Colloquia Schedule: Fall 2000


 
Talks will be delivered on Thursdays at 4:00 p.m. usually in 245 Altgeld Hall
or occasionally in 314 Altgeld Hall when indicated.

Orange & Blue Bar

Thursday, September 14, 245 Altgeld Hall, 4:00 p.m.
Speaker: Professor Vojtech Rödl, Emory University, Atlanta, GA
Title: Independent Sets in Graphs on the Integers
Abstract: If G is a Kr free graph on integers, then there are independent sets in G that contain an arbitrary long arithmetic progression together with its difference. This is a common generalization of theorems by Schur, van der Waerden, and Ramsey. In general, we will discuss the following common generalization of theorems of Rado, Deuber and Ramsey: Any Kr-free graph on the integers contains an independent (m,p,c)-set. (This is a joint work with Imre Leader, David Gunderson and Hans Juergen Proemel.)

Thursday, September 21, 245 Altgeld Hall, 4:00 p.m.
Speaker: Professor Olga Kharlampovich, McGill University
Title: Diophantine Geometry for Free Groups and First-Order Properties
Abstract: This talk will consider different aspects of Diophantine geometry over a free nonabelian group F, such as dimension of algebraic varieties in Fn, Implicit function theorem for F and so on. The talk will also discuss the result (joint with A. Miasnikov) that nonabelian free groups of different ranks have the same elementary theory and this theory is decidable. This is a solution of the old problem raised by A. Tarski.

Thursday, September 28, 245 Altgeld Hall, 4:00 p.m.
Speaker: Heini Halberstam, Professor Emeritus, Department of Mathematics, UIUC
Title: The Riemann Hypothesis
*Abstract: In 1859 Bernhard Riemann delivered an address "On the number of primes less than a given magnitude" to the Prussian Academy of Sciences in Berlin. In the course of the lecture he considered a meromorphic function of the complex variable s = s + it, now known as Riemann's zeta function, whose non-real zeros all lie in the infinite strip 0 < s < 1 and, as Riemann showed, have an intimate connection with the distribution of prime numbers. Riemann expressed the opinion that, "very probably", all these zeros actually lie on the line s = ½. This is the Riemann Hypothesis (RH). The aim of this talk is to sketch what is known about RH and to describe such evidence as there is for and against it. It will touch briefly on the nature of current activity related to RH.

*This is the first in a series of colloquium talks on the background, meaning and importance of the seven "Million Dollar Problems"

Thursday, October 12, 245 Altgeld Hall, 4:00 p.m.
Speaker: Bernd Ulrich, Michigan State University
Title: Generalized Principal Ideal Theorems and the Codimension of Determinantal Loci
Abstract: Estimating the (co)dimension of ideals, rings, or varieties is a fundamental issue in commutative algebra and algebraic geometry. For instance, it is to be expected that the intersection of nhypersurfaces in CN has codimension at most n. This holds in the more general setting of abstract Noetherian rings due to a classical result of dimension theory, the Krull principal ideal (or altitude) theorem.

There are more refined estimates of codimensions that take special features of the variety into account and contain Krull's theorem as a particular case. One of these results is the generalized principal ideal theorem of Eisenbud-Evans and Bruns, which gives information on the codimension of the zero locus of sections of vector bundles. It has been used to estimate the codimension of determinantal loci of a matrix in terms of the size and the rank of the matrix. This in turn gives criteria for a vector bundle to split as a direct sum of line bundles. Although these estimates are sharp in general, one should expect much better results for determinantal loci on smoothvarieties.

This was one of the questions that motivated the recent joint work with David Eisenbud and Craig Huneke I will report on. We extend the Eisenbud-Evans-Bruns theorem and establish bounds for the codimension of determinantal loci that take into account not only the size and the rank rof the matrix, but also the embedding codimension of the ambient ring. This indeed leads to improved estimates over regular rings and generalizes work by Faltings, who had considered the case of rby rminors.

Thursday, October 19, 245 Altgeld Hall, 4:00 p.m.
Speaker: Professor Jeremy Gray, Department of Pure Mathematics, Open University, U.K.
Title: Minimal Surfaces and the Work of Jesse Douglas
Abstract: Few objects in mathematics have the charm of minimal surfaces. Despite important work in the 19th Century by Riemann, Weierstrass and Schwarz, the general study of minimal surfaces remained blocked until the work of the American mathematician Jesse Douglas in the 1930s, for which he received one of the first two Fields Medals in 1936. Yet his work provoked a long-running and painful battle with Tibor Radó and Richard Courant, and even today it is not easy to find out what Douglas actually did. In this talk I shall give an introduction to his work, explain the nature of the controversy, and give some information about Douglas himself, about whom very little is known.
Thursday, October 26, 245 Altgeld Hall, 4:00 p.m.
Speaker: Gregory F. Lawler, Department of Mathematics, Duke University
Title: Universality, Conformal Invariance, and the Dimension of the Brownian Frontier
Abstract: This talk will discuss recent work with Oded Schramm and Wendelin Werner calculating (rigorously) the exact values of the intersection exponents for planar Brownian motion. As a corollary the exact Hausdorff dimension of exceptional sets of Brownian motion is derived: the dimension of the frontier or outer boundary is 4/3 (this establishes a conjecture of Mandelbrot), the dimension of the set of cut points is 3/4 (a conjecture of Duplantier and Kwon), and the dimension of the set of pioneer points is 7/4. Conformal invariance and universality are essential components of the proof as well as a new process, Stochastic Loewner Evolution, introduced recently by Schramm. Implications of this result to other models from statistical physics (percolation, self-avoiding walks) will also be discussed.
Thursday, November 2, 245 Altgeld Hall, 4:00 p.m.
Speaker: Professor Vitaly Bergelson, Department of Mathematics, Ohio State University
Title: Many Facets of the Poincaré Recurrence Theorem
Abstract: After reviewing some history and mystery surrounding the celebrated Poincaré's King Oscar prize-winning paper (in which Poincaré's recurrence theorem plays a prominent role), we shall concern ourselves with its numerous applications and ramifications. In particular we will try to reveal the links between the principles lying behind the Poincaré recurrence theorem and modern theorems belonging to combinatorics and number theory. The talk is intended for a general audience.
Thursday, November 9, 245 Altgeld Hall, 4:00 p.m.
Speaker: Professor Steffen Rohde, Department of Mathematics, University of Washington
Title: Random Sets and the Loewner Differential Equation
Abstract: The Loewner differential equation, classically used by C. Loewner to study questions from geometric function theory, has recently been used by Lawler, Schramm and Werner to settle some outstanding problems in stochastics (such as the determination of the Hausdorff dimension of the Brownian frontier). In this talk I will explain the Loewner equation, its relation to stochastic growth models (such as loop erased random walk, percolation, Brownian motion) and will discuss some recent developments. The talk will be aimed at the non-specialist.
Cairns Memorial Lectures
November 28-30, 314 Altgeld Hall
Speaker: Professor Frank Quinn, Department of Mathematics, Virginia Polytechnic Institute
Title: Why is the Fourth Dimension so Weird?
Abstracts

Items for inclusion in the Weekly Calendar can be submitted using this submission form, or send complete information to Hilda Britt. Deadline for inclusion in the weekly calendar is noon every Wednesday. Speakers are encouraged to provide abstracts. Please send items for inclusion in "Announcements" and "Conferences" to the webmaster.

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Updated October 25, 2000