# Cohomology Theories, Categories, and Applications March 25, 2017 - 9:00am - March 26, 2017 - 5:00pm

This workshop is on the interactions of topology and geometry, motivated by mathematical physics. The main focus will be cohomology theories with their various flavors, the use of higher structures via categories, and applications to geometry.

**Organizer:** *Hisham Sati.*

**Location: **704 Thackeray

**Speakers and schedule:**

*1. SATURDAY, MARCH 25, 2017*

10:00 am** - Ralph Cohen**, Stanford University

11:15 am - **Daniel Grady**, New York University

12:15 - Lunch

2:00 pm - **Daniel Berwick-Evans**, University of Illinois

3:15 pm - **Chris Kapulkin**, University of Western Ontario

2. **SUNDAY, MARCH 26, 2017**** **

9:00 am - **Vitaly Lorman**, John Hopkins University

10:15 am - **Byungdo Park**, CUNY City, University of New York

11:30 am - **Matt Wheeler**, University of Arizona

**ABSTRACTS:**

**1. Ralph Cohen**

*Title: Calabi-Yau categories, the string topology of a manifold, and the Floer theory of its cotangent bundle*

*Abstract:* In this talk I will discuss joint work with S. Ganatra in which we prove an equivalence between two chain complex valued 2D topological field theories: the String Topology of a manifold M, and the symplectic Floer theory of its cotangent bundle, T^*M. This generalizes results of Viterbo, Abbondandolo and Schwarz, Abouzaid, and others. We use recent work of Kontsevich, Soibelman, and Vlassopolous which describes two notions of duality among A-infinity-algebras and categories over a field k, and show how they give rise to field theories. These are referred to as Calabi-Yau" structures. We prove that the string topology category defined by Blumberg, Cohen, and Teleman and the wrapped Fukaya category of the cotangent bundle, dened by Fukaya, Seidel and Smith both possess Calabi-Yau structures in natural, geometric ways, and we show that these categories are equivalent by an equivalence that preserves these Calabi-Yau structures. Finally we show how Koszul duality affects Calabi-Yau structures, and how it implies a duality relationship between field theories.

**2. Daniel Grady**

*Title: Parametrized geometric cobordism and smooth Thom stacks*

*Abstract:* I report on ongoing work with Hisham Sati in which we propose a theory of parametrized geometric cobordism by introducing smooth, motivic style analogues of Thom spaces via stacks. This requires identifying and constructing a smooth representative of the Thom functor, acting on vector bundles equipped with extra geometric data, and leads us to an abstract version of the Pontrjagin-Thom construction in stacks. We explain how this theory can be used to generalize the theorem of Galatius-Madsen-Tillman-Weiss on the Homotopy type of the cobordsim category. The resulting theory is also versatile enough to allow for the inclusion of families of various geometric data, such as metrics and connections, thus including aspects of the geometric cobordisms of Cohen-Kitchloo-Galatius and Ayala. We also provide a generalization to include higher tangential structures, including variations on String, Fivebrane etc. structures.

**3. Daniel Berwick-Evans**

*Title: Hamiltonian and Lagrangian perspectives on elliptic cohomology*

*Abstract:* The Hamiltonian and Lagrangian formalisms offer two complementary approaches to quantum field theory. When applied to the supersymmetric sigma model, ingredients of these perspectives can be used to construct two versions of elliptic cohomology: elliptic cohomology at the Tate curve over the integers and the universal elliptic cohomology theory over the complex numbers, respectively. Quantization procedures give analytic constructions of wrong-way maps in these cohomology theories, using similar techniques to the physics proof of the Atiyah-Singer index theorem. After reviewing the ideas from physics, I will compare the wrong-way maps in elliptic cohomology with the Ando-Hopkins-Strickland-Rezk string orientation of topological modular forms.

**4. Chris Kapulkin**

*Title: Elementary theory of higher toposes*

*Abstract:* This talk will be an introduction to higher topos theory, taking a somewhat different approach to the topic. In particular, I will try to argue that the arguments using methods of higher topos theory have been used since the very beginning of algebraic topology.

**5. Vitaly Lorman**

*Title: Real Johnson-Wilson theories: computations toward applications*

*Abstract:* Complex cobordism and its relatives, the Johnson-Wilson theories, E(n), carry an action of C_2 by complex conjugation. Taking fixed points of the latter yields Real Johnson-Wilson theories, ER(n). They are generalizations of real K-theory and are similarly amenable to computations. We will describe their properties, survey recent work on the ER(n)-cohomology of some well-known spaces, and describe how this brings new information to bear on the immersion problem for real projective spaces. This is joint work with Nitu Kitchloo and W. Stephen Wilson.

**6. Byungdo Park**

*Title: A geometric model of twisted differential K-theory*

*Abstract:* A twisted vector bundle is a weaker notion of an ordinary vector bundle whose cocycle condition is o by a U(1)-valued Cech 2-cocycle (the cycle data of a U(1)-gerbe) called a topological twist. We will introduce a geometric model of a differential extension of twisted complex K-theory using twisted vector bundles with connection as cycles and U(1)-gerbes with connection as differential twists. Here a U(1)-gerbe with connection is a total degree 2 cocycle in the Cech-de Rham double complex. We will give an introduction to the Chern-Weil theory of twisted vector bundles, define a twisted differential K-theory, and introduce a hexagon diagram of twisted differential K-theory.

**7. Matt Wheeler**

*Title: Rational Structures related to O(n)*

*Abstract: *The Whitehead tower corresponding to O(n) gives rise to a sequence of increasingly connected groups. From a homotopy viewpoint, one proceeds up this tower by killing the lowest non-zero homotopy group. However in doing this, it becomes increasingly difficult to study these groups. I will discuss research which looks at the rationalization of these groups as spaces. This approach has many benets. For one the corresponding tower in the homotopy category gives a sequence of abelian groups. One can classify bundles admitting these groups as structure groups. I will show that, unlike the integral case, these bundles can be classified as cohomology classes on a Spin bundle. I will also discuss some of the homotopy types of the corresponding gauge groups.

*Supported by the Mathematics Research Center, University of Pittsburgh.*

#### Location Information

**Location: **704 Thackeray