- Time:
- 11:30am-12:30pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- Commutative Algebra seminar I.

Speaker: Soumi Tikader.

Affiliation: ISI Kolkata.

Date and Time: Monday 21 October, 11:30 am - 12:30 pm.

Venue: Ramanujan Hall, Department of Mathematics.

Title: Orbit spaces of unimodular rows over smooth real affine algebras.

Abstract: In this talk we will discuss about the group structure on orbit

spaces of unimodular rows over smooth real affine algebras. With a few

definition and some results to start, we will prove a structure theorem of

elementary orbit spaces of unimodular rows over aforementioned ring with

the help of similar kind results on Euler class group. As a consequences,

we will prove that :

Let $X=Spec(R)$ be a smooth real affine variety of even dimension $d > 1$,

whose real points $X(R)$ constitute an orientable manifold. Then the set

of isomorphism classes of (oriented) stably free $R$ of rank $d > 1$ is a

free abelian group of rank equal to the number of compact connected

components of $X(R)$.

In contrast, if $d > 2$ is odd, then the set of isomorphism classes of

stably free $R$-modules of rank $d$ is a $Z/2Z$-vector space (possibly

trivial). We will end this talk by giving a structure theorem of Mennicke

symbols.

- Time:
- 3:00pm-4:00pm
- Location:
- Room No 216 Department of Mathematics
- Description:
- Combinatorics seminar.

Speaker: Projesh Nath Choudhury.

Affiliation: IISc Bengaluru.

Date and Time: Monday 21 October, 3:00 pm - 4:00 pm.

Venue: Room 216, Department of Mathematics.

Title: Distance matrices of trees: invariants, old and new.

Abstract: In 1971, Graham and Pollak showed that if $D_T$ is the distance

matrix of a tree $T$ on $n$ nodes, then $\det(D_T)$ depends only on $n$,

not $T$. This independence from the tree structure has been verified for

many different variants of weighted bi-directed trees. In my talk:

1. I will present a general setting which strictly subsumes every known

variant, and where we show that $\det(D_T)$ - as well as another graph

invariant, the cofactor-sum - depends only on the edge-data, not the

tree-structure.

2. More generally - even in the original unweighted setting - we

strengthen the state-of-the-art, by computing the minors of $D_T$ where

one removes rows and columns indexed by equal-sized sets of pendant nodes.

(In fact, we go beyond pendant nodes.)

3. We explain why our result is the "most general possible", in that

allowing greater freedom in the parameters leads to dependence on the

tree-structure.

4. Our results hold over an arbitrary unital commutative ring. This uses

Zariski density, which seems to be new in the field, yet is richly

rewarding.

We then discuss related results for arbitrary strongly connected graphs,

including a third, novel invariant. If time permits, a formula for

$D_T^{-1}$ will be presented for trees $T$, whose special case answers an

open problem of Bapat-Lal-Pati (Linear Alg. Appl. 2006), and which extends

to our general setting a result of Graham-Lovasz (Advances in Math. 1978).

(Joint with Apoorva Khare)

- Time:
- 11:00am
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:

Speaker: Samarpita Ray.

Affiliation: IISc Bengaluru.

Date and Time: Tuesday 22 October, 11:00 am - 12:00 noon.

Venue: Ramanujan Hall, Department of Mathematics.

Title: Some results on spectral spaces and spectral sequences.

Abstract: In this talk, I will present an overview of my research works

and further plans. As part of my thesis work, I have worked on two

different topics which straddle the fields of commutative algebra,

algebraic geometry and category theory. One of my problems is related to

the area of algebraic geometry over the "field with one element"

($\mathbb{F}_1$), several notions of which has been developed in the last

twenty years. It is in this context that monoids became topologically and

geometrically relevant objects of study. Spectral spaces, introduced by

Hochster, are topological spaces homeomorphic to the spectrum of a ring

and are widely studied in the literature. In our work, we present several

naturally occurring classes of spectral spaces using commutative algebra

on pointed monoids. For this purpose, our main tools are finite type

closure operations and continuous valuations on monoids which we introduce

and study in this work.

The other problem involves categorical generalization of certain Hopf

algebra results and a study of their cohomology using Grothendieck

spectral sequence.

It builds on B. Mitchel's famous "ring with several objects" viewpoint of

an arbitrary small preadditive category. In this respect, for a Hopf

algebra H, an H-category will denote an "H-module algebra with several

objects" and a co-H-category will denote an "H-comodule algebra with

several objects". Modules over such Hopf categories were first considered

by Cibils and Solotar. We present a study of cohomology in such module

categories using Grothendieck spectral sequences. I will briefly talk

about these thesis projects and also my further works in this direc

- Time:
- 2:15pm-3:15pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- Mathematics Colloquium I.

Speaker: Vishal Vasan.

Affiliation: ICTS Bengaluru.

Date and Time: Wednesday 23 October, 2:15 pm - 3:15 pm.

Venue: Ramanujan Hall, Department of Mathematics.

Title: Two unexpected applications of boundary value problems.

Abstract: Partial differential equations (PDE) and their boundary value

problems (BVP) arise naturally in a number of applications. Typically the

system of interest is modelled by a PDE/BVP. In this talk, I will present

two unexpected applications of BVPs where the original system does not

immediately indicate their importance. The first application comes from

the study of a particle moving in a fluid whose motion is modelled by a

finite dimensional system. The analysis will imply a natural

interpretation to the half derivative in terms of boundary-value problems.

The second application comes from the classical study of dispersive shock

waves (DSWs). DSWs are specific solutions to nonlinear dispersive

equations. However, I will present a BVP for a linear equation which

reproduces a number of DSW features. This raises an important question on

how to match experimental DSWs with particular nonlinear models:

qualitative comparisons do not suffice.

