Workshop on Group Theory 2022

Organisers: Anupam Singh, IISER Pune, India

Manoj Kumar, HRI, India

Amit Kulshrestha, IISER Mohali, India

Link to the talks: Zoom (Please ask one of the organisers to get the link to the talks)

Time-table: All times in IST (Indian Standard Time)

Speakers with title and abstracts of talks:

1. Sushil Bhunia (IISER Mohali)

Title: Conjugacy and Reversibility Problems in Groups Abstract

2. Kanika Singla (IISER Mohali)

Title: Splittings of Differential Central Simple Algebras Abstract

3. Parul Gupta (IISER Pune)

Title: Splitting fields of differential symbol algebras

Abstract: Let m ≥ 2 and k be a field containing a primitive m-th root of unity. In this talk, we describe derivations on symbol algebras of degree m over k. Symbol algebras are central simple algebras. Analogous to the notion of usual splitting of central simple algebras, Juan and Magid (2008) introduced a notion of splitting for differential central simple algebras and showed that a differential central simple algebra over k is split by a finitely generated field extension of k. Unlike the usual splitting, there are finite-dimensional differential central simple algebras which are not split by any algebraic extension of k. For certain derivations on symbol algebras, we provide explicit construction of differential splitting fields and give bounds on their algebraic and transcendental degrees. This is a joint work with Y. Kaur and A. Singh.

4. Andrea Caranti (Università degli Studi di Trento, Trento, Italy)

Title: Regular subgroups, skew braces, and gamma functions.

Abstract: Skew braces, a novel algebraic structure introduced only in 2015, have already spawned a sizeable literature.

The skew braces with a given additive group structure correspond to the regular subgroups of the permutational holomorph of such a group. These regular subgroups can in turn be described in terms of certain so-called gamma functions from the group to its automorphism group, which are characterised by a functional equation.

We will survey some results for which gamma functions have proved a useful addition to the available methods for studying skew braces.

5. Saikat Panja (IISER Pune)

Title: Classification of Skew Braces Corresponding to Zn ⋊ Z2 Abstract

6. Namrata Arvind (IISER Pune)

Title: On Zn ⋊ Z2-Hopf-Galois Structures Abstract

7. Tushar Kanta Naik (IISER Mohali)

Title: Breadth Type of (Nilpotent) Lie Algebras Abstract

8. Cocke, William (Augusta University, Georgia, USA)

Title: Open Questions about Word Maps 1.

Abstract: An n-variable word w is an element of the free group of rank n. Given a word w, we can define a map on a group G by substituting values of G in for the variables of w. These maps, called word maps, can be used to study G itself. For example, if the commutator word evaluates to 1 for all elements of G, then the group is abelian.

We will present some recent results and open questions related to word maps on finite groups. Some of these questions, e.g., Amit's conjecture, have been stated elsewhere, others are new. A few questions involve explicitly computational questions.

9. Anirban Bose (SRM University AP)

Title: Twisted Conjugacy in Linear Algebraic Groups Abstract

10. Dilpreet Kaur (IIT Jodhpur)

Title : Classifying z-classes of Weyl groups.

Abstract: It is well known that the Weyl group of type A_n is isomorphic to the symmetric group S_{n+1}. The Weyl group of type B_n and C_n are isomorphic to the wreath product of cyclic group C_2 and symmetric group S_n, whereas Weyl group of type D_n is an index 2 subgroup of wreath product of cyclic group C_2 and symmetric group S_n. In this talk, we classify the z-classes of these Weyl groups.

11. Harish Kishnani (IISER Mohali)

Title: Word Images on Niloptent Groups of Class 2 Abstract

12. Krishna Kishore (IIT Tirupati)

Title: Matrix Waring Problem

Abstract: The classical Waring's problem states that every natural number is a sum of 4 squares, 9 cubes, and 19 biquadrates etc. In general, one may ask the same question over arbitrary rings. In this talk we shall discuss Waring's problem over finite fields. We prove that for every positive integer k, there exists a constant C_k depending only on k such that for all q > C_k, every matrix in M_2(F_q) is a sum of two kth powers and every matrix in M_n(F_q) with n greater than or equal to 3 is a sum of three kth powers.

13. Rahul Kaushik (HRI)

Title: Commutators and commutator subgroups in finite groups Abstract

14. Sumana Hatui (IISc)

Title: Projective representations of discrete Heisenberg groups

Abstract: The study of projective representations has a long history starting with the pioneering work of Schur for finite groups (1904). It involves understanding the homomorphisms from a group into the projective general linear groups. Two essential ingredients to study the group’s projective representations are describing its Schur multiplier and representation group. We shall describe these for discrete Heisenberg groups.