The Algebra & Combinatorics Seminar meets on Fridays from 3–4
pm, in Lecture Hall LH1 of the IISc Mathematics Department. The
organizers are Apoorva Khare and R. Venkatesh.


Abstract.
Let $\mathfrak{O}$ be the ring of integers of a nonArchimedean local
field such that the residue field has characteristic $p$. Then the
abscissa of convergence of representation zeta function of Special Linear
group $\mathrm{SL}_2(\mathfrak{O})$ is $1.$ The case $p\neq 2$ is already
known in the literature. For $p=2$ we need more tools to prove the
result. In this talk I will discuss the difference between those cases
and give an outline of the proof for $p=2.$
Let $\mathfrak{p}$ be the maximal ideal of $\mathfrak{O}$ and
$\mathfrak{O}/\mathfrak{p}=q.$ It is already shown in literature that
for $r \geq 1,$ the group algebras $\mathbb
C[\mathrm{GL}_2(\mathfrak{O}/\mathfrak{p}^{r})]$ and $\mathbb
C[\mathrm{GL}_2(\mathbb F_q[t]/(t^{r}))]$ are isomorphic. Also for
$2\nmid q,$ the group algebras $\mathbb
C[\mathrm{SL}_2(\mathfrak{O}/\mathfrak{p}^{r})]$ and $\mathbb
C[\mathrm{SL}_2(\mathbb F_q[t]/(t^{r}))]$ are isomorphic. In this talk I
will also show that if $2\mid q$ and $\mathrm{char}(\mathfrak{O})=0$
then, the group algebras,
$\mathbb C[\mathrm{SL}_2(\mathfrak{O}/\mathfrak{p}^{2\ell})]$ and
$\mathbb C[\mathrm{SL}_2(\mathbb F_q[t]/(t^{2\ell}))]$ are not
isomorphic for $\ell > \mathrm{e}$, where $\mathrm{e}$ is the
ramification index of $\mathfrak{O}.$


Abstract.
Consider the following three properties of a general group $G$:
1. Algebra: $G$ is abelian and torsionfree.
2. Analysis: $G$ is a metric space that admits a "norm", namely, a
translationinvariant metric $d(.,.)$ satisfying: $d(1,g^n) = n d(1,g)$
for all $g \in G$ and integers $n$.
3. Geometry: $G$ admits a length function with "saturated"
subadditivity for equal arguments: $\ell(g^2) = 2 \; \ell(g)$ for all $g
\in G$.
While these properties may a priori seem different, in fact they turn out
to be equivalent. The nontrivial implication amounts to saying that there
does not exist a nonabelian group with a "norm".
We will discuss motivations from analysis, probability, and geometry;
then the proof of the above equivalences; and if time permits, the
logistics of how the problem was solved, via a PolyMath
project that began on a blogpost
of Terence Tao.
(Joint – as D.H.J. PolyMath – with Tobias Fritz, Siddhartha
Gadgil, Pace Nielsen, Lior Silberman, and Terence Tao.)


Abstract.
While the enumeration of linear extensions of a given poset is a
wellstudied question, its cyclic counterpart (enumerating extensions to
total cyclic orders of a given partial cyclic order) has been subject to
very little investigation. In this talk I will introduce some classes of
partial cyclic orders for which this enumeration problem is tractable.
Some cases require the use of a multidimensional version of the classical
boustrophedon construction (a.k.a. Seidel–Entringer–Arnold
triangle). The integers arising from these enumerative questions also
appear as the normalized volumes of certain polytopes.
This is partly joint work with Arvind Ayyer (Indian Institute of Science)
and Matthieu JosuatVergès (Laboratoire d'Informatique Gaspard
Monge / CNRS).




Talk 1: The Polynomial Method in Combinatorics
– Two illustrative theorems 
2:30
pm 

Abstract.
Since Dvir proved the Finite Kakeya Conjecture in 2008, the Polynomial
Method has become a new and powerful tool and a new paradigm for
approaching extremal questions in combinatorics (and other areas too). We
shall take a look at the main philosophical principle that underlies this
method via two recent (2017) theorems. One is the upper bound for
Capsets by Ellenberg–Gijswijt, and the other, a function field
analogue of a theorem of Sarkozy, due to Ben Green.

