Topic: The mathematics of financial risk management March 22

Presenter: Luis Seco, University of Toronto

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

Presenter: Michael Goldstein, University of Toronto

Topic: Instability in Parallel Flows Revisited March 23

Presenter: Susan Friedlander, University of Illinois - Chicago and IAS

Abstract: One of the most classical problems in fluid dynamics is the stability/instability of plane parallel shear flow and it was examined mathematically and experimentally by such 19th century "giants" as Rayleigh, Kelvin and Reynolds. However there remain open questions associated with this problem. We address the question of inviscid instabilities which means studying the eigenvalues of an ODE known as the Rayleigh equation. We show how continued fractions can be used to determine all the unstable eigenvalues in the special case where the shear flow has a sinusoidal profile.

We pose open problems : for example, how can the eigenvalues of the Rayleigh equation be determined for more general profiles ? All this is a special case of of a very difficult problem, namely the nature of the spectrum of the Euler equation for the flow of an inviscid fluid.

Topic: The mathematics of financial risk management March 23

Presenter: Luis Seco, University of Toronto

Topic: Derivation of homogenous hydrostatic equations March 23

Presenter: Emmanuel Grenier, Ecole Normale Superior - Lyon, France

Topic: Diophantine, algebraic, deformation properties and patterns for power series March 23

Presenter: Dinesh Thakur, University of Arizona

Topic: The mathematics of financial risk management March 24

Presenter: Luis Seco, University of Toronto

Topic: Dissipation through dispersion March 24

Presenter: Avy Soffer, Rutgers University

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: Carleson's Theorem March 25

Presenter: Slava Rytchkov, Princeton University

Abstract: In 1966, Lennart Carleson proved his famous theorem: the Fourier series of any square integrable function f converges to f almost everywhere.

This talk will be an introduction to Carleson's original proof of the theorem. I will apply Carleson's method in its simplest form to prove that the Fourier partial sums do not grow faster than loglog(n) almost everywhere. This is already much better than the classical Kolmogorov-Seliverstov-Plessner bound (log n)^{1/2}. Then I will indicate modifications needed to get the full strength of Carleson's result.

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The seminar is informal; please ask questions. Pizza will be provided (thanks to Steve Miller). We encourage self contained talks which are accessible and appeal to a wide audience of graduate students. If there is a topic which interests you and which you would like to present, please contact one of the organizers (Jeff Schenker, Jade Vinson).

Topic: The mathematics of financial risk management March 25

Presenter: Luis Seco, University of Toronto

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

nonhyperbolic systems

Presenter: Michiko Yuri, Sapporo University

Abstract (see attached)

Topic: Character multiplicity duality for the metaplectic group March 25

Presenter: Peter Trapa, IAS

Abstract: Suppose that G_R is a linear reductive Lie group with complexification G and complexified maximal compact subgroup K. In the representation theory of G_R, there are several fundamental instances where one would like to reverse the variance of certain constructions. For instance, the `transfer' of L-packets (which are parametrized by dual group orbits) is a purely formal consequence of Langlands' definitions, while the transfer of representations within an L-packet (which, roughly speaking, are parametrized by local systems dual to the orbit itself) is the difficult problem of endoscopy. A related observation is that in Langlands' theory, the orbits of the dual group on some parameter space play a central role, but in the Beilinson-Bernstein theory there is no dual group present and it is not obvious where to look for it. These issues of variance are elegantly explained by Vogan's character multiplicity duality theorem. In turn, Vogan's theorem can be interpreted a fundamental symmetry of the Kazhdan-Lusztig algorithm for linear real groups -- for

instance, in the highest weight category, the symmetry is encoded in a well-known identity between the coefficients of the Kazhdan-Lusztig polynomials P_{x,y} and those of P_{w_ox,w_oy}. In this talk, we review all these ideas, explain a new duality theorem for the metaplectic group, and discuss its importance in the circle of ideas described above. This is joint work with David Renard.

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

Floer homology groups

Presenter: Kim Froyshov, Harvard University

Topic: Loss of smoothness and intrinsic instability of ideal flows March 25

Presenter: Victor Yudovich, Rostov State University, Russia

Date: Thursday, March 25, 1999

Time: 4 p.m.

Room: Fine 110

Topic: The mathematics of financial risk management March 26

Presenter: Luis Seco, University of Toronto

Topic: Collapsing vs. positive pinching March 26

Presenter: Xiaochun Rong, Rutgers University

Topic: On the instability of boundary layers March 26

Presenter: Emmanuel Grenier, Ecole Normale Superior - Lyon, France

Topic: The Optical Parametric Oscillator: Dynamics, Bifurcations and Stability March 29

Presenter: J. Nathan Kutz, University of Washington

ABSTRACT: We consider the dynamics associated with topological solitons (localized structures) of the optical

parametric oscillator which models the parametric exchange of energy between optical fields at a fundamental and second

harmonic frequency. Simulations show that this nonlinear interaction can support stable front structures as well as

localized, bistable solitary wave solutions. We perform a systematic study of the bifurcation structure and stability analysis

of both solitary wave and front solutions which arise. The stability analysis is carried out for the onset of instability which

arises from a Ginzburg-Landau description as well as a modified Swift-Hohenberg description at resonance. The analysis,

which is carried out in 1-D, can be utilized in predicting the dynamical behavior in 2-D systems. Further, the theoretical

conclusions provide important practical predictions which are verified via extensive numerical simulations.**Please note

new time

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

For complete details

Topic: Moduli spaces of curves with marked points March 30

Presenter: Adam Logan, Harvard University

Topic: Some remarks on non-perturbative QED March 31

Presenter: Elliott Lieb, Princeton University

Topic: Four-manifolds, Symplectic geometry and Mirror symmetry March 31

Presenter: Nikita Nekrasov, Harvard University

Abstract: Some of the old problems in algebraic geometry, as well as relatively new problems in the theory of quantization were solved using topological sigma models. The sigma models deal with maps of a manifold $\Sigma$ to a target space $X$. It is very well-known that no sensible theory of rigid maps exists for the dimensionality of $\Sigma$ being greater then two. In my talk I will try to argue in favor of existence of the interesting theory of maps in case where $\Sigma$ is four dimensional Riemannian manifold and $X$ is a classifying space of some compact Lie group (or its finite-dimensional model). To get there we will need to introduce/develop certain aspects of Donaldson theory and higher-dimensional analogues of Whitham hierarchies. No knowledge of what Donaldson theory is or what Whitham hierarchies are is necessary.

Topic: Global Perspectives on Dynamical Systems April 1

Presenter: Jacob Palis, IMPA, Brazil

Topic: TBA April 6

Presenter: Igor Rivin, Warwick University

Topic: Localization of Eigen Functions of Quasi-Periodic Equations (continuation) April 5

Presenter: Michael Goldstein, University of Toronto

Topic: Three-manifold invariants and the Theta-Divisor April 7

Presenter: Zoltan Szabo, Princeton University

Topic: From Representation Theory to Homotopy Groups April 8

Presenter: Don Davis, Lehigh University

Topic: TBA April 9

Presenter: Gerhard Huisken, Princeton University

Topic: TBA April 13

Presenter: Dietmar Salamon, ETH University, Zurich Switzerland

Topic: On the rank of elliptic curves April 29

Presenter: Joseph Silverman, Brown University