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)




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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.

