## Online research seminar: Algebra, Geometry & Physics

Date: Tuesdays 2.00pm-3.00pm

Venue: NEW ROOM: 1.023 (BMS Room, Haus 1, ground floor), Rudower Chaussee 25, Adlershof, 12489 Berlin

Organisers: Gaëtan Borot (HU Berlin), Yuri Manin (MPIM)

Connection link: https://hu-berlin.zoom.us/j/61686623112

To be in the mailing list, please write to Kristina Schulze (schulze@math.hu-berlin.de)

For HU students in Maths, or Physics P27 or P28, this is 2SWS and you can get credits by regular attendance (>50%) and writing at least one report on a talk of your choice during the term. If you intend to do so, please contact me at the beginning of the semester.

Archive of talks Sept. 2020 - July 2021

**Wintersemester 2022/23**

**18. Okt. 2022**

Ralph Kaufmann (Purdue University)

Slides*Universal operations on the Tate-Hochschild complex*

_{∞}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 m

_{3}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.

**25. Okt. 2022**

Ran Tessler (Weizmann Institute)

Slides*New open r-spin theories*

**1. Nov. 2022**

*r-th roots: better negative than positive*

**8. Nov. 2022**

*Solutions of the Yang-Baxter equation: groups, algebras and braces*

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.

**15. Nov. 2022**

*The shapes of complementary subsurfaces to simple closed hyperbolic multi-geodesics*

**22. Nov. 2022**

*Local wild mapping class groups*

**29. Nov. 2022**

*Loop-tree duality for integrals on the moduli space of graphs*

**6. Dez. 2022**

**13. Dez. 2022**

*On the topology of character varieties of once-punctured torus bundles*

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.

**3. Jan. 2023**

*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.

**10. Jan. 2023**

*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.

**17. Jan. 2023**

**24. Jan. 2023**

*Museum of visual Asymptotic Representation Theory*

**31. Jan. 2023**

**7. Feb. 2023**

*Scattering amplitudes and hidden symmetries in supersymmetric gauge theory*

**21. Feb. 2023**

3.00-4.00 pm: Matilde Marcolli (Caltech)

Not at HU, just in MPIM Lecture Hall and Online