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Autumn Semester 2022

Date / Time Speaker Title Location
22 September 2022
15:15-16:15 		Marco Volpe
Universität Regensburg
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Talks in Mathematical Physics

Title Verdier duality on conically smooth stratified spaces
Speaker, Affiliation Marco Volpe, Universität Regensburg
Date, Time 22 September 2022, 15:15-16:15
Location HG G 43
Abstract Verdier duality is a key feature of derived categories of constructible sheaves on well-behaved stratified spaces. In this talk we will explain how to extend the duality theorem to constructible sheaves on conically smooth stratified spaces and with values in a general stable bicomplete infinity-category. Our proof relies on two main ingredients, one categorical and one geometric. The first one is an equivalence between sheaves and cosheaves proven by Lurie in Higher Algebra. Lurie's theorem will appear in our discussion both as a fundamental building block for the six functor formalism in a very general setting and as a factor of the duality functor on constructible sheaves. The second is the unzip construction introduced by Ayala, Francis and Tanaka, which provides a functorial resolution of singularities to smooth manifolds with corners. This will be used to prove that the exit path infinity-category of any compact conically smooth stratified space is finite.
Verdier duality on conically smooth stratified spacesread_more
HG G 43
29 September 2022
15:15-16:15 		Raschid Abedin
ETH Zürich
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Talks in Mathematical Physics

Title Algebraic geometry of the classical Yang-Baxter equation
Speaker, Affiliation Raschid Abedin, ETH Zürich
Date, Time 29 September 2022, 15:15-16:15
Location HG G 43
Abstract Solutions of the classical Yang-Baxter equation (CYBE) are important elements in the theory of integrable systems and the theory of quantum groups, notably for their connection with Lie bialgebra structures. In this talk we will present a procedure that assigns a coherent sheaf of Lie algebras on a projective curve to any non-degenerate solution of the CYBE. This approach enables the use of algebro-geometric methods in the study of the CYBE. We will explain how these methods can be used to give a new proof of the Belavin-Drinfeld trichotomy, which states that non-degenerate solutions of the CYBE are either elliptic, trigonometric, or rational.
Algebraic geometry of the classical Yang-Baxter equationread_more
HG G 43
6 October 2022
15:15-16:15 		Nino Scalbi
University of Lisbon
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Talks in Mathematical Physics

Title Parallel transports as field theories?
Speaker, Affiliation Nino Scalbi, University of Lisbon
Date, Time 6 October 2022, 15:15-16:15
Location HG G 43
Abstract In higher gauge theory, gerbes with connection are the higher structures categorifying the role of principal bundles with connection in classical gauge theory. Gerbes have been extensively studied in the past and the notion of a 2-connection on a non-abelian gerbe has been introduced via transport functors in the work of Schreiber-Waldorf. At the heart of this construction lies the familiar concept that a connection can be seen equivalently as a parallel transport system. The functorial nature of parallel transport suggests that non-abelian gerbes with connection can be realized as field theories from the geometric cobordism category introduced by Grady-Pavlov. We will focus on describing the construction of Grady-Pavlov’s geometric cobordism category, with an outlook on a possible construction of a field theory encoding parallel transport.
Parallel transports as field theories?read_more
HG G 43
27 October 2022
15:15-16:15 		Volodymyr Lyubashenko
National Academy of Science of Ukraine and University of Zurich
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Talks in Mathematical Physics

Title Categories enriched over closed symmetric multicategories
Speaker, Affiliation Volodymyr Lyubashenko, National Academy of Science of Ukraine and University of Zurich
Date, Time 27 October 2022, 15:15-16:15
Location HG G 43
Abstract We construct a machine which takes as input a locally small symmetric closed complete multicategory V. And its output is again a locally small symmetric closed complete multicategory V-Cat, the multicategory of small V-categories and multi-entry V-functors. An example of such V is provided by short spaces (vector spaces with a system of seminorms) and short maps. When the ground multicategory V is set we obtain strict 2-categories by iterating the construction of categories.
Categories enriched over closed symmetric multicategoriesread_more
HG G 43
3 November 2022
15:15-16:15 		Gabriele Rembado
Universität Bonn
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Talks in Mathematical Physics

