# Topology Festival

May 10–12, 2019

## Abstracts of Talks (2017)

### Factorization Homology and TQFTs

This talk will be a report on an ongoing program, joint with John Francis. I'll explain how factorization homology constructs TQFTs (in the sense of observables and partition functions) from higher categories. I'll elaborate on tantalizing, though speculative, sources of examples of higher categories that arise from algebraic stacks.

### The Kunneth Theorem for Topological Periodic Cyclic Homology

Motivated by Deninger’s proposal for a cohomological interpretation of the Hasse-Weil zeta function, Hesselholt has advocated the study of a certain cohomology theory for schemes over a finite field, the $S^1$-Tate spectrum of the topological Hochschild homology ($THH$) of the scheme. In analogy with the description of periodic cyclic homology in terms of Tate cohomology, one might think of this as topological periodic cyclic homology (or in general, since it is not necessarily periodic when the scheme is not over a finite field, a higher de Rham spectrum).

In this talk, I will explain joint work with Mandell that establishes a strong Kunneth theorem for topological periodic cyclic homology ($TP$) of smooth and proper $dg$-categories over a finite field. One application of this result is that $TP$ can be thought of as a “noncommutative Weil cohomology” theory, which has interesting consequences in the theory of noncommutative motives.

### Computer Driven Questions, Pre-theorems and Theorems in Geometry

Several numbers can be associated to free homotopy class $X$ of closed curves on a surface $S$ with boundary and negative Euler characteristic. Among these,

• the self-intersection number of $X$
• the word length of $X$
• the length of the geodesic corresponding to $X$
• the number of free homotopy classes of a given word length the mapping class group orbit of $X$.

The interrelations of these numbers exhibit many patterns when explicitly determined or approximated by running a variety of algorithms in a computer.

We will discuss how these computations lead to counterexamples to existing conjectures and to the discovery of new patterns. Some of these new patterns, so intricate and unlikely that they are certainly true (even if not proven yet), are “pre-theorems.” Many of these pre-theorems later became theorems. An example of such a theorem states that the distribution of the self-intersection of free homotopy classes of closed curves on a surface, appropriately normalized, sampling among given word length, approaches a Gaussian when the word length goes to infinity. An example of a counterexample (no pun untended!) is that there exists pairs of length equivalent free homotopy classes of curves on a surface $S$ that have different self-intersection number. (Two free homotopy classes $X$ and $Y$ are length equivalent if for every hyperbolic metric $M$ on $S$, $M(X)=M(Y)$).

### Remarks About the Interface of Topology and Physics (introductory talk)

In this survey talk I will begin by telling some general features of topology which make it applicable to problems in physics. Then I will describe some physics problems in which topology has played a large role and conversely mention some developments in topology which grew out of ideas in physics.

### Bordism and Topological Phases of Matter

Topological ideas have at various times played an important role in condensed matter physics. Last year's Nobel Prize recognized the origins of a particular application of great current interest: the classification of phases of a quantum mechanical system. Mathematically, we would like to describe them as path components of a moduli space, but that is not rigorously defined as of now. In joint work with Mike Hopkins we apply stable homotopy theory (Adams spectral sequence) to compute the group of topological phases of invertible field theories, which should be the correct answer to the original problem. We use the Axiom System for field theory initiated by Segal and Atiyah, and various refinements, to prove a theorem about reflection positivity which underlies the computations.

### The Ground State Transformation

In the beginning, quantum field theory was about harmonic oscillators, sometimes finitely many but usually infinitely many. In this lecture I will describe the quantum Hilbert state space for both circumstances. Infinite constants will be subtracted with your approval. The transition to Yang-Mills quantum fields will then be sketched, followed by a conjecture with a large reward. (PDF slides of talk)

### Configuration Spaces of Products

I will explain the construction of a new model for the configuration space of a product of two closed manifolds in terms of the configuration spaces of each factor separately. The key to the construction is the lifted Boardman-Vogt tensor product of modules over operads, developed earlier in joint work with Dwyer. (Joint work with Bill Dwyer and Ben Knudsen.)

### Caratheodory's Metrics on Teichmüller Spaces

One of the most important results in Teichmüller theory is the theorem of Royden which says that the Teichmüller and Kobayashi metrics agree on any Teichmüller spaces. In this talk, I will discuss the recent result that the Teichmüller and Caratheodory metrics disagree on Teichmüller spaces of a closed surface of genus at least two. In addition, I shall explain the role of some difficult theorems from Teichmüller dynamics in the remaining open problem of characterizing Teichmüller discs where the two metrics agree.

### Fatgraphs for String Topology

String topology studies operations on the loop space of a manifold or on the Hochshild complex of an algebra. Different spaces of fatgraphs have been used for some time to organize these operations. In this talk we describe two such spaces. One is a space of metric fatgraphs, defined in joint work with G.C. Drummond-Cole and N.Rounds to parametrize operations on the singular chains of the free loop space. Another is a space of directed fatgraphs, defined by Tradler and Zeinalian to parametrize operations on the Hochschild complex of a $V$-infinity algebra. We establish a relationship between these two spaces of fatgraphs and formulate a conjecture relating them to the moduli space of Riemann surfaces. This is joint work with T. Tradler.

### Bringing More Homotopy Theory to Symplectic Geometry

The mirror symmetry conjecture (inspired by physics) has spurred a lot of development in symplectic geometry. In the last few years, a wave of modern homotopy theory has also entered the symplectic landscape, and begun to present new questions about the structure of symplectic manifolds. In this talk, we’ll explain a basic invariant in symplectic geometry (the Fukaya category) and, as time allows, give a survey of new inroads being opened through Lagrangian cobordisms, derived geometry, and deformation theory.

### Non-Hamiltonian Circle Actions with Isolated Fixed Points

Let the circle act on a closed manifold $M$, preserving a symplectic form $\omega$. We say that the action is Hamiltonian if there exists a moment map, that is, a map $\Psi \colon M \to R$ such that $\iota_\xi \omega = - d \Psi$, where $\xi$ is the vector field that generates the action. In this case, a great deal of information about the manifold is determined by the fixed set. Therefore, it is very important to determine when symplectic actions are Hamiltonian. There has been a great deal of research on this question, but it left the following question, usually called the “McDuff conjecture”: Does there exists a non-Hamiltonian symplectic circle action with isolated fixed points on a closed, connected symplectic manifold? If so, how many fixed points? I will answer the first question by constructing such an example with 32 fixed points. (This construction relies in part on joint work with J. Watts.) I will also discuss work in progress with D. Jang, we plan to construct an example with $2n$ fixed points for any $n \geq 5$.

### On the Geometric Quantization of (Some) Poisson Manifolds

We review geometric quantization in the symplectic case, and show how the program of formal geometric quantization can be extended to certain classes of Poisson manifolds equipped with appropriate Hamiltonian group actions. These include $b$-symplectic manifolds, where the quantization turns out to be finite dimensional, as well as more singular examples ($b^k$-symplectic manifolds) where the quantization is finite dimensional for odd $k$ and infinite dimensional, with a very simple asymptotic behavior, where $k$ is even. If time permits we will also discuss some preliminary results for pseudoconvex domains. (Joint work with Victor Guillemin and Eva Miranda.)