The Algebra & Combinatorics Seminar meets from 3–4 pm, on
Fridays (usually, and on Wednesdays as and when needed), in Lecture Hall
LH1 of the IISc Mathematics Department. The current organizer is Apoorva
Khare.

Terrence George
(Brown University, USA) 
Aug 7, 2019 
Dimers and the
Beauville integrable system 
(3
pm, Wed) 

Abstract.
To any convex integral polygon $N$ is associated a cluster integrable
system that arises from the dimer model on certain bipartite graphs on a
torus. The large scale statistical mechanical properties of the dimer
model are largely determined by an algebraic curve, the spectral curve
$C$ of its Kasteleyn operator $K(x,y)$. The vanishing locus of the
determinant of $K(x,y)$ defines the curve $C$ and coker $K(x,y)$ defines
a line bundle on $C$. We show that this spectral data provides a
birational isomorphism of the dimer integrable system with the Beauville
integrable system related to the toric surface constructed from
$N$.
This is joint work with Alexander Goncharov and Richard Kenyon.



Apoorva Khare (IISc
Mathematics) 
Aug 23, 2019 
Density: How
Zariski helped Schur, Cayley, and Hamilton 
(3
pm, Fri) 

Abstract.
Computing the determinant using the Schur complement of an invertible
minor is wellknown to undergraduates. Perhaps less wellknown is why
this works even when the minor is not invertible. Using this and the
Cayley–Hamilton theorem as illustrative examples, I will gently
explain one "practical" usefulness of Zariski density outside commutative
algebra.

Soumik Pal
(University of Washington, Seattle, USA) 
Aug 30, 2019 
Entropic
relaxations of Monge–Kantorovich optimal transports
(speaking in the Eigenfunctions
Seminar; no Alg–Comb today) 
(3
pm, Fri) 


Guhan Venkat
(Université Laval, Quebec, Canada;
and Morningside Center of Mathematics, Beijing, China) 
Sep 4, 2019 
Stark–Heegner cycles for Bianchi modular
forms 
(3
pm, Wed) 

Abstract.
In his seminal paper in 2001, Henri Darmon proposed a systematic
construction of padic points, viz. Stark–Heegner points, on
elliptic curves over the rational numbers. In this talk, I will report on
the construction of padic cohomology classes/cycles in the
Harris–Soudry–Taylor representation associated to a Bianchi
cusp form, building on the ideas of Henri Darmon and Rotger–Seveso.
These local cohomology classes are conjectured to be the restriction of
global cohomology classes in an appropriate Bloch–Kato Selmer group
and have consequences towards the Bloch–Kato–Beilinson
conjecture as well as Gross–Zagier type results. This is based on a
joint work with Chris Williams (Imperial College London).

Chandan
Dalawat (HarishChandra Research Institute, Allahabad) 
Sep 6, 2019 
Two footnotes
to Galois's Memoirs 
(3:30
pm, Fri) 

Abstract.
We will review the history of solvability of polynomial equations by
radicals, concentrating on the two Memoirs of Evariste Galois. We will
show how the first Memoir allows us to determine all equations of prime
degree which are solvable by radicals, and the second Memoir similarly
leads to the determination of all primitive equations which are solvable
by radicals. A finite separable extension $L$ of a field $K$ is called
primitive if there are no intermediate extensions, and solvable if the
Galois group of its Galois closure is a solvable group. Galois himself
proved in his Second Memoir that if $L$ is both primitive and solvable
over $K$, then the degree $[L:K]$ has to be the power of a prime. We
parametrise the set of all primitive solvable extensions in terms of
other more computable things attached to $K$. Thus, when $K$ is a local
field with finite residue field of characteristic $p$, we can explicitly
write down all primitive extensions! This involves the determination of
all irreducible $\mathbb{F}_p$representations of the absolute Galois
group of $K$.

