefly describe two applications:
one of the consequences is
a generalization of the Chowla-Selberg formula, and another is a
conjecture of Lauter on Igusa invariants.

Informally speaking, a compact set in the plane is computable if
there exists an algorithm to draw it on a screen with an arbitrarily
high magnification. We investigate the question of computability
of a Julia set of a quadratic polynomial and obtain some surprising
answers.
(joint work with Mark Braverman)

Our main goal is to study the behavior of the error in finite element approximations of partial differential equations. The error typically has two components -- local error and global (pollution) error. We also will discuss the very interesting phenomena of superconvergence, and in particular, how to determine superconvergence points, and what advantages can be derived from them.

An r-uniform hypergraph is a generalization of graphs, we consider a
subset of all r-element subsets of a given vertex set. A Hamiltionian
chain is a generalization of hamiltonian cycles for hypergraphs, it is a
"cycle" that contains all vertices. Among the several possible ways of
generalizations this is probably the most strong one, it requires the
strongest structure. Since there are many interesting questions about
hamiltonian cycles in graphs, we can try to answer these questions
for hypergraphs, too. I give a survey on
results about such questions.

Usually, a limit theorem of Probability Theory is a theorem that concerns convergence of a sequence of distributions $P_n$ to a distribution $P$. However, there is a number of works where the traditional setup is modified, and the object of study is two sequences of distributions, $P_n$ and $Q_n$, and the goal consists in establishing conditions implying the convergence
$P_n - Q_n ->0 (1)$
In particular problems,$P_n$ and $Q_n$ are, as a rule, the distributions of the r.v.'s $f(X_1,...,X_n)$ and $f(Y_1,...,Y_n)$, where $f(.)$ is a function, and $X_1,X_2$,... and $Y_1,Y_2$,... are two sequences of r.v.'s. The aim here is rather to show that different random arguments $X_1,...,X_n$ may generate close distributions of $f(X_1,...,X_n)$ , than to prove that the distribution of $f(X_1,...,X_n)$ is close to some fixed distribution (which, above else, may be not true). Clearly, such a framework is more general than the traditional one. First, as was mentioned, the distributions $P_n$ and $Q_n$, themselves do not have to converge. Secondly, the sequences $P_n$ and $Q_n$ are not assumed to be tight, and the convergence in $(1)$ covers situations when a part of the probability mass or the whole distributions "move away to infinity'", while the distributions $P_n$ and $Q_n$, are approaching each other.
We consider a theory on this point, including the very definition of convergence $(1)$, and a particular example of the invariance principle in the general non-classical setup.

In this talk we will develop the concept of arbitrage and discuss the theory of options pricing. Time permitting, we will present the famous Black-Scholes option pricing formula.

We will present an integral representation for automorphic
representations on $U(3) x GL(2)$. It involves parabolic induction to $U(4)$.
The resulting formula can be applied to determine the cuspidal
automorphic representations of $U(3)$ that occur in the restriction of
the Siegel induced residual spectrum of $U(4)$.

We propose variational models for image inpainting in wavelet domain,
which aims to filling in missing or damaged wavelet coefficients
in image reconstruction. The problem is motaviated by error concealment
in image processing and communications. It is closely related to
classical image inpainting, with the difference being that the
inpainting regions are in the wavelet domain. This
brings new challenges to the reconstructions. The new variational
models, especially total variation minimization in conjunction
with wavelets lead to PDE's,in the wavelet domain and can be solved
numerically. The proposed models have effective and automatic control
over geometric features of the inpainted images including sharp edges, even
in the presence of substantial loss of wavelet coefficients, including in
the low frequencies. This work is joint with Tony Chan (UCLA) and
Jackie Shen (Minnesota).

After introducing the concept of Maximal Regularity for parabolic problems and illustrating its usefulness for dealing with (fully) nonlinear problems, a brief introduction to Free Boundary Problems will be given. The focus will then shift to a class of Free Boundary Problems. Maximal regularity results as well as elliptic regularity results will be presented which are needed in the analysis of the Free Boundary Problems of interest.

What does a compact Riemannian manifold with unsolvable word problem look like from within? I will discuss Nabutovsky's work on the subject.

Ordinary forms of weight at least 2 give rise to locally
reducible Galois representations. Greenberg has asked
whether these representations are semi-simple. One
expects this to be the case exactly when the underlying
form has CM. We shall speak about various results towards
this expectation that use p-adic families of forms
and deformation theory. This is joint work with Vatsal.

Numerical relativity has undergone a revolution during the past two years, with several groups now routinely performing accurate simulations of binary black hole systems with multiple orbits, mergers, and ringdown of the holes to a final single hole equilibrium state. This talk will discuss some of the mathematical developments that made this revolution possible. In particular new insights will be discussed into how the gauge degrees of freedom may be specified in the Einstein equations, and how this changes the behavior of the constraints of the theory.

The Robinson-Schensted correspondence is one between the
set of permutations and pairs of same-shape standard Young tableaux.
I'll recall a few of the combinatorial aspects of this.

The Steinberg scheme (for $GL_n$) is a set of triples, one nilpotent matrix
and two flags invariant under the nilpotent, whose components correspond
to permutations. I'll recall why this is (for those who haven't been
coming to the seminar), and show that they also correspond to pairs
of standard Young tableaux. The basic linkage between the linear algebra
and the combinatorics is that Jordan canonical forms of nilpotent matrices
correspond to partitions.

This talk will only require linear algebra, and a willingness to talk
about the ``components'' of an algebraic set.

We will have three panelists who have recently found jobs: Aaron Wong, Assistant Professor, tenure track at Nevada State College, Henderson, Nevada, Mark Colarusso, Visiting Assistant Professor, University of Notre Dame, Indiana, and Dan Rogalski, Assistant Professor, UCSD. 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?
The discussion will be followed by a question and answer period.

For two graphs $S$ and $T$, the constrained Ramsey number $f(S, T)$ is the minimum $n$ such that every edge coloring of the complete graph on
$n$ vertices (with any number of colors) has a monochromatic subgraph isomorphic to $S$ or a rainbow subgraph isomorphic to $T$. Here, a
subgraph is said to be rainbow if all of its edges have different
colors. It is an immediate consequence of the Erd\H{o}s-Rado
Canonical Ramsey Theorem that $f(S, T)$ exists if and only if $S$ is a
star or $T$ is acyclic. Much work has been done to determine the rate
of growth of $f(S, T)$ for various types of parameters. When $S$ and
$T$ are both trees having $s$ and $t$ edges respectively, Jamison,
Jiang, and Ling showed that $f(S, T) \leq O(st^2)$ and conjectured
that it is always at most $O(st)$. They also mentioned that one of
the most interesting open special cases is when $T$ is a path. We
study this case and show that $f(S, P_t) = O(st\log t)$, which differs
only by a logarithmic factor from the conjecture. This substantially
improves the previous bounds for most values of $s$ and $t$.

Highway traffic flows are generally modeled by partial differential equations
(PDEs). These models are used by traffic engineers for
road design, planning or management. However, they
often fail to capture important features of
empirical traffic flow studies, particularly at small
scales. In this talk, I will propose a fairly simple stochastic model for
highway traffic flows in the form of a nonlinear stochastic partial differential
equation (SPDE) with random
coefficients driven by a Poisson random measure. I will discuss the
well posedness of the proposed equation as well as the
corresponding inverse problem that I will illustrate by its
calibration to high resolution traffic data from highway
101 in Los Angeles. I will also present a more sophisticated spde
in the form of a system of coupled hyperbolic-parabolic equations.

Varieties were originally studied by comparing them with
the hypersurfaces that are their generic projections--curves
in the plane and surfaces in three-space, for example. In
low dimensions, the fibers of these generic projections are
pretty well understood, but there are serious obstructions
to extending this understanding to all dimensions.
I'll survey what's known, explain some examples, and present
a new conjecture about these fibers. A connection with the
regularity of powers of an ideal (asymptotic regularity)
plays an interesting role.

The computational approximation of solutions of complex systems such as the Navier-Stokes equations is often a formidable task. For example, in feedback control settings where one often needs solutions of the complex systems in real time, it would be impossible to use large-scale finite element or finite-volume or spectral codes. For this reason, there has been much interest in the development of low- dimensional models that can accurately be used to simulate and control complex systems. We review some of the existing reduced-order modeling approaches, including reduced-basis methods and especially methods based on proper orthogonal decompositions techniques. We also discuss a new approach based on centroidal Voronoi tessellations. We discuss the relative merits and deficiencies of the different approaches and also the inherent limitations of reduced-order modeling in general.

