Discrete Mathematics Seminar

Date and time: Monday, 23rd Jan 2023 at 2.30 pm

Venue: Ramanujan Hall

Host:Sudhir R Ghorpade

Speaker:  Rakhi Pratihar

Affiliation: IIIT Delhi

Title: Matroids, Euler characteristics, Möbius functions, and q-analogs

Abstract: For a co-loopless matroid M of rank r, the reduced Euler characteristic of the corresponding matroid complex S_M is determined by a Mobius function via the relation χ(S_M) = (−1)^{r−1} |μ_{L_M} (\hat{0}, \hat{1})|, where L_M is the lattice of cycles of M. The relation can be seen as a link between the poset of independent sets of M, and the geometric lattice of flats of the dual matroid M^*, which has a very interesting application to coding theory. It has been shown that the generalized Hamming weights of a linear code can be determined by the Betti numbers of the Stanley-Reisner ring of an associated matroid. In this talk, I will present a q-analogue of this relation where one consider the Euler characteristic of the order complex associated to a q-matroid. I will also briefly discuss its potential application to the theory of rank metric codes.

Lecture series on Lie groups 

Date and Time: Six Mondays at 4 pm
Tea: 3.50 pm
Venue: A1-A2, CDEEP, Mathematics Department
Host: Dipendra Prasad

Speaker: M. S. Raghunathan, CEBS, Mumbai

Title: Compact Lie groups and their representations

Abstract: In this course I will first talk about the structure theory of compact Lie groups, beginning with the fact that a compact connected Lie group is an almost direct product of the identity connected component of its centre and its commutator subgroup (which is closed subgroup) conjugacy of maximal tori and the fact that every element is contained in a maximal torus. In the course of proving these results, some results on the topology of compact Lie groups which will also be proved. I will then establish Weyl's theorem which asserts that if G is a compact connected Lie group and [G, G]=G, π_1(G,e) is finite (and hence the universal covering of a compact group whose abelianisation is trivial is  compact. 

Then I will introduce roots and weights and the Dynkin diagram of the compact group and sketch a proof of the fact that the Dynkin diagram determines the group locally. The remaining lectures will be devoted to representation theory. I will establish the bijective correspondence between 'Dominant Weights' and irreducible representations. The course will end with the Weyl Character Formula for the character of an irreducible representation corresponding to a 'dominant' weight. The entire theory is essentially the same as the representation theory of reductive algebraic groups. I will off and on indicate how the two are related.  

I will be assuming some familiarity with basic theory of Lie groups such as the correspondence between Lie sub-algebras of the Lie group and Lie subgroups of the Lie groups, also with some basic results from algebraic topology.
 

Mathematics Colloquium
Tuesday, 24th Jan 2023 at 2.30 pm
Venue: Ramanujan Hall
Host: Sudhir R Ghorpade
Speaker: Mrinmoy Datta
Affiliation: IIT Hyderabad
Title: Codes from Schubert varieties and their parameters
Abstract: The codes from Schubert varieties, a natural generalization of the codes from Grassmann
varieties, have drawn the attention of several mathematicians in the recent past. In this talk, we will
review the known results regarding some basic parameters of these codes (such as dimension and
minimum distance) and present recent developments on a conjecture related to the characterization
of minimum distance codewords of these codes by Ghorpade and Singh. We will also present recent
results on the generalized Hamming weights of the codes from Schubert varieties. This is a joint work
with Sudhir Ghorpade and Avijit Panja.

Algebraic Groups Seminar

Date and time: Tuesday, 24 January 2023, 4 pm

Venue: Room 215

Host:Shripad M. Garge

Speaker:  Arghya Pramanik

Affiliation: IIT Bombay

Title: Some algebraic geometry. III

Abstract: We continue to follow the first chapter of Tonny A. Springer's book. We discuss projective varieties in this lecture.

Combinatrorics seminar
Date: 25th Jan 2023
Time: 14:30-15:30 hrs.
Venue: Room-113, Maths Dept.

Speaker: Dr. Shivani Goel, IISc Bangalore
Host: S. Krishnan

Title: Resistance matrices of balanced directed graphs
Abstract: Attached as a pdf file

Mathematics Colloquium

Date and time: Wednesday 25th January, 2023, 4:00 pm

Venue: Ramanujan Hall

Host: Sanjeev Sabnis

Speaker:  Prof. Sujit Ghosh

Affiliation: Department of Statistics, North Carolina State University, Raleigh

Title: A Gambler's Journey through Monte Carlo

Abstract: In the era of 'data science' we often require solving large dimensional optimization and integration problems for parameter estimation and/or for predictive analytics. Even in moderately large dimensions, the gradient-based deterministic numerical optimization and deterministic grid-based numerical integration methods suffer from the so-called 'curse of dimensionality.' In such cases stochastic or so-called Monte Carlo methods can be shown to work reasonably well even when the dimensions become much larger. This presentation will provide a brief tour of a Gambler's journey starting with basic notions of Monte Carlo methods with simple geometry based examples to more contemporary data science problems on the use of Bayesian methods for variable selection problems.

Virtual commutative algebra semnar
27 January 2023, 5:30 pm
For getting the Google meet link
Please contact the organisers and register by filling a form at the site
https://sites.google.com/view/virtual-comm-algebra-seminar/home
Host: J. K. Verma
Speaker: Alessio Caminata
Affiliation: University of Genoa, Italy
Title: Determinantal varieties from point configurations on hypersurfaces
Abstract: Point configurations appear naturally in different contexts, ranging from the study
of the geometry of data sets to questions in commutative algebra and algebraic geometry
concerning determinantal varieties and invariant theory. In this talk, we bring these
perspectives together: we consider the scheme X_{r,d,n} parametrizing n ordered points in r-
dimensional projective space that lie on a common hypersurface of degree d. We show that
this scheme has a determinantal structure and, if r>1, we prove that it is irreducible, Cohen-
Macaulay, and normal. Moreover, we give an algebraic and geometric description of the
singular locus of X_{r,d,n} in terms of Castelnuovo-Mumford regularity and d-normality.

This yields a complete characterization of the singular locus of X_{2,d,n} and X_{3,2,n}.
This is joint work with Han-Bom Moon and Luca Schaffler.