Kolmogorov's work and similarities between chaotic and rigid dynamical systems

Speaker: 

Professor Don Ornstein

Institution: 

Stanford

Time: 

Thursday, May 29, 2008 - 4:00pm

Location: 

MSTB 254

This will be a non-technical talk. I will start by describing how Kolmogorov models dynamical systems and stationary processes so as to put them both into the same mathematical framework and some results that are made possible by this point of view.

These results lie on the chaotic side of dynamics.

On the rigid side of dynamics, I will describe the Kolmogorov-Moser twist theorem (part of KAM theory) and a generalization of that theorem.

I will discuss the very strong similarities between the stability properties of
chaotic and rigid systems.

Enumerative Geometry: from Classical to Modern

Speaker: 

Professor Aleksey Zinger

Institution: 

Stony Brook

Time: 

Thursday, February 28, 2008 - 4:00pm

Location: 

MSTB 254

The subject of enumerative geometry goes back at least to the middle
of the 19th centuary. It deals with questions of enumerating geometric
objects, e.g.
(a) how many lines pass through 2 points or
through 1 point and 2 lines in 3-space?
(b) how many conics in 2-space are tangent to k lines and
pass through 5-k points?

There has been an explosition of activity in this field over the past
twenty years, following the development of Gromov-Witten invariants in
sympletic topology and string theory. The idea of counting parameterizations
of curves in order to count curves themselves has led to solutions of
whole sets of long-standing classical problems. At the same time, string
theory has generated a multitude of predictions for the structure of
GW-invariants, as well as for the behavior of certain natural families
of Laplacians. It has in particular suggested that there is a diality
between certain symplectic and complex manifolds and that in some cases
GW-invariants see some geometric objects, that are yet to be fully
discovered mathematically.

In this talk I hope to give an indication of what enumerative geometry
is about and of the shift in the paradigm that has occured over the past
two decades.

Onsager's Conjecture and a Model for Turbulence

Speaker: 

Professor Susan Friedlander

Institution: 

USC and UIC

Time: 

Thursday, February 21, 2008 - 4:00pm

Location: 

MSTB 254

We discuss properties of a shell type model for the inviscid fluid equations. We prove that the forced system has a unique equilibrium which is an exponential global attractor. Every solution blows up in H^5/6 in finite time . After this time, all solutions stay in H^s, s

Quantum information and classical percolation

Speaker: 

Professor Janek Wehr

Institution: 

University of Arizona at Tucson

Time: 

Thursday, January 31, 2008 - 4:00pm

Location: 

MSTB 254

Sending quantum information over a distance is a challenging task and the challenge increases, if the task is to be performed several times, to send a message over a large distance. In general, this can only be done with a certain probability. On several quantum networks, a recently introduced procedure, which uses special quantum measurements, enhances this probability, as follows from the properties of related classical percolation models. ALL of the above terms will be explained in the talk, which is aimed at a general mathematical audience. While mathematically rigorous, the presented results were obtained in collaboration with a physics group and are of current physical interest, leading to a number of open problems, which will be mentioned.

Understanding singular algebraic varieties via string theory

Speaker: 

Professor David Morrison

Institution: 

UC Santa Barbara

Time: 

Thursday, March 6, 2008 - 4:00pm

Location: 

MSTB 254

String theory has helped to formulate two major new insights in the study of singular algebraic varieties. The first -- which also arose from symplectic geometry -- is that families of Kaehler metrics are an important tool in uncovering the structure of singular algebraic varieties. The second, more recent insight -- related to independent work in the representation theory of associative algebras -- is that one's understanding of a singular (affine) algebraic variety is enhanced if one can find a non- commutative ring whose center is the coordinate ring of the variety. We will describe both of these insights, and explain how they are related to string theory.

Chaoticity of the Teichmuller flow

Speaker: 

Professor Artur Avila

Institution: 

CNRS/Paris & IMPA

Time: 

Thursday, February 7, 2008 - 4:00pm

Location: 

MSTB 254

A non-zero Abelian differential in a compact Riemann surface of genus $g \geq 1$ endows the surface with an atlas (outside the zeroes) whose coordinate changes are translations. There is a natural ``vertical flow'' (moving up with unit speed) associated with the translation structure, generalizing the genus $1$ case of irrational flows on the torus.

The Teichm\"uller flow in the moduli space of Abelian differentials can be seen as the renormalization operator of translation flows. In this talk, we will discuss how the chaoticity of the Teichm\"uller flow dynamics reflects on the (non-chaotic) dynamics of the associated vertical flows (for typical parameters), and the closely related interval exchange transformations.

Asymptotic-Preserving Schemes for Multiscale Problems

Speaker: 

Professor Shi Jin

Institution: 

Wisconsin

Time: 

Thursday, October 25, 2007 - 4:00pm

Location: 

MSTB 254

We survey the general methodology in developing asymptotic preserving schemes for physical problems with multiple spatial and temporal scales. These schemes are first-principle based, and automatically become macroscopic solvers when the microscoipic scales are not resolved numerically. They avoid the coupling of models of different
scales, thus do not face the difficult task of transfering data from one scale to the other as in most multiscale methods. These schemes are very effective for the coupling of kinetic and hydrodynamic equations, and problems with fast reactions.

Contact Homology via Legendrian curves, an overview

Speaker: 

Professor Abbas Bahri

Institution: 

Rutgers University

Time: 

Thursday, November 8, 2007 - 4:00pm

Location: 

MSTB 254

After the seminal work of Paul Rabinowitz on periodic orbits of Hamiltonian Systems on starshaped surfaces in |R^n, Contact Structures have become a natural object of study for analysts. The search for invariants for these contact forms/structures benefited very much from the deeper understanding of the much more general associated variational problem used in the work of Paul Rabinowitz and of Conley-Zehnder. Contact Homology has then been defined using pseudo-holomorphic curves, but also via Legendrian curves. After broadly recalling the main steps in the formulation and the development of these tools, we present a more detailed account of the contact homology via Legendrian curves, including its definition, its compactness properties and the value of this homology for odd indexes.

Funding Opportunities in the Mathematical Sciences at the National Science Foundation

Speaker: 

Professor Henry A. Warchall

Institution: 

NSF

Time: 

Thursday, October 11, 2007 - 4:00pm

Location: 

MSTB 254

I will describe current opportunities for funding in mathematics and statistics at the National Science Foundation, as well as issues that arise in proposal preparation. There will be ample opportunity for questions from the audience.

Mathematical theory of solids: From atomic to macroscopic scales

Speaker: 

Professor Weinan E

Institution: 

Princeton

Time: 

Thursday, November 29, 2007 - 4:00pm

Location: 

MSTB 254

There are no analogs of Navier-Stokes equations for solid mechanics. One reason is that information at the atomic scale seems to play a much more important role for solids than for fluids. A satisfactory mathematical theory for solids has to taken into account the behavior of solids at different scales, from electronic to atomic, to macroscopic scales.

I will discuss some of the fundamental problems that we have to resolve in order to build such a theory. I will start by reviewing the geometry of crystal lattices, the quantum as well as classical atomistic models of solids. I will then focus on a few selected problems:

(1) The crystallization problem -- why the ground states of solids are crystals and which crystal structure do they select?

(2) the microscopic foundation of elasticity theory;

(3) stability and instability of crystals;

(4) the electronic structure and density functional theory.

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