\n| October 14, 2004<\/b> <\/th>\n | \nBruce Ayati, Southern Methodist University<\/b><\/center>
\n
\nGalerkin Methods for PDE Models of
\nAge- and Space-structured Biological Systems
\n<\/font><\/strong><\/center>
\n
\nAbstract<\/u>: We discuss a class of numerical methods for partial
\ndifferential equations that take into account age as well as space in
\nmodeling the dynamics of a biological system. The equations and their
\nbiological meaning will be presented. These new numerical methods and
\ntheir utility will then be explored in the case of colony growth of the
\nbacteria Proteus mirabilis.<\/p>\n Contact person<\/u>: Glenn Webb<\/font><\/p>\n<\/td>\n<\/tr>\n\n| October 28, 2004<\/b> <\/th>\n | \nChris Phillips, University of Oregon<\/b><\/center>
\n
\n Rational cohomology for Banach algebras
\n<\/font><\/strong><\/center>
\n
\nAbstract<\/u>: Let A be a commutative unital Banach algebra. Its maximal
\nideal space Max (A) is a compact Hausdorff space which plays a role
\nsomewhat similar to that of the space Spec (R) in algebraic geometry.
\nIn particular, A can be represented, not necessarily faithfully, as an
\nalgebra of continuous functions on Max (A).
\n The Taylor problem asks for a construction of the (Cech)
\ncohomology of Max (A) “directly from A”. For example, H^1 (Max (A); Z)
\nis the quotient of the invertible group of A by the image of the
\nexponential map. This, and related descriptions of H^0 (Max (A); Z)
\nand H^2 (Max (A); Z), have been known since the 1970s, but the program
\nseemed to stop there.
\n In this talk, after giving some background, we describe a
\nsolution for the _rational_ (Cech) cohomology H^s (Max (A); Q) for
\narbitrary s, in terms of the rational homotopy groups of the spaces
\nof last columns of invertible n by n matrices over A for suitable n.
\nThe approach has the promise of giving something interesting for
\nnoncommutative Banach algebras as well.
\n This is joint work with Greg Lupton, Claude Schochet, and
\nSamuel Smith. <\/p>\n Contact person<\/u>: Dietmar Bisch<\/font><\/p>\n<\/td>\n<\/tr>\n\n| November 4, 2004<\/b> <\/th>\n | \nMarek Kimmel, Rice University<\/b><\/center>
\n
\nMathematical model of linear or tubular tumor growth with diffusion of
\ngrowth factor molecules
\n<\/font><\/strong><\/center>
\n
\nAbstract<\/u>: We consider a system composed of a tubular sheet of early tumor cells,
\noccupying the surface of a structure existing in the organism. We assume
\nthat the cells have a potential for proliferation in response to a growth
\nfactor. This model can be thought of as representing an early stage (pre-in
\nsitu) of tumor evolution. A biomedical example of such process might be the
\nAtypical Adenomatous Hyperplasia in the lung. Destabilization of the
\nequilibrium in such system represents an initial invasion of cancer. We are
\nlooking for a transition from a slightly perturbed equilibrium state to
\nuncontrolled and irregular growth. This approach is different from other
\napproaches present in the literature. We examine a mathematical model of a
\npopulation of cells distributed over a linear or tubular structure. Growth
\nof cells is regulated by a growth factor, which can diffuse over the
\nstructure. Aside from this, production of cells and of the growth factor is
\ngoverned by a pair of ordinary differential equations. Equation for the cell
\nnumber follows from an accepted model of cell cycle. Equation for the
\nbounded receptor particle number follows from a time-continuous Markov
\nprocess. We demonstrate existence of the solutions of the complete model,
\nusing the method of invariant rectangles. We find conditions under which
\ndiffusion causes destabilization of the spatially homogeneous steady state,
\nleading to exponential growth and apparently chaotic spatial patterns,
\nfollowing a period of almost constancy. This phenomenon may serve as a
\nmathematical explanation of “unexpected” rapid growth and invasion of
\ntemporarily stable structures composed of cancer cells.<\/p>\n Contact person<\/u>: Glenn Webb<\/font><\/p>\n<\/td>\n<\/tr>\n\n| November 11, 2004<\/b> <\/th>\n | \nRobert K. Meyer, The Australian National University<\/b><\/center>
\n
\nMust Arthmetic Be Consistent?
