Default Timings:
2:30pm, Friday
|
Default Venue:
AG-77
Next

## Non vanishing of Riemann Zeta function on the line and Prime Number Theorem

Speaker:
T. N. Shorey (IIT, Mumbai)

Date: Friday, 24th May, 2013
Time: 02:30 pm
Venue: AG-77
Abstract: The Riemann Zeta function does not vanish on the line parallel to Y-axis and passing through (1,0). This is an analytic statement but it is equivalent to Prime Number Theorem, which is an arithmetic statement. We shall elaborate on this equivalence in this talk.
Past

## Contractible algebraic curves in the affine plane

Speaker:
R. V. Gurjar (TIFR, Mumbai)

Date: Friday, 17th May, 2013
Time: 02:30 pm
Venue: AG-77
Abstract: Around 1971, S.S. Abhyankar-T.T. Moh, and independently M. Suzuki, proved the following ramarkable result. Let k be a field of char. 0, and f(T), g(T) polynomials in one indeterminate T with k-coefficients such that T is a polynomial in f, g with coefficients in k. Then either degree f divides degree g or degree g divides degree f. As a consequence they proved that any embedding of the affine line in the affine plane, as an algebraic subvariety, is equivalent to the standard embedding {X=0} after an automorphism of the affine plane. This consequence is called the AMS Theorem.

Another very useful consequence is that any automorphism of k[X,Y] is a composition of "obvious" automorphisms.

Now there are at least ten different proofs of the AMS Theorem (two of them were given by me, one jointy with M. Miyanishi). As a beautiful generalization of the AMS Theorem V.Y. Lin-M. Zaidenberg proved that any contractible irreducible curve in the affine plane can be mapped to the curve {X^a-Y^b=0} by an automorphism of the affine plane for some relatively prime integers a, b.

I will discuss these results. Recently I have found a third proof of both the AMS and Lin-Zaidenberg results from a "higher" viewpoint using some classical results from Riemann Surface Theory and the Theory of Non-complete Algebraic Surfaces. I will give some idea of this proof.

## The moduli problem

Speaker:
Sanjay Kumar Singh (TIFR, Mumbai)

Date: Friday, 10th May, 2013
Time: 02:00 pm
Venue: AG-77
Abstract: I will informally start with the moduli problem. After that I will give some definitions to describe the problem in a mathematical way. I will also discuss some positive results and open problems.

## A brief introduction to the Poincare series of a finitely generated module over a local ring

Speaker:
Anjan Gupta (School of Maths, TIFR)

Date: Friday, 26th April, 2013
Time: 2:30pm
Venue: AG-77

#### A brief introduction to the Poincare series of a finitely generated module over a local ring (Continuation Talk)

Speaker: Anjan Gupta (School of Maths, TIFR)
Date: Friday, 26th April, 2013
Time: 2:30pm
Venue: AG-77

## Counting points on varieties

Speaker:
Ritabrata Munshi (TIFR, Mumbai)

Date: Thursday, 18th April, 2013
Time: 02:00 pm
Venue: AG-77
Abstract:Counting number of rational solutions of polynomial equations have a long and rich history. In most cases deciding whether there are any rational solutions turns out to be extremely difficult, a notorious example being the Fermat's Last Theorem. In many other situations the existence of the point is more or less clear, but counting how many is the main issue. In the early 20th century Hardy and Littlewood adapted Ramanujan's circle method and devised an analytic machine to count rational points. Over the years this has turned out to be very effective. In this talk I will discuss various interesting problems about rational points on varieties, and explain how to use the circle method to count them.

## Iwasawa theory of elliptic curves and residual Galois representation

Speaker:
Sudhanshu Shekhar (TIFR, Mumbai)

Date: Friday, 12th April, 2013
Time: 04:00 pm
Venue: AG-77
Abstract:Let E be an elliptic curve over a number field K. For a prime number p the absolute Galois group of K acts on the the p-torsion points E[p] of E. In this talk, we shall discuss about the various arithmetic (and if time permits, then analytic) properties of the elliptic curve E, determined by the Galois module structure of E[p].

## Dani correspondence

Speaker:
Lovy Singhal (TIFR, Mumbai)

Date: Friday, 05th April, 2013
Time: 02:30 pm
Venue: AG-77
Abstract:A real number $\alpha$ is said to be badly approximable if there exists a $\delta > 0$ such that $| \alpha - p/q | > \delta / q^2$ for all $p/q \in \mathbb{Q}$. In 1985, S G Dani gave a geometric condition for determining when is a number $\alpha$ badly approximable. In the talk, we shall try to quickly run through the machinery required to state this result before illustrating the proof.

