The Algebra & Combinatorics Seminar has traditionally met on Fridays
from 3–4 pm, in Lecture Hall LH1 of the IISc Mathematics
Department – or online since the Autumn 2021 semester. The
organizers are R. Venkatesh and Apoorva Khare.

Shushma Rani (IISER, Mohali) 
Jan 11, 2023 
Free root
spaces of Borcherds–Kac–Moody Lie
superalgebras 
(LH1 – 3
pm, Wed)


Abstract.
Let $\mathfrak g$ be a Borcherds–Kac–Moody Lie superalgebra
(BKM superalgebra in short) with the associated graph $G$. Any such
$\mathfrak g$ is constructed from a free Lie superalgebra by introducing
three different sets of relations on the generators:
(1) Chevalley relations,
(2) Serre relations, and
(3) Commutation relations coming from the graph $G$.
By Chevalley relations we get a triangular decomposition $\mathfrak g =
\mathfrak n_+ \oplus \mathfrak h \oplus \mathfrak n_{}$, and each root
space $\mathfrak g_{\alpha}$ is either contained in $\mathfrak n_+$ or
$\mathfrak n_{}$. In particular, each $\mathfrak g_{\alpha}$ involves
only the relations (2) and (3). In this talk, we will discuss the root
spaces of $\mathfrak g$ which are independent of the Serre relations. We
call these roots free roots of $\mathfrak g$. Since these root spaces
involve only commutation relations coming from the graph $G$ we can study
them combinatorially using heaps of pieces and construct two different
bases for these root spaces of $\mathfrak g$.


Abstract.
The intersection theory of the Grassmannian, known as Schubert calculus,
is an important development in geometry, representation theory and
combinatorics. The Quot scheme is a natural generalization of the
Grassmannian. In particular, it provides a compactification of the space
of morphisms from a smooth projective curve C to the Grassmannian. The
intersection theory of the Quot scheme can be used to recover
VafaIntriligator formulas, which calculate explicit expressions for the
count of maps to the Grassmannian subject to incidence conditions with
Schubert subvarieties.
The symplectic (or orthogonal) Grassmannian parameterizes isotropic
subspaces of a vector space endowed with symplectic (or symmetric)
bilinear form. I will present explicit formulas for certain intersection
numbers of the symplectic and the orthogonal analogue of Quot schemes.
Furthermore, I will compare these intersection numbers with the
Gromov–Ruan–Witten invariants of the corresponding
Grassmannians.

Starting this semester, the Algebra & Combinatorics Seminar is
(mostly) offline / inperson.

Ilias S. Kotsireas (Wilfrid Laurier University,
Waterloo, Canada) 
Dec 16, 2022 
Legendre Pairs:
old and new results/conjectures and the road ahead
[Video] 
(LH1 – 2:30
pm, Fri)


Abstract.
We shall discuss Legendre Pairs, an interesting combinatorial object
related to the Hadamard conjecture. We shall demonstrate the exceptional
versatility of Legendre Pairs, as they admit several different
formulations via concepts from disparate areas of Mathematics and
Computer Science. We shall mention old and new results and conjectures
within the past 20+ years, as well as potential future avenues for
investigation.

Rohini Ramadas (Warwick Mathematics Institute,
UK) 
Dec 14, 2022 
Complex
dynamics: degenerations, and irreducibility problems  (LH1 – 2 pm,
Wed)


Abstract.
$\mathrm{Per}_n $ is an affine algebraic curve, defined over $\mathbb Q$,
parametrizing (up to change of coordinates) degree$2$ selfmorphisms of
$\mathbb P^1$ with an $n$periodic ramification point. The $n$th Gleason
polynomial $G_n$ is a polynomial in one variable with $\mathbb
Z$coefficients, whose vanishing locus parametrizes (up to change of
coordinates) degree$2$ selfmorphisms of $\mathbb C$ with an
$n$periodic ramification point. Two longstanding open questions in
complex dynamics are: (1) Is $\mathrm{Per}_n$ connected? (2) Is $G_n$
irreducible over $\mathbb Q$?
We show that if $G_n$ is irreducible over $\mathbb Q$, then
$\mathrm{Per}_n$ is irreducible over $\mathbb C$, and is therefore
connected. In order to do this, we find a $\mathbb Q$rational smooth
point of a projective completion of $\mathrm{Per}_n$. This $\mathbb
Q$rational smooth point represents a special degeneration of degree$2$
morphisms, and as such admits an interpretation in terms of tropical
geometry.
(This talk will be pitched at a broad audience.)