\n<\/font><\/strong><\/center>
\n
\nAbstract<\/u>: Goedel’s 1st Theorem states that any formal system S that contains “enough arithmetic” must be either inconsistent or
\nincomplete. Conventional wisdom concludes (1) this rests on an INCOMPLETENESS proof (which SMELLS like an INCONSISTENCY proof) and
\nanyway (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
\nS to something STRONGER, in view of Goedel’s 2nd theorem. Well, it’s a little distressing that, in view of some POSSIBLE trick
\ninvolving, 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
\nremedy: formulate S using a RELEVANT logic, and appeal to INCONSISTENT MODELS of S in that logic. The talk will supply details. This is
\njoint work with J. Michael Dunn, Chris Mortensen and Graham Priest.<\/p>\n Contact person<\/u>: Eric
\nSchechter<\/font><\/p>\n<\/td>\n<\/tr>\n\n| December 9, 2004<\/b> <\/th>\n | \nVictor Nistor, Penn State University<\/b><\/center>
\n
\nAnalysis on polyhedral domains
\n<\/font><\/strong><\/center><\/p>\nAbstract<\/u>: The analysis of elliptic partial differential
\noperators on smooth, bounded domains is well understood
\nand has numerous applications, many outside mathematics.
\nBy contrast, the behavior of these operators on non-smooth
\ndomains can be quite different from the one on smooth
\ndomains and is much less understood. In my talk, I will
\nfirst review some known results on boundary value problems
\non non-smooth domains. Then I will present an approach
\nto analysis on polyhedral domains that is based on a
\nmodification of the usual Sobolev spaces, yielding the
\nso called “Sobolev spaces with weights.” One can,
\nfor instance, obtain a regularity theorem within
\nthese Sobolev spaces with weights that is completely
\nanalogous to the usual elliptic regularity on smooth domains.
\nThis result, joint work with C. Bacuta and L. Zikatanov,
\nhas potential applications to numerical methods. I will
\nalso briefly discuss at the end some connections with
\nOperator Algebras, more precisely, with groupoid
\nC^*-algebras.
\nThe talks is meant to be accessible to a mathematical
\nliterate audience, including graduate students. <\/p>\n Contact person<\/u>: Guoliang Yu<\/font><\/p>\n<\/td>\n<\/tr>\n<\/table>\n
\n
\nColloquium Chair (Fall 2004): Doug Hardin<\/b><\/p>\n","protected":false},"excerpt":{"rendered":" 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…<\/p>\n","protected":false},"author":637,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[],"class_list":["post-55","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/my.vanderbilt.edu\/pastcolloquia\/wp-json\/wp\/v2\/pages\/55","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/my.vanderbilt.edu\/pastcolloquia\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/my.vanderbilt.edu\/pastcolloquia\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/my.vanderbilt.edu\/pastcolloquia\/wp-json\/wp\/v2\/users\/637"}],"replies":[{"embeddable":true,"href":"https:\/\/my.vanderbilt.edu\/pastcolloquia\/wp-json\/wp\/v2\/comments?post=55"}],"version-history":[{"count":1,"href":"https:\/\/my.vanderbilt.edu\/pastcolloquia\/wp-json\/wp\/v2\/pages\/55\/revisions"}],"predecessor-version":[{"id":56,"href":"https:\/\/my.vanderbilt.edu\/pastcolloquia\/wp-json\/wp\/v2\/pages\/55\/revisions\/56"}],"wp:attachment":[{"href":"https:\/\/my.vanderbilt.edu\/pastcolloquia\/wp-json\/wp\/v2\/media?parent=55"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/my.vanderbilt.edu\/pastcolloquia\/wp-json\/wp\/v2\/tags?post=55"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}} | | | | |