## A quick and crude overview of Tate's thesis

Speaker:
Sandeep Varma (TIFR, Mumbai)

Date: Thursday, 28th March, 2013
Time: 02:30 pm
Venue: AG-77
Abstract:I will give a very brief overview of an elementary and well known work known as Tate's thesis. I hope hearing a sketch over a course of just one lecture might help register some of the key points quickly, without being cluttered with much detail. I hope that will not be too weird an experiment!

For those who may not be familiar with what Tate's thesis is about : Riemann had proved the functional equation and analytic continuation for the Riemann zeta function. Hecke generalized this to zeta functions of number fields, and more generally to L-functions associated to what are called Hecke characters. Tate reproved Hecke's results in adelic language. Tate's approach, which introduced harmonic analysis over adeles, made the proof simpler and more elegant, and rendered aspects such as the Euler product structure of L and epsilon factors more transparent. Tate's approach turned out to guide the theory of "integral representations" for many L-functions - for instance ones associated to automorphic representations of GL_n (among other examples), which have all been important steps in the Langlands program. I may make a few remarks about these if time permits. If the audience wishes, I may spend the first ten minutes or so motivating L-functions with very simple examples. Depending on feedback during the lecture, I can try to adapt the presentation accordingly and decide to cover more/less.

## Syzygies and Free Resolutions

Speaker:
Anand Sawant (TIFR, Mumbai)

Date: Friday, 08th March, 2013
Time: 02:30 pm
Venue: AG-77
Abstract:Syzygies of a module M are modules that describe linear relations among the generators of M. These are computed and studied using free resolutions. These determine many other important invariants of the module M, such as its Betti numbers and its Hilbert polynomial. The Betti numbers of a module can be arranged in a table, called the Betti table.
In this lecture, we aim to describe a few naturally arising questions about these invariants and state some recent results. In particular, we shall discuss a nice duality between Betti tables of graded modules over a polynomial ring and cohomology tables of vector bundles on the corresponding projective space. The lecture is aimed at non-specialists.

## K-theory of C*-algebras

Speaker:
Paul Baum (Pennsylvania State University)

Date: Friday, 01st March, 2013
Time: 02:30 pm
Venue: AG-77
Abstract:A theorem of I. Gelfand states that the category of locally compact Hausdorff topological spaces and the category of commutative C* algebras are equivalent. Thus a non-commutative C* algebra can be viewed as a "non-commutative locally compact Hausdorff topological space". Atiyah-Hirzebruch K-theory for locally compact Hausdorff topological spaces extends in a straightforward way to become K-theory for C* algebras. If G is a locally compact Hausdorff topological group, then there is a C* algebra associated to G which can be viewed as the non-commutative topological space having one point for each distinct (i.e. non-equivalent) irreducible unitary representation of G which is weakly contained in the regular representation of G. The BC (Baum-Connes) conjecture proposes an answer to calculating the K-theory of this C* algebra.

## New interpretation of chromatic polynomials using Kac-Moody theory.

Speaker:
R. Venkatesh (School of Maths, TIFR)

Date: Friday, 22nd February, 2013
Time: 02:30 pm
Venue: AG-77
Abstract:I will start with the definition of the chromatic polynomial of a given graph, and prove some of its basic properties. I will then discuss a result jointly obtained with S. Viswanath on a new interpretation of chromatic polynomials using the Kac-Moody theory of Lie algebras. No prior knowledge of Kac-Moody theory will be assumed.

## Holomorphic connections on Riemann surfaces.

Speaker:
Indranil Biswas (School of Maths, TIFR)

Date: Friday, 8th February, 2013
Time: 02:30 pm
Venue: AG-77
Abstract: We give a proof of the criterion of Atiyah and Weil that a holomorphic vector bundle E on a compact Riemann surface admits a holomorphic connection if and only if each indecomposable component of E is of degree zero.

## On the Fundamental Group of Genus-2 fibrations.

Speaker:
Sagar Kolte

Date: Friday, 1st February, 2013
Time: 02:30 pm
Venue: AG-77
Abstract:We will compute the fundamental group of a certain class of algebraic surfaces and use this to prove the Shafarevich conjecture for such surfaces.

## How to have a stable relationship with your girlfriend/boyfriend using linear programming.

Speaker:
Arnab Mitra (School of Maths, TIFR)

Date: Friday, 21st December, 2012
Time: 02:30 pm
Venue: AG-77
Abstract:A stable matching is an assignment of n men to n women so that no two people prefer each other to their respective partners. In the talk, we will discuss the Gale-Shapley algorithm which gives a stable matching in the above situation. In the end we will give a linear programming formulation of the problem which allows us to achieve a useful generalization.

