## DEPARTMENTAL PHD THESIS EXAM – Leonid Shartser

Wednesday, October 20, 2010, 12:10 - 1:00 p.m.,
in BA 1200, 40 St. George Street

PhD Candidate:  Leonid Shartser

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.