# Activities in Leeds

Throughout the Programme Grant, we will organise several workshops, seminars and research activities in Leeds.

## Algebra Seminar

During term-time, we hold an Algebra Seminar at 3pm on Tuesdays, click here for information on the 2023/24 series (organised by Emine Yıldırım and Francesca Fedele).

Since June 2024, the information on the seminar (organised by Francesca Fedele and Cristina Palmer-Anghel) will appear here, see below the table for abstracts. Unless otherwise specified, Algebra seminars take place during term time on Tuesdays at 3.00pm in the MALL, School of Mathematics, University of Leeds.

You can also open the table below in a new tab by clicking here.

Please note that in the week 28/10-01/11 there will be two seminars (Tuesday 29/10 and Friday 01/11) while the week after there will be no seminar.

Abstracts

Senne Trappeniers (Vrije Universiteit Brussel): Skew braces and related Algebraic structures

Skew braces are algebraic structures that naturally arise in the study of solutions of the set-theoretic Yang--Baxter equation, which is equivalent with the braid equation. In this talk however, we will focus on its relation with other algebraic structures, namely Hopf--Galois structures and post-Lie rings. In Hopf--Galois theory, skew braces play the same role that groups play in classical Galois theory. We will describe precisely how one obtains a Hopf--Galois structure from a skew brace, building upon a fundamental result by Greither-Pareigis and later work by Childs and Byott. This is based on joint work with Lorenzo Stefanello. Post-Lie algebras appear, among others, naturally in the study of (simply transitive) affine actions of Lie groups, already hinting at a more general connection with skew braces. Indeed, results by Rump and Smoktunowicz give concrete settings where a (partial) correspondence exists between braces and pre-Lie rings. We will illustrate how skew braces are the natural group theoretic counterpart of post-Lie rings and how this perspective can be used to obtain a Lazard correspondence between the two structures.

Leandro Vendramin (Vrije Universiteit Brussel): Nichols Algebras over groups

Nichols algebras appear in several areas of mathematics, from Hopf algebras and quantum groups to Schubert calculus and conformal field theories. In this talk, I will review the main problems related to Nichols algebras and discuss some recent classification theorems.

Atabey Kaygun (Istanbul Technical University): A Homotopical Dold-Kan for Crossed-Simplicial Groups

There is an equivalence between the category of simplicial abelian groups and the category of differential graded abelian groups called the Dold-Kan equivalence. There is also a class of curious objects called Crossed-Simplicial Groups defined by a distributive law between a collection of groups indexed by the natural numbers and the simplicial category \Delta. There have been attempts to extend the Dold-Kan to crossed-simplicial setting by explicitly constructing extensions of the differential graded side. But these are few and far between. In this talk, I'll start by a new but a weakened analogue of the Dold-Kan by using certain induction and restriction functors, and by passing to the homotopy categories on both sides. Then show that this homotopical version of the equivalence extends to the crossed-simplicial setting.

## Internal Algebra Seminar & Research Discussions

On Wednesdays, typically biweekly, we have our group research discussions to be updated on what everyone is up to. Sometimes, instead of the discussions we have an extra seminar or a series of seminars for few weeks. The following is a list of these talks.

Amit Hazi (8/05/2024, 22/05/2024, 29/05/24)

Michelle Daher (1/05/24, 15/05/24): Smoothing theory

Michael Tsironis, Vrije Universiteit Amsterdam, Netherlands (22/11/23): Skein relations for punctured surfaces

Juan Pablo Maldonado, Mar del Plata, Argentina (08/02/23): Determinants arising from frieze patterns (Slides)

Emine Yıldırım (19/10/22, 26/10/22, 02/11/22): In the landscape of algebra and combinatorics: Cluster Categories, Associahedra, The Coxeter Transformation. (Seminar notes)

Francesca Fedele (09/11/22, 16/11/22, 23/11/22): An introduction to higher homological algebra. (Seminar notes)

Özgür Esentepe (30/11/22, 11/01/22, 25/01/23): Cluster tilting for Cohen-Macaulay modules (Seminar notes)

## Programme Grant Workshops

The first Workshop: Combinatorial Representation Theory in Leeds, organised by Esentepe, Faber, Fedele and Parker took place 10-14 July 2023 at the University of Leeds. See the webpage of the workshop for further information and registration.