The Algebra & Combinatorics Seminar has traditionally met on Fridays
in Lecture Hall LH1 of the IISc Mathematics Department – or online
in some cases. The organizers are Apoorva Khare and R. Venkatesh.

Rakesh Pawar
(UMPA, École Normale Supériure de Lyon, France) 
Jul 24, 2024 
Milnor–Witt cycle modules over excellent
DVR 
(LH1 – 2
pm, Wed)


Abstract.
I will briefly recall Milnor cycle modules over a field as defined by
Rost (1996) and their significance and properties. Recently, 'modules'
over Milnor–Witt Ktheory or alternatively Milnor–Witt cycle
modules over fields have been formalized by N. Feld (2020).
I will talk about recent joint work with Chetan Balwe and Amit Hogadi,
where we considered the Milnor–Witt cycle modules over excellent
DVR and studied a subclass of these that satisfy certain lifting
conditions on residue maps associated with horizontal valuations. As an
important example, Milnor–Witt Ktheory of fields belongs to this
subclass.
Moreover, this condition is sufficient to deduce the local acyclicity
property and $A^1$homotopy invariance of the associated Gersten complex.


Abstract.
According to a wellknown result in geometric topology, we have
$(S^2)^n/Sym(n) = \mathbb{CP}^n$, where $Sym(n)$ acts on $(S^2)^n$ by
coordinate permutation. We use this fact to explicitly construct a
regular simplicial cell decomposition of $\mathbb{CP}^n$ for each $n >
1$. In more detail, we take the standard two triangle crystallisation
$S^2_3$ of the 2sphere $S^2$, in its $n$fold Cartesian product. We
then simplicially subdivide, and prove that naively taking the $Sym(n)$
quotient yields a simplicial cell decomposition of $\mathbb{CP}^n$.
Taking the first derived subdivision of this cell complex produces a
triangulation of $\mathbb{CP}^n$. To the best of our knowledge, this is
the first explicit description of triangulations of $\mathbb{CP}^n$ for
$n > 3$. This is a joint work with Jonathan Spreer, University of Sydney.

Vaibhav
Pandey
(Purdue University, West Lafayette, USA) 
Jul 12, 2024 
When do the
natural embeddings of classical invariant rings split? 
(LH1 – 3
pm, Fri)


Abstract.
Consider a reductive linear algebraic group $G$ acting linearly on a
polynomial ring $S$ over an infinite field. Objects of broad interest in
commutative algebra, representation theory, and algebraic geometry like
generic determinantal rings, Plücker coordinate rings of
Grassmannians, symmetric determinantal rings, rings defined by Pfaffians
of alternating matrices etc. arise as the invariant rings $S^G$ of such
group actions.
In characteristic zero, reductive groups are linearly reductive and
therefore the embedding of the invariant ring $S^G$ in the ambient
polynomial ring $S$ splits. This explains a number of good
algebrogeometric properties of the invariant ring in characteristic
zero. In positive characteristic, reductive groups are typically no
longer linearly reductive. We determine, for the natural actions of the
classical groups, precisely when $S^G$ splits from $S$ in positive
characteristic.
This is joint work with Melvin Hochster, Jack Jeffries, and Anurag K.
Singh.

Sunil Chebolu
(Illinois State University, Normal, USA) 
Jul 3, 2024 
Towards a
refinement of the Bloch–Kato conjecture 
(LH1 –
11 am, Wed)


Abstract.
Let $F$ be a field that has a primitive $p$th root of unity. According
to the Bloch–Kato conjecture, now a theorem by Voevodsky and Rost,
the normresidue map
$k_*(F)/pk_*(F) \rightarrow H^*(F, \mathbb{F}_p)$
from the reduced Milnor $K$theory to the Galois cohomology of $F$ is an
isomorphism of $\mathbb{F}_p$algebras.
This isomorphism gives a presentation of the rather mysterious Galois
cohomology ring through generators and relations. In joint work with Jan
Minac, Cihan Okay, Andy Schultz, and Charlotte Ure, we have obtained a
second cohomology refinement of the Bloch–Kato conjecture. Using
this we can characterize the maximal $p$extension of $F$, as the
"decomposing field" for the cohomology of the absolute Galois group.

