Topic: Lower bounds for Selmer groups via Euler systems March 5 (note time)

Presenter: Karl Rubin, Stanford University

Topic: On the lower order of the asymptotic expansion of Zelditch March 5,

Presenter: Lu Zhiqin, Columbia University at New York

Time: 1:30 p.m.

Topic: Minimal Lagrangian Surfaces as a completely integrable system

Presenter: Helein Frederic, France

Time: 2:30 p.m.

Room: Fine 1201 (This week only)

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: A Reaction-Diffusion System with Periodic Front Dynamics March 8

Presenter: Georgiy Medvedev, Boston University

ABSTRACT: A model motivated by {\it Proteus mirabilis} bacterial colony development is presented and analyzed in this work. Mathematically, we study a system of a degenerate parabolic partial differential equation and an ordinary differential equation. The most interesting feature of the model is that it generates interface dynamics that are periodic in time, just as seen in the biological experiments. We analyze the dynamics of different phases of colony evolution, as well as switches and transition between them, using the method of matched asymptotic expansions and parabolic equation techniques. Finally, we present rigorous estimatesfor the inner and outer solutions developed in the matched asymptotic analysis, and for their domains of validity.

Topic: A dispersive analogue of turbulence March 8

Presenter: Peter Lax, Courant Institute for Mathematical Sciences, NYU

RECEPTION TO FOLLOW IN THE COMMON ROOM

Topic: Majorization and a sharp Lieb-Thirring bound in one dimension March 9

Presenter: Dirk Hundertmark, Princeton University

ABSTRACT: I will review the notion of Majorization. This is an old but maybe not so well-known concept in the theory

of matrices: it is used to compare two positive definit matrices or, more generally, two positive compact operators on a

Hilbert space. As a novel application of this concept we give an astonishingly simple proof for a bound on an eigenvalue

moment of a Schroedinger operator. This proof gives the best constant in that inequality and confirms part of an old

conjecture of Lieb and Thirring.

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: The Notorious Piston and Other Problems in March 10

Nonequilibrium Statistical Mechanics

Presenter: Joel Lebowitz, Rutgers University

Topic: Some problems in the theory of trigonometric series March 10

Presenter: Sergei V. Konyagin, University of North Carolina and

Moscow State University

Abstract: A natural way to represent a periodic function is a trigonometric series. But whether that series represents the

function adequately depends on its convergence, or divergence, properties. We propose to present selected results and

problems on one-dimensional trigonometric series. The following questions will be discussed.

1. Convergence of trigonometric Fourier series almost everywhere.

2. The almost everywhere order of growth of the sequence of partial sums of Fourier series.

3. How can a rearrangement of Fourier series improve its convergence properties?

4. (If time permits.) Representation of functions by general trigonometric series.

Topic: Choosability for Ax=y March 11

Presenter: Matt Devos, Princeton University

Abstract: Let F be a finite field with p^c elements and let A be an n x n matrix over F. We are interested in solving the

equation Ax=y for column vectors x and y with certain coordinate-wise restrictions. In particular, for each i we will

restrict the ith coordinate of x to be in some set Xi and the ith coordinate of y to be in some set Yi. For an integer k, when

is it true that for every set of restrictions with |Xi| = k+1 and |Yi| = p^c - k, there exists a solution to Ax=y meeting these

requirements? Clearly if det(A) is nonzero then A has this property for k = p^c - 1. The permanent lemma of Noga Alon

proves that if perm(A) is nonzero then A has this property for k = 1. We will present a theorem which generalizes both of

these facts, and then apply it to prove a "choosability" generalization of Jaeger's 4-flow and 8-flow theorems.

Topic: Semi-invariants of Quivers and Klyachko's Saturation Problem March 11

Presenter: Harm Derksen, MIT

Abstract: For a quiver without oriented cycles, we consider the ring of semi-invariants of all representations of that quiver

of a fixed dimension. Our main result is that the semi-invariants constructed by Schofield always generate the

semi-invariant ring. Other results of Schofield now imply that the set of weights of the semi-invariants is given by linear

inequalities, and, in particular, this set is saturated. Applied to the triple flag quiver, our result gives a short proof of results

of Klyachko and Knutson-Tao about the set of nonzero Littlewood-Richardson coefficients.

Topic: Exotic smooth structures on 3(CP^2)#n(-CP^2) March 11

Presenter: Bahn Doug Park, Princeton University

Topic: On the distribution of spacings between quadratic March 11

residues, (work of Kurlberg and Rudnick)

Presenter: Wilhelm Schlag, Princeton University

Topic: Non-vanishing of quadratic Dirichlet L-functions at s=1/2 March 11

Presenter: Soundararajan Kannan, Princeton University

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

Presenter: Mikhail Vishik, University of Texas, Austin

Topic: Dissipation through dispersion March 24

Presenter: Avy Soffer, Rutgers University

Topic: Localization of Eigen Functions of Quasi-Periodic Equations March 22

Presenter: Michael Goldstein, University of Toronto

Topic: Analysis and Spectral Theory on Graphs and Symplectic Geometry March 24

Presenter: Sergei Novikov, University of Maryland at College Park and

Landau Institute for Theoretical Physics

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 March 25

Floer homology groups

Presenter: Kim Froyshov, Harvard University

Topic: The degree counting formulas for scalar curvature equation on S^n March 29

Presenter: Chang-Shou Lin, Chung-Cheng University, Taiwan

Date: March 29, 30 and April 1, 1999

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