Geometry and Topology

Speaker: Oliver Goertsches (Hamburg)

"Equivariant cohomology of cohomogeneity one and K-contact manifolds"

Time: 15:30

Room: MC 108

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)

Ph.D. Public Lecture

Speaker: Enxin Wu (Western)

"A homotopy theory for diffeological spaces"

Time: 14:30

Room: MC 107

Smooth manifolds are important objects in mathematics. But the category of smooth manifolds are not closed under many useful operations. In 1980, J. Souriau introduced a generalization of smooth manifolds called diffeological spaces, and the theory was further developed by P. Igleisas-Zemmour, etc. Our goal is to develop a homotopy theory (also called a model category structure) on the category of diffeological spaces, which encodes the usual homotopy theory of smooth manifolds, and the bundle theory of Iglesias.

Ph.D. Public Lecture

Speaker: Mehdi Mousavi (Western)

"A Convexity Theorem for Symplectomorphism Groups"

Time: 10:00

Room: MC 108

The study of infinite-dimensional Lie groups is pioneered by Arnold in a celebrated paper. Ebin and Marsden wrote Arnold's idea in a rigorous language. The theory of infinite-dimensional Lie groups is somehow unsatisfactory, although it has been an active area of research in the past 40 years. An analog of a maximal torus has been studied previously by Bao-Raiu for the volume preserving diffeomorphisms of finite cylinder which motivated a later work by El Hadrami on complex projective spaces. Later on, Bloch-Flaschka-Ratiu motivated by Bao-Ratiu's result obtained an analog of Schur-Horn-Kostant convexity theorem. In this talk we will see that there is an analog of maximal torus in the symplectomorphism group of toric manifolds. We also study the existence of an analog of Schur-Horn-Kostant convexity theorem.

Ph.D. Presentation

Speaker: Ali Fath (Western)

"Isospectral Noncommutative Geometries"

Time: 15:00

Room: MC 108

The primary examples of noncommutative dierential geometry are the non-commutative tori [1]. The algebra in these examples can be constructed from the algebra of smooth functions on the ordinary torus Tl by deforming the product with rotation action of the torus on itself. There is also an action of Tl on the resulting deformed algebra. Beginning with a spin structure on Tl there is a Dirac operator 6Dacting on the Hilbert space H of spinors, which is invariant under the lifted action of Tl. Thus the deformed algebra is also represented on this Hilbert space, resulting in a spectral triple with the same Dirac operator. So we have an isospectral deformation of the canonincal commutative spectral triple (C1(Tl);H;D6 ). This example was generalized by Connes and Landi [2] by deforming any compact Riemannian manifold M that admits an isometric action of the torus Tl, for l  2. This method was further rened by Connes and Dubois-Violette [3] to include the noncompact case but still with a torus action. In all these cases, the Dirac operator is invariant under the lifted action and (C1(M);H; 6D) is a noncommutative spin geometry that is isospectral to the undeformed case, because D6 is unchanged. In this presentation we will go through these results and discuss some possible directions for studying certain noncommutative geometries in this setting. References [1] A. Connes, Noncommutative Geometry. Academic Press, London, 1994. [2] A. Connes and G. Landi, Noncommutative manifolds, the insanton algebra and isospectral deformation, Commun. Math. Phys. 221(2001), 141-159. [3] A. Connes and M. Dubois-Violette, Noncommutative nite-dimensional manifolds. I. Spherical manifolds and related examples, Commun. Math. Phys. 230 (2002), 539-579. 1

Noncommutative Geometry

Speaker: Ali Fathi (Western)

"Isospectral deformations of Riemannian manifolds I"

Time: 14:00

Room: MC 107

Isospectral deformations of Riemannian manifolds I