
Bharathwaj Palvannan
(National Center for Theoretical Sciences, Taiwan) 
Aug 28, 2020 
Codimension two
cycles in Iwasawa theory 
(3
pm, Fri) 

Abstract.
In classical Iwasawa theory, one studies a relationship called the
Iwasawa main conjecture, between an analytic object (the padic
Lfunction) and an algebraic object (the Selmer group). This relationship
involves codimension one cycles of an Iwasawa algebra. The topic of
higher codimension Iwasawa theory seeks to generalize this relationship.
We will describe a result in this topic using codimension two cycles,
involving an elliptic curve with supersingular reduction. This is joint
work with Antonio Lei.

Sandeep Varma (TIFR, Mumbai) 
Sep 4, 2020 
Some Bernstein
projectors for $SL_2$ 
(3
pm, Fri) 

Abstract.
Let $G$ be the group $SL_2$ over a finite extension $F$ of
$\mathbb{Q}_p$, $p$ odd. I will discuss certain distributions on $G(F)$,
belonging to what is called its Bernstein center (I will explain what
this and many other terms in this abstract mean), supported in a certain
explicit subset of $G(F)$ arising from the work of A. Moy and G. Prasad.
The assertion is that these distributions form a subring of the Bernstein
center, and that convolution with these distributions has very agreeable
properties with respect to orbital integrals. These are 'depth $r$
versions' of results proved for general reductive groups by J.F. Dat, R.
Bezrukavnikov, A. Braverman and D. Kazhdan.

Charanya Ravi (Universität Regensburg,
Germany) 
Sep 8, 2020 
Algebraic
Ktheory of varieties with group actions 
(unusual day: 3
pm, Tue) 

Abstract.
Cohomology theories are one of the most important algebraic invariants of
topological spaces and this has inspired the definition of several
different cohomology theories in algebraic geometry. In this talk, we
focus on algebraic Ktheory, which is one such classical cohomological
invariant of algebraic varieties. After motivating and introducing this
notion, we discuss several fundamental properties of algebraic Ktheory
of varieties with algebraic group actions. Wellknown examples of
varieties with group actions include toric varieties and flag varieties.

Apoorva Khare (IISc Mathematics) 
Sep 18, 2020 
Totally
positive matrices, Pólya frequency sequences, and Schur
polynomials
(Joint with the APRG Seminar) 
(3
pm, Fri) 

Abstract.
I will discuss totally positive/nonnegative matrices and kernels,
including Polya frequency (PF) functions and sequences. This includes
examples, history, and basic results on total positivity, variation
diminution, sign nonreversal, and generating functions of PF sequences
(with some proofs). I will end with applications of total positivity to
old and new phenomena involving Schur polynomials.

Ignazio Longhi (IISc
Mathematics) 
Sep 25, 2020 
Densities on
Dedekind domains, completions and Haar measure 
(3
pm, Fri) 

Abstract.
A traditional way of assessing the size of a subset X of the integers is
to use some version of density. An alternative approach, independently
rediscovered by many authors, is to look at the closure of X in the
profinite completion of the integers. This for example gives a quick,
intuitive solution to questions like: what is the probability that an
integer is squarefree? Moreover, in many cases, one finds that the
density of X can be recovered as the Haar measure of the closure of X. I
will discuss some things that one can learn from this approach in the
more general setting of rings of integers in global fields. This is joint
work with Luca Demangos.

Manish Mishra (IISER Pune) 
Oct 9, 2020 
Regular
Bernstein blocks 
(3
pm, Fri) 

Abstract.
Let $G$ be a connected reductive group defined over a nonarchimedean
local field $F$. The category $R(G)$ of smooth representations of G(F)
has a decomposition into a product of indecomposable subcategories called
Bernstein blocks and to each block is associated a nonnegative real
number called Moy–Prasad depth. We will begin with recalling all
this basic theory. Then we will focus the discussion on 'regular' blocks.
These are 'most' Bernstein blocks when the residue characteristic of $F$
is suitably large. We will then talk about an approach of studying blocks
in $R(G)$ by studying a suitably related depthzero block of certain
other groups. In that context, I will explain some results from a joint
work with Jeffrey Adler. One of them being that the Bernstein center
(i.e., the center of a Bernstein block) of a regular block is isomorphic
to the Bernstein center of a depthzero regular block of some explicitly
describable another group. I will give some applications of such results.

R. Venkatesh (IISc Mathematics) 
Oct 16, 2020 
Fusion product
decomposition of $\mathfrak{g}$stable affine Demazure
modules 
(3
pm, Fri) 

Abstract.
The affine Demazure modules are the Demazure modules that occur in a
level $\ell$ irreducible integrable representation of an affine Lie
algebra. We call them $\mathfrak{g}$stable if they are stable under the
action of the standard maximal parabolic subalgebra of the affine Lie
algebra. We prove that such a $\mathfrak{g}$stable affine Demazure
module is isomorphic to the fusion (tensor) product of smaller
$\mathfrak{g}$stable affine Demazure modules, thus completing the main
theorems of Chari et al. (J. Algebra, 2016) and Kus et al. (Represent.
Theory, 2016). We obtain a new combinatorial proof for the key fact that
was used in Chari et al. (op cit.), to prove the decomposition of
$\mathfrak{g}$stable affine Demazure modules. Our proof for this key
fact is uniform, avoids the casebycase analysis, and works for all
finitedimensional simple Lie algebras.