Spectral graph theory has enjoyed much success in using eigenvalues of
matrices associated with a graph to understand some structural property or
bound various kinds of behavior of the graph. When two graphs share many
eigenvalues in common it can often be traced to some sort of common
structure that they share. Well known examples of this are common
coverings or equitable partitions.

We will consider another variation of this where (for the normalized
Laplacian) two graphs do not project to a common graph but share a common
``anti-covering'' (which we will define). We will also consider
anti-covers for the adjacency matrix and use it to establish the following
linear algebra result (among others): {\it Let $M$ be an $n{\times}n$
real symmetric matrix and $|M|$ be the $n{\times}n$ matrix found by taking
(entrywise) the absolute values of $M$; then there exists a nonnegative
symmetric $2n{\times}2n$ matrix ${\cal N}$ such that the spectrum of
${\cal N}$ is the union of the spectrums of $M$ and $|M|$.}

Singularities offer the opportunity to study some interesting mathematics
from a relatively simple point of view. First I'll describe what an
affine variety is and what it means for one to have a singularity. Then,
I'll discuss the work of Milnor, Brieskorn and others on the topology of a
complex hypersurface near one of its singular points, which has
connections to knot theory and exotic spheres. If I have time, I'll talk
about a special class of singularities coming from invariant theory that
were discovered by Klein, and the McKay correspondence, which gives an
unexpected connection between resolutions of singularities and the Dynkin
diagrams used to classify compact Lie groups.

Operator decomposition methods are an attractive solution strategy
for computing complex phenomena involving multiple physical processes,
multiple scales or multiple domains. The general strategy is to
decompose the problem into components involving simpler physics
over a relatively limited range of scales, and then to seek the
solution of the entire system through an iterative procedure
involving solutions of the individual components.

We construct an operator decomposition finite element method for a
conjugate heat transfer problem consisting of a fluid and a
solid coupled through a common boundary. Accurate a posteriori
error estimates are then developed to account for both local
discretization errors and the transfer of error between fluid and
solid domains. These estimates can be used to guide adaptive mesh
refinement. We show that the order of convergence of the operator
decomposition finite element method is limited by the accuracy of
the transferred gradient information, and demonstrate how a simple
boundary flux recovery method can be used to regain the optimal
order of accuracy in an efficient manner.

This is joint work with Don Estep and Tim Wildey.

We will discuss our current research which describes and constructs polarizations of regular adjoint orbits for certain classical groups. This research generalizes recent work of Bertram Kostant and Nolan Wallach. Kostant and Wallach construct polarizations of regular adjoint orbits in the space of $n\times n$ complex matrices $M(n)$. They accomplish this by defining an $\frac{n(n-1)}{2}$ dimensional abelian complex Lie group $A$ that acts on $M(n)$ and stabilizes adjoint orbits. Note that the dimension of this group is exactly half the dimension of a regular adjoint orbit in $M(n)$. This fact allows $A$ orbits of dimension $\frac{n(n-1)}{2}$ contained in a given regular adjoint orbit to form the leaves of a polarization of an open submanifold of that orbit. We study the $A$ orbit structure on $M(n)$ and generalize the construction to complex orthogonal Lie algebras $\mathfrak{so}(n)$. In the case of $M(n)$, we obtain complete descriptions of $A$ orbits of dimension $\frac{n(n-1)}{2}$ and thus of leaves of polarizations of all regular adjoint orbits. For $\mathfrak{so}(n)$, we construct polarizations of certain regular semi-simple adjoint orbits.

In 1890 Stickelberger published his eponymous theorem in Math. Annalen giving an explicit annihilator for the `minus' (or imaginary) part of the class group of a cyclotomic field as a Galois module. However, Stickelberger's wonderful theorem raises more questions than it answers. And strangely, many obvious ones have only begun to receive serious attention - let alone answers - in the late 20th and early 21st centuries. For instance:

Is there a similar result for an arbitrary (abelian) extension of number fields?

Is the `Stickelberger ideal' the full annihilator of minus part the class group?

What about the `plus' (or real) part?

The first question leads to Brumer's Conjecture. The answer to the second question is certainly negative, for several different reasons which we shall try to disentangle. This leads to comparisons with the Fitting ideal of the class group and of its Pontrjagin dual, and so to very recent work by Greither, the speaker and others, which we shall survey.

If time allows we should like to report on some recent approaches to the third question.

There is a rich interplay between combinatorial and differential
geometry.
We will give first a geometric proof of a combinatorial result, and
then
a combinatorial analysis of a geometric moduli space. The first is
joint
work with Ivan Izmestiev, Rob Kusner, Guenter Rote, and Boris
Springborn;
the second with Karsten Grosse-Brauckmann, Nick Korevaar and Rob
Kusner.

In any triangulation of the torus, the average vertex valence is 6.
Can there be a triangulation where all vertices are regular (of valence 6) except for one of valence 5 and one of valence 7? The answer is no.
To prove this, we give the torus the metric where each triangle is
equilateral and then explicitly analyze its holonomy. Indeed,
techniques
from Riemann surfaces can characterize exactly which euclidean cone
metrics have full holonomy group no bigger than their restricted
holonomy group (at least when the latter is finite).

Next we consider the moduli space $M_k$ of Alexandrov-embedded surfaces
of constant mean curvature which have k ends and genus 0 and are contained in a slab. We showed earlier that $M_k$ is homeomorphic to an open manifold
$D_k$ of dimension $2k-3$, defined as the moduli space of spherical metrics
on an open disk with exactly k completion points. In fact, $D_k$ is the
ball $B^{2k-3}$; to show this we use the Voronoi diagram or Delaunay
triangulation of the k completion points to get a tree, labeled by
logarithms of cross-ratios. The combinatorics of the tree are tracked
by the associahedron, and the labels give us a complexification of the
cone over its dual. We note similarities to the spaces of labeled
trees
used in phylogenetic analysis.

Maxwell (1866) and Boltzmann (1872) developed a recipe to go from certain Newtonian laws of molecular dynamics to the Navier-Stokes system of gas dynamics. This recipe was controversial at the time. Mathematicians such as Hilbert, Klein, Poincare, and Zermelo were drawn into the debate. Hilbert featured it at the 1900 ICM in the articulation of his sixth problem, and made important contributions towards its resolution. The problem however remains largely open. Recent significant advances start with the DiPerna-Lions (1990) theory of global solutions to Boltzmann equations and lead to the Golse-Saint Raymond (2004) proof of the incompressible Navier-Stokes limit. This lecture will introduce the Boltzmann equation and survey some "new" connections to linear and weakly nonlinear gas dynamics that are the focus of recent research.

The differential equation $\dot{x}(t) = Ax(t) + Bu(t)$ coupled with the
algebraic equation $y(t) = Cx(t) + Du(t)$ where $A\in\mathbb{C}^{n\times n}$,
$B\in\mathbb{C}^{n\times m}$, $C\in\mathbb{C}^{p\times n}$ is
called a state space system and commonly employed to represent
a linear operator from an input space to an output space in control
theory. One major challenge with such a representation is that
typically $n$, the dimension of the intermediate state function $x(t)$,
is much larger than $m$ and $p$, the dimensions of the input
function $u(t)$ and the output function $y(t)$. To reduce the order of
such a system (dimension of the state space) the traditional
approaches are based on minimizing the $H_{\infty}$ norm of the
difference between the transfer functions of the original system and
the reduced-order system. We pose a backward error minimization
problem for model reduction in terms of the norms of the
perturbations to the coefficients $A$, $B$ and $C$ such that the
perturbed systems are equivalent to systems of order $r<n$. It follows
from the fact that singular values are insensitive to perturbations that
a system with a small backward error has a small forward error, that
is the difference between the transfer functions is small in $H_{\infty}$
norm. We derive a singular value characterization for a simplified
version of the backward error minimization problem. The singular
value characterization is a generalization of a formula recently
derived for the Wilkinson distance problem, the norm of the smallest
perturbation to a matrix so that the perturbed matrix has a multiple
eigenvalue. We suggest methods to estimate the Wilkinson distance
and minimize the backward error for model reduction.

The Riemannian Penrose inequality in dimensions less than 8
Abstract: The Positive Mass Theorem states that a complete asymptotically flat manifold of nonnegative scalar curvature has nonnegative mass. The Riemannian Penrose inequality provides a sharp lower bound for the mass when black holes are present. More precisely, this lower bound is given in terms of the area of an outermost minimal surface, and equality is achieved only for Schwarzschild metrics. The Riemannian Penrose inequality was first proved in three dimensions in 1997 by G. Huisken and T. Ilmanen for the case of a single black hole. In 1999, H. Bray extended this result to the general case of multiple black holes using a different technique. In this talk we extend Bray's technique to dimensions less that 8. This is joint work with H. Bray.

