Colloquium:
Speaker: Prof. Bart De Bruyn (Ghent University, Belgium)
Title: Divisible design graphs from symplectic graphs
Time, day and date: 4:00:00 PM, Wednesday, January 29
Venue: Ramanujan Hall

Abstract:

Colloquium:
This talk is based on joint work with Sergey Goryainov, Willem Haemers and Leonid Shalaginov. A regular graph of degree k on v vertices is called a divisible design graph with parameters (v, k, λ1, λ2,m, n) if its vertex-set can be partitioned into m classes of size n such that any two vertices from the same class have λ1 common neighbours and any two vertices from different classes have λ2 common neighbours. Divisible design graphs were introduced because of their connection with divisible designs: the adjacency matrix of any such graph is an incidence matrix of such a design. In the talk, new families of divisible design graphs are constructed that are related to the symplectic graphs Sp(2e, q), e ≥ 2. Starting from a 2e dimensional vector space V over the finite field Fq that is endowed with a nondegenerate alternating bilinear form b(·, ·), the vertices of Sp(2e, q) are the one-dimensional subspaces
of V , where two distinct one-dimensional subspaces ⟨v1⟩ and ⟨v2⟩ are adjacent whenever b(v1, v2) = 0. We define and discuss a family of divisible design graphs based on a partition of Sp(4,q), q odd, in subgraphs isomorphic to K_{q+1,q+1}, and show that there is an example in this family for every odd prime power q. We have classified by computer all examples in this family for q ∈ {3, 5, 7} and we discuss the computational challenges that we faced during this process. The divisible design graphs in this family also give rise to additional examples of divisible design graphs. Finally, we also describe some families of divisible design graphs based on so-called symplectic spreads of Sp(2e, q).

Speaker: Sabyasachi Dhar, IIT Kanpur

Date & Time: Thursday, 30 January 2025, 5 pm

Meet Link: meet.google.com/wqq-hjhc-fqc

Title: Tate cohomology and base change for generic representations of GL(n)

Abstract: Principle of functoriality is a central problem in Langlands
program, proposed by R. P. Langlands. D. Treumann and A. Venkatesh in their
foundational work established a functoriality lifting of mod-l Hecke
eigenclasses of H to the mod-l Hecke eigenclasses of G, where G is a
connected reductive algebraic group defined over a number field, and H is
the connected component of the fixed points of G under an automorphism T of
G of prime order l. They made some conjectures for representation theory of
p-adic groups which predict that the functoriality for local
representations is realized via Tate cohomology under the action of the
cyclic group generated by T. In this talk, we discuss the conjecture in the
context of local cyclic base change for generic representations of the
general linear group GL(n,F), where F is a finite extension of the field of
p-adic numbers.