Workshop on Group Theory 2024

Organiser: Anupam Singh, IISER Pune, India 

Nature of the meeting: All talks will be offline held on IISER Pune campus.

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

Venue: Madhava Hall, main building maths floor

Speakers (title and abstracts of talks):

1. Krishnendu Gongopadhyay, IISER Mohali.  Title: Conjugacy classes of automorphisms of the unit ball in a complex Hilbert space.

Abstract: We consider the ball model of an infinite dimensional complex hyperbolic space, i.e. the open unit ball of a complex Hilbert space centered at the origin equipped with the Caratheodory metric. We shall discuss the conjugacy classes in the group of holomorphic automorphisms of the ball. This is based on a joint work with Rachna Aggarwal and Mukund Madhav Mishra. 

2. Saikat Panja, HRI Prayagraj.  Title: Acceptability of classical groups in non-zero characteristic.

Abstract: A group G is said to be acceptable if, for any finite group H, two element-conjugate homomorphisms are globally conjugate. After giving some examples of acceptable groups, we will see proof that any semisimple algebraic group over an algebraically closed field of non-zero characteristics is not acceptable. This talk will be based on 'https://doi.org/10.1016/j.laa.2023.03.025'.

3. Harish Kishnani, IISER Mohali. Title: Identically Distributed Words On Nilpotent Groups.

Abstract

4. Ayush Udeep, IISER Mohali. Title: Minimal Embedding of p-groups in Symmetric Groups.

Abstract

5. Gurleen Kaur, IISER Mohali. Title: Representation dimension of some finite groups.

Abstract

6. Pushpendra Singh, IIT Jodhpur. Title: Quandle rings.

Abstract: The concept of quandles was first introduced by David Joyce and Sergei Matveev independently as an invariant of knots. A quandle is a set with binary operation satisfying three conditions induced by the three Reidemeister moves of a diagram of a knot. Analogous to group rings, quandle rings are defined. We discuss the description of quandle ring decomposition for dihedral and Takasaki quandles.

7. Sushil Bhunia, BITS Pilani Hyderabad. Title: Twisted conjugacy in groups: Geometric context.

Abstract

8. Namrata Arvind, IMSc Chennai. Title: Some skew braces of size p^nq.

Abstract: A skew brace is a set X with two binary operations which interact in a certain way. In this talk I will talk about my latest work on enumeration of skew braces of size p^nq. Here p and q are distinct primes and we assume that both the additive and the multiplicative group of the skew brace have a cyclic sylow p subgroup.

9. Archita Mondal, ISI Bangalore. Title: Rational equivalence on adjoint classical groups over fields of virtual cohomological dimension 2.

Abstract

10. Shiv Prasad, IIT Goa. Title: z-classes in the Mapping class group.

Abstract: Let  G be a group. Two elements in G are said to be z-equivalent or in the same z-class if their centralizers are conjugate subgroups in G.  The notion of z-equivalence was introduced by R. Kulkarni to study dynamical types of groups. In this talk, we will discuss this notion when G is the mapping class group of a closed oriented surface of the genus at least two. We will see that the notion of z-equivalence completely determines the dynamical types of elements in the mapping class group modulo some restriction on the periodic elements. 

11. Rahul Kaushik, IISER Pune. Title: Commutators and commutator subgroups in finite groups.

      Abstract

12. Prachi Saini, IISER Pune. Title: Surjectivity of polynomial maps.

Abstract

13. Amit Kulshrestha, IISER Mohali. Title: Words in finite p-groups.

Abstract: This talk is an introduction to the work of Iniguez and Sangroniz that assists in classifying word maps on certain finite p-groups. This classification is useful in dealing with Alon Amit conjecture that concerns bounds on fibres of word maps.

14. Krishna Kaipa, IISER Pune. Title:  Classification of binary quartic forms over a finite field up to PGL_2 equivalence.

Abstract: The group PGL_2 acts on the projective space of binary quartic forms i.e. homogeneous polynomials f(X, Y) of degree 4 in two variables: The element  g(t) = (c+dt)/(a+bt) of PGL_2 sends the quartic form f(X,Y) to  f(dX-bY, aY-cX). The problem is to classify quartic forms having no repeated factors into PGL_2 orbits. Over an algebraically closed field, the orbits are parametrized by the j-invariant of the 4 roots of the quartic. We will discuss the solution of the problem over a finite field of characteristic different from 2 and 3. Time permitting, I will discuss the application of this classification to a problem in finite geometry: the classification of lines of the projective 3-space PG(3,q) with respect to the symmetry group  PGL_2(q) of the twisted cubic in PG(3,q).