Ngo
Viet Trung
(Institute of Mathematics, Vietnam Academy of Science and Technology,
Hanoi, Vietnam) 
Jul 1, 2024 
Buchsbaumness
and Castelnuovo–Mumford regularity of nonsmooth monomial
curves 
(LH1 – 2
pm, Mon)


Abstract.
Projective monomial curves correspond to rings generated by monomials of
the same degree in two variables. Such rings always have finite
Macaulayfication. We show how to characterize the Buchsbaumness and the
Castelnuovo–Mumford regularity of these rings by means of their
finite Macaulayfication, and we use this method to study the
Buchsbaumness and to estimate the Castelnuovo–Mumford regularity of
large classes of nonsmooth monomial curves in terms of the given
monomials.

Sridhar
Venkatesh
(University of Michigan, Ann Arbor, USA) 
Jun 27, 2024 
Local vanishing
for toric varieties 
(LH1 – 2
pm, Thu)


Abstract.
Let $f:Y \to X$ be a log resolution of singularities which is an
isomorphism over the smooth locus of $X$, and the exceptional locus $E$
is a simple normal crossing divisor on $Y$. We prove vanishing (and
nonvanishing) results for the higher direct images of differentials on
$Y$ with log poles along $E$ in the case when $X$ is a toric variety. Our
consideration of these sheaves is motivated by the notion of $k$rational
singularities introduced by FriedmanLaza. This is joint work with Anh
Duc Vo and Wanchun Shen.

Debaditya Raychaudhury
(University of Arizona, Tucson, USA) 
Jun 5, 2024 
Ulrich
subvarieties and nonexistence of low rank Ulrich bundles on complete
intersections 
(LH1 – 2
pm, Wed)


Abstract.
We characterize the existence of an Ulrich vector bundle on a variety
$X\subset{\bf P}^N$ in terms of the existence of a subvariety satisfying
certain conditions. Then we use this fact to prove that
$(X,\mathcal{O}_X(a))$ where $X$ is a complete intersection of dimension
$n\geq 4$, which if n = 4, is either ${\bf P}^4$ with $a\geq 2$, or very
general with $a\geq 1$ and not of type (2), (2, 2), does not carry any
Ulrich bundles of rank $r\leq 3$. Work in collaboration with A.F. Lopez.

Martin
Andler
(Université de Versailles, France) 
May 22, 2024 
Equivariant
cohomology in a tensor category 
(LH1 – 2
pm, Wed)


Abstract.
In the 1950s, topologists introduced the notion of equivariant cohomology
$H_G(E)$ for a topological space $E$ with an action by a compact group
$G$. If the action is free, $H_G(E)$ should be $H(E/G)$, and be computed
using de Rham cohomology. In 1950, even before the concept of equivariant
cohomology had been formulated, Henri Cartan introduced a complex of
equivariant differential forms for a compact Lie group acting on a
differential manifold $E$, and proved a result amounting to stating that
the cohomology of that complex computes $H_G(E)$. In 1999, Guillemin and
Sternberg reformulated Cartan's work in terms of a supersymmetric
extension of the Lie algebra of $G$.
Our aim is to reconsider such considerations, by replacing vector spaces
by a $k$linear symmetric monoidal category, requiring that this category
contain an odd unit to account for the supersymmetric dimension plus some
further properties, and considering modules of a rigid Lie algebra object
in that category. In that context, we obtain a version of Koszul's
homotopy isomorphism theorem, and recover as a consequence some known
results as the acyclicity of the Koszul resolution. (Joint work with
Siddhartha Sahi.)

