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.

Bharatram
Rangarajan
(Einstein Institute of Mathematics, Hebrew University of Jerusalem,
Israel) 
Oct 17, 2024 
Expansion, Group Homomorphism Testing, and
Cohomology 
(LH1 –
4 pm, Thu)


Abstract.
Expansion in groups (or their Cayley graphs) is a valuable and
wellstudied notion in both mathematics and computer science, and
describes a robust form of connectivity of graphs (a gap property of
fixed points of representations of groups). It can also be interpreted as
a graph on which connectivity is efficiently locally testable.
Group stability, on the other hand, is concerned with another robustness
property – but of homomorphisms (or representations). Namely, is an
almosthomomorphism of a group necessarily a small deformation of a
homomorphism? This too can be interpreted as a local testability property
of group homomorphisms in the right settings.
Expansion in groups (or property (T)) had been classically reformulated
in the language of algebraic topology – in terms of the vanishing
of the first cohomology of the group. In this talk we will see approaches
in capturing group stability in terms of the vanishing of a second
cohomology of the group, motivating higherdimensional generalizations of
expansion.
Based on joint (previous and ongoing) works with Monod, Glebsky,
Lubotzky, FournierFacio, Dogon.

Prateek Kumar
Vishwakarma
(IISc Mathematics) 
Oct 9, 2024 
Plücker inequalities for weakly separated
coordinates in the TNN Grassmannian 
(LH1 –
4:15 pm, Wed)


Abstract.
We show that the partial sums of the long Plücker relations for
pairs of weakly separated Plücker coordinates oscillate around 0 on
the totally nonnegative part of the Grassmannian. Our result generalizes
the classical oscillating inequalities by Gantmacher–Krein (1941)
and recent results on totally nonnegative matrix inequalities by
Fallat–Vishwakarma (2024). In fact we obtain a characterization of
weak separability, by showing that no other pair of Plücker
coordinates satisfies this property.
Weakly separated sets were initially introduced by Leclerc and Zelevinsky
and are closely connected with the cluster algebra of the Grassmannian.
Moreover, our work connects several fundamental objects such as weak
separability, Temperley–Lieb immanants, and Plücker relations,
and provides a very general and natural class of additive determinantal
inequalities on the totally nonnegative part of the Grassmannian. This is
joint work with Daniel Soskin.


Abstract.
We discuss an algebraic version of Schoenberg's celebrated theorem
[Duke Math. J., 1942] characterizing entrywise matrix transforms
that preserve positive definiteness. Compared to the classical real and
complex settings, we consider matrices with entries in a finite field and
obtain a complete characterization of such preservers for matrices of a
fixed dimension. When the dimension of the matrices is at least 3, we
prove that, surprisingly, the positivity preservers are precisely the
positive multiples of the field's automorphisms. Our work makes crucial
use of the wellknown charactersum bound due to Weil, and of a result of
Carlitz [Proc. Amer. Math. Soc., 1960] that leads to
characterizing the automorphisms of Paley graphs. This is joint with
Dominique Guillot and Himanshu Gupta.

Prateek Misra
(Technical University of Munich, Germany) 
Sep 5, 2024 
Graphical models, causality and algebraic
perspectives 
(LH3 –
4 pm, Thu)


Abstract.
Algebraic Statistics is a relatively new field of research where tools
from Algebraic Geometry, Combinatorics and Commutative Algebra are used
to solve statistical problems. A key area of research in this field is
the Gaussian graphical models, where the dependence structure between
jointly normal random variables is determined by a graph. In this talk, I
will explain the algebraic perspectives on Gaussian graphical models and
present some of my key results on understanding the defining equations of
these models. In the end, I will talk about the problem of structural
identifiability and causal discovery and how algebraic techniques can be
implemented to tackle them.

2023–24
2020–23
2019–20
2018–19
