| October 14, 2004 |
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 |
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| October 28, 2004 |
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 |
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 |
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 Schechter |
| December 9, 2004 |
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 C^*-algebras. The talks is meant to be accessible to a mathematical literate audience, including graduate students. Contact person: Guoliang Yu |