- Time:
- 3:00pm - 4:00pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- Probability and Statistics seminar (It will be a talk by videoconference).

Speaker: Priyanka Majumder.

Affiliation: Indian Institute of Engineering Science and Technology, Shibpur.

Date and Time: Tuesday 01 October, 3:00 pm - 4:00 pm.

Venue: Ramanujan Hall, Department of Mathematics.

Title: On Certain Probabilistic and Inferential Aspects of Reliability

Theory.

Abstract: Several nonparametric ageing classes have been established in

literature based on the various reliability characteristics. The talk will

cover certain probabilistic issues such as, reliability bound, moment

bound, closure under the formation of weak limits, characterisation

theorem, etc., which have been hitherto unknown in the literature, for the

New Better than Average Failure Rate (NBAFR) class of life distributions.

We further explore the validity of these results in the context of a more

general ageing class related to NBAFR family.

In the inferential part of my presentation, I will discuss a problem of

testing exponentiality against an alternative which is dened based on the

Laplace-Stieltjes transform, namely the so-called L-class. The asymptotic

distributions of our scale-invariant test statistics are derived and

consistency of the test established. General expressions of the local

approximate Bahadur eciencies for the test statistics are obtained and

evaluated for typical alternatives. The performance of the test is

assessed by means of a simulation study and through application to some

real life data sets.

- Time:
- 4:30pm - 5:30pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- Coding Theory seminar.

Speaker: Prasant Singh.

Affiliation: Arctic University of Norway, Tromso.

Date and Time: Thursday 03 October, 4:30 pm - 5:30 pm.

Venue: Ramanujan Hall, Department of Mathematics.

Title: Decoding Grassmann codes using lines in Grassmannians.

Abstract: The talk is based on my joint work with Prof Peter Beelen. In

this talk, we recall the notion of Majority logic decoding for binary

codes introduced by Reed in the late 60s. We also recall some basic

notions of coding theory, construction of Grassmann codes and its basic

parameters. Lines in Grassmannians are closely related to parity checks of

Grassmann codes. We exploit this property of Grassmann codes together with

several other properties of point-line geometry of Grassmannian to

construct certain kind of parity checks for these codes. In the end, we

use these parity checks and the idea of Reed's majority logic decoding to

correct certain errors for Grassmann codes.

- Time:
- 5:15pm - 6:15pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- IITB Mathematics Colloquium via videoconference.

Speaker: Jean Dolbeault.

Affiliation: University of Paris IX - Dauphine.

Date and Time: Monday 07 October, 5:15 pm - 6:15 pm.

Venue: Ramanujan Hall, Department of Mathematics.

Title: Hypocoercivity.

Abstract: The purpose of hypocoercivity is to obtain rates for solutions

of non-purely diffusive equations, in asymptotic regimes. This is a very

useful technique for kinetic equations. After reviewing some easy results

based on hypo-ellipticity, the lecture will focus on linear kinetic

equations without regularizing effects and the L2 hypocoercivity method.

Some motivations will be introduced, with a toy model. The core of the

lecture will be a theoretical result based on a joint work with C. Mouhot

and C. Schmeiser. Initially intended for systems with compactness or

confinement in position space and simple local equilibira, the method has

been extended to various local equilibria in velocities and non-compact

situations in positions. It is also flexible enough to include non-local

transport terms associated with Poisson coupling. Some recent results rely

on various, deep functional inequalities. An application to the linearized

Vlasov-Poisson-Fokker-Planck system will also be briefly presented.

- Time:
- 10:15am - 11:15am
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- Combinatorics seminar

Speaker: Sudeep Stephen.

Affiliation: National University of Singapore.

Date and Time: Wednesday 09 October, 10:15 am - 11:15 am.

Venue: Ramanujan Hall, Department of Mathematics.

Title: Zero Forcing in Graphs.