Title Wild mapping class groups
Speaker, Affiliation Gabriele Rembado, Universität Bonn
Date, Time 3 November 2022, 15:15-16:15
Location HG G 43
Abstract The standard mapping class groups are fundamental groups of moduli spaces/stacks of pointed Riemann surfaces: they thus encode much information about the topology of the deformations of such surfaces. Recently this story has been extended to wild Riemann surfaces, which generalise pointed Riemann surface by adding local moduli at each marked point -- the irregular classes. The new parameters control the polar parts of meromorphic connections with wild/irregular singularities, defined on principal bundles, and importantly provide an intrinsic viewpoint on the `times' of isomonodromic deformations.
In this talk we will explain how to compute the fundamental groups of (universal) spaces of deformations of irregular classes, related to cabled versions of braid groups, which thus play the role of `wild' mapping class groups. This is joint work with P. Boalch, J. Douçot and M. Tamiozzo. If time allows we will sketch a relation with bundles of irregular conformal blocks in the Wess-Zumino-Witten model, in joint work with G. Felder (past) and G. Baverez (in progress).
Wild mapping class groupsread_more
HG G 43
10 November 2022
15:15-16:15 		Philippe Mathieu
Universität Zürich
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Talks in Mathematical Physics

Title Extensions of the Abelian Turaev-Viro construction and U(1) BF theory to any finite dimensional smooth oriented closed manifold
Speaker, Affiliation Philippe Mathieu, Universität Zürich
Date, Time 10 November 2022, 15:15-16:15
Location HG G 43
Abstract In 1992, V. Turaev and O. Viro defined an invariant of smooth oriented closed \(3\)-manifolds consisting of labelling the edges of a triangulation of the manifold with representations of \(\mathcal{U}_{q}\!\left(\mathfrak{sl}_{2}\!\left(\mathbb{C}\right)\right)\) (\(q\) being a root of unity), associating a (quantum) \(6j\)-symbol to each tetrahedron of the triangulation, taking the product of the \(6j\)-symbols over all the tetrahedra of the manifold, then summing over all the admissible labelling representations. It is commonly admitted that this construction is a regularization of a path integral occurring in quantum gravity, the so-called "Ponzano-Regge model", which is a kind of \(\mathrm{SU}\!\left(2\right)\) BF gauge theory. A naive question is: Is it possible to define an abelian version of this invariant? If yes, is there a relation with an abelian BF gauge theory? These questions were answered positively in 2016, and the corresponding Turaev-Viro invariant is built from \(\mathbb{Z}/k\mathbb{Z}\) labelling representations (the equivalent of \(6j\)-symbols being "modulo \(k\)" Kronecker symbols) while the associated gauge theory is a particular \(\mathrm{U}\!\left(1\right)\) BF theory (with coupling constant \(k\)). This \(\mathrm{U}\!\left(1\right)\) BF theory can be straightforwardly extended to any finite dimensional closed oriented manifold, and so can be the Turaev-Viro construction built from \(\mathbb{Z}/k\mathbb{Z}\) labelling representations. A natural question is thus: Are these extensions still related? I will answer this question during the talk.
Extensions of the Abelian Turaev-Viro construction and U(1) BF theory to any finite dimensional smooth oriented closed manifoldread_more
HG G 43
17 November 2022
15:15-16:15 		Marco De Renzi
Universität Zürich
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Talks in Mathematical Physics

Title Quantum and homological representations of mapping class groups of surfaces
Speaker, Affiliation Marco De Renzi, Universität Zürich
Date, Time 17 November 2022, 15:15-16:15
Location HG G 43
Abstract Quantum topology provides a wealth of highly organized invariants, produced by machinery that operates in very general contexts. For some of them, a more classical homological reformulation is known. This often allows us to better understand the topological content of the resulting invariants, as witnessed by Bigelow’s spectacular proof of the linearity of braid groups. For the mapping class group Mod(Σ) of a surface Σ, we will explain how to recover the family of quantum representations associated with the small quantum group of sl(2) by a classical construction, with Mod(Σ) acting on twisted homology groups of configuration spaces of Σ. This is a joint work with Jules Martel.
Quantum and homological representations of mapping class groups of surfacesread_more
HG G 43
1 December 2022
15:15-16:15 		Christian Kassel
CNRS & Université de Strasbourg
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Talks in Mathematical Physics

Title The Hilbert scheme of n points on a torus and modular forms
Speaker, Affiliation Christian Kassel, CNRS & Université de Strasbourg
Date, Time 1 December 2022, 15:15-16:15
Location HG G 43
Abstract In joint work with Christophe Reutenauer (UQAM) we explicitly computed the zeta function of the Hilbert scheme of n points on the two-dimensional torus, which amounts to the same as computing the number of ideals of codimension n of the algebra of two-variable Laurent polynomials over a finite field. On the way we found a family Pn(q) of polynomials with nice properties: they are palindromic, their coefficients are non-negative integers and their values at 1 and at roots of unity of order 2, 3, 4 and 6 can be expressed in terms of modular forms related to Dedekind's eta function.
The Hilbert scheme of n points on a torus and modular formsread_more
HG G 43

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