VU Math Colloquia
Fall 2003
Thursdays 4:10 pm in 1206 Stevenson unless otherwise noted.
Tea at 3:30 pm in 1425 Stevenson.

Nicolas Monod, University of Chicago. Geometric group theory leads naturally to the notion of Measure Equivalence. This concept generalizes the classical Orbit Equivalence studied in ergodic theory. I will introduce these topics and present new superrigidity statements. We are in particular interested in "negatively curved groups", e.g. hyperbolic in Gromov's sense. This illustrates a new approach initiated in collaboration with Y. Shalom.

Dan Archdeacon, University of Vermont. The most often-cited result in graph theory is Kuratowski's Theorem, which characterizes planar graphs in terms of two excluded subgraphs. This provides the "theme" in the title: finding obstructions that prevent a graph from having a particular type of drawing. In this talk I'll survey variations of Kuratowski's Theorem. I'll examine both finite and infinite graphs, surfaces and pseudosurfaces, and generalizations of outer-planarity.

Maxim Olshansky, Moscow University. The multigrid method is a powerful tool to solve algebraic systems of equations arising in many applications and it is known to be among a few methods to provide an optimal complexity in terms of arithmetic operations per unknown. Pioneered in the 70's, multigrid soon become a crucial ingredient in engineering software for numerical solution of PDEs and integral equations. A mathematical theory of multigrid methods is nowadays well established in application to such "nice" problems as elliptic equations with full regularity. However it is still in an infant stage for many other problems of interest, even for some of those where a long evidence of successful calculations exists in the engineering applications. In the talk we first recall how and why the multigrid methods work. Further convergence analysis for a model problem is outlined. Finally we present some recent results for multigrid methods for singular-perturbed problems.

Steve Ferry, Rutgers University. A compact connected metric space such that every pair of points disconnects the space is homeomorphic to a circle. A compact connected metric space containing more than one point such that no pair of points separates but such that every embedded circle separates is homeomorphic to a 2-sphere. We will discuss how efforts to extend these characterizations into higher dimensions have led to surprising positive and negative results.

Vaughan Jones, University of California, Berkeley. Quantum groups were invented largely to provide solutions of the Yang-Baxter equation and hence solvable models in 2-dimensional statistical mechanics and one-dimensional quantum mechanics. They have been hugely successful. But not all Yang-Baxter solutions fit into the framework of quantum groups. We shall explain how other mathematical structures, especially subfactors, provide a language and examples for solvable models. The prevalence of the Connes tensor product of Hilbert spaces over von Neumann algebras leads us to speculate concerning its potential role in describing entangled or interacting quantum systems.