Humboldt-Universität zu Berlin - Mathematisch-Naturwissenschaftliche Fakultät - Institut für Mathematik

The Tate-Hochschild complex is a complex stitched together from Hochschild homology and cohomology of an associative Frobenius algebra. It appears naturally in the study of singularities and in representation theory. There are known operations on this complex which are extensions the cup product, Gerstenhaber bracket and their duals which include the Goresky-Hingston (co)product, the existence of which is already non-trivial. There are also mixed products, which yield and m_3 multiplication which is part of an A_∞ structure with all m_i= 0 for i >4, as was shown by Rivera and Wang. Together with Rivera and Wang, we show that these operations are part of a universal family of operations obtained analogously as the operations on the Hochschild complex we previously defined. This allows us to identify a series of higher bracket operations of which the bi-bracket is dual to the m₃ operation and the tri-bracket guarantees the higher associativity. Other operations guarantee the Poisson property of the bi-bracket. We will introduce this formalism and comment on how this leads to a new type of bordification of the Chas-Sullivan string topology space.

Ran Tessler (Weizmann Institute)
Slides
New open r-spin theories

In his 92' work, Witten has defined the r-spin intersection theory. This theory have found an open

counterpart, in genus 0, in a recent joint work with Buryak and Clader. In the closed and open setting there are (internal) marked points, which are allowed to carry any twist from the set 0,1,...,r-1, and in the open setting there are also boundary markings whose twist is restricted to be r-2. My talk will describe a new construction of genus=0 open r-spin theories, which allows different collections of boundary states, as well as the relations satisfied by the resulting intersection numbers. If time permits I will also say a few words on higher genus. Based on a joint work with Yizhen Zhao.

Nitin Kumar Chidambaram (University of Edinburgh)
Slides

I will talk about the construction and properties of a cohomological field theory (without a flat unit) that parallels the famous Witten r-spin class. In particular, one can view it as the negative r analogue of the Witten r-spin class. For r=2, it was constructed by Norbury in 2017 and called the Theta class, and we generalize this construction to any r. By studying certain deformations of this class, we prove relations in the tautological ring, and in the special case of r=2 they reduce to relations involving only Kappa classes (which were recently conjectured by Norbury-Kazarian). In the second part of this talk, we will exploit the relation between cohomological field theories and the Eynard-Orantin topological recursion to prove W-algebra constraints satisfied by the descendant potential of the class. Furthermore, we conjecture that this descendant potential is the r-BGW tau function of the r-KdV hierarchy, and prove it for r=2 (thus proving a conjecture of Norbury) and r=3. This is based on joint work with Alessandro Giacchetto and Elba Garcia-Failde.

 

Arne van Antwerpen (Vrije Universiteit Brussels)
Slides

In 1992, Drinfel'd suggested the study of set-theoretic solutions of the Yang-Baxter equation. The seminal papers of Etingof, Schedler and Soloviev, and Gateva-Ivanova and Van den Bergh studied the structure group G(X,r) and structure monoid M(X,r) for the subclass of involutive non-degenerate solutions and their monoid algebras. These algebraic structures encode the combinatorial structure of the solution (X,r) and are of importance as their monoid algebra is a quadratic algebra. In recent joint works with I. Colazzo, E. Jespers, L. Kubat and C. Verwimp, we study the structure monoid for the larger class of left non-degenerate solutions. Furthermore, we obtain results on the finiteness properties of the associated quadratic algebras. 

In the second part of the talk, we discuss skew left braces. These algebraic structures generate and govern non-degenerate set-theoretic solutions and were recently introduced by W. Rump, and L. Guarnieri and L. Vendramin. Intuivitely, a skew left brace is a set with two group operations that are related via a skew left distributivity condition.

We discuss some recent works, joint with E. Jespers, L. Kubat and L. Vendramin. In particular, we discuss radicals of skew left braces. Last, to illustrate that the study of skew left braces is a melting pot of different techniques, we present a recently unexpected connection (by A. Smoktunowicz) between pre-Lie algebras and skew left braces.

