## Tropical geometry

## Verifying (or not) the simple loop conjecture

The simple loop conjecture places very strong restrictions on the way a closed surface can map into a 3-manifold. Even if true, however, there are difficulties in constructing an algorithm to verify that any given map of a surface satisfies the conjecture. This has to do with the observation that certain questions about regular covers of surface groups are undecidable.

## tba

## Multiple Dedekind Zeta Values are Periods of Mixed Tate Motives

Recently, the author defined multiple Dedekind zeta values associated to a

number K field and a cone C. These object are number theoretic analogue of

multiple zeta values. We prove that every multiple Dedekind zeta value over

totally real number field is a period of a mixed Tate motive. The mixed

motive is defined over K in terms of a the Deligne-Mumford compactification

of the moduli space of curves of genus zero with n marked points.

## Geometric Langlands: Comparing the views from CFT and TQFT

The goal of my talk will be to discuss the relation between two approaches to the geometric Langlands program. The first has been proposed by Beilinson and Drinfeld, using ideas and methods from conformal field theory (CFT). The second was initiated by Kapustin and Witten based on a topological version of four-dimensional maximally supersymmetric Yang--Mills theory and its reduction to a two-dimensional topological sigma model. After discussing some issues complicating a direct comparison we will formulate a proposal for a precise relation between two main ingredients in the two approaches.

## The Duistermaat-Heckman distribution for the based loop group (Livestream from PI)

The based loop group is an infinite-dimensional manifold equipped with a Hamiltonian action of a finite dimensional torus. This was studied by Atiyah and Pressley. We investigate the Duistermaat--Heckman distribution using the theory of hyperfunctions. In applications involving Hamiltonian actions on infinite-dimensional manifolds, this theory is necessary to accommodate the existence of the infinite order differential operators which aries from the isotropy representation on the tangent spaces to fixed points. (Joint work with James Mracek)

**(Livestream from Perimeter Institute)**

## Geometric TQFTs and parallel transport

I will discuss a definition of bordism categories where bordisms are equipped with general geometric structures. This is motivated by applications such as the representation of cohomology theories through field theories. Then I will discuss the case of 1-dimensional TQFTs over a manifold X. It turns out, as one would hope, that these are nothing but vector bundles with connection over X. I'll explain how the difficulties entailed in this statement and their resolution are related to the problem of representing cohomology theories. This is joint work with Matthias Ludewig and Stephan Stolz.

## An algebraic locality principle to renormalise higher zeta functions

According to the principle of locality in physics, events taking place at different locations should behave independently of each other, a feature expected to be reflected in the measurements. We propose an algebraic locality framework to keep track of the independence, where sets are equipped with a binary symmetric relation we call a locality relation on the set, this giving rise to a locality set category.

In this algebraic locality setup, we implement a multivariate regularisation, which gives rise to multivariate meromorphic functions. In this case, independence of events is reflected in the fact that the multivariate meromorphic functions involve independent sets of variables. A minimal subtraction scheme defined in terms of a projection map onto the holomorphic part then yields renormalised values.

This multivariate approach can be implemented to renormalise at poles, various higher multizeta functions such as conical zeta functions (discrete sums on convex cones) and branched zeta functions (discrete sums associated with rooted trees). This renormalisation scheme strongly relies on the fact that the maps we are renormalizing can be viewed as locality algebra morphisms. This talk is based on joint work with Pierre Clavier, Li Guo and Bin Zhang. \\

## Higher operations in supersymmetric field theory (Livestream from PI)

I will review the construction of ''higher operations'' on local and extended operators in topological field theory, and some applications of this construction in supersymmetric field theory. In particular, the higher operation on supersymmetric local operators in a 3d N=4 theory turns out to be induced by the holomorphic Poisson structure on the moduli space of the theory. This leads to a new way of establishing the non-renormalization properties of this Poisson structure, and also to a simple topological reason for the appearance of its deformation quantization when the theory is placed in Omega-background. This is an account of joint work with Christopher Beem, David Ben-Zvi, Mathew Bullimore, and Tudor Dimofte.

**(Livestream from Perimeter Institute)**

## A categorified Dold-Kan correspondence

Various recent developments, in particular in the context of topological Fukaya

categories, seem to be glimpses of an emerging theory of categorified homotopical and homological

algebra. The increasing number of meaningful examples and constructions make it desirable to develop

such a theory systematically. In this talk, we discuss a step towards this goal: a categorification

of the classical Dold--Kan correspondence.

## Bundle Gerbes, D-Branes, and Smooth Open-Closed Field Theories

Bundle gerbes are the geometric objects which describe B-fields in string theory. Their sections, in turn, are the (twisted) Chan-Paton bundles that model the K-theory charges of D-branes. While this describes the topological part of a spacetime geometry in string theory, the configuration space of strings consists of loop and path spaces. On these spaces, the same geometry takes a different form; we show that it translates to bundles of algebras and bimodules that generalise coloured, knowledgeable Frobenius algebras. From a yet different perspective, the perturbative interactions of strings are encoded in a smooth, open-closed functorial field theory on the background manifold.

In this talk, based on a collaboration with Konrad Waldorf, we will employ the 2-categorical structure of bundle gerbes to provide concrete constructions that relate the spacetime, path space, and functorial field theory perspectives on B-fields and D-branes in bosonic string theory.

