Workshop on Group Theory 2023

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. Maneesh Thakur, ISI Bangaluru. Title: R-equivalence on algebraic groups.

Abstract: We will define the notion of R-equivalence on algebraic groups and give a survey on what is known, ending with some recent results. The talk would be expository in nature, accessible to PhD students.

2. Dipendra Prasad, IIT Bombay. Title: Whittaker and degenerate Whittaker models for finite groups of Lie type.

Abstract: The notion of a generic representation plays an important role in representation theory. Although there is no similarly useful concept for representations which are not generic, there is the notion of degenerate Whittaker model due to Kawanaka, Moeglin-Waldspurger, as well as Lusztig. The talk will be meant as an exposition of the concept together with some illustrative examples.

3. Arvind Ayyer, IISc. Title: Combinatorics of character factorisations.

Abstract: I will discuss joint work with my student Nishu Kumari where we prove formulas for irreducible characters of polynomial representations of the classical groups (types A, B, C and D). Time permitting, I will mention further work of Nishu in this direction.

4. Krishnendu Gongopadhyay, IISER Mohali. Title: Reversibility of affine transformations.

Abstract: In this talk, we shall discuss a joint work with Tejbir Lohan and Chandan Maity on reversibility of affine transformations. Our primary approach is based on the notion of an infinitesimal version of the classical reversibility, namely adjoint reality in the Lie algebra set-up. Using this new approach, we also revisit such classification for the general linear groups.

5. Varadharaj Ravi Srinivasan, IISER Mohali. Title: Galois theory of matrix differential equations.

Abstract: In this talk, I will first sketch the Galois theory of matrix differential equations. I will then discuss the direct and inverse problems in this context and their connections to the study of differential central simple algebras.

6. Amit Kulshrestha, IISER Mohali. Title: Chirality and gamma-chirality in groups

Abstract: The notion of chiral groups was introduced by Cocke while studying word images with respect to inversion of group elements. He showed the existence of chiral groups, thus negatively answering a question of Ashurst. In this expository talk we will discuss about chirality and the notion of gamma-chirality that appeared in a survey article of Gordeev, Kunyavskii and Plotkin.

7. Rijubrata Kundu, IISER Mohali. Title: Coprime commutators in finite simple groups and related problems. Abstract

8. Krishna Kishore, IIT Tirupati. Title: Matrix Waring Problem.

Abstract: We discuss the following statement: Let q be a prime power. For every k \geq 1 there exists a constant C_k such that for all q > C_k and for all n \geq 1 every matrix in M_n(F_q) is a sum of two k-th powers. Here F_q is a finite field with q elements.

9. Saikat Panja, HRI Prayagraj. Title: Images of Word maps on groups: Old and new.

Abstract: By "images of word maps" we mean the evaluation of words by plugging group values. This topic has its origin in the work of the Swiss mathematician Armand Borel. Since then it has been an intriguing topic of research. For example, Ore's conjecture (concerning commutator in the finite simple non-abelian group) which was posed in 1951, took over half a century to have a complete solution. In this talk, we will discuss some old results and a few new developments in this area.

10. Viji Thomas, IISER Tvm. Title: On the Exponent and the Size of Schur Multiplier of certain classes of groups.

Abstract: We will survey the literature on the exponent and size of Schur Multiplier for certain classes of finite p-groups. If time permits, we will outline some connections to maximum number of triangles in a graph.

11. Harish Kishnani, IISER Mohali. Title: Images of word maps on finite groups. Abstract

12. Sushil Bhunia, BITS Hyderabad. Title: Twisted Conjugacy Classes in Groups.

Abstract: Let φ be an automorphism of a group G. Two elements x and y of G are said to be φ-twisted conjugate if y=gxφ(g)^{-1} for some g in G. Clearly this is an equivalence relation on G, and the equivalence classes are called the φ-twisted conjugacy classes. If φ = Id, then the φ-twisted conjugacy classes are the ordinary conjugacy classes. A group G is said to have the R_{∞} -property if the number of its φ-twisted conjugacy classes is infinite for every automorphism φ of G. The problem of determining which classes of groups have the R_{∞}-property is an active area of research.

In this talk, we will consider groups that are coming from geometry. The mapping class group MCG(S) of a surface S is the group of isotopy classes of orientation-preserving homeomorphisms of S. If S is an infinite-type surface (i.e. the fundamental group π_1(S) of S is not finitely generated) then MCG(S) is called the big mapping class group of S. I will show that the big mapping class group MCG(S) possesses the R_{∞} -property if S contains a "non-displaceable" subsurface. This is joint work with Swathi Krishna.

13. Sumit Chandra Mishra, IISER Mohali. Title: Alternating groups as powers of cycle classes is symmetric groups. Abstract

14. Prashant Arote, IISER Pune. Title: Prasad’s Conjecture about dualizing involutions. Abstract

15. Uday Bhaskar Sharma, UPES Dehradun. Title: A Frobenius-Type Formula for Compact Lie Groups.

Abstract: Let G be a finite group, and alpha: G x G \rightarrow G denote the commutator map. The cardinalities of the fibres \alpha^{-1}(g) can be computed by a formula due to Frobenius, using the character values \chi(g) of irreducible characters \chi of G. We derive the generalized formula for connected compact lie groups, orthogonal groups, as well as FC groups.

16. Steven Spallone, IISER Pune. Title: Characteristic Classes of Representations of Lie Groups.

Abstract: To a complex representation of a group G is associated a sequence of cohomological invariants called Chern classes. We have proven for a large class of Lie groups, including the special linear groups, that the Chern class of an irreducible representation is a polynomial in the highest weight. This is joint work with Rohit Joshi.

17. Parul Gupta, IISER Pune. Title: Splitting of differential quaternion algebras

Abstract: A quaternion algebra is a 4-dimensional central simple algebra over a field k and is always split by a quadratic extension of k. However, a differential quaternion algebra need not be split over any algebraic extension of k. We show that the minimal transcendence degree of splitting fields of a differential quaternion algebra is at most 3. For 0 ≤ r ≤ 3, we provide examples of differential quaternion algebras whose splitting fields have transcendence degree at least r over k. This is a joint work with Y. Kaur and A. Singh.

18. Sagar Kalane, IISER Pune. Title: Free and discrete groups generated by two parabolic maps.

Abstract: In this talk, we consider a group generated by two unipotent parabolic elements of SU(2, 1) with distinct fixed points. We give several conditions that guarantee the group is discrete and free. We also give a result on the diameter of a finite R-circle in the Heisenberg group. This is joint work with Prof.John Parker

Participants:

19. Manish Mishra, IISER Pune

20. Dilpreet Kaur, IIT Jodhpur

21. Rahul Kaushik, IISER Pune

22. Parul Gupta, IISER Pune

23. Yashpreet Kaur, IISER Pune

24. Prachi Saini, IISER Pune

25. Sagar Kalane, IISER Pune

26. Manujith K Michel, IISER Mohali

27. Pushpendra Singh, IIT Jodhpur

28. Sutirtha Datta, IISER Pune

29. Saptarshi Dandapat, IISER Pune

30. Kaustabh Mondal, IISER Pune

31. Srijan Das, IISER Pune

32. Priyanka Dey, IISER Pune

33. Sakshi Suresh Manmode, IISER Pune

34. T I Darsan, IISER Pune

35. Aniketh Sivakumar, IISER Pune

36. Kunal Arora, IISER Pune

37. Divyasree C R , IISER Pune

38. Dhrubajyoti Das, IISER Pune

39. Ayon Roy, IISER Pune

40. Biswanath Samanta, IISER Pune

41. Arup Mandal, IISER Pune