Talk 2: Bisections and bicolorings of
hypergraphs 
4
pm 

Abstract.
Given a hypergraph $\mathcal{H}$ with vertex set $[n]:=\{1,\ldots,n\}$, a
bisecting family is a family $\mathcal{A}\subseteq\mathcal{P}([n])$ such
that for every $B\in\mathcal{H}$, there exists $A\in\mathcal{A}$ with the
property $A\cap BA\cap\overline{B}\in\{1,0,1\}$. Similarly, for a
family of bicolorings $\mathcal{B}\subseteq \{1,1\}^{[n]}$ of $[n]$ a
family $\mathcal{A}\subseteq\mathcal{P}([n])$ is called a System of
Unbiased Representatives for $\mathcal{B}$ if for every $b\in\mathcal{B}$
there exists $A\in\mathcal{A}$ such that $\sum_{x\in A} b(x) =0$.
The problem of optimal families of bisections and bicolorings for
hypergraphs originates from what is referred to as the problem of
Balancing Sets of vectors, and has been the source for a few interesting
extremal problems in combinatorial set theory for about 3 decades now. We
shall consider certain natural extremal functions that arise from the
study of bisections and bicolorings and bounds for these extremal
functions. Many of the proofs involve the use of polynomial methods (not
the Polynomial Method, though!).
(Joint work with Rogers Mathew, Tapas Mishra, and Sudebkumar Prasant
Pal.)




Abstract.
We say a unital ring $R$ has the Invariant Basis Number (IBN) property in
case, for each pair of positive integers $i,j$ if the left $R$modules
$R^i$ and $R^j$ are isomorphic, then $i=j$. The first examples of non IBN
rings were studied by William Leavitt in the 1950s and he defined (what
are now known as) Leavitt algebras which are 'universal' with non IBN
property. In 2004 the algebraic structures arising from directed
(multi)graphs known as Leavitt path algebras (LPA for short) were
initially developed as algebraic analogues of graph $C^*$ algebras. LPAs
generalize a particular class of Leavitt algebras.
During the intervening decade, these algebras have attracted significant
interest and attention, not only from ring theorists, but from analysts
working in $C^*$algebras, and symbolic dynamicists as well. The goal of
this talk is to introduce the notion of Leavitt path algebras and to
present some results on LPAs arising from weighted Cayley graphs of
finite cyclic groups.


Abstract.
Valiant (1979) showed that unless P = N P, there is no polynomialtime
algorithm to compute the number of perfect matchings of a given graph
– even if the input graph is bipartite. Earlier, the physicist
Kasteleyn (1963) introduced the notion of a special type of orientation
of a graph, and we refer to graphs that admit such an orientation as
Pfaffian graphs. Kasteleyn showed that the number of perfect matchings is
easy to compute if the input graph is Pfaffian, and he also proved that
every planar graph is Pfaffian. The complete bipartite graph $K_{3,3}$ is
the smallest graph that is not Pfaffian. In general, the problem of
deciding whether a given graph is Pfaffian is not known to be in N
P.
Special types of minors, known as conformal minors, play a key role in
the theory of Pfaffian orientations. In particular, a graph is Pfaffian
if and only if each of its conformal minors is Pfaffian. It was shown by
Little (1975) that a bipartite graph $G$ is Pfaffian if and only if $G$
does not contain $K_{3,3}$ as a conformal minor (or, in other words, if
and only if $G$ is $K_{3,3}$free); this places the problem of deciding
whether a bipartite graph is Pfaffian in co – N P. Several years
later, a structural characterization of $K_{3,3}$free bipartite graphs
was obtained by Robertson, Seymour and Thomas (1999), and independently
by McCuaig (2004), and this led to a polynomialtime algorithm for
deciding whether a given bipartite graph is Pfaffian.
Norine and Thomas (2008) showed that, unlike the bipartite case, it is
not possible to characterize all Pfaffian graphs by excluding a finite
number of graphs as conformal minors. In light of everything that has
been done so far, it would be interesting to even identify rich
classes of Pfaffian graphs (that are nonplanar and nonbipartite).
Inspired by a theorem of Lovasz (1983), we took on the task of
characterizing graphs that do not contain $K_4$ as a conformal minor
– that is, $K_4$free graphs. In a joint work with U. S. R. Murty
(2016), we provided a structural characterization of planar $K_4$free
graphs. The problem of characterizing nonplanar $K_4$free graphs is much
harder, and we have evidence to believe that it is related to the problem
of recognizing Pfaffian graphs. In particular, we conjecture that every
graph that is $K_4$free and $K_{3,3}$free is also Pfaffian. The talk
will be mostly selfcontained. I will assume only basic knowledge of
graph theory. For more details, see: https://onlinelibrary.wiley.com/doi/full/10.1002/jgt.21882.


Abstract.
We investigate the properties of a random walk on the alternating group
$A_n$ generated by $3$cycles of the form $(i,n1,n)$ and $(i,n,n1)$. We
call this the transpose top$2$ with random shuffle. We find the spectrum
of the transition matrix for this shuffle. We mainly use the
representation theory of alternating group. We show that the mixing time
is of order $\left(n\frac{3}{2}\right)\log n$ and prove that there is a
total variation cutoff for this shuffle.