## Topological Twists of Supersymmetric Factorization Algebras

The idea of topologically twisting a supersymmetric field theory was introduced in the physics literature in order to generate interesting new examples of topological field theories. The idea is very general, but systematically realising the examples it produces using mathematical models for topological quantum field theory (such as the functorial axioms of Atiyah-Segal or the theory of $E_n$-algebras) is not always possible. In this talk I’ll explain what it means to twist a supersymmetric field theory in the factorization algebra framework developed by Costello and Gwilliam, and address the question of just how topological these topologically twisted theories really are. This is based on joint work with Pavel Safronov.

## Equivariant higher Hochschild homology and topological field theories

We present a version of higher Hochschild homology for spaces equipped with principal G-bundles. As coefficients we allow $E_\infty$-algebras with G-action. For this homology theory we establish an equivariant version of excision to prove that it extends to an equivariant topological field theory with values in the $(\infty,1)$-category of co-spans in $E_\infty$-algebras. As an example we construct equivariant Dijkgraaf-Witten theories. This is joint work in progress with Lukas Woike.

## Homotopy types and geometries below Spec$(\mathbb{Z})$ (Livestream from PI)

This talk is based on joint work with Yuri Manin. The idea of a "geometry over the field with one element $\mathbb{F}_1$'' arises in connection with the study of properties of zeta functions of varieties defined over $\mathbb{Z}$. Several different versions of $\mathbb{F}_1$ geometry (geometry below Spec$(\mathbb{Z})$ have been proposed over the years (by Tits, Manin, Deninger, Kapranov--Smirnov, etc.) including the use of homotopy theoretic methods and ``brave new algebra'' of ring spectra (Toën--Vaquié). We present a version of $\mathbb{F}_1$ geometry that connects the homotopy theoretic viewpoint, using Zakharevich's approach to the construction of spectra via assembler categories, and a point of view based on the Bost--Connes quantum statistical mechanical system, and we discuss its relevance in the context of counting problems, zeta-functions and generalised scissors congruences.

**(Livestream from Perimeter Institute)**

## Moduli of connections on open varieties

This is a join work with T. Pantev. In this talk, we will discuss moduli of flat bundles on smooth algebraic varieties, with possibly irregular singularities at infinity. For this, we use the notion of ''formal boundary'', previously studied by Ben Bassat-Temkin, Efimov and Hennion-Porta-Vezzosi, as well as the moduli of flat bundles at infinity. We prove that the fibers of the restriction map to infinity are representable. We also prove that this restriction map has a canonical Lagrangian structure in the sense of shifted symplectic geometry.

## Elliptic quantum groups and their finite-dimensional representations (Livestream from PI)

I will describe joint work with Sachin Gautam where we give a definition of the category of finite-dimensional representations of an elliptic quantum group which is intrinsic, uniform for all Lie types, and valid for numerical values of the deformation and elliptic parameters. We also classify simple objects in this category in terms of elliptic Drinfeld polynomials. This classification is new even for ${\eufm sl}(2)$, as is our definition outside of type A.

**(Livestream from Perimeter Institute)**

## Vertex models and $\mathbb{E}_n$-algebras

I will explain and state a conjecture of Kontsevich, that relates vertex models from statistical mechanics to $\mathbb{E}_n$-algebras.

I will also give the main ingredients of the proof of Kontsevich's conjecture, which is a joint work in progress with Damien Lejay.

## Invertible topological field theories are SKK manifold invariants (Livestream from PI)

Topological field theories in the sense of Atiyah-Segal are symmetric monoidal functors from a bordism category to the category of complex (super) vector spaces. A field theory E of dimension d associates vector spaces to closed (d-1)-manifolds and linear maps to manifolds of dimension d. It turns out that if E is invertible, i.e., if the vector spaces associated to (d-1)-manifolds have dimension one, then the complex number E(M) that E associates to a closed d-manifold M, is an SKK manifold invariant. Here these letters stand for schneiden=cut, kleben=glue and kontrolliert=controlled, meaning that E(M) does not change when modifying the manifold by cutting and gluing along hypersurfaces in a controlled way. The main result of this joint work with Matthias Kreck and Peter Teichner is that the map described above gives a bijection between topological field theories and SKK manifold invariants.

**(Livestream from Perimeter Institute)**

## Spectral problems for the E6 Minahan-Nemeschansky theory

According to Nekrasov and Shatashvili the Coulomb vacua of four-dimensional N=2 theories of "class S'', subjected to the Omega background in two of the four dimensions, correspond to the eigenstates of a quantisation of the Hitchin integrable system. The vacua may be found as the intersection between two Lagrangian branes in the Hitchin moduli space, one of which is the space of opers (or quantum Hamiltonians) and one is defined in terms of a system of Darboux coordinates on the corresponding moduli space of flat connections. I will introduce such a system of Darboux coordinates on the moduli space of SL(3) flat connections on the three-punctured sphere through a procedure called abelianization and describe the spectral problem characterising the corresponding quantum Hitchin system. This talk is based on work to appear with Andrew Neitzke.

## Cutting and gluing branes (Livestream from PI)

I'll discuss some results and expectations about the behavior of branes in Betti geometric Langlands under cutting and gluing Riemann surfaces.

**(Livestream from Perimeter Institute)**

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