We got yer moduli spaces right here.

Introduced by Whitney in 1935, matroids are a combinatorial abstraction of linear independence. Beginning in the 70's the study of real and complex hyperplane arrangements brought matroids to geometry and topology. We will examine these applications and more recent ones involving quotients of spheres by elementary abelian p-groups, and arrangements of homotopy spheres.

This talk will introduce Lie groups, one of the cornerstones of modern geometry, and the exponential map.

Several systematic deviations from Bayesian updating have been discussed in the psychology literature and some of these have been incorporated into modeling exercises in behavioral finance. This paper presents a unifying axiomatic framework for a broad range of updating behavior including some of these disparate specifications. The key is to study preference on (contingent) menus of acts. Given suitable axioms, the functional form for utility features two measures, where one is the 'usual prior' p and the other determines how updating deviates from Bayesian updating of p. A dynamic version of the Anscombe-Aumann model is obtained as the special case where the two measures coincide.

Traveling waves in reaction-diffusion equations have been very successful in modeling various phenomena in physics, biology and other sciences. Recently there has been a lot of mathematical works on the possible effect of a fluid flow or wind on front propagation. A flow may siginificanlty enhance the front speed as well as extinguish a flame. We will review our results that exhibit and quantify these phenomena as well as discuss some progress in the analysis of the reactive Boussinesq system.

After briefly describing the geometric meaning of Oseledec's multiplicative ergodic theorem as it applies to volume-preserving flows, we will present a proof of the theorem appearing in a paper of David Ruelle's that is due to M. S. Raghunathan. The proof contains both analytic and geometric ideas; being a seminar on geometry, we will highlight the geometric ideas and suppress (that is, state, but not prove) some of the analytic detail.

It is well known that strange attractors occur in the H\'enon family of maps. Benedicks and Carleson proved that such attractors, called H\'enon-like, coincide with the closure $\overline{W^u(O)}$ of the unstable manifold of some hyperbolic periodic orbit $O=\{p_1,\dots,p_n\}$ of saddle type. On the other hand, numerical studies show the occurrence of attractors with a quasi-periodic component in the dynamics, called quasi-periodic H\'enon-like. As models for such attractors, we study families of diffeomorphisms of the cylinder $\mR^2 \times \mS^1$. Our main result shows the occurrence of H\'enon-like strange attractors when is a skew-product obtained by driving a H\'enon-like family $H$ in $\mR^2$ by the Arnold family of circle maps. The dynamics of the Arnold family on $\mS^1$ is assumed to be of Morse-Smale type. The attractors we obtain are H\'enon-like and occur on a set of parameter values of positive measure. Furthermore, occurrence is persistent under sufficiently small $C^3$-perturbations of the H\'enon-like family $H$. We also consider the case when the family has an invariant circle . Under suitable hypotheses, we show that the closure $\overline{W^u(S)}$ attracts an open set of initial states.

I will discuss main requirements for quantum computation. Elementary quantum gates, swap operation, creation of entanglement, teleportation will be described. A perturbation theory for scalable solid-state quantum computation with many qubits will be discussed. The ideas of error correction will be explained. Finally, I will describe the unwanted effect of decoherence for quantum bits.

CANCELLED

The level set method for capturing moving fronts was introduced in 1987 by Osher and Sethian. It has proven to be phenomenally successful as a numerical device. For example, typing in "Level Set Methods" on Google's search engine gives roughly 3300 responses. Applications range from capturing multiphase fluid dynamical flows, to special effects in Hollywood to visualization, image processing, control, epitaxial growth, computer vision and many more. In this talk we shall give an overview of the numerical and analytical technology and a few new applications.

This is the speaker's candidacy talk. Abstract: This talk concerns recent work on evaluations of L-series related to Zagier's Kronecker limit formula for real quadratic fields. More specifically, I will define three L-series related to a certain universal function appearing in Zagier's Kronecker limit formula, and show that each of these series has a meromorphic continuation to the complex plane. I will then derive a limit formula which determines the values of these L-series at the negative even integers.

