Distinguished Visitors Archives
Academic Year 2006-2007
| October 2-6, 2006 (Ida Beam Distinguished Visitor) | |
| Speaker | Prof. Willard Miller(University of Minnesota) |
| Titles | What is separation of variables?; Superintegrability and Quasi-Exactly Solvable Eigenvalue Problems in Quantum Mechanics; Truth and Beauty in Science: An orbital mechanics case study; Second order superintegrable systems in two and three dimensions. (Solving a system in multiple ways.); |
| Abstracts |
| November 13-17, 2006 | |
| Speaker | Prof. James Cogdell (Ohio State University) |
| Titles | Converse Theorems, Functoriality, and Applications; Derivatives and L-functions for GL(n) |
| Abstracts | Converse Theorems, Functoriality, and Applications: Converse Theorems traditionally have provided a way to characterize Dirichlet series associated to modular forms in terms of their analytic properties. The prototypical Converse Theorem was due to Hamburger who characterized the Riemann zeta function in terms of ts analytic properties. More familiar may be the Converse Theorems of Hecke and Weil. Hecke first proved that Dirichlet series associated to modular forms enjoyed “nice” analytic properties and then proved “Conversely” that these analytic properties in fact characterized modular Dirichlet series. Weil extended this Converse Theorem to Dirichlet series associated to modular forms with level. In their modern formulation, Converse Theorems are stated in terms of automorphic representations instead of modular forms. Jacquet, Piatetski-Shapiro, and Shalika have proved that the L-functions associated to automorphic representations have nice analytic properties similar to those of Hecke. The relevant “nice” properties are: analytic continuation, boundedness in vertical strips, and functional equation. Converse Theorems in this context invert this process and give a criterion for automorphy, or modularity, in terms of these L-functions being “nice.” To use Converse Theorems for applications, proving that certain objects are automorphic or modular, one must be able to show that certain L-functions are “nice”. However, essentially the only way to show that an L-function is nice is to have it associated to an automorphic form. Hence the most natural application of these Converse Theorems is to Functoriality, or the transfer of automorphic forms. Functoriality itself is a manifestation of Langlands’ vision of a non-abelian class field theory. Recently, this approach via the Converse Theorem has proved successful and many cases of Functoriality have now been established. These results are not isolated results in the theory of automorphic forms as they might first appear. Modular forms and automorphic forms have often played a central role in arithmetic questions. These new cases of Functoriality have had applications to several classical problems in number theory including the Ramanujan Conjecture, Hilbert’s eleventh problem, and the inverse Galois problem. In this series of lectures I would like to present an overview of this circle of ideas. In the first lecture I will discuss the classical results of Hecke and Weil, the problems they were interested in at the time, and then the modern formulations of the Converse Theorem. The second lecture I will devote to the “what” and “why” of Functoriality as well as the use of the Converse Theorem in establishing cases of Functoriality. In the final lecture I would like to turn to applications of these results to various questions in number theory including their relation to Hilbert’s eleventh problem, the inverse Galois problem and to general questions of “modularity”. My hope is to present an introduction to this material suitable for a general mathematical audience. The lectures will be somewhat colloquial in tone and will (purposefully) avoid many technical details. |
Derivatives and L-functions for GL(n): Let F be a p-adic field and  an irreducible admissible generic representation of GL(n,F).Bernstein and Zelevinsky have attached to a sequence of representations of GL(m,F), for m less than or equal to n, called the derivatives of . In these talks we will investigate the relationship between the derivatives of a representation and its L-function. In essence we will show that the Rankin-Selberg convolution L-function of a pair of representations can be understood in terms of invariant pairings between their derivatives. We will also discuss the exterior square L-function in this light. |
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| March 19-23, 2007 | |
| Speaker | Prof. Carlos Kenig (University of Chicago) |
| Titles | Recent progress on the well-posedness of dispersive equations; On the Schrodinger map system; The energy critical focusing nonlinear Schrodinger equation; The energy critical focusing nonlinear wave equation |
| Abstracts | Recent progress on the well-posedness of dispersive equations: We will give an expository account of the development, in the last 18 years, of the well-posedness theory for dispersive equations. We will focus on the Korteweg-de Vries equation to illustrate this. We will then explain some of the current challenges in the field and illustrate them with a description of recent work with Ionescu on the Benjamin-Ono equation. Both the Korteweg-de Vries and the Benjamin-Ono equations are models for water wave propagation. |
| On the Schrodinger map system: The Schrodinger map system is a geometric flow which arises in the context of Kahler manifolds. It also arises as a model in ferromagnetism. I will discuss recent works, with Ionescu and Bejenaru-Ionescu on the global in time behavior of solutions in the critical regularity spaces. |
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| The energy critical focusing nonlinear Schrodinger equation: I will discuss recent joint work with Merle, in which we introduce a new method for the study of energy critical dispersive problems, through the use of concentration-compactness ideas. For radial solutions we are able to establish, for the energy critical focusing NLS case, that for data with energy smaller than that of the ground-state, we have global wellposedness and scattering, while for bigger energy we have blow-up in finite time. Our method also applies in the defocusing case, yielding an alternative approach to the works of Bourgain and Tao. | |
| The energy critical focusing nonlinear wave equation: I will discuss work with Merle, in which we prove results analogous to the ones we obtained for the Scrodinger equation, for the wave equation, but this time without the radial assumption, thus establishing for the first time in a critical dispersive problem, the so called ”ground-state energy conjecture.” |
