Mathematics Department and Graduate School Colloquium Archive 1998-1999 May 24, 1999 James Renegar (Cornel University) A Mathematical View of Interior-Point Methods in Convex Optimization Abstract: The most important concepts in the general theory of interior-point methods will be presented. The study of these algorithms has dominated the continuous optimization literature for nearly 15 years, beginning with the work of Karmarkar. In that time, the theory has matured tremendously, most notably due to the path-breaking, broadly-encompassing and enormously influenetial work of Nesterov and Nemirovskii. Much of the literature on the general theory of interior-point methods is written in a way that makes it very difficult to understand, even for specialists. We approach the theory from a different angle, one that makes the key concepts evident.   April 27, 1999 Charles W. Curtis (University of Oregon) Some early history of group representation theory Abstract: Frobenius invented characters of finite groups in a series of papers published in 1896, in connection with the problem of factoring the group determinant of a finite group. He developed a theory of representations of groups the following year. This talk contains historical comments, and a sketch of the mathematics involved, for the solution of the next two problems taken up in the new theory. Frobenius's calculation of the characters of the symmetric groups (1900), and Schur's construction and classification of the polynomial representations of the general linear group (1901).   April 20, 1999 Panagiota Daskalopoulos (University of California, Irvine) C∞ Regularity of the Free-Boundary in Nonlinear Degenerate Parabolic Problems Abstract: We will discuss the regularity of the interface in free boundary problems arising from the degeneracy of various nonlinear parabolic equations, including the porous medium equation, the p-Laplacian equation, the Gauss curvature flow with flat sides, and the mean curvature flow for a 3-dimensional surface of revolution. We will show that, under certain assumptions on the initial data, the free-boundary becomes C∞-smooth.   April 15, 1999 S. Ivaschkovitch (Lille-I) Complex curves in almost-complex manifolds and analytic continuation of meromorphic functions Abstract: The aim of this talk is to show how recent results and techniques of symplectic geometry can be applied to the study of classical questions in complex analysis. We shall recall for the convenience of the audience the basic notions of the Gromov theory of "pseudoholomorphic curves," give a formulation of Gromov compactness theorem for stable curves, introduce a $\bar\partial$-operator and give some results describing the moduli space of compact complex curves in almost-complex manifold. Then we shall briefly recall the notion of Banach analytic set, give examples and explain a theorem about Banach analytic structure of the moduli space in the neighborhood of noncompact complex curve in stable topology. Finally, we are going to apply results, mentioned above to thge construction of envelopes of meromorphy of symplectic surfaces in complex Kähler manifolds of dimension two. In particular we shall show that any function meromorphic in the neighborhood of symplectic sphere in complex projective plane CP2 is in fact rational.   April 13, 1999 Louis Nirenberg (NY Courant Institute) Degree theory beyond continuous maps Abstract: In recent years it has been found useful to extend degree theory even in finite dimensions to certain classes of maps which need not be continuous. The talk will be an expository one describing recent results including the use of BMO.   March 30, 1999 Eric Sommers (Harvard University) Hyperplane Arrangements and Lie Theory Abstract: The usual exponents of a Weyl group W arise in many different ways in Lie theory. For example, they appear in the topology of the corresponding compact Lie group, the invariant theory of W, the representation theory of W, and the representation theory of the corresponding complex Lie algebra. In this talk, I will discuss a generalization of the usual exonents that arise in the work of Orlik and Solomon on hyperplane arrangements (called OS exponents). After introducing the combinatorics in Orlik and Solomon's work, I will present two or three appearances of OS exponents in Lie theory. I will also mention recent work of Broer and Douglass which proves (using Lie theory methods) that certain hyperplane arrangements are free. If time permits, I will mention some conjectural generalizations of these results.   March 9, 1999 Robert Hardt (Rice University) The Nodal and Critical Sets of a Solution of an Elliptic Equation Abstract: Suppose u is a solution of a linear elliptic equation with smooth coefficients in n-space. The nodal set of u is the set where u vanishes, and the singular set is the set where both u and its gradient vanish. We discuss the finiteness and bounds for the n-1 dimensional measure of the nodal set and the n-2 dimensional measure of the singular set. The proofs involve various lemmas from algebraic and integral geometry. We also mention some conjectures relating to geometric applications. These results are joint works with L. Simon, Q. Han, F. Lin, M. & T. Hoffman-Ostenhof, and N. Nadirashvili.   February 9, 1999 Persi Diaconis (Stanford University) Mathematics of Solitaire Abstract: One of the embarrassments of applied probability is that we can not analyze the original game of solitaire. I will show that a simplified solitaire leads to fascinating mathematics and complete analysis. The mathematics involves combinatorics, random matrices, and Riemann-Hilbert theorem. This is a joint work with David Aldous. This colloquium talk is accessible to a general audience including undergraduate students.   January 29, 1999 Allen Knutson (Brandeis University) Hermitian matrices, tensor products, and honeycombs Abstract: In 1912 Weyl first studied the following question: given two N x N  Hermitian matrices with known eigenvalues, what might the eigenvalue spectrum of the sum be? This is the "classical mechanical" version of a problem with a "quantum mechanical" counterpart: given two finite dimensional irreducible representations of GLn({\mathbb{C}}), which irreducible representations occur in the tensor product? In 1962 Horn conjectured an explicit (albeit recursive) answer to the first question, and recently Klyachko proved a (less explicit) answer, in the form of a list of inequalities among the eigenvalues. We present a model for studying both these problems simultaneously, the honeycomb model. The existence of a solution to the classical question, a triple of Hermitian matrices A+B=C with given spectra, is equivalent to the existence of honeycombs with certain boundary conditions. WIth this, we prove Horn's conjecture, and determine exactly which of Klyachko's inequalities are essential (nonredundant).   