We sketch, in as much detail as possible in 50 minutes, a proof of Oseledec's multiplicative ergodic theorem. We follow the presentation in a paper of David Ruelle's of a version of the proof that is due to M. S. Raghunathan. The proof contains both analytic and geometric ideas; being a seminar on geometry, we will highlight the geometric ideas and suppress some of the analytic detail. The presenter's notes for this talk, which can be found at http://www.ma.utexas.edu/users/kelliher/Geometry/Geometry.html contain all the details of the proof.

A mathematical analysis of the ``time reversal mirror" is given. the emphasis is put on phenomena which are described by the genuine acoustic equation with Dirichlet or ``impedance" boundary conditions. An ideal situation is first considered then relation between the question of the local decay of energy and the accuracy of the method are exploited. At variance with other approaches no randomness is introduced and only deterministic phenomena are considered. However instead randomness deterministic ergodicity is analyzed and it positive effects are explained with the (now classical) theorems on Quantum Chaos.

We define the reciprocal Mahler's measure to be Mahler's measure restricted to the subspace of reciprocal polynomials with complex coefficients of fixed degree. We view the reciprocal Mahler's measure as a symmetric, homogeneous distance function in the sense of the geometry of numbers. The distribution function associated with the monic reciprocal Mahler's measure encodes statistical information about the range of values of Mahler's measure takes on the reciprocal subspace. We take the Mellin transform of this distribution function to encode this statistical information into a function analytic in a half-plane. This analytic function extends to a rational function with simple poles at small integers and a zero of high multiplicity at zero. From this rational function we explicitly compute the distribution function, as well as the volume of the star body (non-convex unit ball) associated with the reciprocal Mahler's measure. In particular the distribution function is a reciprocal Laurent polynomial, and the volume of the reciprocal star body is a power of pi times a rational number.

After proving his revolutionary hyperbolization theorem for Haken 3-manifolds in the late 1970s, W. Thurston proposed a conjectural classification and parameterization for infinite volume hyperbolic 3-manifolds with finitely generated fundamental group. His conjectural picture, which has topological and geometric components, predicts that such a hyperbolic 3-manifold is determined by simple topological, combinatorial, and conformal data. I will discuss recent progress on the classification problem from various points of view, including topological, geometric, and dynamical perspectives. The speaker is an Assistant Professor candidate

We investigate the behavior of the Gibbs measure for point vortices as their number goes to infinity and we analyze the resulting mean-field equations. We also discuss an extension of the theory to nearly parallel three dimensional vorticity filaments.

An introduction to de Rham cohomology, possibly hitting homotopy invariance. The goal of this string of talks is to get to equivariant de Rham theory.

Paradifferential calculus, created in the early 1980's by J. M. Bony, is a tool used to give a limited meaning to and bounds on products of tempered distributions, and is useful for obtaining regularity results for PDEs, including those appearing in the study of fluids. I present an overview of paradifferential calculus based on Jean-Yves Chemin's text "Perfect Incompressible Fluids." This is my candidacy exam talk.

Our plan this semester is to work through Thierry Aubin's "Some Nonlinear Problems in Riemannian Geometry," getting through at least Section 4.2 on the Green's function of the Laplacian. To start off, Jerry will cover the background material on Riemannian geometry we will need throughout the semester. [Note that this seminar will meet Mondays at 4:00 in 10.176 for the rest of the semester.]

Second Lecture: see description for Tuesday's lecture.

Jerry will finish his overview of Riemannian Geometry, then Jim will introduce the Sobolev spaces on a Riemannian manifold and begin Aubin's proof of the Sobolev embedding theorem for compact Riemannian manifolds.

In this talk I will define a class of multiple zeta functions, called multiple L-series, and investigate their analytic properties. In particular, I will establish a sufficient condition for these series to have a meromorphic continuation to C^{d}, and will locate the hyperplanes containing the possible singularities. I will then give several examples of multiple L-series satisfying this sufficient condition, including multiple L-series with periodic and quasi-periodic coefficients, multiple L-series associated to cusp forms of weight 2k, and multiple Hurwitz zeta functions.