Amartya Kumar
Dutta (ISI Kolkata) 
Apr 9, 2024 
Epimorphism
theorems and allied topics 
(LH1 –
11:30 am, Tue)


Abstract.
In the area of Affine Algebraic Geometry, there are several problems on
polynomial rings which are easy to state but difficult to investigate.
Late Shreeram S. Abhyankar was the pioneer in investigating a class of
such problems known as Epimorphism Problems or Embedding Problems. In
this nontechnical survey talk, we shall highlight some of the
contributions of Abhyankar, Moh, Suzuki, Sathaye, Russell, Bhatwadekar
and other mathematicians.

Debsoumya
Chakraborti (Mathematics Institute, University of Warwick,
UK) 
Apr 3, 2024 
Approximate
packing of independent transversals in locally sparse
graphs 
(Joint with the APRG
Seminar) 
(LH1 –
4 pm, Wed)


Abstract.
Consider a multipartite graph $G$ with maximum degree at most $no(n)$,
parts $V_1,\ldots,V_k$ have size $V_i=n$, and every vertex has at most
$o(n)$ neighbors in any part $V_i$. Loh and Sudakov proved that any such
$G$ has an independent set, referred to as an 'independent transversal',
which contains exactly one vertex from each part $V_i$. They further
conjectured that the vertex set of $G$ can be decomposed into pairwise
disjoint independent transversals. We resolve this conjecture
approximately by showing that $G$ contains $no(n)$ pairwise disjoint
independent transversals. As applications, we give approximate answers to
questions on packing list colorings and multipartite
HajnalSzemerédi theorem. We use probabilistic methods, including
a 'twolayer nibble' argument. This talk is based on joint work with Tuan
Tran.

Ratheesh T V
(IMSc Chennai) 
Feb 23, 2024 
Monomial
expansions for $q$Whittaker polynomials 
(LH1 –
3 pm, Fri)


Abstract.
We consider the monomial expansion of the $q$Whittaker polynomials given
by the fermionic formula and via the inv and quinv
statistics. We construct bijections between the parametrizing sets of
these three models which preserve the $x$ and $q$weights, and which are
compatible with natural projection and branching maps. We apply this to
the limit construction of local Weyl modules and obtain a new character
formula for the basic representation of $\widehat{\mathfrak{sl}_n}$.

Manish Patnaik
(University of Alberta, Edmonton, Canada) 
Feb 9, 2024 
Whittaker
functions on covers of padic groups and quantum groups at roots of
unity 
(Joint with the Number
Theory Seminar) 
(LH1 –
2 pm, Fri)


Abstract.
Since the work of Kubota in the late 1960s, it has been known that
certain Gauss sum twisted (multiple) Dirichlet series are closely
connected to a theory of automorphic functions on metaplectic covering
groups. The representation theory of such covering groups was then
initiated by Kazhdan and Patterson in the 1980s, who emphasized the role
of a certain nonuniqueness of Whittaker functionals.
Motivated on the one hand by the recent theory of Weyl group multiple
Dirichlet series, and on the other by the socalled "quantum" geometric
Langlands correspondence, we explain how to connect the representation
theory of metaplectic covers of $p$adic groups to an object of rather
disparate origin, namely a quantum group at a root of unity. This gives
us a new point of view on the nonuniqueness of Whittaker functionals and
leads, among other things, to a Casselman–Shalika type formula
expressed in terms of (Gauss sum) twists of
"$q$"Littlewood–Richardson coefficients, objects of some
combinatorial interest.
Joint work with Valentin Buciumas.