## Stokes' Theorem

Speaker:
Sudarshan Gurjar (School of Maths, TIFR)

Date: Friday, 14th December, 2012
Time: 02:30 pm
Venue: AG-77

## Higher dimensional local fields

Speaker:
Supriya Pisolkar (School of Maths, TIFR)

Time: Friday 5/03/10, 2:00pm-3:00pm
Venue: Seminar Room - A 369
Abstract: A very interesting direction of generalizations of class field theory is to develop a theory for higher dimensional fields. One of the first ideas in higher class field theory is to work with Milnor K-groups instead of multiplicative groups. Higher local class field theory contains classical local class field theory as 1-dimensional version.
I will first recall important notions/statements in local class field theory (1-dimensional case). Then we will see some basic notions for higher dimensions. I won't be going into the details of class field theory in this case , and thus this talk will be very elementary.

#### Higher dimensional local fields (Continuation Talk)

Speaker: Supriya Pisolkar (School of Maths, TIFR)
Time: Tuesday , 9/03/10, 11:30am-12:30am
Venue: Seminar Room

## Measuring Randomness

Speaker:
Sarang Sane (School of Maths, TIFR)

Time: Tue, 23rd Feb, 2:30pm-3:30pm
Venue: AG-77
Abstract: Consider throwing a "fair" coin as opposed to throwing a coin with two heads. It is natural to think that the experiment/action/event of throwing a "fair" coin is "more random" than that of throwing the coin with two heads. This takes us to the natural question of whether we can somehow measure randomness.
For the major portion of this talk, we will try to answer the question, "How random is this action/event/experiment/phenomenon"? This part is intended to be elementary, precise and completely self-contained. The answer will be an expression which occurs in many other contexts as well. Towards the end, I will try and make an ambitious attempt to try and tie this up with applications in science (this part might be very vague).

## Grothendieck's splitting theorem for vector bundle on projective line

Speaker:
Sanjay Kumar Singh (School of Maths, TIFR)

Time: Tue, 16th Feb, 2:30pm-3:30pm
Venue: AG-77
Abstract: I will try to explain all definitions and results needed to prove the theorem. I will give definition of a divisor and a vector bundle,and the correspondence between line bundles and divisors, and then finally give proof of the main theorem.

## Langlands functoriality principle

Speaker:
Debargha Banerjee (School of Maths, TIFR)

Time:Tue, 9th Feb, 2:30pm-3:30pm
Venue:AG-77
Abstract: In this talk, I will try to explain the concept of dual groups and functoriality principle. We will try to understand the functoriality in the simplest possible examples. We will give emphasis on the particular examples like base changes and adjoint lifts.

## Mathematics and Logic

Speaker:
Nitin Nitsure (School of Maths, TIFR)

Time: Tue, 2nd Feb, 2:30pm-3:30pm
Venue: AG-77

#### Proofs of Godel's theorem (Continuation Talk)

Speaker: Nitin Nitsure (School of Maths, TIFR)
Time: Fri, 19th Feb, 2:30pm-3:30pm
Venue: AG-77
Abstract: This lecture will focus on explaining three different proofs of G\"odel's first incompleteness theorem: the semantic proof, the syntactic proof, and the computability proof. The main technical points are the diagonalization argument, the G\"odel encoding and the basics of primitive recursion.

## Tate Curves

Speaker:
Somnath Jha (School of Maths, TIFR)

Time: Fri, 29th Jan, 2:00pm-3:15pm
Venue: AG-77
Abstract: An elliptic curve over C can be viewed as C/Λ for some lattice Λ in C. Tate has fomulated analogous description for elliptic curves over p-adic number fields with non-integral j-invariant. Starting from the definition of elliptic curve we will give an overview of the structure of elliptic curves over the field of Complex Numbers, Real Numbers and then talk about the Tate parametrisation of elliptic curves over p-adic number fields

## On the Shafarevich Conjecture and Genus 2 fibrations

Speaker:
R.V.Gurjar (School of Maths, TIFR)

Time: Tue, 19th Jan, 2:30pm
Venue: Seminar room
Abstract: The famous shaferavich conjecture asserts that the universal cover of a smooth projective variety over complex numbers is holomorphically convex, that is given any sequence of points going to infinity in the universal cover there is a holomorphic function which is unbounded on that sequence. This conjecture is the correct generalization of the uniformization theorem of Riemann. I will talk about the conjecture in general, the known cases when the conjecture is proved, its consequences which I have proved, and my results with Shastri and Purnaprajna.