Deniz Kus (RuhrUniversity Bochum, Germany) 
Oct 23, 2020 
Polytopes,
truncations of representations and their characters 
(3
pm, Fri) 

Abstract.
Generators and relations of graded limits of certain finitedimensional
irreducible representations of quantum affine algebras have been
determined in recent years. For example, the representations in the
HernandezLeclerc category corresponding to cluster variables appear to
be certain truncations of representations for current algebras and tensor
products are related to the notion of fusion products. In this talk, we
will discuss some known results on this topic and study the characters of
arbitrary truncated representations.

Michael J. Schlosser (University of Vienna) 
Nov 6, 2020 
A
weightdependent inversion statistic and Catalan numbers 
(3
pm, Fri) 

Abstract.
We introduce a weightdependent extension of the inversion statistic,
a classical Mahonian statistic on permutations.
This immediately gives us a new weightdependent extension of $n!$.
By restricting to $312$avoiding permutations our extension happens
to coincide with the weighted Catalan numbers that were considered
by Flajolet in his combinatorial study of continued fractions.
We show that for a specific choice of weights the weighted
Catalan numbers factorize into a closed form, hereby yielding a new
$q$analogue of the Catalan numbers, different from
those considered by MacMahon, by Carlitz, or by Andrews.
We further refine the weighted Catalan numbers by introducing
an additional statistic, namely a weightdependent extension of
Haglund's bounce statistic, and obtain a new family of biweighted
Catalan numbers that generalize Garsia and Haiman's $q,t$Catalan
numbers and appear to satisfy remarkable properties.
This is joint work with Shishuo Fu.

Shifra Reif (BarIlan University, Israel)  Nov 13, 2020 
Denominator
identities for the periplectic Lie superalgebra
p(n) 
(3
pm, Fri) 

Abstract.
We present the denominator identities for the periplectic Lie
superalgebras and discuss their relations to representations of
$\mathbf{p}(n)$ and $\mathbf{gl}(n)$.
Joint work with Crystal Hoyt and Mee Seong Im.

Sanoli Gun (IMSc, Chennai)  Nov 20, 2020 
Distinguishing
newforms by their Hecke eigenvalues 
(3
pm, Fri) 

Abstract.
In this talk, we will discuss some of the existing techniques for
distinguishing newforms. We will also report on a recent joint work with
Kumar Murty and Biplab Paul.

Arvind Ayyer (IISc Mathematics)  Dec 4, 2020 
Toppleable
permutations, excedances and acyclic orientations 
(3
pm, Fri) 

Abstract.
Recall that an excedance of a permutation $\pi$ is any position
$i$ such that $\pi_i > i$. Inspired by the work of Hopkins, McConville
and Propp (arXiv:1612.06816) on sorting using toppling, we say that a
permutation is toppleable if it gets sorted by a certain sequence of
toppling moves. For the most part of the talk, we will explain the main
ideas in showing that the number of toppleable permutations on $n$
letters is the same as those for which excedances happen exactly at
$\{1,\dots, \lfloor (n1)/2 \rfloor \}$. Time permitting, we will give
some ideas showing that this is also the number of acyclic orientations
with unique sink of the complete bipartite graph $K_{\lceil n/2 \rceil,
\lfloor n/2 \rfloor + 1}$.
This is joint work with P. Tetali (GATech) and D. Hathcock (CMU), and is
available at arXiv:2010.11236

Ravindranathan Thangadurai (HarishChandra
Research Institute, Allahabad)  Dec 7, 2020 
On the
simultaneous approximation of algebraic numbers 
(moved to unusual
day: 3 pm, Mon) 

Abstract.
In 2004, Corvaja and Zannier proved an extension of Roth's theorem on
rational approximation of algebraic numbers. With a collaboration of Dr.
Veekesh Kumar, we proved a simultaneous version of Corvaja and Zannier's
result. These results are applications of a strong form of the Subspace
Theorem. In this talk, we shall discuss the motivation of Corvaja and
Zannier's result and our generalization.



Sudhanshu Shekhar (IIT Kanpur)  Dec 18, 2020 
Multiplicities
in Selmer groups and root numbers for Artin twists 
(3
pm, Fri) 

Abstract.
Given a Galois extension of number fields $K/F$ and two elliptic curves
$A$ and $B$ with equivalent residual Galois representation mod $p$, for
an odd prime $p$, we will discuss the relation between the $p$parity
conjecture of $A$ twisted by $\sigma$ and that of $B$ twisted by $\sigma$
for an irreducible, self dual, Artin representation $\sigma$ of the
Galois group of $K/F$.
This is a joint work with Somnath Jha and Tathagata Mandal.

Pooja Singla (IIT Kanpur)  Jan 8, 2021 
Projective
representations of discrete nilpotent groups 
(3
pm, Fri) 

Abstract.
The study of projective representations of a group has a long history
starting from the work of Schur. Two essential ingredients to study the
group's projective representations are describing its Schur multiplier
and representation group. In this talk, we describe these for the
discrete Heisenberg groups. We also include a few general results
regarding projective representations of finitely generated discrete
nilpotent groups. This talk is based on the joint work with Sumana Hatui
and E.K. Narayanan.