JeanMarie
De Koninck (Université Laval, Quebec, Canada) 
Sep 11, 2019 
Consecutive
integers divisible by a power of their largest prime
factor 
(4:30
pm, Wed) 

Abstract.
The connection between the multiplicative and additive structures of an
arbitrary integer is one of the most intriguing problems in number
theory. It is in this context that we explore the problem of identifying
those consecutive integers which are divisible by a power of their
largest prime factor. For instance, letting $P(n)$ stand for the largest
prime factor of $n$, then the number $n=1294298$ is the smallest integer
which is such that $P(n+i)^2$ divides $n+i$ for $i=0,1,2$. No one has yet
found an integer $n$ such that $P(n+i)^2$ divides $n+i$ for $i=0,1,2,3$.
Why is that? In this talk, we will provide an answer to this question and
explore similar problems.

Prasad
Tetali (Georgia Tech, Atlanta, USA and IISc CSA) 
Sep 13, 2019 
Counting
Independent Sets in Graphs and Hypergraphs and Spectral
Stability 
(3
pm, Fri) 

Abstract.
In 2001, Jeff Kahn showed that a disjoint union of $n/(2d)$ copies of the
complete bipartite graph $K_{d,d}$ maximizes the number of independent
sets over all $d$regular bipartite graphs on n vertices, using Shearer's
entropy inequality. In this lecture I will mention several extensions and
generalizations of this extremal result (to graphs and hypergraphs) and
will describe a stability result (in the spectral sense) to Kahn's
result.
The lecture is based on joint works with Emma Cohen, David Galvin, Will
Perkins, Michail Sarantis and Hiep Han.

Rekha Biswal (Max Planck
Institut, Bonn, Germany) 
Sep 16, 2019 
Macdonald
polynomials and level two Demazure modules for affine
$\mathfrak{sl}_{n+1}$ 
(unusual
day: 3:05 pm, Mon) 

Abstract.
Macdonald polynomials are a remarkable family of orthogonal symmetric
polynomials in several variables. An enormous amount of combinatorics,
group theory, algebraic geometry and representation theory is encoded in
these polynomials. It is known that the characters of level one Demazure
modules are nonsymmetric Macdonald polynomials specialized at $t=0$. In
this talk, I will define a class of polynomials in terms of symmetric
Macdonald polynomials and using representation theory we will see that
these polynomials are Schurpositive and are equal to the graded
character of level two Demazure modules for affine $\mathfrak{sl}_{n+1}$.
As an application we will see how this gives rise to an explicit formula
for the graded multiplicities of level two Demazure modules in the
excellent filtration of Weyl modules. This is based on joint work with
Vyjayanthi Chari, Peri Shereen and Jeffrey Wand.

Santosh
Nadimpalli (Radboud University, Nijmegen, Netherlands) 
Sep 20, 2019 
Linkage
principle and Tame cyclic base change for ${\rm GL}_n(F)$ 
(3
pm, Fri) 

Abstract.
Let $F$ be a finite extension of $\mathbb{Q}_p$, and let $E$ be a finite
cyclic Galois extension of $F$. The theory of base change associates to
an irreducible smooth $\overline{\mathbb{Q}}_l$representation $(\pi_F,
V)$ of ${\rm GL}_n(F)$ an irreducible
$\overline{\mathbb{Q}}_l$representation $(\pi_E, W)$ of ${\rm GL}_n(E)$.
The ${\rm GL}_n(E)$representation $\pi_E$ extends as a representation of
${\rm GL}_n(E)\rtimes {\rm Gal}(E/F)$. Assume that the central character
of $\pi_F$ takes values in $\overline{\mathbb{Z}}_l^\times$, and $l\neq
p$. When $\pi_E$ is cuspidal, for any ${\rm GL}_n(E)\rtimes {\rm
Gal}(E/F)$ stable lattice $\mathcal{L}$ in $\pi_E$, Ronchetti supporting
the linkage principle of Treumann and Venkatesh conjectured that the
zeroth Tate cohomology of $\mathcal{L}$ with respect to ${\rm Gal}(E/F)$
is the Frobenius twist of mod$l$ reduction of the representation
$\pi_F$, i.e.,
$$\widehat{{\rm H}}^0({\rm Gal}(E/F), \mathcal{L})\simeq
\overline{\pi}_F^{(l)}.$$ This conjecture is verified by Ronchetti when
$\pi_F$ is a depthzero cuspidal representation using compact induction
model. We will explain a proof in the case where $n=2$ and $\pi_F$ has
arbitrary depth, using Kirillov model. If time permits, we will discuss
the general case by local Rankin–Selberg convolutions.


