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 that 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: The remarkable mathematics of some simple number-recreations March 25

Presenter: John Conway, Princeton University

Abstract: Iterating a simple function N -> f(N) can lead to arbitrarily difficult mathematics. I'll sketch a proof of this using my "FRACTRAN" calculus, and briefly describe how it was extended to prove the universality of "LIFE" and very recently, of "LINELIFE" (which has given us the best-known universality results for Turing machines). I also want to discuss some iteration-problems whose solutions are simpler but still intriguing.

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, Institute for Advanced Studies

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 Floer March 25

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

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

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

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.

Date: March 29, 30 and April 1, 1999

For complete details see IAS Homepage.

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: Type A graded tensor product multiplicities April 1

Presenter: Mark Shimozono, Virginia Polytechinic

Abstract: We will discuss certain type A graded tensor product multiplicities which appear in affine crystal theory, modules supported in nilpotent conjugacy class closures, the Bethe Ansatz, and as certain parabolic Kazhdan-Lusztig polynomials for affine type A.

Topic: Holomorphy and boundedness of the third symmetric power April 1

L-functions for GL(2)

Presenter: Freydoon Shahidi, Purdue University

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

Presenter: Michael Goldstein, University of Toronto

Topic: A World of Fluid Instailities April 5

Presenter: Susan Friedlander, University of Illinois, Chicago and Institute

for Advanced Study

ABSTRACT: The issue of stability/instability of fluid flows presents an important example of a physical problem which may be addressed through sophisticated mathematical techniques. The answers have direct physical interpretations: stable flows are robust under inevitable disturbances in the environment while unstable flows may break up rapidly. The question of stability/instability of a fluid flow is a classical one, however there remain many open problems that are mathematically challenging. In this talk we will introduce the concept of a ``fluid Lyapunov exponent'' and describe an effective sufficient condition for detecting instabilities in an inviscid fluid. We use this tool to show that in some sense ``most'' steady flows of ideal fluid are unstable. We illustrate the instability with particular examples including smoke rings and so called ``chaotic flows''. This is joint work with Misha Vishik.

Topic: TBA April 6

Presenter: Igor Rivin, Warwick University

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

Presenter: Zoltan Szabo, Princeton University

Abstract: In this talk I will discuss an invariant for three-manifolds, which we found recently with Peter Ozsvath. The invariant is defined by using a Heegaard decomposition of the three-manifold along a Riemannian surface and studying how the Theta-Divisor of behaves when the surface is degenerated along some curves that are naturally associated to the Heegaard decomposition. We prove that this invariant is independent of the Heegaard decomposition, and so it gives a topological invariant. We also relate this invariant with more classical invariants: Alexander polynomial, Turaev torsion and Casson invariant. The close relationship between our invariant and the Seiberg-Witten invariant for 3-manifolds will also be discussed.

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