


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 qanalogs
Abstract: For a coloopless 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 StanleyReisner ring of an associated matroid. In this talk, I will present a qanalogue of this relation where one consider the Euler characteristic of the order complex associated to a qmatroid. 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: A1A2, 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 subalgebras of the Lie group and Lie subgroups of the Lie groups, also with some basic results from algebraic topology.