Flavors of Cohomology

Schedule

Speakers:

• Carl McTague, Johns Hopkins
  1. The Cayley Plane and String Bordism
  2. tmf Is Not a Ring Spectrum Quotient of String Bordism
  3. Towards Higher Genera: A Synergy Between Elliptic and Witten Genera
• Eric Peterson, Berkeley
  An invitation to Morava E-theory
  Abstract for day 1: Starting from a review of the notion of a complex orientation, we will construct the cohomology theories referred to as "Morava E-theories" and uncover their most essential properties.
  Abstract for day 2: With Morava E-theories now in hand, we will get a sense of how they're used by exploring one of their main applications: the structure theorems of Devinatz, Hopkins, and Smith for finite spectra.
  Abstract for day 3: Having now seen a success story with Morava E-theory, we turn to less well-understood behaviors and open questions, selected so as to be of interest to other attendees.
• Corbett Redden, Long Island University
  Applications of differential cohomology
  1. Introduction to differential cohomology
  Differential cohomology is a non-trivial hybrid of cohomology and differential forms.  It is the natural home for many geometric invariants, including a refinement of the Chern-Weil theory for principal bundles with connection.  It allows the language of algebraic topology to be used in geometric problems.  We will explain this history in more detail, and we will show the importance of cocycle models.
  2. Differential equivariant cohomology
  For a manifold acted on by a compact Lie group, we construct "differential equivariant cohomology” groups, refining Borel’s construction of equivariant cohomology and the Cartan-Weil complex of equivariant differential forms.  The Chern-Weil homomorphism for equivariant bundles with connection naturally factors through differential equivariant cohomology.
  3. Geometric string structures and trivializations
  String structures arise as a higher analog of spin structures.  Unfortunately, working with them explicitly must involve infinite-dimensional spaces or higher categories.  We explain how the language of trivializations in differential cohomology can be used to describe geometric string structures in a more elementary way.
• Urs Schreiber, Charles University, Prague
  Structure theory for higher WZW terms
  I will present results characterizing the behaviour higher WZW terms, with some applications to string theory. Following sections 5.3.9-5.3.14 of this document. Lecture notes available here.

Organized by Hisham Sati and Jason DeBlois

Sponsored by the Mathematics Research Center