Throughout the talk, we will mention open problems and avenues for further research.

Francisco Arana-Herrera (University of Maryland)
Slides

Cutting a hyperbolic surface along a simple closed multi-geodesic yields a hyperbolic structure on the complementary subsurface. We study the distribution of the shapes of these subsurfaces in moduli space as boundary lengths go to infinity, showing that they equidistribute to the Kontsevich measure on a corresponding moduli space of metric ribbon graphs. In particular, random subsurfaces look like random ribbon graphs, a law which does not depend on the initial choice of hyperbolic surface. This result strengthens Mirzakhani’s famous simple close multi-geodesic counting theorems for hyperbolic surfaces. This is joint work with Aaron Calderon.

 

Gabriele Rembado (HCM Bonn)

The standard mapping class groups can be constructed as fundamental groups of moduli spaces/stacks of genus-g Riemann surfaces: they thus encode much information about the topology of the deformations of such surfaces. This can be extended to pointed Riemann surfaces, adding the braid groups into the picture (related to the fundamental group of the configuration space of the marked points). 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 arbitrary singularities, defined on principal bundles over Riemann surfaces, 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 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.

 

Marko Berghoff (HU Berlin)

Loop-tree duality is a method for (numerical) computation of momentum space Feynman integrals. In this talk I will present a variant for Feynman integrals in the parametric representation. It is based on a geometrical decomposition of the integration locus, subject to the combinatorics of the Feynman graph under consideration. Since this procedure does not depend on the specific form of the integrand, it also applies to similar integrals, for instance integrals of Francis Brown's canonical forms on the moduli space of graphs. This space is a geometric incarnation of Kontsevich's commutative graph complex, and canonical forms can be used to study its homology. Moreover, it contains a subcomplex, called its spine, which is a classifying space for the outer automorphism group of a free group. In this setting, loop-tree duality arises as integration along the fibers of a map from the moduli space of graphs onto its spine. I will discuss how this can be used to construct forms on the spine, by pushing-forward the above mentioned canonical forms.

 

No seminar

 

Stephan Tillmann (University of Sydney/MPIM Bonn)

Let M be a complete hyperbolic 3-manifold of finite volume. The seminal work of Thurston and Culler-Shalen established the SL(2,C)-character variety of the fundamental group of M as a powerful tool in the study of the topology of M. This talk focusses on the particular class of manifolds that are hyperbolic once-punctured torus bundles. These are generally very well understood. Yet, there are some interesting open questions regarding their character varieties, especially concerning their topology and how much topological information can be obtained from them about the bundles.
This talk gives a quick overview of Culler-Shalen theory, introduces the manifolds in the title, and explains some work with Youheng Yao (arXiv:2206.14954) concerning their character varieties.

Gurbir Dhillon (MPIM Bonn)
Phantom minimal series and the Peter--Weyl theorem for loop groups
Let G be a complex reductive group. The celebrated Peter--Weyl theorem decomposes the algebra of functions on G as a G x G module with respect to left and right translations. In this talk we introduce a natural analogue for the loop group G((z)). A key role is played by a family of G((z)) representations at negative level, the phantom minimal series. These are dual, in a precise but somewhat subtle homological sense, to the more familiar positive energy representations at positive level. Time permitting, we will discuss the existence of phantom minimal series for many related vertex algebras, and some interesting analytic properties of their characters.

 

Maxim Smirnov (Augsburg Universität)

Quantum cohomology and derived categories of coadjoint varieties

We will discuss properties of quantum cohomology, both small and big, of coadjoint varieties of simple algebraic groups and how they relate to the structure of Lefschetz collections in the derived categories of these varieties.Some general conjectures pertaining to this will be formulated. The talk is based on the joint works with Alexander Kuznetsov and Nicolas Perrin.

 

no seminar