Vanderbilt Mathematics Colloquia Fall 2000

Colloquia are listed in reverse chronological order. The top of the list is subject to change, since more colloquia are still being planned. Our colloquia, as well as our seminars and other activities, feature speakers not only from our own department but also from other departments all over the world. You may also want to consult our weekly calendar and past calendars. Some additional information about this year's colloquia may be available at the web page maintained by this year's Colloquium Chairperson.

    December 7. Marek Kimmel, of Rice University. TBA.

    November 30. Van Vu, of Microsoft. On a refinement of Waring's assertion. In 1770 Waring asserted (without proof) that for every fixed k, there is a number s such that every natural number can be represented as sum of s k^th powers. For instance, every natural number can be written as sum of four squares, 9 cubes and so on. This has become the famous Waring's conjecture in number theory. ¶ Waring's conjecture was proved in its full generality by several famous mathematicians (including Hilbert, Hardy-Littlewood, Vinogradov, Hua etc) in the beginning of the 20th century. ¶ In this talk, we consider a more "economical" version of Waring's assertion and investigate the following question: Given k and s, do we really need all k^th powers to guarantee that Waring's assertion holds? Is it possible that only a small fraction still suffices? If yes, then how small? ¶ The answer will be given during the talk.

    November 13. Avner Friedman, of the University of Minnesota. A simple model of tumor growth. We shall describe a phenomenological model of tumor growth with inhibitors. Mathematically the model consists of a system of partial differential equations in a region with free boundary - the moving boundary of the tumor. We shall prove that the model produces dormant states that have non-radially symmetric solutions.

    November 9. Chris Godsil, of the University of Waterloo. Spin Models and Knots. Jones introduced two ways to get knot invariants - algebraically, using a class of traces on braid groups and combinatorially, using partition functions on planar graphs. He observed that these methods were closely related. Subsequent work has extended the combinatorial view by introducing so-called "four-weight spin models". However this theory is complicated and not obviously related to braid groups. I will describe a new approach which shows that these spin models are related to representations of the braid groups and have a surprisingly rich structure.

    November 2. Larry Taylor, of the University of Notre Dame. Multivariable Calculus Reprised. I will begin by explaining the strong sense in which the calculus of n-variables is unique if n is not 4, a result known since the early 70's. In the last 20 years we have begun to understand just how bizarre the 4-variable case really is. Using results of mine and others, I will explore some of the exciting results that have emerged, many of which should be accessible to anyone with an understanding of multi-variable calculus and the elementary theory of smooth manifolds. These results range from basic existence results from the 80's to recent constraints on the differential geometry of exotic 4-variable calculus.

    October 26. Igor Mineyev, of the University of South Alabama. Hyperbolicity and cohomology. This talk will be accessible to a general audience. We review basic properties of hyperbolic groups introduced by Gromov. Following Gersten and Gromov, we discuss how geometric properties of hyperbolic groups (and more generally, of finitely presented groups) can be described homologically. We also give the following characterization:

Theorem. A finitely presentable group G is hyperbolic if and only if the map from bounded cohomology H²_b(G,V) to H²(G,V) is surjective for all bounded G-modules V.