On the Numbers of the Form $x^2+11y^2$, Part 1:
Modular Class Groups 
(unusual room:
LH5, 11 am, Wed) 

On the Numbers of the Form $x^2+11y^2$, Part 2:
Number Rings 
(3:30 pm,
Wed) 

Abstract.
A famous result of Leonhard Euler says that his socalled "convenient
numbers" $N$ have the property that a positive integer $n$ has a unique
representation of the form $n=x^2+Ny^2$ with $\gcd(x^2,Ny^2)=1$ if and
only if $n$ is a prime, a prime power, twice one of these, or a power of
2. The set of known 65 convenient numbers is $\{
1,2,3,4,5,6,7,8,9,10,12,13,15,\dots,1848 \}$, and it is conjectured that
these are all of them. So, when we look at this set, we see that 11 is
the first "inconvenient" number, and therefore we consider the natural
question which positive integers have a representation of the form
$n=x^2+11 y^2$ with $\gcd(x,11y)=1$.
Our approach is split into two parts. First we introduce the modular
class group $G_{11}$ of level 11 and give a detailed description of its
structure. We show that there are four conjugacy classes of elliptic
elements of order 2, we provide concrete matrices representing these
elliptic elements, and we give an explicit representation of $G_{11}$
using them. Then we conjugate the first of these matrices, namely
$t_1=\binom{0\; 1}{1\;0}$, by the elements of $G_{11}$ and get matrices
whose top right entry is of the form $x^2+11 y^2$. Conversely, we
construct elliptic elements $A_n(\ell)$ of order 2 in $G_{11}$ which are
conjugate to one of the generators. Then the matrices conjugate to $t_1$
are the ones we are interested in, and we find a set of candidate numbers
$C$ such that $C=S_1 \cup S_2$, where $S_1$ is the set we want to
characterise. Thus the task is reduced to distinguishing between $S_1$
and $S_2$.
This problem is addressed in the second part of the talk using number
rings in $K=\mathbb{Q}(\sqrt{11})$. The ring of integers of this number
field is $\mathcal{O}_K=\mathbb{Z}[(11+\sqrt{11})/2]$, and the more
natural ring $\mathbb{Z}[\sqrt{11}]$ is its order of conductor 2. By
realizing the elements of $S_1$ and $S_2$ as norms of elements in
$\mathcal{O}_K$, we get some of their basic properties. The main theorem
provides a precise description of the primitive representations $n=x^2+11
y^2$ into four classes, where cubic numbers and prime numbers are two
classes which admit separate, detailed descriptions. For the prime
numbers in $S_1$, we need to use some consequences of ring class field
theory for $\mathbb{Z}[\sqrt{11}]$, but all other results are largely
selfcontained.




Abstract.
In this talk, we shall discuss a combinatorial characterization of the
family of secant lines of the 3dimensional projective space $PG(3,q)$
which meet a hyperbolic quadric in two points (the so called secant
lines) using their intersection properties with points and planes. This
is joint work with Puspendu Pradhan.

Atul Dixit (IIT,
Gandhinagar) 
Oct 9, 2019 
Recent
developments in the theory of the restricted partition function $p(n,
N)$ 
(3:30
pm, Wed) 

