Reading Seminar
Thursday, 24 February · 2:30 – 3:45 pm
Google Meet joining info
Video call link: https://meet.google.com/auv-mwkn-ixh
Website
The speaker is Sarjick Bakshi. He will give the 4th talk on the following topics.
Title: Modular representations of Algebraic groups,
Abstract:
We will discuss a few important and classical theorems in the representation theory of reductive algebraic groups like the Borel-Weil-Bott theorem, Kempf's vanishing theorem. The main reference would be Jantzen's book ``Representation theory of Algebraic Groups'' and a note by Andersen ``Modular representation of Algebraic groups and Relations to Quantum groups''.
Speaker: Pierre-Damien Thizy (Institut Camille Jordan, Université Claude
Bernard Lyon 1, France)
Time: February 24, Thursday, 5:30 pm
Title: Large blow-up sets for Q-curvature equations
Abstract: On a bounded domain of the Euclidean space $\mathbb{R}^{2m}$,
m>1, Adimurthi, Robert and Struwe pointed out that, even assuming a volume
bound $\int e^{2mu} dx\le C$, some blow-up solutions for prescribed
Q-curvature equations $(-\Delta)^m u= Q e^{2m u}$ without boundary
conditions may blow-up not only at points, but also on the zero set of
some nonpositive nontrivial polyharmonic function. This is in striking
contrast with the two dimensional case (m=1). During this talk, we will
discuss the construction of such solutions which involve (possible
generalizations of) the Walsh-Lebesgue theorem and some issues about
elliptic problems with measure data.
Google Meet joining info
Video call link: https://meet.google.com/efe-orej-zet
Or dial: (US) +1 651-560-3980 PIN: 954 349 109#
Time:
11:30am-12:30pm
Description:
Speaker: Ankit Mishra, IIT Bombay
February 25 (Friday), 11:30-12:30.
Link : https://meet.google.com/jvr-izyy-ngd?authuser=0
Title : Hilbert series of CM local rings with
$e_2=e_1-e+1$ and MCM modules over hypersurface rings.
Time:
2:30pm-3:30pm
Description:
Date: February 25 (Friday), 2.30 - 3.30 pm
Link: meet.google.com/pbb-odky-xvs
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.
Time:
4:00pm-5:00pm
Description:
February 25 (Friday), 4:00-5:00.
Link : https://meet.google.com/jvr-izyy-ngd?authuser=0
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.
Time:
6:30pm
Description:
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: meet.google.com/rco-ewra-xmh
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: https://sites.google.com/view/virtual-comm-algebra-seminar