Oct

08

Wednesday, October 20, 2010, 12:10 - 1:00 p.m., in BA 1200, 40 St. George Street PhD Candidate: Leonid Shartser PhD Advisor: Pierre Milman Thesis Title: De Rham Theory and Semialgebraic Geometry (http://www.math.toronto.edu/shartl/shartser-thesis.pdf) Thesis Abstract: The thesis consists of six chapters and deals with four topics related to De Rham Theory on semialgebraic sets. The first topic deals with a proof of Poincare type inequality for differential forms on compact manifolds. We prove the latter inequality by means of a constructive 'globalization' method of a local Poincare inequality on convex sets. The second topic is a construction of a Lipschitz deformation retraction on a neighborhood of a point in a semialgebraic set with estimates on its derivatives. Such a deformation retraction is the key to the results of the remaining two topics. The third topic deals with L^\infty cohomology on semialgebraic sets. We introduce smooth L^\infty differential forms on a singular (semialgebraic) space X in R^n. Roughly speaking, a smooth L^\infty differential form is collection of smooth forms on disjoint smooth subsets (stratification) of X with matching tangential components on the adjacent strata and bounded size (in the metric induced from R^n). We identify the singular homology of X as the homology of the chain complex generated by semialgebraic singular simplices, i.e. continuous semialgebraic maps from the standard simplices into X. Singular cohomology of X is defined as the homology of the Hom dual to the chain complex of the singular chains. Finally, we prove a De Rham type theorem establishing a natural isomorphism between the singular cohomology and the cohomology of smooth L^\infty forms. The last topic is related to Poincare inequality on a semialgebraic set. We study Poincare type L^p inequality on a compact semialgebraic subset of R^n for p >> 1. First we derive a local inequality by using a Lipschitz deformation retraction with estimates on its derivatives. Then, we extend the local inequality to a global inequality by employing a technique developed in the first topic. As a consequence we obtain an isomorphism between L^p cohomology and singular cohomology of a normal compact semialgebraic set.

## no comment as of now