The Positive Mass Theorem states that a complete asymptotically flat manifold of nonnegative scalar curvature has nonnegative mass. The Riemannian Penrose inequality provides a sharp lower bound for the mass when black holes are present. More precisely, this lower bound is given in terms of the area of an outermost minimal surface, and equality is achieved only for Schwarzschild metrics. The Riemannian Penrose inequality was first proved in three dimensions in 1997 by G. Huisken and T. Ilmanen for the case of a single black hole. In 1999, H. Bray extended this result to the general case of multiple black holes using a different technique. In this talk we extend Bray's technique to dimensions less that 8. This is joint work with H. Bray.

I will begin with some elementary remarks about Hermitian
symmetric functions on complex manifolds. I will introduce
various positivity condtions for such functions and discuss
the relationships among them. Examples include globalizable metrics on Hermitian
line bundles. A version of the Cauchy-Schwarz inequality for
Hermitian symmetric functions and how it relates to complex geometry will
be a major part of the talk. To conclude I will pose several accessible problems

I will discuss Feynman's path integral interpretation of quantum mechanics over curved configurations spaces, i.e. Riemannian manifolds. We will see how curvature of the configuration space enters in the interpretation (and the ambiguity) of Feynman's path integral prescription.

An optimization problem is ill-posed if its solution is not unique or is acutely sensitive to data perturbations. A common approach to such problems is to construct a related problem with a well-behaved solution that deviates only slightly from the original solution set. The strategy is often used in data fitting applications, and also within optimization algorithms as a means for stabilizing the solution process.

This approach is known as regularization, and deviations from solutions of the original problem are generally accepted as a trade-off for obtaining solutions with other desirable properties.

In fact, however, there exist necessary and sufficient conditions such that solutions of the regularized problem continue to be exact solutions of the original problem. We present these conditions for general convex programs, and give some applications of exact regularization.

(Joint work with Paul Tseng.)

We present the first analysis of the Join-the-Shortest-Queue (JSQ) routing policy for Web server farms. Web server farms involve a collection of Processor-Sharing (PS) servers, whereas prior analyses of JSQ have always assumed First-Come-First-Serve (FCFS) servers. This work introduces a new technique: Single-Queue-Approximation (SQA), and uses the technique to prove some interesting insensitivity properties for Web server farms.
Based on joint work with: Varun Gupta, Karl Sigman, and Ward Whitt.

We consider a 2-person perfect information ``liar'' game, often called a R\'enyi-Ulam game. The basic game is that of ``twenty questions'' played between questioner Paul and responder Carole; Paul searches for a distinguished element $x$ in a search space $[n]$ by asking Yes-No questions of the form ``is $x\in A$'', where $A\subseteq [n]$. Carole responds `Yes' or `No', lying in up to $k$ responses. The fully off-line game is equivalent to $k$-error-correcting codes.

We extend this game to a general channel $\mathcal{C}$ which governs the manner in which Carole may lie. Specifically, given the alphabet $[t]:=\{1,\ldots,t\}$, Paul searches for $x\in[n]$ by partitioning $[n]=A_1\cup \cdots \cup A_t$ and asking for $a$ such that $x\in A_a$. A lie is a tuple $(a,b)\in[t]\times [t]$ with $a\neq b$. The channel $C$ specifies an arbitrary set of lie strings of bounded length $\leq k$ from which Carole may choose a string and intersperse its lies, in order, among her responses. For example, when $t=2$, Carole lies with $(1,2)$ when she responds with 2 (``No'') when the correct response is 1 (``Yes''). We further restrict Paul to ask his questions in two off-line batches. We show that the maximum size of the search space $[n]$ for which Paul can guarantee finding the distinguished element is $t^{q+k}/(E_k(C){q \choose k})$ as $q\rightarrow\infty$, where $E_k(C)$ is the number of lie strings in $\mathcal{C}$ of maximum length $k$, generalizing previous work of Dumitriu and Spencer, and of Cicalese, Deppe, and Mundici. We similarly solve the pathological liar variant. This is joint work with Kathryn Nyman (Loyola University-Chicago).

Suppose n pursuers starting at the origin chase a single prey
starting at 1, all doing standard independent Brownian motions on the real line. Bramson and Griffeath (1991) showed that the expected capture time is infinite for three or fewer pursuers and, after simulations, conjectured that it is finite for four or more. Li and Shao (2001) proved it for five or more pursuers. In recent work with Ratzkin, we show that it finite for four, completing the proof. We use the idea of Li and Shao to reduce the problem to an estimate of the first Dirichlet eigenvalue of a domain in the sphere.

I'll discuss eigenvalues, describe the reduction, the eigenvalue estimates, and some related numerics.

We consider the problem of estimating a set S from a random sample of
points of S, which amounts at estimating the support of the
underlying probability density. Set estimation has applications in
various situations, including medical diagnosises, image analysis,
and quality control for example. We focus on the simple set estimator
defined as the union of balls centered at the random points. Using
tools from Riemannian geometry, and under mild analytic conditions on
the underlying density of the data, we derive the exact rate of
convergence of this set estimator.
In closed connection with the problem of set estimation, we study the
estimation of the number of connected components of a level set of a
multivariate probability density. This allows one to assess the
number of clusters of a statistical population, which is an essential
problem of unsupervised learning. We introduce an estimator based on
a graph, and using similar geometrical tools, we establish the
asymptotic consistency of the methodology.

For a fixed polynomial $f \in \mathbb Z_k[X]$, let $\rho_k(N)$ denote the
maximum size of a set $A \subset \{1,2,\dots,N\}$ such that no product of
$k$ distinct elements of $A$ is in the value set of $f$. This problem was
studied by Erd\H{o}s and Erd\H{o}s, S\'{o}s and S\'{a}rk\"{o}zy

A graph with independence number alpha is called (alpha,k)-balanced
if every induced subgraph on k vertices has independence number alpha
as well. We will discuss the maximum number of vertices in an
(alpha,k)-balanced graph for fixed k and alpha, a problem with obvious
connections to Ramsey Theory. We focus specifically
on the case k=2alpha which is motivated from polyhedral combinatorics.

(Joint work with A. Brieden, Z. Furedi and R. Ramamurthi)

n this talk, I will first recall a mean field approximation of
electrostatic free energy for an ionic solution, and discuss two
issues: (1) Rigorous mathematical justification of the existence
of equilibrium concentrations and their Boltzmann relations; (2)
The effect of inhomogeneous Dirichlet boundary condition to the
solution of the related Poisson-Boltzmann equation for the
electrostatic potential. I will then consider a class of variational
implicit solvent models for the solvation of biomolecules, and
present a formal derivation of the first variation of the
electrostatic free energy with respect to the location change of the
dielectric boundary. This result is needed for level-set relaxation
and force calculations of biomolecular structures and dynamics.

http://www.math.ucsd.edu/~bli/research/MBBseminar/

A computational framework is presented for the continuum modeling of
cellular biomolecular diffusion influenced by electrostatic driving
forces. This framework is developed from a combination of numerical
methods, geometric meshing and computer visualization tools. In
particular, a hybrid of (adaptive) finite element and boundary element
methods is adopted to solve the Smoluchowski equation (SE), the Poisson
equation (PE), and the Poisson-Nernst-Planck equation (PNPE) in order
to describe electrodiffusion processes. The finite element method is
used because of its flexibility in modeling irregular geometries and
complex boundary conditions. The boundary element method is used due
to the convenience of treating the singularities in the source charge
distribution and its accurate solution to electrostatic problems on
molecular boundaries. Nonsteady-state diffusion can be studied using
this framework, with the electric field computed using the densities
of charged small molecules and mobile ions in the solvent. A solution
for mesh generation for biomolecular systems is supplied, which is an
essential component for the finite element and boundary element computations.
The partially coupled Smoluchowski equation and Poisson-Boltzmann equation
(PBE) are considered as special cases of the PNPE in numerical algorithm,
and therefore can be solved in this framework as well. The possible
extensions of the physical model in this frame are also discussed. Some
example computations are reported for: reaction-diffusion rate coefficient,
ion density distribution, time-dependent diffusion process of the
neurotransmitter consumption.

The properties of adaptive non-parametric kernel estimators for the multivariate probability density $f(x)$ (and its derivatives) of identically distributed random vectors $\varepsilon_n,\ n\geq 1$ at a given point are studied. It is supposed that the vectors $\varepsilon_n,\ n\geq 1$ form a martingale-difference process $(\varepsilon_n)_{n\ge 1}$ and the function to be estimated belongs to a class of densities slightly narrower than the class of densities with the following condition on the highest derivatives of the order $\nu:$

$$
|f^{(\nu)}(y)-f^{(\nu)}(x)|\leq \Delta(\|x-y\|),\qquad x,y\in {R^m},
$$

where $\Delta(t),\ t\geq 0,$ is some positive, bounded from above, monotonously decreasing for $t,$ small enough unknown function.

