# Higher Structures and Cohomology Theories

## Schedule

### Saturday, March 27, 2015

9:30 - 10:00 - Welcome and Coffee

10:00 – 11:00 – Igor Kriz, University of Michigan
Remarks on Derived Representation Theory

Abstract: I will report on the current stage of my joint project with Hu and Somberg on S-module categorification of representations of $SL_k$. I will talk about S-module analogues of certain concepts of Bernstein, Frenkel, Khovanov and Sussan, including $gl_n(S)$-Verma modules. In the process, I will discuss developments and questions in the theory of S-module operads.

11:00 – 11:30 – Coffee Break

11:30 – 12:30 – Benjamin Antieau, University of Illinois at Chicago
Localization sequences in K-theory and a question of Rognes

Abstract: After giving background on the algebraic K-theory of ring spectra, I will discuss recent joint work with Barthel and Gepner on the algebraic K-theory of truncated Brown-Peterson spectra and our answer to a question of Rognes about higher chromatic analogues of the famous fiber sequence $K(\mathbb{F}_p)\rightarrow K(\mathbb{Z}_p)\rightarrow K(\mathbb{Q}_p)$ of Quillen.

12:30 – 2:00 – Lunch

2:00 – 3:00 – Ralph Cohen, Stanford University
Comparing Topological Field Theories: the string topology of a manifold and the symplectic cohomology of its cotangent bundle

Abstract: I will describe joint work with Sheel Ganatra, in which we prove an equivalence between two chain complex valued topological field theories: the String Topology of a manifold M, and the Symplectic Cohomology of its cotangent bundle, $T^*M$. I will also discuss how the notion of Koszul duality appears in the study of TFT's.

3:30 – 4:30 – Craig Westerland, University of Minnesota
Topological T-duality as a form of twisted Atiyah duality

Abstract: Bouwknegt-Evslin-Mathai began the investigation of the topological aspects of T-duality some 12 years ago.  They studied pairs (E, H) consisting of a circle E bundle over a manifold M and a 3-dimensional cohomology class H on E.  Remarkably, they showed that these data came in dual pairs, whose twisted K-theories are isomorphic (with a shift).  In this talk, we will re-examine the T-duality isomorphism from an "intersection-theory" point of view, and cast it as a form of Atiyah-duality for modules over the K-theory spectrum.

### Sunday, March 29, 2015

9:00 – 10:00 – Mahmoud Zeinalian, Long Island University
Towards a definition of loop differential K-theory

Abstract: A differential cohomology is a refinement of a cohomology theory that encodes information about the space beyond its homotopy type. A prototypical example is the Cheeger-Simons ring of differential characters, which serves as the receptacle for the primary and secondary characteristic classes of bundles. Differential K-theory encodes information about the space, beyond its K-theory in the sense that its classes know about the Chern character at the chain level. These classes however don't know about the Wilson lines and traces of higher holonomies. In this talk, we will discuss
how one may construct a variant of differential cohomology that retains information about trace of holonomy. One of the main tools is the Bismut Chern character of a connection which is an equivariant extension of the Chern character. I will also discuss the extension of these ideas to the holonomy of superconnections. This is joint work with Scott Wilson and Thomas Tradler.

10:00 – 11:00 – Chris Kapulkin, University of Western Ontario
Simplicial localization of fibration categories

Abstract: Given a fibration category (which is a category equipped with two classes of maps: fibrations and weak equivalences, subject to some axioms), one can form its quasicategory of frames. I will show a proof that this assignment gives an explicit (i.e., not involving fibrant replacement) construction of the simplicial localization of the given category. This is joint work with Karol Szumilo.

11:00 – 11:30 – Coffee Break

11:30 – 12:30 – Dan Grady, University of Pittsburgh

### Transport

Pittsburgh International Airport (PIT) is 20.5 miles away from the University of Pittsburgh (25-45 minutes driving time depending on the time of day). Here are some options to get to campus from the airport.

• Yellow Cab (412-321-8100). Average fare from the airport to campus is approximately $50. • SuperShuttle (1-800-258-3826). Fare is$25 each way between Oakland and the airport.
• City Bus: the 28X Airport Flyer. The fare is \$3.75 each way; exact change is required. To take it to the hotel, get off at Forbes Ave at Schenley Drive and walk toward the Cathedral of Learning (big tower across the street; can't miss it).

### Sustenance

Here is a Google map with a few food and coffee options, focusing on some local landmarks (your uncle who worships Guy Fieri really wants you to go to Primanti Bros). There are many others; in particular, the stretch of Forbes between Meyran Ave and Bouquet Street is crawling with restaurants.

Organized by Hisham Sati

Thanks for the MRC for support