Topic: Effective Interactions in Lattice Systems Due to Quantum Fluctuations February 24

Presenter: Daniel Ueltschi, Rutgers University

Topic: Orthogonal Geometry and Quantum Error Correction February 24

Presenter: Robert Calderbank, AT&T Research

Abstract: Quantum effects are seldom evident in today's electronic devices since the quantum states of many millions of atoms are averaged together blurring their discreteness. But in quantum computing the foundations of quantum mechanics are finding direct and visible application in information processing. The unreasonable effectiveness of quantum computing is founded on coherent quantum superposition or entanglement which allows a large number of calculations to be performed simultaneously. This coherence is lost as a quantum system interacts with its environment and an important challenge today is to devise means of preserving it.

A quantum error correcting code is a way of encoding quantum states into qubits so that error or decoherence in a small number of individual qubits has little or no effect on the encoded data. This talk will describe a beautiful group theoretic framework that simplifies the presentation of known quantum error correcting codes and greatly facilitates the construction of new examples.

Topic: Existence and Uniqueness for 2-D and 3-D Navier Stokes Equations February 25

Presenter: Vadim Y. Kaloshin, Princeton University

Abstract: We shall present Mattingly-Sinai's proof of a global existence and uniqueness theorem for the 2-D Navier-Stokes

equation. In order to make the talk a bit different from the one Sinai gave last week, we also describe how this proof can be extended to prove local existence and uniqueness for the 3-D Navier-Stokes equation.

Topic: On some new applications of the semi-random (or nibble) method February 25

Presenter: Van Vu, Institute for Advanced Study

Abstract: The semi-random method is a useful probabilistic tool which has found many interesting applications in the last decade. In this talk we describe the method and discuss its applications in the following two areas:

(1) (List-) Coloring of locally sparse graphs.

(2) Nearly perfect matchings in hypergraphs.

Besides several earlier results, we will present two new theorems:

Theorem 1. If a graph G has degree at most d and the neighborhood of every vertex has at most f edges, then the choice number of G is O(d/ log f).

Theorem 2. Assume that H is a (k+1)-uniform, D-regular hypergraph on n points. Assume furthermore that there are positive numbers

D_1= D, D_2,..., D_s and x such that:

(1) For all j \le s, every set of j vertices is contained in at most D_j edges,

(2) D_j / D_{j+1} > x^3 for all j < s-1,

(3) D_{s-1}/ D_s > x^{k-s +2}.

Then H contains a matching which covers all but O( x^{-1} n log ^3 d ) vertices.

These theorems improve and generalize results by several researchers. If time allows, we will also present the key idea of the proofs, which is a new technique of proving large deviation bounds of independent interest.

Topic: Applications of Homogeneous Dynamics to Diophantine Approximation February 25

on Manifolds

Presenter: Dmitry Kleinbock, Rutgers University

Topic: Crystal bases for quantum superalgebras February 25

Presenter: Seok-Jin Kang, Seoul National University and MIT

Abstract: In this talk, I will review some of recent developments in the theory of crystal bases for quantum superalgebras which include osp(1,2n), gl(m,n), and some affine superalgebras. The main topics will be

1) Tensor product rule for Kashiwara operators

2) Tableaux and crystals

3) Decomposition of tensor product of crystals

4) Perfect crystals and crystals with core

If time permits, I will discuss the crystal basis theory for generalized Kac-Moody algebras and superalgebras.

Topic: The Nash Conjective for Threefolds February 25

Presenter: Janos Kollar, University of Utah

Topic: Geodesic on a class of non-hyperbolic space of nonpositive curvature February 26

Presenter: Chris Croke, University of Pennsylvania

Topic: Weak solutions of the Euler equations February 26

Presenter: Alexander I. Shnirelman, Princeton University

Topic: On Besicovitch sets in three dimensions March 1

Presenter: Izabella Laba, Princeton University

Topic: Stochastic Models for perception and possible implications about March 3 Fine Hall

the way we think

Presenter: David Mumford, Brown University

Abstract: The development of computer vision in particular and AI in general has led further and further from logic-based deductions and more and more towards Bayesian statistical methods. But how can such algorithms work in the face of the exponential explosion of variables and their interactions? I want to describe some of the latest computational experiments and some of the mathematical issues they have raised. Very intriguing for us is to ask: does this suggest something about what goes on in our heads?