An asymptotic mean square criterion is proposed. The optimality, in asymptotically minimax sense of adaptive estimators of density derivatives, is proved for a class of the Bartlett kernel estimators with a random data-driven bandwidth.

It's well-known that the optimization of the asymptotic value of
the mean squared error for the Bartlett kernel density estimators leads to the optimal bandwidth depending on unknown functions. Therefore it is not quite simple to apply these estimators to practice.

The paper proposes an adaptive approach to this problem, which
is based on the idea of changing the unknown functions in optimal bandwidth by a sequence of estimators converging to the unknown values of these functions. It is shown, that the constructed adaptive kernel estimators keep all the asymptotic
properties of the sharp-optimal non-adaptive Bartlett estimators.

An example of the adaptive estimator, optimal in the sense of the introduced criterion is considered. This estimator has simple structure and may be easily of practical usage in real statistical
problems. The proposed estimators possess the property of uniform asymptotic normality and almost sure convergence.

A (1+1)-dimensional Topological Quantum Field Theory is a tensor functor from the category of 2-dimensional cobordisms to the category of vector spaces. It is easy to give a characterisation of such functors: they are determined by the vector space associated to a single circle together with the structure maps it inherits from the disc and the pair of pants, which make it into a finite-dimensional Frobenius algebra.

A (1+1)-dimensional Conformal Field Theory is a much more subtle thing, being a functor from the category of Riemann surfaces (2-dimensional cobordisms equipped with complex structures or "moduli") to a category of Hilbert spaces.

Somewhere between lies the idea of Topological Conformal Field Theory, which is a "chain level" version of a CFT. It is determined by a chain complex on which the spaces of chains of the (morphism spaces of the) category of Riemann surfaces act. Such a structure arises in several places in modern topology, most notably in the theory of Gromov-Witten invariants of symplectic manifolds and in the Sullivan-Chas string topology of a loop space.

This term we aim to read Kevin Costello's paper "Topological Conformal Field Theories and Calabi-Yau categories" (math.QA/0412149), which gives an algebraic characterisation of TCFTs analogous to the "Frobenius algebra" classification of TQFTs. In the first talk in the series I will try to give an overview of what the paper says, and we will organise talks for the rest of the term.

We describe a domain decomposition algorithm
for use in several variants of the parallel
adaptive meshing paradigm of Bank and Holst.
This algorithm has low communication, makes
extensive use of existing sequential solvers,
and exploits in several important ways data
generated as part of the adaptive meshing paradigm.
We show that for an idealized version of the
algorithm, the rate of convergence is independent
of both the global problem size N and the number
of subdomains p used in the domain decomposition
partition. Numerical examples illustrate the
effectiveness of the procedure.

In this talk, I will present some new and old connections
between the eigenvalues of a graph and its structure.

We'll give a broad introduction to the subject of non-linear dispersive equations by starting with an overview of some of the most basic examples of non-linear systems, and then discussing specific linear and non-linear estimates which have been used to study these equations.

The Angel lives on an infinite chessboard, and plays a game with the Devil. Each turn, the Devil removes a square; the Angel then flies up to 1000 king moves away, to any square still remaining on the board. Can the Devil trap the Angel?

Berlekamp, Conway, and Guy introduced this game in 1982, and the Angel's fate remained conspicuously unknown until recent papers by four independent authors all showed that the Devil's cause is hopeless. One of the papers, by Andr\'{a}s M\'{a}th\'{e}, achieves this by considering a ``Nice Devil.'' This paper will be presented after some background is explored.

The solutions of many important problems in population genetics require
tools from probability theory because the evolution of a population over
time is modeled as a random process. In this talk, we focus on the question
of how to describe the effect that beneficial mutations have on the
genealogy of a population. We present two approximations. The first, which
involves coin tossing, is simple but not very accurate. The second, which
is based on a stick-breaking construction, leads to much more accurate
results.

I will present an introduction to zeta and L-functions of graphs by
comparison with the zeta and L-functions of number theory. Basic
properties will be discussed, including: the Ihara formula saying that the
zeta function is the reciprocal of a polynomial. I will then explore graph
analogs of the Riemann hypothesis, the prime number theorem, Chebotarev's
density theorem, zero (pole) spacings, and connections with expander
graphs and quantum chaos. References include my joint papers with Harold
Stark in Advances in Mathematics. There is also a book I am writing on my
website: http://math.ucsd.edu/\%7Eaterras/newbook.pdf

Let $R$ be a ring generated by $l$ elements with stable
range $r$. Assume that the group $EL_d(R)$ has Kazhdan constant
$\epsilon_0>0$ for some $d \geq r+1$. We prove that there exist
$\epsilon(\epsilon_0,l) >0$ and $k \in \mathbb{N}$, s.t. for every
$n \geq d$, $EL_n(R)$ has a generating set of order $k$ and a
Kazhdan constant larger than $\epsilon$. As a consequence, we
obtain for $SL_n(\mathbb{Z})$ where $n \geq 3$, a Kazhdan constant
which is independent of $n$ w.r.t generating set of a fixed size.

In this talk we'll continue with linear and non-linear equations, focusing on several "monotonicity principles". These include the so called Morawetz and local smoothing estimates.

In the formulation of practical optimization methods,
it is often the case that the choice of numerical
linear algebra method used in some inherent calculation
can determine the choice of the whole optimization
algorithm. The numerical linear algebra is particularly
relevant in large-scale optimization, where the linear
equation solver has a dramatic effect on both the
robustness and the efficiency of the optimization.

We review some of the principal linear algebraic issues
associated with the design of modern optimization
algorithms. Much of the discussion will concern the use
of direct and iterative linear solvers for large-scale
optimization. Particular emphasis will be given to some
recent developments in the use of regularization.

Given a directed graph $G$ with girth at least $m+1$ (and no parallel edges),
let $\beta(G)$ denote the size of the smallest subset $X \subseteq E(G)$
so that $G \setminus X$ has no directed cycles, and let $\gamma(G)$ be the
number of non-edges. Prior joint work with Maria Chudnovsky and Paul Seymour
showed that when $m = 3$, $\beta(G) \leq \gamma(G)$, and we conjectured
$\beta(G) \leq \frac{1}{2}\gamma(G)$.
Can one say anything stronger if $m > 3$? In this talk, I will discuss
a new conjecture giving a ratio between $\beta(G)$ and $\gamma(G)$, namely
$\beta(G) \leq \frac{2}{m^2-m-1}\gamma(G)$, for $m \geq 3$. The talk will
also cover two new results in this direction: the bound
$\beta(G) \leq \frac{1}{3}\gamma(G)$ when $m=4$, and for
circular interval graphs, a generalization of previous methods which
gives a new bound for all $m$.

I'll try to explain how one may enumerate manifolds endowed with
some structure. For example, we may want to list all almost complex or
parallelizable manifolds. This problem is too general as posed, so we will
work up to cobordism (to be defined). The result will be a very elegant
framework which was first explored by R. Thom, and now belongs to the
toolbox of any self-respecting algebraic topologist.

Based on large N Chern-Simons/topological string duality, in a series of papers, J.M.F. Labastida, M. Marino, H. Ooguri and C. Vafa conjectured certain remarkable new algebraic structure of link invariants and the existence of infinite series of new integer invariants. In this lecture, I will describe a proof of this conjecture. Moreover, I will show that these new integer invariants vanish at large genera. In the end of the talk, some application in the knot theory and related problems (e.g., the famous volume conjecture), will also be discussed.

\center A complete schedule for this two day workshop can be found at: \\
http://math.ucsd.edu/~lni/OctWorkshop.html

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.

Electrostatic forces play a crucial role in biomolecular interactions such as
protein-ligand association, protein induced membrane deformation or DNA curvature
and deformation in protein-DNA complexes. Although the electrostatic forces
derived from the potential solution of the Poisson-Boltzmann equation (PBE)
have been widely used in studying these interactions with Brownian dynamics
simulations, the modeling and computation of these forces in the continuum
framework are not well established. In this talk I will propose the models of
the electrostatic body force and surface force compatible with the PBE, and
discuss the stable computation methods of these forces using a new stable
regularization scheme of the PBE. Numerical experiments will be presented to
verify these models and their usefulness of in studying the rigid body motion
of biomolecules.

I will discuss canonical metrics on Kahler manifolds,
including Kahler-Einstein and constant scalar curvature Kahler
metrics. By conjectures of Yau, Tian and Donaldson, the existence of
such a metric should be equivalent to the properness of the relevant
energy functional and to the algebraic `stability' of the manifold.
I will describe some work on energy functionals in Kahler geometry
and methods for finding canonical metrics.