More de Rham goodness.

We will prove the Beale-Kato-Majda blow-up criterion.

Many consider Francis Crick and James Watson's discovery from 1952 of the double helix structure of DNA as one of the greatest scientific discoveries of the 20th century. One may ask why this form? Why is it so natural? It is not clear whether Crick and Watson were aware that also the internal ear, the cochlea, has the form of a double spiral staircase. In fact, in the Renassaince Leonardo di Vinci and others were fascinated by double spiral staircases. For instance, in the Loire valley one finds one of the most famous double spiral staircases. The staircase is in the Chateau de Chambord and was designed by Leonardo di Vinci. When the pope in 1527 fled his Roman empire in the middle of the night disguised as a fruit vendor and settled near Florence in the hill town of Orvieto, why did he build a well with the form of a double spiral staircase?

I am going to prove a Theorem that establishes sufficient conditions for a function between certain vectorial spaces to be linear. The Theorem can be re-interpreted in terms of the kernel of a certain finite Radon transform.

We will discuss some recent developments of the theory of multilinear singular integrals and their maximal Carleson type analogues. The speaker is an assistant professor candidate.

The Hamilton-Jacobi-Bellman equation resulting from a Merton problem with incomplete information can be linearized and thus transformed into a Zakai-type equation, for which numerical algorithms exist. We use this idea to extend indifference pricing to informationally incomplete markets and to test for the existence of asymmetric information.

We will give Aubin's proof of the Sobolev embedding theorem for compact Riemannian manifolds. Notes for the proof can be found at http://www.ma.utexas.edu/users/kelliher/Analysis/Analysis.html

Aby will describe an improvement of the Agrawal, Kayal, and Saxena (AKS) primality test done by H.W. Lenstra Jr. Also, he will answer some frequently asked questions about the AKS primality test.

A (mostly) nonsense-free introduction to principal bundles.

The title of the talk is a reference to http://arxiv.org/abs/hep-th/0301006

A (mostly) abstract nonsense free introduction to the Ext groups and their role in understanding extensions of abelian groups (or modules).

Jonathan Pila is an assistant professor candidate. Abstract: A classic theorem of P\'olya shows that $2^z$ is, in a strong sense, the "smallest" transcendental entire function that is integer valued on $\NN$. An analogous result of Gel'fond concerns entire functions that are integer valued on the set $X_a=\&ob;a^n, n\in\NN\&cb;$ where $a\in\ZZ, |a|\ge 2$. Let $X=\NN$ or $X=X_a$ and $k\in\NN$ or $k=\infty$. I will discuss analogous results for entire functions having the following property: on any finite subset of with $\#D\le k+1$, the values $f(z), z\in D$ admit interpolation by an element of $\ZZ[z]$.

In topology and geometry singularities tend to occur because of global reasons. For example, the projective plane can not be mapped into 3-space without triple points, the reason is the global topology of the projective plane. A differential form on a manifold may on may not have points where the form degenerates in various senses. A generic section of a quiver bundle may or may not hit the points corresponding to certain rank conditions---this depends on the global topology of the bundle. A family of Riemann surfaces with vector bundles over them may or may not contain one over which the vector bundle is non-stable. In the lecture we will show that these global singularity theoretic phenomena can be formulated in a general framework: in the theory of Thom polynomials (a.c.a. degeneracy loci) for Lie group representations. We will define and discuss the Thom polynomial as the equivariant Poincare dual of an orbit closure, and show how it computes the cohomology class represented by various degenerated objects. Most importantly, we will give an effective algorithm to compute Thom polynomials, leading to topological and enumerative applications. Our method will be based on the `local symmetry governs global geometry' principle. In the second part some applications of Thom polynomials in the combinatorics of Schur and Schubert polynomials will be shown. As a consequence we present a formula for the "quiver cycles" of Buch and Fulton. We will also discuss the relations with Geometric Invariant Theory.