Speaker: Prof. John McCarthy, Mathematics Department, Washington University, Host: Li/Landsberg
Title: Extending analytic functions without increasing their norm

Abstract: Suppose V is a subset of the bidisk, the set {(z,w) \in C² : |z| < 1, |w| < 1 }. If V has the property that any polynomial on V can be extended to a bounded analytic function on the whole bidisk with the same sup-norm, what does this say about V? In a recent paper with Jim Agler of UCSD, available at http://www.math.wustl.edu/~mccarthy/papers.html, we showed that the polynomially convex hull of V must be a retract of the bidisk. Without the norm restriction, H. Cartan showed that, for V any subvariety, all functions can be extended to the bidisk. I shall discuss a circle of ideas related to this problem, including its connections with operator theory and complex geometry.

Speaker: Prof. Tamas Erdelyi, Department of Mathematics, Texas A&M University, Host: Lubinsky
Title: Excursions in Unimodular Polynomials

Abstract: Unimodular polynomials are polynomials with complex coefficients of modulus one. In 1957 Erdos asked two natural questions about unimodular polynomials. The easier one was disproved by Kahane some twenty five years later. The correct answer to the other one is still a mystery today. Kahane's result gave rise to the study of ultraflat unimodular polynomials. The talk discusses some conjectures about ultraflat unimodular polynomials. A special interest is paid to a few ones formulated by Saffari. Some questions about Barker polynomials are also touched. Some caution ought to be exercised here since despite the fact (published in 1961) that Barker polynomials of even degree higher than twelve do not exist, the existence of Barker polynomials of odd degree is still under investigation.

Speaker: Prof. Michael Larsen, Department of Mathematics, Indiana University, Host: Landsberg
Title: Can you hear the shape of a group?

Abstract: In various mathematical settings, a geometric object X determines a spectrum S(X), that is, a non-decreasing sequence of real numbers tending to infinity. The inverse problem, to what extent S(X) determines X, has been intensively studied in a number of different settings. I will discuss what happens when X is a number field or a Riemannian manifold and describe my own recent work in which X is a compact semisimple Lie group.

Speaker: Prof. Christopher Sogge, Department of Mathematics, John Hopkins University, Host: Metcalfe
Title: Eigenfunctions of the Laplacian

Abstract: I shall discuss estimates for eigenfunctions of the Laplacian on compact Riemannian manifolds. It has been known for some time that if the eigenfunctions are L^2 normalized then their sup-norms grow at most like \lambda^{(n-1)/2}, where \lambda^2 is the eigenvalue. In most cases there is slower growth, while if this growth rate is achieved, we say that he manifold has "scarring". We shall attempt to classify Riemannian manifolds that have "scarring". We shall also say something about estimates for the more difficult case where there is a boundary. In all these cases, the "quantum correspondence principle" plays a big role. Recall that the correspondence principle says that there should be a relationship between properties of the geodesic flow and eigenfunction concentration.

Speaker: Prof. Andrei Gabrielov, Department of Mathematics and Earth and Atmospheric Sciences, Purdue University, Host: Landsberg
Title: Rational functions with real critical points, the Catalan numbers, and the Schubert calculus

Abstract: How many rational functions of degree d have a given set of 2d-2 points as their set of critical points? If we identify functions that differ by a fractional-linear transformation in the target space (all such functions have the same critical points) the answer is u_d, the Catalan number. This is equivalent to a problem in the Schubert calculus: How many codimension 2 affine subspaces in C^d intersect a given set of 2d-2 affine lines? The answer (Schubert, 1886) is u_d, the Catalan number.

Theorem. Suppose that all 2d-2 points are real. Then all u_d equivalence classes of rational functions with these critical points contain real functions.

The corresponding result in the Schubert calculus: Consider tangent lines to a real rational normal curve of degree d in the d-dimensional space at a given set of 2d-2 real points on this curve. All codimension 2 subspaces intersecting these 2d-2 lines are real. This proves a special case of a general conjecture of B. and M. Shapiro in real enumerative geometry.

Speaker: Prof. Constantine Dafermos, Division of Applied Mathematics, Brown University, Host: Pan
Title: Hyperbolic Balance Laws with Dissipation

Abstract: By combining the random choice method with entropy estimates, solutions will be constructed to the Cauchy problem for systems of balance laws with weak dissipation induced by relaxation mechanisms.

Speaker: Kannan Soundararajan, University of Michigan Host: Pan
Title: An Uncertainty Principle for Arithmetic Sequences

Abstract: We show that interesting arithmetic sequences cannot be simultaneously well distributed in both arithmetic progressions and short intervals.

Fall 2003
Spring 2003
Fall 2002

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