Talk

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.

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.

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)

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. \\
 

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.

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.

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)

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)

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)