A large part of this talk will be devoted to symmetric bilinear forms and inner product spaces, which should be of interest to geometers and algebraists alike. I'll outline some of the theory over general (commutative) rings and then turn to the classification of non-degenerate symmetric bilinear forms over the integers. The case of indefinite forms is completely understood and not too hard. On the contrary, the case of positive definite forms is quite difficult and, surprisingly, turns out to be related to the question of packing oranges in Euclidean boxes.

In the second part of the talk, we will see that symmetric bilinear forms come up as invariants in the theory of $4$-dimensional manifolds (their intersection forms). Freedman showed that a (topological) $4$-manifold is (almost) classified by its associated bilinear form and that indeed every possible form arises in this way. On the other hand, Donaldson showed that the intersection form of a \emph{smooth} $4$-manifold has a very special structure. Combining the two results led to examples of many topological $4$-manifolds that do not admit a smooth structure, a revolution of the (previously almost non-existent) theory of $4$-manifolds, two fields medals, lots of other interesting research, and this talk.

Suppose we observe a security network composed of
sensors with each sensor returning a value indicating whether the
sensor is at risk (high value) or not (low value). A typical attack
leaves a trail where the sensors return higher-than-normal values.
The goal is to detect a possible attack. Within a simplified
framework, we will see that if the sensor do not return high-enough
values (we will quantify that), then detection is impossible.

Formal abstract: Consider the complete regular binary tree of depth M
oriented from the root to the leaves. To each node we associate a
random variable and those variables are assumed to be independent.
Under the null hypothesis, these random variables have the standard
normal distribution while under the alternative, there is a path from
the root to a leaf along which the nodes have the normal distribution
with mean A and variance 1, and the standard normal distribution away
from the path. We show that, as M increases, the hypotheses become
separable if, and only if, A is larger than the square root of 2 ln 2. We obtain corresponding results for other graphs and other
distributions. The concept of predictability profile plays a crucial
role in our analysis.

Joint work with Emmanuel Candes, Hannes Helgason and Ofer Zeitouni.

I will explain the cohomological approach to local class
field theory, and how it generalizes to higher local fields. For
example, a 2-local field $K$ is a complete discrete valuation field with
residue field a local field; class field theory for K states, in part,
that there is an Artin map from $K_2(K)$ to the abelianized absolute
Galois group of $K$.

I will show how the cohomological approach can be used to construct a
Weil group for any finite extension of 2-local fields.

Molecular surface and volumetric meshes are important for molecular
modeling and simulation. I'll talk about mesh generation using a
so-called "soft-model" approach, meaning that each atom is treated as
a smoothly decaying Gaussian function instead of a hard sphere, and
the molecular surfaces are given as level sets of the summation of
such functions from all atoms in a molecule. The mesh generation
toolchain consists of several steps: (1) from molecules to 3D volumes;
(2) from volumes to initial surface meshes; (3) surface mesh
post-processing; and (4) volumetric mesh generation and smoothing. A
number of examples on molecules taken from Protein Data Bank (PDB)
will be demonstrated.

Since they were introduced by Macdonald 20 years ago,
Macdonald polynomials have been widely studied and have been found to
have applications in such areas as representation theory, algebraic
geometry, group theory, statistics, and quantum mechanics. The
Macdonald integral form may be defined as the unique function
satisfying certain triangularity and orthogonality conditions, from
which symmetry follows. The Macdonald Positivity Conjecture (now
Theorem) states that the coefficients of the Macdonald integral form
expanded into Schur functions are non-negative integers. The original
proof, due to Haiman in 2001 building on joint work with Garsia, uses
difficult machinery in algebraic geometry and does not provide a
combinatorial understanding of the coefficients. In this talk we
present a purely combinatorial proof of Macdonald positivity and give
a combinatorial interpretation for the Schur coefficients. The proof
utilizes an elegant monomial expansion for Macdonald polynomials
discovered by Haglund in 2004 and a new combinatorial tool called a
dual equivalence graph.

This talk surveys fundamental concepts in computability theory, including undecidability (noncomputability) and the G\"odel incompleteness theorems. The motivation and proof sketches are based on the Berry paradox."

I will explain the cohomological approach to local class
field theory, and how it generalizes to higher local fields. For
example, a 2-local field K is a complete discrete valuation field with
residue field a local field; class field theory for K states, in part,
that there is an Artin map from $K_2(K)$ to the abelianized absolute
Galois group of K.

I will show how the cohomological approach can be used to construct a
Weil group for any finite extension of 2-local fields.

This week we'll talk about proving $L^p$ estimates for solutions to constant coefficient dispersive equations. This is really a (quite specialized) problem in a much larger field known as "geometric harmonic analysis". We'll discuss some of the simplest tools for proving estimates of this type, and also a bit about the structure of the oscillatory integrals themselves. If time permits, we'll give some applications of these estimates to non-linear problems.

The level set method is a way of tracking moving interfaces
numerically.
Topological changes (splitting or joining) of several moving
interfaces are
captured naturally by the level set method. I will briefly introduce
the
level set method, then discuss some of my current research: the level
set
method in a finite element setting, body fitting a mesh to the
interface,
reinitialization and application to finding equilibrium solute-solvent
interfaces.

Electrostatic forces play a crucial role in biomolecular interactions
such as protein-ligand association, protein induced membrane deformation
or DNA curvature and deformation in protein-DNA complexes. Although the
electrostatic forces derived from the potential solution of the
Poisson-Boltzmann equation (PBE) have been widely used in studying these
interactions with Brownian dynamics simulations, the modeling and
computation of these forces in the continuum framework are not well
established. In this talk I will propose the models of the electrostatic
body force and surface force compatible with the PBE, and discuss the stable
computation methods of these forces using a new stable regularization scheme
of the PBE. Numerical experiments will be presented to verify these models
and their usefulness of in studying the rigid body motion of biomolecules.

In this food for thought seminar, we will reflect upon the beautiful Euler identity
$$e^{i \pi} = -1 $$
Apparently, Euler viewed in this equation all the body of mathematics. The number $e$ is related to analysis, the number $\pi$ to geometry, the number $i$ to algebra, and the number $-1$ to arithmetic or synonymously number theory.
\\ \\
The following identities are well known:
\begin{itemize}
\item The $x$ and $y$ coordinates of the third roots of unity on the unit circle
\begin{itemize}
\item $\sin(\frac{2 \pi}{3}) = \frac{\sqrt{3}}{2}$ and $\cos(\frac{2 \pi}{3}) = -\frac{1}{2}$
\item $\sin(\frac{4 \pi}{3}) = -\frac{\sqrt{3}}{2}$ and $\cos(\frac{4 \pi}{3})= -\frac{1}{2}$
\end{itemize}
\item The $x$ and $y$ coordinates of the fourth roots of unity on the unit circle
\begin{itemize}
\item $\sin(\frac{2 \pi}{4}) = 1$ and $\cos(\frac{2 \pi}{4}) = 0$
\item $\sin(\frac{4 \pi}{4}) = 0$ and $\cos(\frac{4 \pi }{4}) = -1$
\item $\sin(\frac{6 \pi}{4}) = -1$ and $\cos(\frac{6 \pi}{4}) = 0$
\end{itemize}
\end{itemize}
The following ones are less well known:
\begin{itemize}
\item The $x$ and $y$ coordinates of the fifth roots of unity on the unit circle
\begin{itemize}
\item $\sin(\frac{2 \pi}{5})= \frac{\sqrt{5 + 2 \sqrt{5}}}{\sqrt[5]{176 + 80\sqrt{5}}}$ and $\cos(\frac{2 \pi}{5}) = \frac{1}{\sqrt[5]{176 + 80\sqrt{5}}}$
\item $\sin(\frac{4 \pi}{5}) = {\frac{\sqrt{5 - 2\sqrt{5}}}{\sqrt[5]{-176 + 80\sqrt{5}}}}$ and $\cos(\frac{4 \pi}{5}) = -\frac{1}{\sqrt[5]{-176 + 80\sqrt{5}}}$
\item $\sin(\frac{6 \pi}{5}) = - \frac{\sqrt{5 - 2\sqrt{5}}}{\sqrt[5]{-176 + 80 \sqrt{5}}}$ and $\cos(\frac{6 \pi}{5}) = - \frac{1}{\sqrt[5]{-176 + 80\sqrt{5}}}$
\item $\sin(\frac{8 \pi}{5}) = -\frac{\sqrt{5 + 2\sqrt{5}}}{\sqrt[5]{176 + 80\sqrt{5}}}$ and $\cos(\frac{8\pi}{5}) = \frac{1}{\sqrt[5]{176 + 80\sqrt{5}}}$
\end{itemize}
\end{itemize}
We shall explain a painless way of deriving these formulas. For that, we will do a little bit of analysis, geometry, algebra and arithmetic.