Rob Silversmith (Warwick Mathematics Institute,
UK) 
Dec 14, 2022 
Crossratios
and perfect matchings 
(LH1 – 10
am, Wed)


Abstract.
Given a bipartite graph $G$ (subject to a constraint), the "crossratio
degree" of G is a nonnegative integer invariant of $G$, defined via a
simple counting problem in algebraic geometry. I will discuss some
natural contexts in which crossratio degrees arise. I will then present
a perhapssurprising upper bound on crossratio degrees in terms of
counting perfect matchings. Finally, time permitting, I may discuss the
tropical side of the story.

Subhajit Ghosh (BarIlan University, RamatGan,
Israel) 
Nov 4, 2022 
A $q$analog of
the adjacency matrix of the $n$cube 
(LH1 – 2:30
pm, Fri)


Abstract.
Let $q$ be a prime power and define $(n)_q:=1+q+q^2+\cdots+q^{n1}$, for
a nonnegative integer $n$. Let $B_q(n)$ denote the set of all subspaces of
$\mathbb{F}_q^n$, the $n$dimensional $\mathbb{F}_q$vector space of all
column vectors with $n$ components.
Define a $B_q(n)\times B_q(n)$ complex matrix $M_{q,n}$ with entries given by
\begin{equation}
M_{q,n}(X,Y):=
\begin{cases}
1&\text{ if }Y\subseteq X, \dim(Y)=\dim(X)1,\\
q^{\dim(X)}&\text{ if }X\subseteq Y, \dim(Y)=\dim(X)+1,\\
0&\text{ otherwise.}
\end{cases}
\end{equation}
We think of $M_{q,n}$ as a $q$analog of the adjacency matrix of the
$n$cube. We show that the eigenvalues of $M_{q,n}$ are
\begin{equation}
(nk)_q(k)_q\text{ with multiplicity }\binom{n}{k}_q,\quad k=0,1,\dots,n,
\end{equation}
and we write down an explicit canonical eigenbasis of $M_{q,n}$. We give
a weighted count of the number of rooted spanning trees in the $q$analog
of the $n$cube.
This talk is based on a joint work with M. K. Srinivasan.

Junaid Hasan (University of Washington, Seattle,
USA) 
Sep 2, 2022 
Zeta functions
on graphs 
(LH1 – 4
pm, Fri)


Abstract.
This talk is based on the work of Stark and Terras (Zeta functions of
Finite graphs and Coverings I, II, III). In this talk we start with an
introduction to zeta functions in various branches of mathematics. Our
focus is mainly on zeta functions on finite undirected connected graphs.
We obtain an analogue of the prime number theorem, but for graphs, using
the Ihara Zeta Function. We also introduce edge and path zeta functions
and show interesting results.

S. Venkitesh (IIT, Bombay) 
Aug 26, 2022 
Closures,
Coverings, and Complexity 
(LH1 – 4
pm, Fri)


Abstract.
The polynomial method is an everexpanding set of algebraic techniques,
which broadly entails capturing combinatorial objects by algebraic means,
specifically using polynomials, and then employing algebraic tools to
infer their combinatorial features. While several instances of the
polynomial method have been part of the combinatorialist's toolkit for
decades, development of this method has received more traction in recent
times, owing to several breakthroughs like
(i) Dvir's solution (2009) to the finitefield Kakeya problem, followed
by an improvement by Dvir, Kopparty, Saraf, and Sudan (2013),
(ii) Guth and Katz (2015) proving a conjecture by Erdös on the
distinct distances problem,
(iii) solutions to the capset problem by Croot, Lev, and Pach (2017), and
Ellenberg and Gijswijt (2017), to name a few.
One of the ways to employ the polynomial method is via the classical
algebraic objects – (affine) Zariski closure, (affine) Hilbert
function, and Gröbner basis. Owing to their applicability in several
areas like computational complexity, combinatorial geometry, and coding
theory, an important line of enquiry is to understand these objects for
'structured' sets of points in the affine space. In this talk, we will
be mainly concerned with Zariski closures of symmetric sets of points in
the Boolean cube.
Firstly, we will look at a combinatorial characterization of Zariski
closures of all symmetric sets, and its application to some hyperplane
and polynomial covering problems for the Boolean cube, over any field of
characteristic zero. We will also briefly look at Zariski closures over
fields of positive characteristic, although much less is known in this
setting. Secondly, we will see a simple illustration of a 'closure
statement' being used as a technique for proving bounds on the complexity
of approximating Boolean functions by polynomials. We will conclude with
some open questions on Zariski closures motivated by problems on these
two fronts.
Some parts of this talk will be based on the works:
 https://arxiv.org/abs/2107.10385
 https://arxiv.org/abs/2111.05445
 https://arxiv.org/abs/1910.02465

