Algebraic Geometry seminar

Date and Time: Thursday, 12 Jan 2023, 11.30-12.45

Venue: Ramanujan Hall

Host: Sudarshan Gurjar

Speaker: Atharva Korde

Affiliation: University of British Columbia, Vancouver, Canada

Title: Gromov Witten invariants and Donaldson Thomas invariants-III

Abstract: I will give an introduction to Gromov-Witten invariants (via algebraic geometry, rather than symplectic geometry) and Donaldson-Thomas invariants. Then I will explain the GW-DT correspondence, which relates these two sets of numbers for a Calabi-Yau threefold. If time permits, I will talk about quantum cohomology later, an idea of Kontsevich which is used to answer the question of counting the following GW-invariant – how many rational curves of degree d pass through 3d-1 general points in the plane? The first talk will be less technical and accessible to anyway with minimal knowledge of algebraic geometry. The second talk will be more technical.

Seminar on Coding Theory

Date and time:: Thursday, 12th January 2023 at 2.30 pm 
Venue: Ramanujan Hall

Speaker: Mahir Bilen Can
Affiliation: Tulane University, New Orleans, USA
Host: Sudhir Ghorpade

Title: Dual Higher Grassman Codes

Abstract: The Grassmann variety of k-dimensional subspaces of an n-dimensional vector space over a finite field with q-elements can be thought of as the ``moduli space'' of all linear q-ary (n,k)-codes. At the same time, each Grassmann variety naturally provides an algebraic geometry code via its Plucker embedding. The structure of Grassmann codes has been parsed by many researchers, most notably by Ghorpade. In this talk, we will discuss a fruitful generalization of the Grassmann codes by using the embeddings of Grassmannians into higher dimensional projective spaces. This new family of ``higher Grassmann codes'' has interesting connections with representation theory of SL_n over finite fields.
 

Commutative algebra seminar

Date and time: Thursday, 12 January 2023, 4 pm
Venue: Ramanujan Hall
Host: Manoj Keshari

Speaker: Soumi Tikedar, Diamond Harbour Women's University

Title: On a question of Moshe Roitman and its applications 
Abstract: Let A be a ring of dimension d and P be a projective A[T]-module of rank n. We say that p ∈ P is a unimodular element if there exists a homomorphism f in P*  such that f(p) = 1. When n > d, then Plumstead proved that P has a unimodular element. But this is not the case for n=d and n< d.

 In this talk, we will discuss the following results:

Theorem:  Let A be a ring of dimension d containing an infinite field k,  P be a projective A[T]-module of rank n such that  2n is not less than d + 3 and singular locus of Spec(A) is a closed set V(J) with ht J  is atleast d − n + 2. If P_f has a unimodular element for some monic polynomial f(T). Then P has a unimodular element.

Next, we will discuss some applications of Roitman's question to define the Euler class group, which serves as an obstruction group to detect the existence of unimodular elements in the Projective module with certain conditions. In this talk, we associate a stably free module to the Euler class group and show that the vanishing of this is the precise obstruction having P unimodular element.