Talk

http://people.mpim-bonn.mpg.de/teichner/Math/4-Manifolds.html

Video: 

20130502_stipsicz1a.mp4

20130502_stipsicz1b.mp4

According to a theorem of Eliashberg and Thurston a C²-foliation on a closed 3-manifold can be C⁰-approximated by contact structures unless all leaves of the foliation are spheres. Examples on the 3-torus show that every neighbourhood of  a foliation can contain
infinitely many non-diffeomorphic contact structures. In this talk we show that this is rather exceptional: In many interesting situations the contact structure in a sufficiently small
neighbourhood of the foliation is uniquely determined up to isotopy. This fact can be
applied to obtain results about the topology of the space of taut foliations.

http://people.mpim-bonn.mpg.de/teichner/Math/4-Manifolds.html

Video: 

20130426_gukov4a.mp4

20130426_gukov4b.mp4

http://people.mpim-bonn.mpg.de/teichner/Math/4-Manifolds.html

The  distribution  character  is  a  complete  invariant  of  Harish-Chandra modules
(it determines the representation up to infinitesimal equivalence.)  It is very
difficult to compute.  There are other invariants, sometimes weaker, which
contain very important information about the representation. An example is
the associated cycle. In the first part of the talk I will introduce some such
invariants and explain relevant information encoded by them. The second part
of the talk, based on work with R. Zierau, will focus on the computation of
associated cycles for a class of representations.

I will give a fairly high level introduction to topological modular forms (Tmf) and their relationship to the Witten genus. Tmf was constructed, in part, as the target of a cohomological lift of Witten genus. The Witten genus assigns a modular form to a string manifold and this assignment is invariant under string bordism. This is analogous to the A-hat genus which assigns to a spin manifold an elliptic operator which, in turn, defines a formal difference of vector spaces. Moreover the A-hat genus is invariant under spin bordism. Aityah, Bott, and Shapiro demonstrated the A-hat genus lifts to a map of cohomology theories MSpin-->KO. The analogous theory for the Witten genus is the cohomology theory tmf which has been constructed, in several ways, by Mike Hopkins, Haynes Miller, Paul Goerss, and Jacob Lurie. These lifts were constructed and shown to preserve an enormous amount of structure (they are E_\infty maps) by Ando, Hopkins, and Rezk.