Fri, February 25, 2022
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February 2022
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11:00am [11:30am] Ankit Mishra, IIT Bombay
Speaker: Ankit Mishra, IIT Bombay February 25 (Friday), 11:30-12:30. Link : Title : Hilbert series of CM local rings with $e_2=e_1-e+1$ and MCM modules over hypersurface rings.

2:00pm [2:30pm] Maria Ann Mathew, IIT Bombay
Date: February 25 (Friday), 2.30 - 3.30 pm Link: Title: Generalization of Serre Splitting to monoid algebras R[M] Abstract: In the search for an answer to his conjecture, Serre gave a splitting theorem which states that for a commutative noetherian ring R and a projective R-module P of rank r, if r > dim(R), then P admits a splitting with a free direct summand. This result, often aptly referred to as Serre splitting theorem, shrinks the class of projective R-modules one needs to study to projective R-modules of rank < dim(R) + 1. One may thus ask if a similar splitting exists for projective R[M]-modules of rank > dim(R), when M is a submonoid of Z^n. This problem will be addressed in two parts. In the first part, when rank(P) coincides with dim(R[M]), the said splitting will be demonstrated. The second part will tackle the problem when rank(P) dips even further, i.e., dim(R) < rank(P) < dim(R[M])-1. For n > 0, we define classes of monoids M_n such that if M in M_n is seminormal and rank(P) > dim(R[M]) – n, then P admits a splitting. As a consequence, it can be shown that for a projective module P over Segre extensions S_mn over R, splitting is possible when rank(P) > dim(S_mn)-[(m+n-1)/min(m,n)]. We will also discuss the possibilities of splitting under monic inversion.

4:00pm [4:00pm] Saumyajit Saha: IIT Bombay
February 25 (Friday), 4:00-5:00. Link : Title : Effects of perturbation on low energy Laplace eigenfunctions Abstract : In this talk, we will discuss the effects of perturbation on certain topological and geometrical properties of the nodal sets/vanishing sets of Laplace eigenfunctions. Our discussion will be centred around a well-known conjecture of Payne which states that: the zero set corresponding to the second Laplace eigenfunction of any bounded planar domain should intersect the boundary at exact two points. We will look into certain stability properties of the nodal sets and obtain some interesting results concerning the conjecture.

6:00pm [6:30pm] K. Sather-Wagstaff, Clemson University, South Carolina.
Speaker: K. Sather-Wagstaff, Clemson University, South Carolina. Date/Time: 25 February 2022, 6:30pm IST/ 1:00pm GMT / 8:00am ET (joining time 5:15pm IST). Gmeet link: Title: Monomial Ideals Arising from Graph Domination Problems. Abstract: Graph domination problems are ubiquitous in graph theory. In the broadest terms, they ask how one can ‘observe’ an entire graph by designating a certain list of vertices, following a proscribed list of rules. An example of this is the vertex covering problem which happens to describe the irredundant irreducible decomposition of the edge ideal of a graph. In this talk, we will survey recent work with various collaborators on other monomial ideal constructions that arise from other graph domination problems, including one coming from electrical engineering. For more information and links to previous seminars, visit the website of VCAS: