The Mathematical and Computational Sciences at Western are represented by four separate departments: Applied Mathematics, Computer Science, Mathematics, and Statistical & Actuarial Sciences. The Department of Mathematics has established research groups in several areas of contemporary mathematics including algebra, analysis and analytic geometry, homotopy theory, and noncommutative geometry.
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The question motivating the first part of this talk is the following:
What information can one deduce about ordinary (de Rham) cohomology of a
manifold using the theory of equivariant cohomology, if the manifold
admits a special type of Lie group action?
The class of group actions we will consider is that of cohomogeneity one
actions (i.e., those that admit an orbit of codimension one). Among
other things, one can derive the following topological obstruction: if a
compact manifold with positive Euler characteristic admits an action of
cohomogeneity one, then all of its odd Betti numbers vanish. (A result
that was previously shown by Grove and Halperin using rational homotopy
theory.)
In the second part we will go into a completely different geometric
situation and show how one can use similar techniques to derive a link
between the basic cohomology of certain Riemannian foliations and the
number of closed leaves of the foliation. The main example here will be
the Reeb foliation of a K-contact manifold.
(The first part is a joint work with Augustin-Liviu Mare, and the second
one with Hiraku Nozawa and Dirk Töben)
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