January 26, 1999 Piotr Chrusciel (Université de Tours) Global problems for some hyperbolic PDE's of mathematical physics Abstract: One of the main streams of research concerning nonlinear partial differential equations of hyperbolic type is the quest for global existence theorems. Being heavily biased by the "interesting physics is geometry" point of view, I will concentrate on the following classes of equations: wave maps, the Yang-Mills equations, and the Einstein equations. I will briefly describe what those equations are, what the Cauchy problem for those equations is, and present various existence and non-existence results (due to D. Christodoulou, D. Eardley, H. Friedrich, S. Klainerman, M. Machedon, V. Moncrief, J. Shatah, S. Tahvildar-Zadeh and others) as well as a few open problems of current interest.   December 10, 1998 Adrian Iovita (University of Washington) On good reduction of Abelian varieties Abstract: Let $K$ be a local field (finite extension of ${\mathbb{Q}}_p$) and $A/K$ an Abelian variety. For every prime number $\ell$, we denote by $T_\ell(A)$ the $\ell$-adic Tate-module of $A$. It is an $\ell$-adic representation of the Galois group $G_K:=Gal(\bar K/K)$. A famous theorem of Néron-Ogg-Shafarevich-Serre-Tate states: Theorem The Abelian variety $A$ has good reduction if and only if for some (all) $\ell\ne p$ the representation $T_\ell(A)$ is unramified. We'll talk about how one can characterize the good reduction of $A$ looking only at the $G_K$-representation $T_p(A)$.   December 1, 1998 Bernd Sturmfels (University of California at Berkeley) Gröbner Deformations of Hypergeometric Differential Equations Abstract: Mutsumi Saito, Nobuki Takayama and I are currently writing a book with the above title. Its subject is the use of Gröbner bases for D-modules and resulting applications to multidimensional hypergeometric functions (as defined by Gel'fand, Kapranov and Zelevinski). The purpose of this lecture is to give an elementary exposition of these topics.   November 17, 1998 Frank Sottile (MSRI) Feedback control of linear systems and the real Schubert calculus Abstract: In 1981, Brockett and Byrnes showed how the static feedback laws which control a given linear system are determined thgrough the Schubert calculus of enumerative geometry. This talk will describe that connection and propose numerical homotopy methods to solve the resulting systems of polynomials. The related questions of finding real feedback laws and of trying to do Schubert's calculus over the real numbers are intertwined with precise conjecture eof Shapiro and Shapiro. We discuss this connection and present some impressive computational evidence in support of this conjecture. We close with a new result, proving a version of this conjecture and showing iit is possible to do Schubert's calculus over the real numbers.   November 5, 1998 J. M. Landsberg (Université Paul Sabatier, Toulouse, France Dual varieties, symmetric degeneracy loci, and a classical problem in linear algebra Abstract: The dual variety $X^*$ of a projective variety $X^n\subset{\mathbb{C}}{\mathbb{P}}^{n+a}$ is the set of hyperplanes tangent to $X$. It is a geometric generalization of the Legendre transform in classical mechanics. I will discuss a result (with B. Ilic) that describes a way in which the global geometry of $X$ restricts the local geometry of $X^*$. I will then explain how this result led us to study and solve a classical problem in linear algebra, namely, to determine the maximum dimension of a linear subspace A of the space of m\times m symmetric matrices having the property that every nonzero $q\inA$ has the same rank r.   October 27, 1998 Pekka Koskela (University of Jyväskylä Analysis on metric spaces Abstract: In general, the concept of a partial derivative is meaningless on a metric space. However, it is natural to call a Borel-measurable function $g\ge 0$ an upper gradient of a function $u$ if $[|u(x)-u(y)|\le\int_{\gamma}g\,ds]$ holds for each pair $x, y$ and all rectifiable curves $\gamma$ joining $x,y$. Thus, in the Euclidean setting, we consider the length of the gradient of a smooth function instead of the actual gradient. Given a metric space X equipped with a doubling measure $\mu$, one would like to obtain good control of the averages of a function u in terms of volume integrals of an upper gradient of u. This is related to recent attempts to obtain generalizations of the classical theory of Sobolev spaces to the setting of a metric space equipped with a measure. Such generalizations are necessary for applications to Carnot-Carathé'odory spaces, subelliptic equations, quasiconformal mappings on Carnot groups and more general Loewner spaces, analysis on topological manifolds, potential theory on infinite graphs, analysis on fractals and the theory of Dirichlet forms. I will try to explain the basics of upper gradients and the desired estimates.   October 20, 1998 Victor Ivrii Spectral Asymptotics with Sharp Remainder Estimates Abstract: The problem of spectral asymptotics, in particular the problem of asymptotic distribution of eigenvalues is one of the central problems in the spectral theory of partial differential equations. The problem originated in the early part of the century when Hermann Weyl published a pioneer paper devoted to asymptotics of the Dirichlet eigenvalues (i.e., the frequencies of vibration of a fixed elastic membrane) in terms of the volume of the domain. In this talk we will consider local semiclassical asymptotics arising in quantum mechanics. In this case the parameter is Planck's constant which tends to zero. We will explain how one can obtain these asymptotics by rescaling and how these asymptotics imply many other spectral asymptotics of a different nature.
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