(funded by the Department of Mathematics, IISER Pune)
Worskhop on Group Theory 2019 Workshop on Group Theory 2020 Group Theory Sangam 2021 Workshop on Group Theory 2022 Workshop on Group Theory 2023 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):
Dilpreet Kaur, IIT Jodhpur
Title: Essential Dimension of small finite groups
Abstract : We discuss the Essential Dimension of finite groups and its connection with representation dimension. We use this connection to compute the essential dimension of some small finite groups.
Yashpreet Kaur, Thapar Institute of Engineering and Technology
Title: D-simple rings and their differential automorphism groups
Abstract: For a differential field (k,d) of characteristic zero, we consider derivations D on a polynomial ring R in two variables over k. If R has no invariant ideal under the map D then R is D-simple. We determine the conditions under which R is simple for certain derivations D when d is a non-zero derivation on k. For such derivations D, we also determine the subgroup of Aut(R) that commutes with D. This is a joint work with Ritika Choudhary.
Parul Gupta, Shiv Nadar University
Title: A local-global principle for quadratic forms over semi-global fields
Sushil Bhunia, BITS Pilani, Hyderabad
Title: Reality in the Lie Group of Type F4(−20)
Saikat Panja, ISI Bangalore
Title: Iteration of Word Maps and Polynomial Maps: A Dynamical Perspective
Abstract: We introduce the notion of the Fatou sets and Julia sets in the setting of word maps on groups and polynomial maps on algebras, drawing a parallel with the theory of several complex variables. The talk will primarily focus on two illustrative cases: the power map on the complex general linear group and the polynomial map induced by a single-variable polynomial, on the full matrix algebra over the complex numbers.
Amit Kulshrestha, IISER Mohali
Title: Enumerating word maps in finite groups
Abstract: This talk presents the work of Chelbus, Cocke and "Turbo" Ho under the same title, where an algorithm to calculate representative of all word maps over a finite group is developed. An interesting consequence of this is an explicit construction of a word that detects generating sets {a,b} in the alternating group A_5.
Harish Kishnani, HRI Prayagraj
Title: Waring-like problems in non-commutative contexts.
Abstract: The classical Waring's problem, resolved by Hilbert in 1909, states that every natural number can be expressed as a sum of a specific number of kth powers. A key area of modern research is exploring non-commutative versions of this problem, such as word maps on groups, polynomial maps on Lie algebras, and the matrix Waring problem. I will provide a brief overview of key results in this direction.
Sumit Chandra Mishra, IIT Indore
Title: Simple derivations of polynomial rings and isotropy groups
Abstract: Derivations of polynomial rings over fields is an important topic in the area of commutative algebra and algebraic geometry. In this talk, we will discuss simple derivations of polynomial rings over fields of characteristic zero and the associated isotropy groups. We will discuss some conjectures in this area and present results from our recent works (joint with Dr. Dibyendu Mondal and Mr. Pankaj Shukla).
Sunil Kumar Prajapat, IIT Bhubaneswar
Title: Wedderburn Decomposition of Rational Group Algebras of certain classes of finite p-groups
Abstract: In this talk, I will discuss the Wedderburn decomposition of rational group algebras of VZ p-groups and Camina p-groups, where p is a prime.
Pushpendra Singh, IIT Jodhpur
Title: Classifying simple quandles of small order
Abstract: Quandles were introduced by David Joyce and Sergei Matveev independently as invariant of knots. A quandle is an algebraic structure satisfying conditions induced by the three Reidemeister moves of a diagram of a knot. In this talk, I will talk about finite simple quandles and their classification.
Tushar Kanta Naik, NISER Bhubaneswar
Title: Automorphisms and Congruence Subgroup Property of Coxeter groups.
Abstract: Coxeter groups form a rich and diverse class of groups that arise naturally in geometry, combinatorics, and algebra. In this talk, which will be elementary in nature, we will survey some of the fundamental results concerning automorphisms of Coxeter groups and the Congruence Subgroup. Then we will discuss some of our recent contributions in this direction. If time permits, we will discuss some open problems.
Pratyusha Chattopadhyay, BITS Pilani, Hyderabad
Title: Results related to elementary symplectic groups
Abstract: In this talk we will consider an elementary symplectic group with respect to an invertible alternating matrix of Pfaffian 1. This group is a generalisation of the elementary symplectic group $ESp_{2n}(R)$. There are known results of equality of orbits of unimodular elements under the action of elementary linear group $E_{2n}(R)$ and $ESp_{2n}(R)$, normality of $ESp_{2n}(R)$ in $Sp_{2n}(R)$, and equality of $ESp_{2n}(R)$ and $Sp_{2n}(R)$ over some special rings. We will talk about generalisation of these results for an elementary symplectic group with respect to an invertible alternating matrix of Pfaffian 1.
Manujith Michel, IISER Mohali
Title: Splitting iterative derivations on central simple algebras
S. A. Katre, SPPU/ Bhaskarachrya Pratishthan
Title: Waring's problem for Matrices
Abstract: This is a survey talk on Waring's problem for matrices. We shall discuss the problem of writing matrices over commutative and noncommutative rings as sums of k-th powers of matrices. We shall also indicate some recent results on the problem of universality of a quadratic form over the matrix ring of 2x2 matrices over Z or a quadratic ring. We shall further state some results on matrices over maximal orders in rational quaternion division algebras and rational cyclic division algebras of odd prime degree as sums of squares and sums of cubes.
Preena Samuel, IIT Kanpur
Title: Making 'classical' invariant theory 'super' again!'
Abstract: Classical invariant theory in its present form owes its beginnings to the foundational work of Hermann Weyl. The conceptual framework laid out by Weyl for studying polynomial invariants under the action of groups, which he termed as `classical groups', has bridged algebra and geometry via invariant theory. Building on this framework, Procesi later looked at the conjugation action of the classical groups on the space of m-tuples of matrices. This work has since been a blueprint for much of the development in classical invariant theory. In this talk, we revisit this problem and generalize it to the setting of mixed tensor spaces. We illustrate why the invariant rings of mixed tensor spaces are particularly interesting. We then recast this problem to the setting of superspaces and illustrate how Procesi's ideas resonate even as the nature of symmetry evolves in mathematics.
Prachi Saini, IISER Pune
Title: Waring-Type Decompositions: From Scalars to Algebra-Valued Coefficients.
Ayon Roy, IISER Pune
Title: Power maps on general linear groups over finite local principal ideal rings of length two.
Participants:
Steven Spallone
Gaurav Kumar
Sagar Kalane
Charu Goel
Sutirtha Datta
Pratham Roy
Antara Manik
Debajyoti Deb
Rishabh Jain
Sayan Thokdar
Dhrubajyoti Das
Ravi Jain
Divyasree
Arup Mondal
Prashant Arote
Srijonee Chaudhury
Akshay Kharade
SubhashRanjan Choudhary
Gaurav Kumar
Biswanath Samanta
Ramandeep Singh Arora
Kaustabh Mondal
Sourav Ghosh
Antik Paul
AbirRangan Datta
M Prashant Krishnan
Nipurn Shakya
Aniket Datta
Trishartadeb Mistri