Colloquia, Fall 2004

Mathematics Colloquia, Fall 2004

Thursdays 4:10 pm in 1206 Stevenson Center, unless otherwise noted

Tea at 3:30 pm in 1425 Stevenson Center


October 14, 2004 

Bruce Ayati, Southern Methodist University

Galerkin Methods for PDE Models of
Age- and Space-structured Biological Systems

Abstract: We discuss a class of numerical methods for partial
differential equations that take into account age as well as space in
modeling the dynamics of a biological system. The equations and their
biological meaning will be presented. These new numerical methods and
their utility will then be explored in the case of colony growth of the
bacteria Proteus mirabilis.

Contact person: Glenn Webb

October 28, 2004 

Chris Phillips, University of Oregon

Rational cohomology for Banach algebras

Abstract: Let A be a commutative unital Banach algebra. Its maximal
ideal space Max (A) is a compact Hausdorff space which plays a role
somewhat similar to that of the space Spec (R) in algebraic geometry.
In particular, A can be represented, not necessarily faithfully, as an
algebra of continuous functions on Max (A).
The Taylor problem asks for a construction of the (Cech)
cohomology of Max (A) “directly from A”. For example, H^1 (Max (A); Z)
is the quotient of the invertible group of A by the image of the
exponential map. This, and related descriptions of H^0 (Max (A); Z)
and H^2 (Max (A); Z), have been known since the 1970s, but the program
seemed to stop there.
In this talk, after giving some background, we describe a
solution for the _rational_ (Cech) cohomology H^s (Max (A); Q) for
arbitrary s, in terms of the rational homotopy groups of the spaces
of last columns of invertible n by n matrices over A for suitable n.
The approach has the promise of giving something interesting for
noncommutative Banach algebras as well.
This is joint work with Greg Lupton, Claude Schochet, and
Samuel Smith.

Contact person: Dietmar Bisch

November 4, 2004 

Marek Kimmel, Rice University

Mathematical model of linear or tubular tumor growth with diffusion of
growth factor molecules

Abstract: We consider a system composed of a tubular sheet of early tumor cells,
occupying the surface of a structure existing in the organism. We assume
that the cells have a potential for proliferation in response to a growth
factor. This model can be thought of as representing an early stage (pre-in
situ) of tumor evolution. A biomedical example of such process might be the
Atypical Adenomatous Hyperplasia in the lung. Destabilization of the
equilibrium in such system represents an initial invasion of cancer. We are
looking for a transition from a slightly perturbed equilibrium state to
uncontrolled and irregular growth. This approach is different from other
approaches present in the literature. We examine a mathematical model of a
population of cells distributed over a linear or tubular structure. Growth
of cells is regulated by a growth factor, which can diffuse over the
structure. Aside from this, production of cells and of the growth factor is
governed by a pair of ordinary differential equations. Equation for the cell
number follows from an accepted model of cell cycle. Equation for the
bounded receptor particle number follows from a time-continuous Markov
process. We demonstrate existence of the solutions of the complete model,
using the method of invariant rectangles. We find conditions under which
diffusion causes destabilization of the spatially homogeneous steady state,
leading to exponential growth and apparently chaotic spatial patterns,
following a period of almost constancy. This phenomenon may serve as a
mathematical explanation of “unexpected” rapid growth and invasion of
temporarily stable structures composed of cancer cells.

Contact person: Glenn Webb

November 11, 2004 

Robert K. Meyer, The Australian National University

Must Arthmetic Be Consistent?

Abstract: Goedel’s 1st Theorem states that any formal system S that contains “enough arithmetic” must be either inconsistent or
incomplete. Conventional wisdom concludes (1) this rests on an INCOMPLETENESS proof (which SMELLS like an INCONSISTENCY proof) and
anyway (2) the usual S (based on a CLASSICAL or related logic) are after all consistent, though (3) we can’t prove that unless we extend
S to something STRONGER, in view of Goedel’s 2nd theorem. Well, it’s a little distressing that, in view of some POSSIBLE trick
involving, say, the HIGHER ordinals as constructed in S we can’t be sure that 2+2 = 5 is unprovable in S. And there is a crystal clear
remedy: formulate S using a RELEVANT logic, and appeal to INCONSISTENT MODELS of S in that logic. The talk will supply details. This is
joint work with J. Michael Dunn, Chris Mortensen and Graham Priest.

Contact person: Eric

December 9, 2004 

Victor Nistor, Penn State University

Analysis on polyhedral domains

Abstract: The analysis of elliptic partial differential
operators on smooth, bounded domains is well understood
and has numerous applications, many outside mathematics.
By contrast, the behavior of these operators on non-smooth
domains can be quite different from the one on smooth
domains and is much less understood. In my talk, I will
first review some known results on boundary value problems
on non-smooth domains. Then I will present an approach
to analysis on polyhedral domains that is based on a
modification of the usual Sobolev spaces, yielding the
so called “Sobolev spaces with weights.” One can,
for instance, obtain a regularity theorem within
these Sobolev spaces with weights that is completely
analogous to the usual elliptic regularity on smooth domains.
This result, joint work with C. Bacuta and L. Zikatanov,
has potential applications to numerical methods. I will
also briefly discuss at the end some connections with
Operator Algebras, more precisely, with groupoid
The talks is meant to be accessible to a mathematical
literate audience, including graduate students.

Contact person: Guoliang Yu


Colloquium Chair (Fall 2004): Doug Hardin

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