Rameez Raja (NIT
Srinagar) 
Jan 24, 2024 
Some
combinatorial structures realized by commutative rings 
(LH1 –
2:30 pm, Wed)


Abstract.
There are many ways to associate a graph (combinatorial structure) to a
commutative ring $R$ with unity. One of the ways is to associate a
zerodivisor graph $\Gamma(R)$ to $R$. The vertices of $\Gamma(R)$
are all elements of $R$ and two vertices $x, y \in R$ are adjacent in
$\Gamma(R)$ if and only if $xy = 0$. We shall discuss Laplacian of a
combinatorial structure $\Gamma(R)$ and show that the representatives of
some algebraic invariants are eigenvalues of the Laplacian of
$\Gamma(R)$. Moreover, we discuss association of another combinatorial
structure (Young diagram) with $R$. Let $m, n \in \mathbb{Z}_{>0}$ be two
positive integers. The Young's partition lattice $L(m,n)$ is defined to
be the poset of integer partitions $\mu = (0 \leq \mu_1 \leq \mu_2 \leq
\cdots \leq \mu_m \leq n)$. We can visualize the elements of this poset
as Young diagrams ordered by inclusion. We conclude this talk with a
discussion on Stanley's conjecture regarding symmetric saturated chain
decompositions (SSCD) of $L(m,n)$.



Bharatram
Rangarajan
(Einstein Institute of Mathematics, Hebrew University of Jerusalem,
Israel) 
Jan 10, 2024 
"Almost"
Representations and Group Stability 
(LH1 –
4 pm, Wed)


Abstract.
Consider the following natural robustness question: is an
almosthomomorphism of a group necessarily a small deformation of a
homomorphism? This classical question of stability goes all the way back
to Turing and Ulam, and can be posed for different target groups, and
different notions of distance. Group stability has been an active line of
study in recent years, thanks to its connections to major open problems
like the existence of nonsofic and nonhyperlinear groups, the group
Connes embedding problem and the recent breakthrough result MIP*=RE,
apart from property testing and errorcorrecting codes.
In this talk, I will survey some of the main results, techniques, and
questions in this area.


Sridhar
Venkatesh
(University of Michigan, Ann Arbor, USA) 
Dec 28, 2023 
The Du Bois complex and some associated
singularities 
(LH1 –
11:30 am, Thu)


Abstract.
For a smooth variety $X$ over $\mathbb{C}$, the de Rham complex of $X$ is
a powerful tool to study the geometry of $X$ because of results such as
the degeneration of the Hodgede Rham spectral sequence (when $X$ is
proper). For singular varieties, it follows from the work of Deligne and
Du Bois that there is a substitute called the Du Bois complex which
satisfies many of the nice properties enjoyed by the de Rham complex in
the smooth case. In this talk, we will discuss some classical
singularities associated with this complex, namely Du Bois and rational
singularities, and some recently introduced refinements, namely $k$Du
Bois and $k$rational singularities. This is based on joint work with
Wanchun Shen and Anh Duc Vo.

Ravindra Girivaru
(University of Missouri, St. Louis, USA) 
Dec 20, 2023 
Matrix factorisations of polynomials 
(LH3 –
4 pm, Wed)


Abstract.
A matrix factorisation of a polynomial $f$ is an equation $AB = f \cdot
{\rm I}_n$ where $A,B$ are $n \times n$ matrices with polynomial entries
and ${\rm I}_n$ is the identity matrix. This question has been of
interest for more than a century and has been studied by mathematicians
like L.E. Dickson. I will discuss its relation with questions arising in
algebraic geometry about the structure of subvarieties in projective
hypersurfaces.

R. Venkatesh
(IISc Mathematics) 
Dec 13, 2023 
A simple proof for the characterization of chordal
graphs using Horn hypergeometric series 
(LH3 –
11:30 am, Wed)


Abstract.
Let $G$ be a finite simple graph (with no loops and no multiple edges),
and let $I_G(x)$ be the multivariate independence polynomial of $G$. In
2021, Radchenko and Villegas proved the following interesting
characterization of chordal graphs, namely $G$ is chordal if and only if
the power series $I_G(x)^{1}$ is Horn hypergeometric. In this talk, I
will give a simpler proof of this fact by computing $I_G(x)^{1}$
explicitly using multicoloring chromatic polynomials. This is a joint
work with Dipnit Biswas and Irfan Habib.