Abstract.
A beautiful $q$series identity found in the unorganized portion of
Ramanujan's second and third notebooks was recently generalized by Maji
and I. This identity gives, as a special case, a threeparameter identity
which is a rich source of partitiontheoretic information allowing us to
prove, for example, Andrews' famous identity on the smallest parts
function $\textrm{spt}(n)$, a recent identity of Garvan, and identities
on divisor generating functions, to name a few. Guo and Zeng recently
derived a finite analogue of Uchimura's identity on the generating
function for the divisor function $d(n)$. This motivated us to look for a
finite analogue of my generalization of Ramanujan's aforementioned
identity with Maji. Upon obtaining such a finite version, our quest to
look for a finite version of Andrews' $\textrm{spt}$identity
necessitated finding finite analogues of rank, crank and their moments.
We could obtain finite versions of rank and crank for vector partitions.
We were also able to obtain a finite analogue of a partition identity
recently conjectured by George Beck and proven by Shane Chern. I will
discuss these and some related results. This is joint work with Pramod
Eyyunni, Bibekananda Maji and Garima Sood.







Tathagata Basak
(Iowa State University, Ames, USA) 
Nov 4, 2019 
Fundamental
group of a complex ball quotient 
(3
pm, Mon) 

Abstract.
Let W be a Weyl group and V be the complexification of its natural
reflection representation. Let H be the discriminant divisor in (V/W),
that is, the image in (V/W) of the hyperplanes fixed by the reflections
in W. It is well known that the fundamental group of the discriminant
complement ((V/W) – H) is the Artin group described by the Dynkin
diagram of W.
We want to talk about an example for which an analogous result holds.
Here W is an arithmetic lattice in PU(13,1) and V is the unit ball in
complex thirteen dimensional vector space. Our main result (joint with
Daniel Allcock) describes Coxeter type generators for the fundamental
group of the discriminant complement ((V/W) – H). This takes a step
towards a conjecture of Allcock relating this fundamental group with the
Monster simple group.
The example in PU(13,1) is closely related to the Leech lattice. Time
permitting, we shall give a second example in PU(9,1) related to the
Barnes–Wall lattice for which some similar results hold.

Chandrasheel
Bhagwat (IISER Pune) 
Nov 6, 2019 
Special Values
of Lfunctions and period relations for motives 
(4 pm,
Wed) 

Abstract.
We will discuss certain rationality results for the critical values of
the degree$2n$ $L$functions attached to $GL_1 \times O(n,n)$ over a
totally real number field for an even positive integer $n$. We will also
discuss some relations for Deligne periods of motives. This is part of a
joint work with A. Raghuram.



Bernhard Heim (German University of Technology,
Oman) 
Nov 14, 2019 
The G.C. Rota
approach and the Lehmer conjecture 
(unusual day: 2:30
pm, Thu) 

Abstract.
Report on joint work with M. Neuhauser. This includes results with C.
Kaiser, F. Luca, F. Rupp, R. Troeger, and A. Weisse.
The Lehmer conjecture and Serre's lacunary theorem describe the vanishing
properties of the Fourier coefficients of even powers of the Dedekind eta
function.
G.C. Rota proposed to translate and study problems in number theory and
combinatorics to and via properties of polynomials.
We follow G.C. Rota's advice. This leads to several new results and
improvement of known results. This includes Kostant's nonvanishing
results attached to simple complex Lie algebras, a new nonvanishing zone
of the Nekrasov–Okounkov formula (improving a result of G. Han), a
new link between generalized Laguerre and Chebyshev polynomials, strictly
signchanges results of reciprocals of the cubic root of Klein's absolute
$j$ invariant, and hence the $j$invariant itself. Finally we give an
interpretation of the first nonsign change of the Ramanujan $\tau(n)$
function by the root distribution of a certain family of polynomials in
the spirit of G.C. Rota.

Pavol Hell (Simon Fraser
University, Canada) 
Nov 22, 2019 
Graphs with
possible loops 
(3 pm,
Fri) 

Abstract.
I will discuss a few examples of concepts that have interesting
extensions if loops are allowed (but not required). I will include
interval graphs, strongly chordal graphs, and other concepts.

Indranath
Sengupta (IIT, Gandhinagar) 
Nov 27, 2019 
Ideals
generated by quadrics defined by determinantal conditions 
(4 pm,
Fri) 

Abstract.
We will present some recent studies on ideals of the form $I_{1}(XY)$,
where $X$ and $Y$ are matrices whose entries are polynomials of degree at
most 1. We will discuss, how a good Groebner basis for these ideals help
us compute primary decompositions and gather various other homological
informations.

