October 21, 2019 - October 25, 2019
Public Access Category: All |
- 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.