- Time:
- 2:30pm-3:30pm
- Location:
- Room No. 113, Department of Mathematics
- Description:
- Number theory seminar I.

Speaker: Vinayak Vatsal.

Affiliation: UBC, Vancouver.

Date and Time: Wednesday 23 October, 2:30 pm - 3:30 pm.

Venue: Room 113, Department of Mathematics.

Title: Iwasawa theory for Artin representations.

Abstract: An Artin representation is simply a finite dimensional complex

representation of the Galois group of a finite extension of the rational

number field. Despite their apparent simplicity, Artin representations are

very complicated and much harder to study than apaprently more complicated

representations such as those attached to elliptic curves, and much of the

theory remains conjectural.

In this talk I will survey an aspect of the theory where Artin

representations are actually simpler and more concrete than other kinds of

representations.

- Time:
- 11:45am-12:45pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- Speaker: Amit Kumar Singh.

Affiliation: IIT Madras.

Date and Time: Thursday 24 October, 11:45 am - 12:45 pm.

Venue: Ramanujan Hall, Department of Mathematics.

Title: Semi-stability of certain vector bundles on elliptic curves.

Abstract: Abstract Let L be a line bundle of degree d on an elliptic curve C and ϕ : C → P

n

is a morphism

given by a sub-linear system of the complete linear system |L| of dimension n + 1. When d = 4, n

= 2, we prove that ϕ

∗TPn is semi-stable if deg(ϕ(C)) > 1. Moreover, we prove that ϕ

∗TPn is isomorphic to direct sum of two isomorphic line bundles if and only if deg(ϕ(C)) = 2. Conversely, for any

rank two semi-stable vector bundle E on an elliptic curve C of degree 4, there is a non-degenerate

morphism ϕ :C → P

n

such that ϕ

∗TPn (−1) = E. More precisely, E is isomorphic to direct sum of two

isomorphic line bundles if and only if deg(ϕ(C)) = 2. Further E is either indecomposable or direct

sum of non-isomorphic line bundles if and only if deg(ϕ(C)) = 4. When d = 5, n = 3, we compute

the Harder-Narasimhan filtration of ϕ

∗TPn .

- Time:
- 3:30pm-5:00pm
- Location:
- Room No. 215 Department of Mathematics
- Description:
- Commutative Algebra seminar II.

Speaker: Tony Puthenpurakal.

Affiliation: IIT Bombay.

Date and Time: Thursday 24 October, 3:30 pm - 5:00 pm.

Venue: Room 215, Department of Mathematics.

Title: Triangulated categories-I,II,III.

Abstract: We define and give elementary properties of triangulated

categories. We also give an application of triangulated categories to

linkage theory in commutative algebra.

- Time:
- 2:30pm-3:30pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- Number theory seminar II.

Speaker: Prasuna Bandi.

Affiliation: TIFR Mumbai.

Date and Time: Friday 25 October, 2:30 pm - 3:30 pm.

Venue: Ramanujan Hall, Department of Mathematics.

Title: Simultaneous density of integer values for an inhomogeneous

quadratic form and a linear form.

Abstract: In 1929 Oppenheim conjectured that for a nondegenerate,

indefinite, irrational quadratic form Q in n ≥ 5 variables, Q(Zn) is

dense in R. It was later strengthened to n ≥ 3 by Davenport and

proved in 1987 by Margulis based on Raghunathan’s conjecture on closures

of unipotent orbits.

Later, Dani and Margulis proved the simultaneous density at integer values

for a pair of quadratic and linear form in 3 variables when certain

conditions are satisfied. We prove an analogue of this for the case of an

inhomogeneous quadratic form and a linear form.

- Time:
- 4:00pm-5:00pm
- Location:
- Room No. 215 Department of Mathematics
- Description:
- Number theory seminar III.

Speaker: Shaunak Deo.

Affiliation: TIFR Mumbai.

Date and Time: Friday 25 October, 4:00 pm - 5:00 pm.

Venue: Room 215, Department of Mathematics.

Title: Effect of level raising on pseudo-deformation rings.

Abstract: Given a prime p, an integer N and a 2 dimensional

pseudo-representation of G_{Q,Np} over a finite field of characteristic p,

we will analyze how the structure of the universal pseudo-deformation ring

changes after allowing ramification at a prime $\ell$ not dividing Np.

This question has been studied by Boston and Bockle for deformation rings

of absolutely irreducible representations and Borel representations,

respectively. As a related question, we will also determine when a

pseudo-representation arises from an actual representation. The talk will

begin with a brief survey of the theory of pseudo-representations.

- Time:
- 4:00pm-5:00pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- Mathematics Colloquium II.

Speaker: Tathagata Basak.

Affiliation: Iowa State University.

Date and Time: Friday 25 October, 4:00 pm - 5:00 pm.

Venue: Ramanujan Hall, Department of Mathematics.

Title: A couple of curious reflection groups.

Abstract: Reflection groups occur all over representation theory and

geometry. We want to begin with a quick survey of finite reflection

groups, talk a little about classifying them and their connections to some

other areas of mathematics.

Then we want to focus on two examples of hyperbolic reflection groups; one

real and one complex. Both examples involve the Leech lattice; the lattice

that produces the best packing of spheres in 24 dimensional Euclidean

space. Both examples are (probably) related to the largest sporadic finite

simple group known as the monster. The connection in the complex case is

still a conjecture.

We will not assume any previous familiarity with hyperbolic reflection

groups or the Leech lattice or the Monster.