Duncan Laurie (University of
Oxford, UK) 
Dec 8, 2023 
The structure and representation theory of quantum
toroidal algebras 
(LH3 –
3 pm, Fri)


Abstract.
Quantum toroidal algebras are the next class of quantum affinizations
after quantum affine algebras, and can be thought of as "double affine
quantum groups". However, surprisingly little is known thus far about
their structure and representation theory in general.
In this talk we'll start with a brief recap on quantum groups and the
representation theory of quantum affine algebras. We shall then introduce
and motivate quantum toroidal algebras, before presenting some of the
known results. In particular, we shall sketch our proof of a braid group
action, and generalise the socalled Miki automorphism to the simply
laced case.
Time permitting, we shall discuss future directions and applications
including constructing representations of quantum toroidal algebras
combinatorially, written in terms of Young columns and Young walls.

Shashank Kanade
(University of Denver, USA) 
Dec 1, 2023 
A glimpse into the world of Rogers–Ramanujan
identities 
(LH3 – 11
am, Fri)


Abstract.
I will give a gentle introduction to the combinatorial
Rogers–Ramanujan identities. While these identities are over a
century old, and have many proofs, the first representationtheoretic
proof was given by Lepowsky and Wilson about four decades ago.
Nowadays, these identities are ubiquitous in several areas of
mathematics and physics. I will mention how these identities arise from
affine Lie algebras and quantum invariants of knots.

Sagar Shrivastava
(TIFR, Mumbai) 
Nov 28, 2023 
Branching multiplicity of symplectic groups as $SL_2$
representations 
(LH1 –
11:30 am, Tue)


Abstract.
Branching rules are a systematic way of understanding the multiplicity of
irreducible representations in restrictions of representations of Lie
groups. In the case of $GL_n$ and orthogonal groups, the branching rules
are multiplicity free, but the same is not the case for symplectic
groups. The explicit combinatorial description of the multiplicities was
given by Lepowsky in his PhD thesis. In 2009, Wallach and Oded showed
that this multiplicity corresponds to the dimension of the multiplicity
space, which was a representation of $SL_2(=Sp(2))$. In this talk, we
give an alternate proof of the same without invoking any partition
function machinery. The only assumption for this talk would be the Weyl
character formula.

Rekha Biswal
(NISER, Bhubaneswar) 
Nov 17, 2023 
Ideals in
enveloping algebras of affine Kac–Moody algebras 
(LH3 –
11:30 am, Fri)


Abstract.
In this talk, I will discuss about the structure of ideals in enveloping
algebras of affine Kac–Moody algebras and explain a proof of the
result which states that if $U(L)$ is the enveloping algebra of the
affine Lie algebra $L$ and "$c$" is the central element of $L$, then any
proper quotient of $U(L)/(c)$ by two sided ideals has finite
Gelfand–Kirillov dimension. I will also talk about the applications
of the result including the fact that $U(L)/(c\lambda)$ for non zero
$\lambda$ is simple. This talk is based on joint work with Susan J.
Sierra.

Subhajit Ghosh (BarIlan University, RamatGan,
Israel) 
Nov 15, 2023 
Aldoustype
spectral gap results for the complete monomial group 
(LH1 – 3
pm, Wed)


Abstract.
Let us consider the continuoustime random walk on $G\wr S_n$, the
complete monomial group of degree $n$ over a finite group $G$, as
follows: An element in $G\wr S_n$ can be multiplied (left or right) by an
element of the form
 $(u,v)_G:=(\mathbf{e},\dots,\mathbf{e};(u,v))$ with rate
$x_{u,v}(\geq 0)$, or

