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: Remarks on Low Dimensional Groups of Matrices and the Bergman April 1

Kernel Function of the Unit Disk

Presenter: Sean Paul, Princeton University

Note: Pizza will be provided

Topic: Efficient testing of large graphs April 1

Presenter: Mario Szegedy, AT&T Labs

Abstract: Let $P$ be a property of graphs. An $\epsilon$-test for $P$ is a randomized algorithm which, given the ability to make queries whether a desired pair of vertices of an input graph $G$ with $n$ vertices are adjacent or not, distinguishes, with high probability, between the case of $G$ satisfying $P$ and the case that it has to be modified by adding and removing more than $\epsilon n^2$ edges to make it satisfy $P$. The property $P$ is called testable, if for every $\epsilon$ there exists an $\epsilon$-test for $P$ whose total number of queries is independent of the size of the input graph. In this talk we study the testability of first order graph properties, showing that properties not containing a quantifier alteration of type ``$\forall\exists$'' are always testable, while some properties containing this alteration are not.

Our results are proven using a combinatorial lemma, a special case of which, that may be of independent interest, is the following. A graph $H$ is called $\epsilon$-unavoidable in $G$ if all graphs that differ from $G$ in no more than $\epsilon|G|^2$ places contain an induced copy of $H$. A graph $H$ is called $\delta$-abundant in $G$ if $G$ contains at least $\delta|G|^{|H|}$ induced copies of $H$. If $H$ is $\epsilon$-unavoidable in $G$ then it is also $\delta(\epsilon,|H|)$-abundant. This work is joint with Noga Alon, Eldar Fischer and Michael Krivelevich.

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: ARTIN L-SERIES AND STARK'S CONJECTURE April 2

Presenter: John Tate, The University of Texas at Austin

Topic: Localization of Eigen Functions of Quasi-Periodic April 5

Equations (continuation)

Presenter: Michael Goldstein, University of Toronto

Topic: A World of Fluid Instailities April 5

Presenter: Susan Friedlander, University of Illinois, Chicago &

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: Geometry of Polyhedron April 6

Presenter: Igor Rivin, Warwick University

Topic: Spectral Properties of Many-Particle Systems in Magnetic Fields April 6

Presenter: Gregory Zhislin, Nijnii Novgorod State University

Topic: Quantum Classical Crossover in the Spin 1/2 XXZ Chain April 7

Presenter: Barry McCoy, SUNY Stony Brook

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: Asymptotics of set partitions and the failure and indirect success April 8

of analytic methods

Presenter: Andrew Odlyzko, AT&T Labs

Abstract: The asymptotic behavior of the number of set partitions of an n-element set into blocks of distinct sizes is determined. This behavior is more complicated than is typical for set partition problems. Although there is a simple generating function, the usual analytic methods for estimating coefficients fail in the direct approach, and elementary approaches combined with some analytic methods are used to obtain most of the results.

Topic: On the distribution of the iterates of a class of dynamical systems April 8

associated with sigma-delta quantization and the

related exponential sums

Presenter: Sinan Gunturk, Princeton University

Topic: Trace of infinite wedge, tori coverings, and random partitions April 8

Speaker: Andrei Okounkov, University of Chicago

Abstract: I shall discuss a formula which describes: 1) characters of the algebra of differential operators on the circle, 2) ramified coverings of a torus, 3) shape and all order fluctuations of a random partition. Talk will be based on joint work with S. Bloch and work in progress with A. Eskin. Talk will be independent from the talk on 04/09, but related to it.

Topic: From Representation Theory to Homotopy Groups April 8

Presenter: Don Davis, Lehigh University

Topic: Rogers-Ramanujan identities: a convergence of q series, partitions, April 8

algebra, function theory and physics

Presenter: Barry McCoy, SUNY

Topic: Dual Pairs and Random matrix Theory April 9 IAS

Presenter: Martin R. Zirnbauer, Princeton University

Abstract: The traditional tool for computing the correlations of random matrix eigenvalues are orthogonal polynomilas. In this talk an alternative method, rooted in the theory of dual pairs and symplectic geometry, is presented. By tensoring the oscillator representation of the metaplectic group with the spinor representation of the spin group, a representation p of an orthosymplectic Lie supergroup OSp is obtained. Its character is the square root of a superdeterminant, and serves as a generating function for the eigenvalue correlations of unitary matrices. We then focus on the circular random matrix ensembles defined over the classical compact Lie groups. Their correlation functions are computed by considering any one of the dual pairs O(N) X OSp(2n|2n), U(N) X GL(n|n), or Sp(N) X OSp(2n|2n) inside the representation p of OSp. On integrating voer the first factor with Haar measure, and using Howe's exposition of classical invariant theory, the correlation functions become characters of an irreducible highest-weight representation of the second factor. In the large-N limit, the latter can be evaluated by a supersymmetric generalization of the Duistermaat-Heckman theorem.

Topic: Random matrices and random permutations April 9 IAS

Presenter: Andrei Okunkov, University of Chicago

Abstract: I shall discuss recent results about the connection between GUE and the Plancherel measure on the set of representations of the symmetric group, including the proof of a version of a conjecture due to Baik, Deift, and Johansson. Talk will be independent from the talk on 04/08, but related to it.

Topic: Nonlinear biharmonic equations with negative exponents April 9

Presenter: Xu Xingwang, Princeton University

Topic: Evolving real hypersurfaces on the trace of Levi form April 9

Presenter: Gerhard Huisken, Princeton University

Topic: Weak solutions of the Euler equations. Part II April 9

Presenter: Alexander I. Shnirelman, Princeton University

Topic: TBA April 13

Presenter: Dietmar Salamon, ETH University, Zurich Switzerland

Topic: An analogue for GL(n) of Serre's conjecture April 13

Presenter: Avner Ash, Ohio State University

Topic: TBA April 27

Presenter: Jeffrey Achter, University of Massachusetts

Topic: On the rank of elliptic curves April 29

Presenter: Joseph Silverman, Brown University