In this talk we'll discuss the phenomena of wave propagation in
elastic membranes and solids. Using a few simplifying assumptions, one can
use Newton's equations to write down a partial differential equation for how
such a medium attempts to return to equilibrium from an initial displacement
or initial impulse.

There are several striking consequences one can derive from these partial
differential equations. The first is that even in an isotropic solid there
are two different kinds of waves, each of which propagates with a distinct
speed! The second is that there is a marked difference between how waves
propagate through a solid as opposed to how they propagate across a
membrane. In the former case the disturbance completely leaves any bounded
region and heads outward unless it is reflected back by some sort of
boundary. But in the latter case there is always some lingering "residue" of
a wave in any bounded region, no matter how far the bulk portion of the wave
has progressed. There are still open research questions concerning this
curious phenomena, and we'll try to discuss these by the end of the talk.

I will present the base-change lifting of automorphic, or local admissible, representations of the unitary group U(3), with respect to an odd degree cyclic field extension.

In the past decade, superfluid ultracold gases and liquids
have become the physicists' preferred domain for exploring
novel phenomena in condensed matter, and for testing many-body
theories of how atoms interact with each other. There is an
explosion of the new quantum fluids discovered, that challenge
both mathematicians and physicists by their fascinating, yet
enigmatic behaviors. The most famous of them is the ability
to flow through narrow gaps without any friction and energy
dissipation and the existence of quantized vortices. Different
problems and approaches in mathematical modeling of the
macroscopic superfluid dynamics will be discussed. Particular
emphasis will be given to the formation of coherent structures
and the quantized vortex dynamics.

Eukaryotic double-stranded DNA achieves cellular compaction through several
hierarchical levels of organization. First, DNA wraps around nucleosomes that
comprise of two copies each of the positively charged core histones H2A, H2B,
H3 and H4. The resulting "bead-on-a-string" nucleoprotein complex folds further
at physiological salt, and in the presence the linker histone, into the 30-nm
chromatin fiber. The thermodynamic and structural details of how histone tails
(N-termini of core histones) and linker histones critically compact and modulate
chromatin structure as well as regulate gene transcription are not well understood.
I present a new mesoscopic model of chromatin that represents nucleosome cores
as rigid bodies with an electrostatic surface, linker DNA as a discrete worm-like
chain model, and histone tails as protein bead chains, to elucidate the physical
role of each histone tail and the linker histone in chromatin folding. An
end-transfer configurational-bias Monte Carlo approach provides the positional
distribution of histone tail and their physical interactions at different salt
milieus. Analyses indicate that the H4 tails mediate the strongest internucleosomal
interactions; the H3 tails crucially screen electrostatic repulsion between the
linker DNAs; and the H2A and H2B tails mediate fiber/fiber interactions. The
primary function of the linker histones is to decrease the nucleosome triple angles,
resulting in highly compact chromatin with a different internucleosomal interaction
pattern than that obtained in linker-histone deficient chromatin. The development
of this model also opens new avenues for studying higher-order structures of
chromatin and the role of posttranslational modifications and variants of histone
tails in gene regulation.

SAGE is a free, open-source computer algebra that aims to
be a replacement for Mathematica, Maple, Matlab, and Magma and was
started by William Stein, a number theorist at the University of
Washington. Recently, a lot of support has been added to SAGE for
working with combinatorial objects such as partitions, permutations,
tableaux, etc. Additionally, SAGE now has fairly good support for
symmetric functions, which are "polynomials" in infinitely many
variables which are invariant under any permutation of those
variables. In this talk, I will give a brief introduction to SAGE and
give an overview of symmetric functions and how to work with them in
SAGE. I will also talk briefly about some generalizations of
symmetric functions including Hall-Littlewood polynomials and
Macdonald polynomials.

Do you ever find yourself disappointed and confused when an
interesting result follows from an algebraic identity that seems to
have come out of nowhere? Do arguments that are "so elegant that they
conceal what is really going on" leave you longing for more? WHAT IF
EVERYTHING YOU EVER WANTED CAME IN ONE FFT TALK?
I'm going to present a short proof of a classical result that
follows from an unmotivated algebraic identity. I will then walk
through an intuitive, Functional Analytic, derivation of the theorem
introducing and explaining definitions as I go. Ideally the long proof
of the theorem will completely illustrate the meaning of the short
one. The emphasis of the talk will be on introducing interesting ideas
in functional analysis in the context of an extended example.

A polynomial $p$ is bigger than $q$ if $p(x)> q(x)$ for all $x$ and a classical area of mathematics (called semialgebraic geometry) focuses on providing algebraic ways to test if such inequalities hold.

Semialgebraic geometry can be generalized to polynomials with matrix variables which is fortunate, since many problems in systems engineering have matrix unknowns. The difficulty here is that the polynomials are in variables which do not commute.
Particularly important (for numerical solutions) is convexity and there has been a recent quest to classify convex noncommutative polynomials. The conclusion turns out to be simple. The talk will be on the mathematics and the motivation.

The best bounds known for the Euclidean sphere packing problem (at least in dimensions 4 through 36) are proved using linear programming bounds due to Cohn and Elkies. These bounds are derived using a simple argument based on Poisson summation, but optimizing the bounds has proved difficult. It is conjectured that the bounds are in fact sharp in 8 and 24 dimensions, and this has been verified to 30 decimal places, but it has not been proved. In this talk, I'll discuss the challenges that must be overcome to prove it as well as some new mysteries that have arisen recently in this area.

For a Gelfand pair $(G,H)$ consisting of a locally compact group $G$ and a compact subgroup $H$
of $G$, the $H$-bi-invariant measures on $G$ can be identified with measures on the double coset space $G/\!/H$. The canonical projection from $G$ onto $G/\!/H$ induces a convolution of measures on $G/\!/H$ such that $G/\!/H$ becomes a commutative hypergroup. Central to the present talk will be the discussion of
L\'evy processes with values in the hypergroup $G/\!/H$ and their characterization in terms of complex-valued martingales.

In order to achieve this goal some harmonic analysis on hypergroups has to be developed: the convolution hemigroups associated with L\'evy processes will be studied with the help of a Fourier transform.

On Sturm-Liouville hypergroups the characterization of Gaussian hemigroups is obtained via martingales involving moments. For convolution semigroups, i.e.\ stationary L\'evy processess, the results yield characterizations of radial Brownian motions on Euclidean and hyperbolic spaces.

Koch-Tataru solutions are global-in-time small data mild
solutions to the 3D NSE emanating from small data in
$BMO^{-1}$. There have been several recent works in which
the regularizing-decay rate estimates for Koch-Tataru
solutions have been obtained. Spatial analyticity of solutions
then follows as a consequence. I will present a different
approach in which the spatial analyticity is obtained
directly together with explicit estimates on the time-evolution
of the domain of analyticity. The regularizing-decay rate
estimates will then follow at once.

This talk is about how reproducing kernels and certain
rearrangements of finite dimensional subspaces of two variable
polynomials can reprove several interesting results. These include
recent results of Geronimo and Woerdeman on orthogonal polynomials on
the two dimensional torus as well as Ando's inequality (and its close
relative, Pick interpolation theorem on the bidisk). Considerable
time will be spent on one variable in order to make the ideas clear.

The locator problem models the following physical situation.
Suppose one lands an unmanned space craft on an unobservable terrain (e.g., under the clouds of Venus or on the backside of the moon) and wishes to determine the location of the landing site. Suppose that the craft can sample the altitude at the landing site and at several other spots (say at 100 meters east and at 100 meters west). However, the
craft does not have a map of the altitudes of the terrain (i.e.,
$a(x,y)$), but only a single function, $p(x,y)$ that approximates the altitude. The locator problem is to find a function p from a family of functions $P$ that minimizes the error between the actual location of the craft and the computed location of the craft using the approximation $p$.
The error is to be minimized over all possible locations that is we seek the $p$ in $P$ to minimize

$$\Arrowvert (x,y)- p^{(-1)}(a(x,y))\Arrowvert$$

This is equivalent to classical approximation questions about existence and uniqueness of best approximations from this (non-linear) family of inverse functions. The question is most interesting when the elements in the setting are the most fundamental and basic: for example, when $P$ is the polynomials of degree n and the norm is the uniform norm or the
$L_1$ norm.

Although this is a rich theoretical setting with five fundamental
elements to define (various metrics, data collections and families of approximating functions) and potentially has useful applications, we
have the only known solutions. These are for $P$ the increasing
polynomials of degree $n$; the domain and range being the unit interval
and the norm being either the uniform norm or the $L_1$ norm. In this
setting there exist best locator functions and they are unique.