$(g)^{(w)}:=(\dots,\mathbf{e},g,\mathbf{e},\dots;\mathbf{id})$ with rate
$y_w\alpha_g\; (y_w \gt 0,\;\alpha_g=\alpha_{g^{1}}\geq 0)$,
such that $\{(u,v)_G,(g)^{(w)}:x_{u,v} \gt 0,\;y_w\alpha_g \gt 0,\;1\leq u \lt v
\leq n,\;g\in G,\;1\leq w\leq n\}$ generates $G\wr S_n$. We also consider
the continuoustime random walk on $G\times\{1,\dots,n\}$ generated by
one natural action of the elements $(u,v)_G,1\leq u \lt v\leq n$ and
$g^{(w)},g\in G,1\leq w\leq n$ on $G\times\{1,\dots,n\}$ with the
aforementioned rates. We show that the spectral gaps of the two random
walks are the same. This is an analogue of the Aldous' spectral gap
conjecture for the complete monomial group of degree $n$ over a finite
group $G$.

V. Sathish Kumar
(IMSc, Chennai) 
Sep 22, 2023 
Unique factorization for tensor products of parabolic
Verma modules 
(LH1 – 3
pm, Fri)


Abstract.
Let $\mathfrak g$ be a symmetrizable KacMoody Lie algebra with Cartan
subalgebra $\mathfrak h$. We prove that unique factorization holds for
tensor products of parabolic Verma modules. We prove more generally a
unique factorization result for products of characters of parabolic Verma
modules when restricted to certain subalgebras of $\mathfrak h$. These
include fixed point subalgebras of $\mathfrak h$ under subgroups of
diagram automorphisms of $\mathfrak g$. This is joint work with
K.N. Raghavan, R. Venkatesh and S. Viswanath.


Abstract.
The Alpha invariant of a complex Fano manifold was introduced by Tian to
detect its Kstability, an algebraic condition that implies the existence
of a Kähler–Einstein metric. Demailly later reinterpreted the
Alpha invariant algebraically in terms of a singularity invariant called
the log canonical threshold. In this talk, we will present an analog of
the Alpha invariant for Fano varieties in positive characteristics,
called the FrobeniusAlpha invariant. This analog is obtained by
replacing "log canonical threshold" with "Fpure threshold", a
singularity invariant defined using the Frobenius map. We will review the
definition of these invariants and the relations between them. The main
theorem proves some interesting properties of the FrobeniusAlpha
invariant; namely, we will show that its value is always at most 1/2 and
make connections to a version of local volume called the Fsignature.

Sutanay Bhattacharya (University of California, San
Diego, USA) 
Aug 21, 2023 
The lattice of nilHecke algebras over reflection
groups 
(LH1 – 11:30
am, Mon)


Abstract.
Associated to every reflection group, we construct a lattice of quotients
of its braid monoidalgebra, which we term nilHecke algebras, obtained
by killing all "sufficiently long" braid words, as well as some integer
power of each generator. These include usual nilCoxeter algebras,
nilTemperley–Lieb algebras, and their variants, and lead to
symmetric semigroup module categories which necessarily cannot be
monoidal.
Motivated by the classical work of Coxeter (1957) and the
Broue–Malle–Rouquier freeness conjecture, and continuing
beyond the previous work of Khare, we attempt to obtain a classification
of the finitedimensional nilHecke algebras for all reflection groups
$W$. These include the usual nilCoxeter algebras for $W$ of finite type,
their "fully commutative" analogues for $W$ of FCfinite type, three
exceptional algebras (of types $F_4$,$H_3$,$H_4$), and three exceptional
series (of types $B_n$ and $A_n$, two of them novel). We further uncover
combinatorial bases of algebras, both known (fully commutative elements)
and novel ($\overline{12}$avoiding signed permutations), and classify
the Frobenius nilHecke algebras in the aforementioned cases. (Joint with
Apoorva Khare.)

2020–23
2019–20
2018–19
