Grouptheory

Workshop on Group Theory

22-23 February 2019 at IISER Pune

Organiser: Dr Anupam Singh

Funded by: IISER Pune

This workshop is funded by IISER Pune and will focus on certain aspects of group theory, such as, conjugacy problem, word maps, generation etc. The lectures will be expository in nature thus accessible to graduate students.

If you are interested in attending this workshop, please send an email to "anupam@iiserpune.ac.in".

Time Table:

Invited Speakers:

    1. B. Sury, ISI Bangaluru Title: Finite and infinite quotients of the modular and Hurwitz groups. Abstract: The modular group PSL(2,Z) - also known as the (2,3)-group and, the Hurwitz group - also known as the (2,3,7) group - have rich structures and admit many interesting quotient groups. These groups hold some wonderful surprises and the methods employed in proving that certain groups are/are not quotients are quite varied. This will be a survey of what is known and what is not; also, some of the methods involved will be described.

    2. Silvio Dolfi, University of Firenze Titile: On the character degree graphs of finite groups. Abstract

    3. Manoj Kumar, HRI Allahabad Title: Left Braces. Abstract: An abelian group (A, +) is said to be a left brace if the set A also admits a group structure under another binary operation '.' such that a.(b+c) + a = a.b + a.c for all a, b, c in A. The notion of braces came into existence a decade ago in the context of set theoretic solutions of the Yang Baxter equation in quantum group theory. It is planned to survey some developments, mainly focussing on group theoretical aspects, in this new exciting area, discuss certain methods used and project some directions for future research.

    4. Shripad Garge, IIT Mumbai Title: Generators for simple groups. Abstract: This is an exposition of a paper by R. Steinberg of the same title where he proves that if G is a finite simple group (of Lie type) then G is generated by 2 elements.

    5. Amit Kulshrestha, IISER Mohali Title : Word maps on groups. Abstract: Over last two decades, there has been considerable interest in word maps on groups. Works of Larsen, Shalev, O'Brien, Tiep, Kunyavskii, Segal and Nikolov have been quite significant in the area. Tools have been widely derived from algebraic group theory, representation theory, probability and geometry. The area is still a source of variety of open problems. In this talk, I will present a short survey on some aspects of the theory of word maps, with a special emphasis on the size of images of such maps, and will highlight some open problems in the area.

    6. Siddhartha Sarkar, IISER Bhopal Title: Symmetries of surfaces of p-groups of co-class 1. Abstract: The genus spectrum of a finite group G is a set of integers g \geq 2 such that G acts on a closed orientable compact surface of genus g preserving the orientation. In this talk we show that for groups G of order beyond p^{p+1} the number ofsuch spectrums is at most five with three generic and two non-generic cases. The three generic cases resemble the similarity of the problem to Dihedral, Quaternion and Semi-dihedral type of groups (i.e. while the prime is 2). This also open up the hope that the number of such spectrums in general must be a bounded function of the co-class alone.

    7. Varadharaj R. Srinivasan, IISER Mohali Title: Central Simple Algebras with Derivations. Abstract: In this talk, I will outline the construction, originally due to Hochschild(1954), of derivations on Central Simple Algebras over differential fields. I will also detail the recent work by Magid-Juan (2007) connecting certain cohomologies with differential central simple algebras. If time permits, I will briefly discuss splitting of differential quaternion algebras, which is a joint work with Amit Kulshrestha.

    8. Sunil Prajapati, IIT Bhubaneswar Title: Simultaneous Conjugacy Classes as Combinatorial Invariants of Finite Groups. Abstract

    9. Supriya Pisolkar, IISER Pune Title: Hilbert theorem-90 for a ring of Witt vectors with p-adic integer coefficients. Abstract: Let L/K be a finite cyclic extension of degree-n and G be it's Galois group. Then Hilbert theorem-90 says that H^1(G, L) = 0. In this talk we will prove an analogue of this result for a ring of Witt vectors of p-adic integers i.e. H^1(G(L/K), W(\sO_L)) = 0 for an extension L/K of p-adic fields. We will also see that this result holds in both mixed characteristic and equi-characteristic case of local fields.

  1. Amritanshu Prasad, IMSc Chennai Title: q-rious positivities in orbit problems. Abstract: We survey counting problems over matrix rings and p-groups where polynomials with non-negative coefficients appear unexpectedly.

Participants:

  1. Uday Bhaskar Sharma, IISER Pune

  2. Dilpreet Kaur, IISER Pune

  3. Saikat Panja, IISER Pune

  4. Rijubrata Kundu, IISER Pune

  5. Bhargavi Parthasarathy, IISER Pune

  6. Yash Arora, IISER Pune

  7. Shubham Jaiswal, IISER Pune

  8. Sunil Choudhary, IISER Pune

  9. Ganesh Kadu, SPPU Pune

  10. Jyotirmoy Ganguly, IISER Pune

  11. Anirban Bose, IISER Mohali

  12. Tushar Kanta Naik, IISER Mohali

  13. Rahul Kitture, IISER Mohali