Viscoelastic flow modeled by the Oldroyd-B equations will be discussed from an analytical and computational perspective. First I will present a local energy decay theorem which applies to a large class of hyperbolic systems including the Oldyoryd-B model. This decay theorem is used to prove that global smooth solutions exist for small initial data. While small solutions are global, the problem for large data is much more complicated. I will present recent computational work on the Oldroyd-B equations which indicates that the system develops singularities exponentially in time at hyperbolic stagnation points in the flow. The singularities arise in the stress field of the flow and the algebraic structure of these singularities depends critically on an important elasticity parameter, the Weissenberg number. A local approximation to the solution at the hyperbolic stagnation point is constructed and there is excellent agreement between the local solution and the simulations. In addition, past a critical Weissenberg number the flow pattern becomes quite sensitive to time periodic perturbations of the background forcing (or changes in initial data) and there is a transition from small scale local mixing around the stable and unstable manifolds to global mixing in the fluid.

In this talk, we study three versions of
adaptive finite element method: r, h and p-adaptive. While
the first two are well developed and widely used, much less
is known about the third one. By looking at some ideas and
techniques used in r-version and h-version, we propose a
tentative plan to construct a p-version of adaptive finite
element method. The key thing makes it possible is the
recently result of Bank, Xu and Zheng on generalizing
gradient recovery technique for linear elements to
derivative recovery for Lagarange elements of order p
arbitrary.

We study nonlinear integro-differential equations. Typical examples
are the ones that arise from stochastic control problems with
discontinuous Levy processes. We can think of these as nonlinear
equations of fractional order. Indeed, second order elliptic PDEs are
limit cases for integro-differential equations. Our aim is to extend
the theory of fully nonlinear elliptic equations to this class of
equations. We are able to obtain a result analogous to the Alexandroff
estimate, Harnack inequality and $C^{1,\alpha}$ regularity. As the
order of the equation approaches two, in the limit our estimates
become the usual regularity estimates for second order elliptic pdes.
This is a joint work with Luis Caffarelli.

Elliptic divisibility sequences are integer recurrence sequences, each
of which is associated to an elliptic curve over the rationals
together with a rational point on that curve. I'll give the background
on these and present a higher-dimensional analogue over arbitrary base
fields. Suppose $E$ is an elliptic curve over a field $K$, and $P_1$, ...,
$P_n$ are points on E defined over $K$. To this information we associate
an $n$-dimensional array of values of $K$ satisfying a complicated
nonlinear recurrence relation. These are called elliptic nets. All
elliptic nets arise from elliptic curves in this manner. I'll explore
some of the properties of elliptic nets and the information they
contain, relating them to generalised Jacobians and to the Tate
and Weil pairings.

Mechanical forces are generated during nearly every facet of the
cell cycle. Recent advances in experimental techniques enable
experimentalists to exert forces on individual molecules and observe
their response in real time. Thus, the single-molecule approach
has changed the way many physical, chemical and biological problems
are addressed. We present a theory for extracting kinetic information
from single-molecule pulling experiments at constant force or
constant loading rate. Our procedure provides estimates of not
only i) the intrinsic rate coefficient and ii) the location of the
transition state, as in the widely used phenomenological approach
based on Bell's formula, but also iii) the free energy of activation.
We illustrate the use of our approach by applying it to sets of data
obtained from nanopore unzipping of individual DNA hairpins and from
unfolding of single protein molecules with the atomic force microscope.

A combinatorial geometry problem, related (in a surprising way)
to the Graham's g.c.d. conjecture, is as follows. Let $n\ge 1$ be
an integer. Given a vector $(a_1 , ... , a_n)\in R^n$, write
$$ a^+ := ( \max(a_1,0) , ... , \max(a_n,0) ) $$
(the "projection of $a$ onto the positive orthant"), and for a set
$A\subset R^n$ put
$$ A^+ := \{ a^+ : a\in A \}. $$
How small $|(A-A)^+|$ can be for a set $A\subset R^n$ of given
cardinality $|A|$? We discuss previously known results and report
on recent developments due to Ron Holzman, Rom Pinchasi, and
the presenter.

Building on work by Hans Lindblad, Daniel Coutand and Steve Shkoller, we prove that the equations of motion of an incompressible, inviscid, self-gravitating fluid with free boundary are well-posed. The methodology consists of the use of a smoothing operator which smoothes along the boundary of the fluid. This then allows the use of a fixed-point-type argument, which provides a solution to a smoothed version of the problem. Estimates subsequently show that these solutions converge to a solution of the full problem.

We will give four proofs of the Cheeger inequality which relates
the eigenvalues of a graph with various isoperimetric variations of the Cheeger constant. The first is a simplified proof of the classical Cheeger inequality using eigenvectors. The second is based on a rapid mixing result for random walks by Lov\\'asz and Simonovits. The third uses PageRank, a quantitative ranking of the vertices introduced by Brin and Page. The fourth proof is by an improved notion of the heat kernel pagerank. The four proofs lead to further improvements of graph partition algorithms and in particular the local partition algorithms with cost proportional to its output instead of in terms of the total size of the graph.

The design of electrodes' and insulators' shapes with respect to
electric field stresses in High Voltage Devices is formulated as a
nonlinear optimization problem under constraints.
We apply a sensitivity equation method in order to achieve gradient
information for the optimal design problem. This enables us to use the
SQP-Algorithm for the optimization task while a FEM software package
accomplishes the electric field computation.
Results are presented for gas-insulated transmission lines and
circuit-breakers in highly integrated switch-gears.

We first introduce the honored guest, Ricci Flow and present some simple examples. We'll then get excited about why it is used: it is essentially a nonlinear heat equation which is used for smoothin' out the rough edges around some baddie metrics. However there are some real rogues who are a hopeless case, and develop singularities when goin' with the flow. These can ``excised'' under certain conditions, and we can continue on our merry way.

Next, we describe why it is called a ``flow,'' and chat about how flows fit into the big picture in solving PDEs in general. We really hope that this will not only (partially) demystify the now famous Ricci Flow, but also give newbie analysts and PDE-ers a big picture of what kinds of totally awesome modern tools and methods they'll be encountering.

The so-called Lattice-Boltzmann-Method is a relatively young numerical
method for the computation of fluid mechanics. In contrary to
'classical' techniques of computational fluid dynamics where the
Navier-Stokes-Equations - a system of partial differential equations
describing the macroscopic behavior of a flow - are discretized and
solved, the Lattice-Boltzmann-Method starts from the microscopic
description of a flow, the Boltzmann-equation. It is a time-explicit
procedure based on a strongly simplified kinetic theory, yet it is
approximating the Navier-Stokes equations.

The presentation gives a short overview about some basic principles of
the Lattice-Boltzmann-Method. Typical engineering applications, in
particular the feasibility of describing aeroacoustic phenomena (i.e.
sound generated by flows which typically requires a time-dependent
calculation of pressure and density within a fluid), are pointed out,
and a partitioned approach for the coupled simulation of flows and
structures is presented.

Suppose we observe a security network composed of
sensors with each sensor returning a value indicating whether the
sensor is at risk (high value) or not (low value). ?A typical attack
leaves a trail where the sensors return higher-than-normal values.?
The goal is to detect a possible attack.? Within a simplified
framework, we will see that if the sensor do not return high-enough
values (we will quantify that), then detection is impossible.

Formal abstract: Consider the complete regular binary tree of depth M
oriented from the root to the leaves.? To each node we associate a
random variable and those variables are assumed to be independent.?
Under the null hypothesis, these random variables have the standard
normal distribution while under the alternative, there is a path from
the root to a leaf along which the nodes have the normal distribution
with mean A and variance 1, and the standard normal distribution away
from the path. We show that, as M increases, the hypotheses become
separable if, and only if, A is larger than the square root of? 2 ln
2.? We obtain corresponding results for other graphs and other
distributions.? The concept of predictability profile plays a crucial
role in our analysis.

Joint work with Emmanuel Candes, Hannes Helgason and Ofer Zeitouni.

Period integrals of automorphic forms, such as Fourier coefficients of modular forms, is a subject of much contemporary interest.
In this talk we survey some of the literature on the subject, and
include some of my own works which relate these global concepts to
analogous local questions.