Abstract.
An ordinary ring may be expressed as a preadditive category with a single
object. Accordingly, as introduced by B. Mitchel, an arbitrary small
preadditive category may be understood as a "ring with several objects".
In this respect, for a Hopf algebra H, an Hcategory will denote an
"Hmodule algebra with several objects" and a coHcategory will denote
an "Hcomodule algebra with several objects". Modules over such Hopf
categories were first considered by Cibils and Solotar. In this talk, we
present a study of cohomology in such module categories.
In particular, we consider Hequivariant modules over a Hopf module
category C as modules over the smash extension C#H. We construct
Grothendieck spectral sequences for the cohomologies as well as the
Hlocally finite cohomologies of these objects. We also introduce
relative (D,H)Hopf modules over a Hopf comodule category D. These
generalize relative (A,H)Hopf modules over an Hcomodule algebra A. We
construct Grothendieck spectral sequences for their cohomologies by using
their rational Hom objects and higher derived functors of
coinvariants.
(Joint work with Abhishek Banerjee and Mamta Balodi.)




Abstract.
The theory of orthogonal polynomials started with analytic continued
fractions going back to Euler, Gauss, Jacobi, Stieltjes... The
combinatorial interpretations started in the late 70's and is an active
research domain. I will give the basis of the theory interpreting the
moments of general (formal) orthogonal polynomials, Jacobi continued
fractions and Hankel determinants with some families of weighted paths.
In a second part I will give some examples of interpretations of
classical orthogonal polynomials and of their moments (Hermite, Laguerre,
Jacobi, ...) with their connection to theoretical physics.


Abstract.
The PASEP (partially asymmetric exclusion process) is a toy model in the
physics of dynamical systems strongly related to the moments of some
classical orthogonal polynomials (Hermite, Laguerre, Askey–Wilson).
The partition function has been interpreted with various combinatorial
objects such as permutations, alternative and treelike tableaux, etc. We
introduce a new one called "Laguerre heaps of segments", which seems to
play a central role in the network of bijections relating all these
interpretations.


Abstract.
We describe how a completion of the factorial row by Suslin led to
showing some ideals in a polynomial ring are settheoretic complete
intersections.

Ravi A. Rao (TIFR, Bombay) 
Mar 15, 2019 
Talk 2:
The study of projective modules over an affine algebra 
(2:30 pm –
note: unusual time) 

Abstract.
The study of projective modules along the lines initiated by J.P. Serre,
H. Bass is outlined; the impetus given by A. Suslin in that direction is
described, and recent progress made along the vision of A. Suslin on it
by Fasel–Rao–Swan is shared briefly.


Abstract.
'Holomorphic eta quotients' are certain explicit classical modular forms
on suitable Hecke subgroups of the full modular group. We call a
holomorphic eta quotient $f$ 'reducible' if for some holomorphic eta
quotient $g$ (other than $1$ and $f$), the eta quotient $f/g$ is
holomorphic. An eta quotient or a modular form in general has two
parameters: weight and level. We shall show that for any positive integer
$N$, there are only finitely many irreducible holomorphic eta quotients
of level $N$. In particular, the weights of such eta quotients are
bounded above by a function of $N$. We shall provide such an explicit
upper bound. This is an analog of a conjecture of Zagier which says that
for any positive integer $k$, there are only finitely many irreducible
holomorphic eta quotients of weight $k/2$ which are not integral
rescalings of some other eta quotients. This conjecture was established
in 1991 by Mersmann. We shall sketch a short proof of Mersmann's theorem
and we shall show that these results have their applications in
factorizing holomorphic eta quotient. In particular, due to Zagier and
Mersmann's work, holomorphic eta quotients of weight $1/2$ have been
completely classified. We shall see some applications of this
classification and we shall discuss a few seemingly accessible yet
longstanding open problems about eta quotients.
This talk will be suitable also for nonexperts: We shall define all the
relevant terms and we shall clearly state the classical results which we
use.




Abstract.
TBA





Abstract.
I will give a gentle introduction to the Diamond Lemma. This is a useful
technique to prove that certain "PBWtype" bases exist of algebras given
by generators and relations. In particular, we will see the PBW theorem
for usual Lie algebras.