Bruno Kahn (Institut de Mathématiques de
JussieuParis Rive Gauche, Paris, France) 
Aug 25, 2022 
The rank
spectral sequence on Quillen's Q construction 
(LH1 – 12
pm, Thu)


Abstract.
I will explain a generalisation of the constructions Quillen used to
prove that the $K$groups of rings of integers are finitely generated. It
takes the form of a 'rank' spectral sequence, converging to the homology
of Quillen's $Q$construction on the category of coherent sheaves over a
Noetherian integral scheme, and whose $E^1$ terms are given by homology
of Steinberg modules. Computing its $d^1$ differentials is a challenge,
which can be approached through the universal modular symbols of
Ash–Rudolph.

This semester, the Algebra & Combinatorics Seminar is subsumed by
the ARCSIN
Seminar$\dots$ for the most part. Here are the (few)
exceptions.

Madhusudan Manjunath (IIT, Bombay) 
May 17, 2022 
Combinatorial
Brill–Noether theory via lattice points and
polyhedra 
(LH1 – 2
pm, Tue)


Abstract.
We start by considering analogies between graphs and Riemann surfaces.
Taking cue from this, we formulate an analogue of Brill–Noether theory
on a finite, undirected, connected graph. We then investigate related
conjectures from the perspective of polyhedral geometry.

Amritanshu Prasad (IMSc, Chennai) 
May 6, 2022 
Splitting
subspaces and the Touchard–Riordan formula 
(LH1 – 3
pm, Fri)


Abstract.
Let $T$ be a linear endomorphism of a $2m$dimensional vector space. An
$m$dimensional subspace $W$ is said to be $T$splitting if $W$
intersects $TW$ trivially.
When the underlying field is finite of order $q$ and $T$ is diagonal with
distinct eigenvalues, the number of splitting subspaces is essentially
the the generating function of chord diagrams weighted by their number of
crossings with variable $q$. This generating function was studied by
Touchard in the context of the stamp folding problem. Touchard obtained a
compact form for this generating function, which was explained more
clearly by Riordan.
We provide a formula for the number of splitting subspaces for a general
operator $T$ in terms of the number of $T$invariant subspaces of various
dimensions. Specializing to diagonal matrices with distinct eigenvalues
gives an unexpected and new proof of the Touchard–Riordan
formula.
This is based on joint work with Samrith Ram.

Jacob P. Matherne (Universität Bonn, Germany) 
Mar 25, 2022 
Singular Hodge
theory for combinatorial geometries 
(LH1 – 3 pm,
Fri) 

Abstract.
Here are two problems about hyperplane arrangements.
Problem 1: If you take a collection of planes in $\mathbb{R}^3$,
then the number of lines you get by intersecting the planes is at least
the number of planes. This is an example of a more general statement,
called the "TopHeavy Conjecture", that Dowling and Wilson conjectured in
1974.
Problem 2: Given a hyperplane arrangement, I will explain how to
uniquely associate a certain polynomial (called its Kazhdan–Lusztig
polynomial) to it. These polynomials should have nonnegative
coefficients.
Both of these problems were formulated for all matroids, and in the case
of hyperplane arrangements they are controlled by the Hodge theory of a
certain singular projective variety, called the Schubert variety of the
arrangement. For arbitrary matroids, no such variety exists; nonetheless,
I will discuss a solution to both problems for all matroids, which
proceeds by finding combinatorial standins for the cohomology and
intersection cohomology of these Schubert varieties and by studying their
Hodge theory. This is joint work with Tom Braden, June Huh, Nicholas
Proudfoot, and Botong Wang.

Maitreyee Kulkarni (Universität Bonn, Germany) 
Mar 25, 2022 
A combinatorial
model for totally nonnegative partial flag varieties 
(LH1 – 2:30
pm, Fri) 

Abstract.
Postnikov defined the totally nonnegative Grassmannian as the part of the
Grassmannian where all Plücker coordinates are nonnegative. This space
can be described by the combinatorics of planar bipartite graphs in a
disk, by affine Bruhat order, and by a host of other combinatorial
objects. In this talk, I will recall some of this story, then talk about
in progress joint work, together with Chris Fraser and Jacob Matherne,
which hopes to extend this combinatorial description to more general
partial flag varieties.

This semester, the Algebra & Combinatorics Seminar is subsumed by
the ARCSIN
Seminar.
This semester, the Algebra & Combinatorics Seminar is subsumed by
the ARCSIN
Seminar.
This semester, the Algebra & Combinatorics Seminar is online on
Microsoft Teams.

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.

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.



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.

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

2019–20
2018–19
