Colloquia, Fall 2000

Vanderbilt Mathematics
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
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.
, of Rice University.


November 30.
Van Vu, of
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
kth 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
(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 kth powers
to guarantee that Waring’s assertion holds? Is it possible
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
, 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:

A finitely presentable group G is hyperbolic if and only if
the map from bounded cohomology H2b(G,V)
to H2(G,V) is
surjective for all bounded G-modules V.

This extends the Gromov’s claim of surjectivity for real coefficients,
and is analogous to the Gersten’s cohomological characterization of
hyperbolic groups and to the Johnson’s characterization of amenable
groups. —
Relations to computations, geometry, and analysis will be discussed.


October 5. Xingxing
, of Georgia Tech.
Long cycles in graphs.
Motivated by the 4-color problem, Whitney (1931) proved
that every 4-connected triangulation of the sphere contains a
Hamilton cycle. Tutte (1956) generalized Whitney’s result to
4-connected planar graphs. However, 3-connected planar graphs
need not contain Hamilton cycles. In this talk, I will briefly
survey some results on long cycles in graphs, and sketch the
proofs of two related conjectures.


September 28. Dr.
Yuri Gurevich
, of Microsoft Research and
University of Michigan.
What is an algorithm? One may think that the title problem was
solved long ago by Church and Turing. It wasn’t: there is more to an
algorithm than the function it computes. (Besides, what function does an
operating system compute?) The interest in the problem is not only
theoretical: applications include modeling, specification, verification
and design of software and hardware systems. However, in this talk, we
will concentrate on the theoretical aspects. We will explain the
sequential Abstract State machine thesis. (See
in this connection.) If time permits, we will mention parallel,
distributed and real-time abstract state machines.


September 21. Jon
, of Texas A&M
. Calculating curvatures in concrete complexes.
Geometric group theorists have been borrowing techniques from
differential geometers for more then 15 years. As a result there is now
a well developed theory of nonpositively curved metrical simplicial
complexes which can be used to study of the “geometry of a group”.
Unfortunately for the working group theorist, there are very few tests
for determining whether a given finite complex will fit into this
framework. … Specifically, given an explicit, finite, piecewise
Euclidean (PE) complex, there does not at present exist an algorithm for
determining whether this particular complex is nonpositively curved. …
After a quick introduction to/review of the theory of metric simplicial
complexes, I will present a recent result (joint with Murray Elder) in
which we provide an algorithm to solve this problem if the PE complex
has dimension less than four.


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