Abstract: Fo a two-colouring of the vertex set of a simple graph G = (V,E), consider the following colour-change rule: a red vertex is converted to blue if it is the only red neighbour of some blue vertex. A vertex set S ⊆ V is called zero-forcing if, starting with the vertices in S blue and the vertices in the complement V \ S red, all the vertices can be converted to blue by repeatedly applying the colour-change rule. The minimum cardinality of a zero-forcing set for the graph G is called the zero-forcing number of G, denoted by Z(G). This concept was introduced by the AIM Minimum Rank –Special Graphs Work Group in [1] as a tool to bound the minimum rank of matrices associated with the graph G. In this talk, I shall give an overview of the zero forcing problem along with some of the results that we have obtained during my Ph.D candidature. To conclude, I shall state few open problems that I intend to tackle along with my mentors. References [1] AIM Minimum Rank –Special Graphs Work Group. Zero forcing sets and the minimum rank of graphs. Linear Algebra and its Applications, 428(7):16281648, 2008.

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

Speaker: Charu Goel.

Affiliation: IIIT Nagpur.

Date and Time: Wednesday 09 October, 4:00 pm - 5:00 pm.

Venue: Ramanujan Hall, Department of Mathematics.

Title: Nonnegative Polynomials and Sums of Squares

Abstract:

Abstract. Sums of squares representations of polynomials is of fundamental importance in real algebraic geometry and goes back to the 1888 seminal paper of Hilbert. The main theorem in his paper is a full characterisation of all pairs (n,d) for which every nonnegative polynomial of a fixed degree d in a given number of variables n is a sum of squares of polynomials. Ninety years later, Choi and Lam asserted that this characterisation remains unchanged for symmetric forms. In this talk first some key observations and problems related to Hilbert’s theorem will be discussed. We then complete the above assertion of Choi-Lam. Along the way, we shall also discuss briefly how test sets for positivity of symmetric polynomials play an important role in establishing this assertion.

- Time:
- 9:00am - 10:00am
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- CACAAG seminar I.

Speaker: Charu Goel.

Affiliation: IIIT Nagpur.

Date and Time: Thursday 10 October, 9:00 am - 10:00 am.

Venue: Ramanujan Hall, Department of Mathematics.

Title:

The analogue of Hilbert’s 1888 Theorem for even symmetric forms

Abstract:

Abstract. Hilbert in 1888 studied the inclusion Pn,2d ⊇ Σn,2d, where Pn,2d and Σn,2d are respectively the cones of positive semidefinite forms and sum of squares forms of degree 2d in n variables. He proved that: “Pn,2d = Σn,2d if and only if n = 2,d = 1, or (n,2d) = (3,4)”. In order to establish that Σn,2d Pn,2d for all the remaining pairs, he demonstrated that Σ3,6 P3,6, Σ4,4 P4,4, thus reducing the problem to these two basic cases. In 1976, Choi and Lam considered the same inclusion for symmetric forms and claimed that Hilbert’s characterisation above remains unchanged. They demonstrated that establishing the strict inclusion reduces to show it just for the basic cases (3,6),(n,4)n≥4. In this talk, we will explain the algebraic geometric ideas behind these reductions and how we extended these methods to investigate the above inclusion for even symmetric forms. We will present our leading tool a “Degree Jumping Principle”, an analogue of reduction to basic cases and construction of explicit counterexamples for the basic pairs. As a complete analogue of Hilbert’s theorem for even symmetric forms, we establish that “an even symmetric n-ary 2d-ic psd form is sos if and only if n = 2 or d = 1 or (n,2d) = (n,4)n≥3 or (n,2d) = (3,8)”. This is a joint work with S. Kuhlmann and B. Reznick.

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

Speaker: Sudarshan Gurjar.

Affiliation: IIT Bombay.

Date and Time: Thursday 10 October, 3:30 pm - 5:00 pm.

Venue: Room 215, Department of Mathematics.

Title: Introduction to Vector Bundles.

Abstract: I will introduce the subject of algebraic vector bundles on

projective varieties in this talk. Vector bundles are used in commutative

algebra in several contexts. For example, they provide geometric

interpretation of tight closure of an ideal. They were used to show that

tight closure does not commute with localization. A subtle notion of the

semistability of vector bundles plays an important role in this subject. I

will try to explain the relevance of this notion and discuss some

examples. This talk will be a prequel to a talk by Prof. Nitin Nitsure on

October 14.

- Time:
- 4:30pm - 5:30pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- CACAAG seminar II.

Speaker: Mrinal Kumar.

Affiliation: Computer Science Department, IIT Bombay.

Date and Time: Friday 11 October, 4:30 pm - 5:30 pm.

Venue: Ramanujan Hall, Department of Mathematics.

Title: Some applications of the Polynomial Method in Combinatorics.

