Workshop on Group Theory and Around

14-20 December 2023

HRI Prayagraj, India

Organiser: Professor Manoj Kumar Yadav

 We are glad to inform you that a group theory activity "Workshop on Group Theory and Around" is planned to take place in HRI Prayagraj during Dec 14-20, 2023. We'll have two courses, one each on combinatorial group theory and representation theory of groups. Details on the courses and speakers are given below. 


Details of the courses


A. Combinatorial Group Theory


Title: Basics of Combinatorial Group Theory and automorphisms of Coxeter groups

Contents: Word combinatorics; Basic techniques, Free groups; Presentation of a group; Nielsen transformations; Free product of groups; Ping-pong lemma; Tietze transformations; Automorphisms of Coxeter groups.


Title: A quick introduction to Bass-Serre theory and further topics

Contents: Amalgamated free products; HNN extensions; Group actions on trees; A survey on group actions on nonpositively curved complexes.


Title: Relation modules and related things

Contents: Generators and relators of groups; Relation modules; Relations among relations; Relation gap problem;  Second homotopy module of group presentation; Group homology and introduction to fr-language.


Title: Hypergeometric groups and their thinness

Abstract: A hypergeometric group is a subgroup of GL(n, C) generated by the companion matrices of two monic coprime polynomials of degree n. It arises as the monodromy group of a hypergeometric differential equation, and if the defining polynomials are also self-reciprocal and form a primitive pair, then its Zariski closure inside GL(n, C) is either a symplectic or an orthogonal group. In this talk, we will discuss the arithmeticity, and thinness (using the ping-pong lemma) of the hypergeometric groups whose defining polynomials also have integer coefficients. 



B. Representation Theory


Title: Review of representations of finite groups

Contents: A quick revision of  the standard material on finite group (ordinary) representation theory; Basics of character theory; A survey on representation dimension of finite groups. 


Title: Introduction to finite Coxeter groups and their representations

Contents:  Finite Coxeter groups (examples: the dihedral groups, the symmetries of regular polyhedra and the symmetric groups); Basic definitions; Description of various properties;  Classification of all irreducible Coxeter systems; Available methods to describe complex irreducible representations of Coxeter groups.


Title: Gelfand-Graev representations

Abstract: Let G be a finite reductive group defined over a finite field F_q, take GL_n(F_q) for example, and let U be a Sylow p-subgroup of G where p|q. If \psi is a non-degenerate character of U, then its induced representation to G is multiplicity-free. We will learn all the terms used in this abstract in these lectures and try to sketch a proof of the main result. 


Title: A story concerning the birth of group representations.

Abstract: This lecture aims to recount the story related to the birth of the representation theory of finite groups. We shall discuss how Dedekind proposed to Frobenius the problem of factoring a certain homogeneous polynomial arising from a determinant (called the "group determinant") associated with a finite group G. In the latter part of the lecture, we will discuss the contributions of William Burnside, and how he enters the scene.


Other Resource persons


How to apply


In case you wish to participate, please write back to grouptheory21@gmail.com with a copy to myadav@hri.res.in with the following data:


Name:

Designation (with seniority if you are a PhD student):

Affiliation:

Gender: 

Whether you need travel support:


Please note that we have limited funds to support travel. So, if you have your own funds, use the same so that needy participants can be helped. 

The last date for receiving participation requests is Oct 08, 2023. 

Confirmation of participation will be sent by Oct 13, 2023.


Selected Participants


List 1: Local Participants 


List 2: Post-docs and Faculty


List 3: PhD and IPhD Students


List 4: Included (against non-confirmation of the above listed)


We got an overwhelming response for this activity. Thank you for showing your interest.  We will release another waitlist by 20 October if we have further available space.  Note that emails are being sent to the above-listed participants.