The main goal in Invariant Theory is to study the ring of all
polynomials that are invariant under an group action. Invariant
rings are not always finitely generated, because of Nagata's
counterexample to Hilbert fourteenth problem.
By results of Hilbert, Nagata and Haboush, invariant rings of
reductive groups are finitely generated. Unfortunately, most
finite generation proofs are not constructive. In particular,
they do not provide algorithms for finding a set of generators
for the invariant ring. In this talk I will discuss
various algorithms for generators of invariant rings. I will also
present recent results of Gregor Kemper and myself:
We found the first algorithm for generators of invariant rings
of reductive groups actions on affine varieties
in arbitrary characteristic.
We also found an algorithm for generators of invariant
rings for unipotent group actions on the polynomial ring.
In that case, the ring of invariants may not be finitely generated,
but the output of the algorithm presents the ring of invariants as the
ring of regular functions on some explicitly given quasi-affine variety.

Combinatorial allocation problems have been a subject of recent interest
due to their role in on-line auctions and electronic commerce. An
allocation problem entails a finite set of "items" that should be
distributed among participating "players" in order to maximize a certain
"social utility" function. A particular case of interest is the Submodular
Welfare Problem, where the utility functions are assumed to be submodular.

Our recent result is that a $(1-1/e)$-approximation can be achieved for
the Submodular Welfare Problem, which is known to be optimal. The
$(1-1/e)$-approximation can be extended to a more general problem for which
a $1/2$-approximation was known since 1978 [Fisher,Nemhauser,Wolsey]. I will
discuss our improvements and the techniques that we use - randomization,
replacing the discrete problem by a continuous one, and approximately
solving a non-linear optimization problem using a continuous greedy
method.

Partly joint work with G. Calinescu, C. Chekuri and M. Pal.

We introduce a new method for analysing certain problems in
extremal combinatorics that involve small forbidden configurations. The
cornerstone of our approach is a quasirandom counting lemma for
quasirandom hypergraphs, which extends the standard counting lemma by
not only counting copies of any particular configuration but also
showing that these copies are evenly distributed. We demonstrate the
power of the method by proving a conjecture of Mubayi on the codegree
threshold of the Fano plane, that any 3-graph on n vertices for which
every pair of vertices is contained in more than $n/2$ edges must contain
a Fano plane, for n sufficiently large. For projective planes over
fields of odd size we show that the codegree threshold is between $n/2-q+1$ and $n/2$, but for $PG_2(4)$ we find the somewhat surprising phenomenon that the threshold is less than $(1/2-c)n$ for some absolute $c>0$.

We review basics of subfactor theory in von Neumann algebras.
We then show how one can construct examples using the notion of
commuting squares. This will be illustrated using actions of finite
groups. These are models for more general examples using tensor categories.

Let $A$ be the first Weyl algebra, in the Euler gradation. We classify graded rings $B$ such that $gr-A$ and $gr-B$ are equivalent (we say that $A$ and $B$ are graded equivalent), and produce some surprising examples. In particular, we show that $A$ is graded equivalent to an idealizer in a localization of $A$.

In the process, we derive a concise new characterization of equivalences of graded module categories that generalizes the classical Morita theorems.

A large part of this talk will be devoted to symmetric bilinear forms and inner product spaces, which should be of interest to geometers and algebraists alike. I'll outline some of the theory over general (commutative) rings and then turn to the classification of non-degenerate symmetric bilinear forms over the integers. The case of indefinite forms is completely understood and not too hard. On the contrary, the case of positive definite forms is quite difficult and, surprisingly, turns out to be related to the question of packing oranges in Euclidean boxes.

In the second part of the talk, we will see that symmetric bilinear forms come up as invariants in the theory of $4$-dimensional manifolds (their intersection forms). Freedman showed that a (topological) $4$-manifold is (almost) classified by its associated bilinear form and that indeed every possible form arises in this way. On the other hand, Donaldson showed that the intersection form of a \emph{smooth} $4$-manifold has a very special structure. Combining the two results led to examples of many topological $4$-manifolds that do not admit a smooth structure, a revolution of the (previously almost non-existent) theory of $4$-manifolds, two fields medals, lots of other interesting research, and this talk.

A region, $X$, (called a cake) is to be "sliced" so that each of a panel of m judges assess that the division as fair. Each judge has his or her own measure, $ui(S)$ of the value of each part, $S$ of the cake and $ui(X)=1$.

There are two settings. In the first the cake is to be distributed to two people so that every judge believes that the portions given to each recipients $(U and X-U)$ is worth exactly $½$ (i.e. $ui(U)= ½ = ui(X-U) for all i=1,2, … , m)$.

In the second setting, the m $(m > 2)$ judges are taking a portion of the cake (i.e., $Ui$) for themselves. They want a division of the cake (i.e., $Uj \cap Uk$ is empty for each $j\neq k$ and $U1 +U2+ … +Um = X$) so that each believes they received more than their fair share of the cake (i.e. $ui(Ui) > 1/m for each i)$.

Both settings have solutions.

The solutions give an introduction to measure theory and a fixed point theorem.

The conjectures in the title were formulated in
the late 1970s as vast generalizations of the classical theorem of
Stickelberger. They make a very subtle connection between the
$\Bbb Z[G(F/k)]$--module structure of the Quillen K-groups K${_\ast}(O_F)$ in an abelian extension $F/k$ of
number fields and the values at negative integers of the
associated $G(F/k)$--equivariant $L$--functions $\Theta_{F/k}(s)$.

These conjectures are known to hold true if the base field $k$ is
$\Bbb Q$, due to work of Coates-Sinnott and Kurihara. In this
talk, we will provide evidence in support of these conjectures
over arbitrary totally real number fields $k$.

The Littlewood-Paley theory is extended to weighted spaces of
distributions on $[-1,1]$ with Jacobi weights
$
w(t)=(1-t)^\alpha(1+t)^\beta, <i>italic</i>
$
and to the unit ball $B^d$
in $R^d$ with weights $W_\mu(x)= (1-|x|^2)^{\mu-1/2}$, $\mu \ge 0$.
Almost exponentially localized polynomial elements (needlets)
$\{\varphi_\xi\}$, $\{\psi_\xi\}$ are constructed
and,
in complete analogy with the classical case on $R^d$,
it is shown that weighted Triebel-Lizorkin and Besov spaces
can be characterized by the size of the needlet coefficients
$\{\langle f,\varphi_\xi\rangle\}$
in respective sequence spaces.

Let $\mu_1, \mu_2, ... , \mu_m$ be non-atomic probability measures on a measurable space $(X, \Sigma)$.

Theorem (Liapanouv 1940) ${\mu(\cup) = (\mu_1(\cup) + \mu_2(\cup) + ... + \mu_m(\cup) ): \cup \ {\text{in}}\ \Sigma }$ is a compact convex set.

If in addition there is a topology on X and ‚$\Sigma$ is the Borel sets (or the Baire sets, respectively) we can ask when the range of the vector valued measure $\mu$ is obtained even when the measure is restricted to the sets $\cup$ which are open (or the support of a non-negative continuous function, resp.).

We will give a couple applications of the Classical Theorem. We will then cast the Liapanouv Theorem in an equivalent form about the range of a vector of integrals on X.

In that form we will give a single theorem that, in addition to proving the Classical Liapanouv Theorem, also characterizes when the open sets (or the supports of continuous functions, resp.) suffice. That is let L be a cone of functions. Let S be the supports of functions in L, and let ‚$\Sigma$ be the sigma-algebra generated by S. The three cases above result when $L = L\infty$, the upper-semi-continuous functions on X, and C(X) respectively.

Several statistical problems where the goal is to minimize an unknown convex risk function, can be formulated in the general framework of stochastic convex optimization. For example the problem of model selection and more generally of aggregation can be treated using the machinery of stochastic optimization in several frameworks including density estimation, regression and convex classification. We describe a family of general algorithms called "mirror averaging algorithms" that yield and estimator (or a classifier) which attains optimal rates of model selection in several interesting cases. The theoretical results are presented in the form of exact oracle inequalities similar to those employed in optimization theory. The practical performance of the algorithms is illustrated on several real and artificial examples and compared to standard estimators or classifiers.

We first prove an exact result for hypergraphs: given $r\ge 2$, let
$p$ be the
smallest prime factor of $r-1$. If $n> (p-1)r$ and $G$ is an $r$-
uniform hypergraph on
$[n]$ such that every $r+1$ vertices contain $0$ or $r$ edges, then
$G$ is either
empty or a star, $\{E\subset [n]: |E|=r, E\ni x\}$ for some $x\in
[n]$.
Then we use it to slightly improve best known bounds for hypergraph
Tur\'an numbers. We show that $\pi(K^r_{r+1})\leq 1- \frac{1}{r} -
\left(1- \frac{1}{r^{p-1}}\right)\frac{(r-1)^2}{2r^p({r+p\choose
p-1}+{r+1\choose 2})}$
when $r\ge 4$ is even.

This is joint work with Linyuan Lu.