RECEPTION TO FOLLOW COMMON ROOM, THIRD FLOOR, FINE HALL

Topic: Cosmology with a March 3

Presenter: Joel Smoller, University of Michigan

Topic: Standard Monomial Theory March 3

Presenter: Peter Littelmann, University of Strasbourg

Abstract: The talk will give an overview on recent developments in Standard Monomial Theory. When Lakshmibai, Musili and Seshadri started the program in the 70's, their aim was to investigate the geometry of Schubert varieties. It turned out actually that the program yields many intriguing connections between representation theory, combinatorics and algebraic geometry which are interesting for themselves. For example, the combinatorics of the path model is closely related to the structure of Verma modules as well as to the structure of certain toric varieties obtained from Schubert varieties by flat deformations.

Topic: Lanford's Program March 4

Presenter: Michael Yampolsky, Yale University

Topic: The honeycomb model for GL(n) tensor products and applications March 4

Presenter: Allen Knutson, Brandeis

Abstract: We introduce honeycombs, which are combinatorial objects that (like Littlewood-Richardson tableaux, Berenstein-Zelevinsky patterns, dominant concatenated Littelmann paths, etc.) count tensor product multiplicities in GL(n). These have two advantages over e.g. L-R tableaux; a reasonable notion of continuous deformation and an "overlay" operation that relates GL(n) and GL(m) to GL(n+m). Our principal applications are (1) a complete determination of the set of triples of irreps (U,V,W) such that U tensor V contains W^* which includes the fact that (2) linear inequalities between the dominant weights suffice to make this determination (the "saturation conjecture", not true for arbitrary groups). This was in particular the missing ingredient needed to prove Horn's 1962 conjecture on the spectrum of the sum of two Hermitian matrices. This work is joint with Terry Tao of UCLA and Chris Woodward of Rutgers/MSRI.

Topic: Characteristic classes for very singular spaces March 4

Presenter: Steve Ferry, Rutgers University

Topic: TBA March 4

Presenter: Kamal Khuri-Makdisi, McGill University

Topic: Instabilities for the Euler equations March 5

Presenter: Susan Friedlander, Institute for Advanced Studies

Topic: On divergence of trigonometric Fourier series everywhere March 8

Presenter: Sergei V. Konyagin, Moscow State & University of South Carolina

Abstract: (see attached)

Topic: Graded algebras and a theorem of p-descent for log-schemes March 9

Presenter: Pierre Lorenzon, of Muenster University

Abstract: After discussing gradings by sheaves of degrees, we associate to any log scheme a canonical invertible sheaf endowed with a certain multiplicative structure, which we call its associated graded algebra. In the relative case we construct a canonical connection on this algebra. In the log smooth case over a base of positive characteristic p, we study integrable and p-integrable graded modules over this algebra, and establish a Cartier type p-descent theorem, generalizing previous results of Ogus. We apply it to give an alternate proof of a result of Tsuji on closed forms fixed by the Cartier operator.

Topic: Incompressible flows of an ideal fluid with unbounded vorticity March 12

Presenter: Mikhail Vishik, University of Texas, Austin

Topic: Statistical properties of weak Gibbs measures for certain March 25

nonhyperbolic systems

Presenter: Michiko Yuri, Sapporo University

Abstract (see attached)

Topic: Finite type of Donaldson polynomials, and the structure of Floer March 25

homology groups

Presenter: Kim Froyshov, Harvard University