John Meakin
(University of Nebraska at Lincoln, USA) 
Dec 3, 2019 
Some remarks on
Leavitt path algebras 
(unusual day:
11 am, Tue) 

Abstract.
The study of Leavitt path algebras has two primary sources, the work of
W.G. Leavitt in the early 1960's on the module type of a ring, and the
work by Kumjian, Pask, and Raeburn in the 1990's on
Cuntz–Krieger graph $C^*$algebras. Given a directed graph
$\Gamma$ and a field $F$, the Leavitt path algebra $L_F(\Gamma)$ is an
$F$algebra essentially built from the directed paths in the graph
$\Gamma$. Reasonable necessary and sufficient graphtheoretic conditions
for two directed graphs to have isomorphic Leavitt path algebras do not
seem to be known. In this talk I will discuss a recent construction, due
to Zhengpan Wang and myself, of a semigroup $LI(\Gamma)$ associated with
a directed graph $\Gamma$, that we call the Leavitt inverse
semigroup of $\Gamma$. The semigroup $LI(\Gamma)$ is closely related
to the corresponding Leavitt path algebra $L_F(\Gamma)$ and the graph
inverse semigroup $I(\Gamma)$ of $\Gamma$. Leavitt inverse semigroups
provide a certain amount of structural information about Leavitt path
algebras. For example if $LI(\Gamma) \cong LI(\Delta)$, then
$L_F(\Gamma) \cong L_F(\Delta)$, but the converse is false. I will
discuss some topological aspects of the structure of graph inverse
semigroups and Leavitt inverse semigroups: in particular, I will provide
necessary and sufficient conditions for two graphs $\Gamma$ and $\Delta$
to have isomorphic Leavitt inverse semigroups.
This is joint work with Zhengpan Wang, Southwest University, Chongqing,
China.

L. Sunil Chandran (IISc
CSA) 
Dec 11, 2019 
Oriented and
Colorful Variants of Gyárfás–Sumner
Conjecture 
(3 pm,
Wed) 

Abstract.
The Gyárfás–Sumner conjecture states the following:
Let $a, b$ be positive integers. Then there exists a function $f$, such
that if $G$ is a graph of clique number at most $a$ and chromatic number
at least $f(a,b)$, then $G$ contains all trees on at most $b$ vertices as
induced subgraphs. This conjecture is still open, though for several
special cases it is known to be true. We study the oriented version of
this conjecture: Does there exist a function $g$, such that if the
chromatic number of an oriented graph $G$ (satisfying certain properties)
is at least $g(s)$ then $G$ contains all oriented trees on at most $s$
vertices as its induced subgraphs. In general this statement is not true,
not even for triangle free graphs. Therefore, we consider the next
natural special class – namely the 4cycle free graphs – and
prove the above statement for that class. We show that $g(s) \le 4s^2$ in
this case.
We also consider the rainbow (colorful) variant of this conjecture. As a
special case of our theorem, we significantly improve an earlier result
of Gyárfás and Sarkozy regarding the existence of induced
rainbow paths in $C_4$ free graphs of high chromatic number. I will also
discuss the recent results of Seymour, Scott (and Chudnovsky) on this
topic.



Ken Ono
(University of Virginia, Charlottesville, USA) 
Dec 18, 2019 
The Riemann
Hypothesis: History and Recent Work
(speaking in the Eigenfunctions
Seminar) 
(3
pm, Wed) 


Sujatha Ramdorai
(University of British Columbia, Vancouver, Canada) 
Dec 20, 2019 
On congruences
in Iwasawa theory 
(3 pm,
Fri) 

Abstract.
The talk will focus on congruences modulo a prime $p$ of arithmetic
invariants that are associated to the Iwasawa theory of Galois
representations arising from elliptic curves. These congruences fit in
the framework of some deep conjectures in Iwasawa theory which relate
arithmetic and analytic invariants.

2018–19