Abstract.
Let $G$ be a finite simple graph. The Lie algebra $\mathfrak{g}$ of $G$
is defined as follows: $\mathfrak{g}$ is generated by the vertices of $G$
modulo the relations $[u, v]=0$ if there is no edge between the vertices
$u$ and $v$. Many properties of $\mathfrak{g}$ can be obtained from the
properties of $G$ and vice versa. The Lie algebra $\mathfrak{g}$ of $G$
is naturally graded and the graded dimensions of the Lie algebra
$\mathfrak{g}$ of $G$ have some deep connections with the vertex
colorings of $G$. In this talk, I will explain how to get the generalized
chromatic polynomials of $G$ in terms of graded dimensions of the Lie
algebra of $G$. We will use this connection to give a Lie theoretic proof
of of Stanley's reciprocity theorem of chromatic polynomials.


Abstract.
The aim of this talk is to give a highlevel overview of the theory of
expander graphs and introduce motivations and possible approaches to
generalizing it to higher dimensions. I shall begin with three
perspectives on expansion in graphs discrepancy, isoperimetry and mixing
time, and show a qualitative equivalence of these notions in defining
expansion for graphs. Next I shall briefly discuss upper and lower bounds
on expansion, and sketch the LubotzkyPhillipsSarnak construction of
Ramanujan graphs. Finally, I hope to motivate highdimensional expanders
using two interesting topics the overlapping problem, and the threshold
problem.




Abstract.
We will study recurrence patterns in decimal expansions of rational
numbers (in any integer base for this talk). After making some initial
observations, we will compute the length of the repeating part of any
fraction. We conclude by explaining this result over a Euclidean
domain.


Abstract.
We will describe Gelfand's criterion for the commutativity of associative
algebras and discuss some of its applications towards the multiplicity
one theorems for the representations of finite groups.


Abstract.
Characters of classical groups appear in the enumeration of many
interesting combinatorial problems. We show that, for a wide class of
partitions, and for an even number of variables of which half are
reciprocals of the other half, Schur functions (i.e., characters of the
general linear group) factorize into a product of two characters of
other classical groups. Time permitting, we will present similar results
involving sums of two Schur functions. All the proofs will involve
elementary applications of ideas from linear algebra.
This is joint work with Roger Behrend.


(Note: The following
Seminar on Nov 2 may be of related interest to those attending this
talk.)


Abstract.
The counts of algebraic curves in projective space (and other toric
varieties) has been intensely studied for over a century. The subject saw
a major advance in the 1990s, due to groundbreaking work of Kontsevich
in the 1990s. Shortly after, considerations from high energy physics led
to an entirely combinatorial approach to these curve counts, via
piecewise linear embeddings of graphs, pioneered by Mikahlkin. I will
give an introduction to the surrounding ideas, outlining new results and
new proofs that the theory enables. Time permitting I will discuss
generalizations, difficulties, and future directions for the
subject.
(Organizer's note: It may help to attend the
preceding Eigenfunctions Seminar on Oct 31, before this talk.)




Abstract.
An ordinary ring may be seen as a preadditive category with just one
object. This leads to the powerful analogy, first formulated explicitly
by Mitchell in 1975, that a small preadditive category should be seen as
a "ring with several objects". We will trace the history and development
of the category of modules over a preadditive category.


Abstract.
Valuations are a classical topic in convex geometry. The volume plays an
important role in many structural results, such as Hadwiger's famous
characterization of continuous, rigidmotion invariant valuations on
convex bodies. Valuations whose domain is restricted to lattice polytopes
are less wellstudied. The Betke–Kneser Theorem establishes a
fascinating discrete analog of Hadwiger's Theorem for latticeinvariant
valuations on lattice polytopes in which the number of lattice points
– the discrete volume – plays a fundamental role. In this
talk, we explore striking parallels, analogies and also differences
between the world of valuations on convex bodies and those on lattice
polytopes with a focus on positivity questions and links to Ehrhart
theory.


Abstract.
Let $\mathfrak{gl}(n)$ denote the Lie algebra of the general linear group
$GL(n)$. Given two finite dimensional irreducible representations
$L(\lambda), L(\mu)$ of $\mathfrak{gl}(n)$, its tensor product
decomposition $L(\lambda) \otimes L(\mu)$ is given by the
LittlewoodRichardson rule.
The situation becomes much more complicated when one replaces
$\mathfrak{gl}(n)$ by the general linear Lie superalgebra
$\mathfrak{gl}(mn)$. The analogous decomposition $L(\lambda) \otimes
L(\mu)$ is largely unknown. Indeed many aspects of the representation
theory of $\mathfrak{gl}(mn)$ are more akin to the study of Lie algebras
and their representations in prime characteristic or to the BGG category
$\mathcal{O}$. I will give a survey talk about this problem and explain
why some approaches don't work and what can be done about it. This will
give me the chance to speak about a) the character formula for an
irreducible representation $L(\lambda)$, b) Deligne's interpolating
category $Rep(GL_t)$ for $t \in \mathbb{C}$ and c) the process of
semisimplification of a tensor category.