Abstract: In the next couple of lectures, we will see some applications of

the so called Polynomial Method to problems in Combinatorics. We will

focus on the following three applications:

1. Joints Problem: For a set L of lines in R^3, a point p in R^3 is said

to be a joint in L if there are at least three non-coplanar lines in L

which pass through p. We will discuss a result of Guth and Katz who showed

an upper bound on the maximal number of joints in an arrangement of N

lines.

2. Lower bounds on the size of Kakeya sets over finite fields: For a

finite field F, a Kakeya set is a subset of F^n that contains a line in

every direction. We will discuss a result of Dvir showing a lower bound

of C_n*q^n on the size of any Kakeya set over F^n, where C_n only depends

on n and F is a finite field of size q.

3. Upper bounds on the size of 3-AP free sets over finite fields: We will

then move on to discuss a recent result of Ellenberg and Gijswijt who

showed that if F is a finite field with three elements, and S is a subset

of of F^n such that S does not that does not contain three elements in an

arithmetic progression, then |S| is upper bounded by c^n for a constant c

< 3.

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

Speaker: Nitin Nitsure.

Affiliation: TIFR Mumbai.

Date and Time: Monday 14 October, 3:30 pm - 5:00 pm.

Venue: Room 215, Department of Mathematics.

Title: Vector bundles, stability and Harder-Narasimhan filtration.

Abstract: Vector bundles in geometry generalize projective modules in

algebra. They are the simplest sort of coherent sheaves. We will introduce

the all important notion of stability of vector bundles. The stable

bundles have nice properties, and luckily, most bundles are stable. But

even those bundles that are not stable can be analysed in terms of stable

bundles. This is done by the notion of the Harder-Narasimhan filtration of

a bundle. We will give a sketch of the theory and illustrate it with

examples.

- Time:
- 5:30pm - 7:00pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- Probability and Statistics seminar.

Speaker: Probal Chaudhuri.

Affiliation: ISI Kolkata.

Date and Time: Tuesday 15 October, 05:30 pm - 07:00 pm.

Venue: Ramanujan Hall, Department of Mathematics.

Title: Learning Mahalanobis distance from C R Rao.

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

Speaker: Probal Chaudhuri.

Affiliation: ISI Kolkata.

Date and Time: Wednesday 16 October, 2:15 pm - 3:15 pm.

Venue: Ramanujan Hall, Department of Mathematics.

Title: Minimization by Monte-Carlo.

- Time:
- 4:15pm - 5:15pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- CACAAG seminar.

Speaker: Mrinal Kumar.

Affiliation: Computer Science Department, IIT Bombay.

Date and Time: Friday 18 October, 4:15 pm - 5:15 pm.

Venue: Ramanujan Hall, Department of Mathematics.

Title: Introduction to the Polynomial Method in Combinatorics II (Lower

bounds on Kakeya sets over finite fields).

Abstract: For a finite field F, a Kakeya set is a subset of F^n that

contains a line in every direction. We will discuss a result of Dvir

showing a lower bound of C_n*q^n on the size of any Kakeya set over F^n,

where C_n only depends on n and F is a finite field of size q.

- 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.

- Time:
- 11:30am - 12:30pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- Probability and Statistics seminar.

Speaker: Bimal Roy.

Affiliation: ISI Kolkata.

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

Venue: Ramanujan Hall, Department of Mathematics.

Title: Overview on test of randomness of a binary sequence and its

application in cryptography.

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

Speaker: Bimal Roy.

Affiliation: ISI Kolkata.

Date and Time: Wednesday 30 October, 4:00 pm - 5:00 pm.

Venue: Ramanujan Hall, Department of Mathematics.

Title: Application of statistics in cryptography.

- Time:
- 11:45am
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- Commutative Algebra seminar (Please note that this talk will be via Skype).

Speaker: Soumi Tikader.

Affiliation: ISI Kolkata.

Date and Time: Thursday 31 October, 11:45 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.

PS: Soumi Tikader is a post doctoral candidate.

- 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 31 October, 3:30 pm - 5:00 pm.

Venue: Room 215, Department of Mathematics.

Title: Triangulated categories - Lecture 2.

Abstract: We define and give elementary properties of triangulated

categories. We also give an application of triangulated categories to

linkage theory in commutative algebra.