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

Speaker: Jishnu Ray.

Affiliation: University of British Columbia, Vancouver.

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

Venue: Ramanujan Hall, Department of Mathematics.

Title: Selmer group of elliptic curves and explicit presentation of

Iwasawa algebras.

Abstract:

The Selmer group of an elliptic curve over a number field encodes

several arithmetic data of the curve providing a p-adic approach to

the Birch and Swinnerton Dyer, connecting it with the p-adic Lfunction via the Iwasawa main conjecture. Under suitable extensions of

the number field, the dual Selmer becomes a module over the Iwasawa

algebra of a certain compact p-adic Lie group over Z_p (the ring of padic integers), which is nothing but a completed group algebra. The

structure theorem of GL(2) Iwasawa theory by Coates, Schneider and

Sujatha (C-S-S) then connects the dual Selmer with the “reflexive

ideals” in the Iwasawa algebra.

We will give an explicit ring-theoretic presentation, by generators

and relations, of such Iwasawa algebras and sketch its implications to

the structure theorem of C-S-S. Furthermore, such an explicit

presentation of Iwasawa algebras can be obtained for a much wider

class of p-adic Lie groups viz. pro- p uniform groups and the pro-p

Iwahori of GL(n,Z_p). If we have time, alongside Iwasawa theoretic

results, we will state results (joint with Christophe Cornut)

constructing Galois representations with big image in reductive groups

and thus prove the Inverse Galois problem for p-adic Lie extensions

using the notion of “p-rational” number fields.

- Time:
- 2:45pm - 3:45pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- Mathematics Colloquium Talk 1.

Speaker: R.V. Gurjar.

Affiliation: IIT Bombay.

Date and Time: Wednesday 7 August, 2:45 pm - 3:45 pm.

Venue: Ramanujan Hall, Department of Mathematics.

Title: Ramification in Commutative Algebra and Algebraic Geometry.

Abstract: We will consider mainly the following situation.

Let R,S be complete normal local domains over an alg. closed field k of

char. 0 such that S is integral over R. Our aim is to describe three

ideals in S; I_N, I_D, I_K (Noether, Dedekind, Kahler differents resp.)

each of which capture the ramified prime ideals in S over R. In general

these three ideals are not equal. An important special case when all are

equal is when S is flat over R.

The case when there is a finite group G of k-automorphisms of S such that

R is the ring of invariants is already very interesting. Then many nice

results are proved.

These include works of Auslander-Buchsbaum, Chevalley-Shephard-Todd,

Balwant Singh, L. Avramov, P. Roberts, P. Griffith, P. Samuel,....

I will try to discuss all these results.

I believe that these results and ideas involved in them will be very

valuable to students and faculty both.

- Time:
- 4:00pm - 5:00pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- Mathematics Colloquium Talk 2.

Speaker: Ameya Pitale.

Affiliation: University of Oklahoma.

Date and Time: Wednesday 7 August, 4:00 pm - 5:00 pm.

Venue: Ramanujan Hall, Department of Mathematics.

Title: Special values of L-functions and congruences between modular forms.

Abstract: In this talk, we will discuss an important object in number

theory : L-functions. A well-known example is the Riemann zeta function.

We will focus on the arithmetic properties of the special values of

L-functions. These have very interesting applications to congruences

between modular forms. We will give a gentle introduction to these

concepts highlighting several examples and important results in the

literature. We will present recent joint research with Abhishek Saha and

Ralf Schmidt regarding special L-values and congruences of Siegel modular

forms.

- Time:
- 2:00pm - 3:30pm
- Location:
- Room No. 215 Department of Mathematics
- Description:
- Commutative Algebra Seminars

Speaker: Dilip Patil.

Affiliation: IISc, Bangalore.

Seminar I - Date and Time: Tuesday 13 August, 2:00 pm - 3:30 pm.

Venue: Room 215, Department of Mathematics.

Title: The Cohen-Structure Theorems.

Abstract: The purpose of these two lectures is to provide the proof of

Cohen’s structure theorem for complete local rings (which Cohen proved in

his PhD thesis 1942, Johns Hopkins University under the guidance of Oscar

Zariski). In these lecture we deal with the equicharacteristic case. We

give a modern and concise treatment by using the notion of formal

smoothness which was introduced by Grothendieck in 1964 in EGA Chapter IV.

It is closely connected with the differentials and throws new light to the

theory of regular local rings and used in proving Cohen’s structure

theorem of complete local rings.

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

Speaker: Uday Bhaskar.

Affiliation: TIFR, Mumbai.

Date and Time: Tuesday 13 August, 4:00 pm - 5:00 pm.

Venue: Ramanujan Hall, Department of Mathematics.

Title: Simultaneous conjugacy classes of commuting tuples of matrices.

Abstract: We discuss the classification of tuples of commuting matrices

over a finite field, up to simultaneous conjugation.

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

Speaker: Saikat Mazumdar.

Affiliation: IIT Bombay.

Date and Time: Wednesday 14 August, 4:00 pm - 5:00 pm.

Venue: Ramanujan Hall, Department of Mathematics.

Title: Yamabe problem and beyond: an interplay of geometry and PDE.

Abstract: Motivated by the theory of compact surfaces, Yamabe wanted to

show that on a given compact Riemannian manifold of any dimension there

always exists a (conformal) metric with constant scalar curvature. It

turns out that solving the Yamabe problem amounts to solving a nonlinear

elliptic partial differential equation (PDE). The solution of the Yamabe

problem by Trudinger, Aubin and Schoen highlighted the local and global

nature of the problem and the unexpected role of the positive mass theorem

of general relativity. In the first part of my talk, I will survey the

Yamabe problem and the related issues of the compactness of solutions.

In the second part of the talk, I will discuss the higher-order or

polyharmonic version of the Yamabe problem: "Given a compact Riemannian

manifold (M, g), does there exists a metric conformal to g with constant

Q-curvature?" The behaviour of Q-curvature under conformal changes of the

metric is governed by certain conformally covariant powers of the

Laplacian. The problem of prescribing the Q-curvature in a conformal class

then amounts to solving a nonlinear elliptic PDE involving the powers of

Laplacian called the GJMS operator. In general the explicit form of this

GJMS operator is unknown. This together with a lack of maximum principle

makes the problem difficult to tackle. I will present some of my results

in this direction and mention some recent progress.

- Time:
- 2:00pm - 3:30pm
- Location:
- Room No. 215 Department of Mathematics
- Description:
- Commutative Algebra Seminars

Speaker: Dilip Patil.

Affiliation: IISc, Bangalore.

Seminar II - Date and Time: Friday 16 August, 2:00 pm - 3:30 pm.

Venue: Room 215, Department of Mathematics.

Title: The Cohen-Structure Theorems.

Abstract: The purpose of these two lectures is to provide the proof of

Cohen’s structure theorem for complete local rings (which Cohen proved in

his PhD thesis 1942, Johns Hopkins University under the guidance of Oscar

Zariski). In these lecture we deal with the equicharacteristic case. We

give a modern and concise treatment by using the notion of formal

smoothness which was introduced by Grothendieck in 1964 in EGA Chapter IV.

It is closely connected with the differentials and throws new light to the

theory of regular local rings and used in proving Cohen’s structure

theorem of complete local rings.

- Time:
- 4:00pm - 5:00pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- Speaker: Sampat Kumar Sharma.

Affiliation: ISI, Kolkata.

Date and Time: Friday 16 August, 4:00 pm - 5:00 pm.

Venue: Ramanujan Hall, Department of Mathematics.

Title: On a question of Suslin about completion of unimodular rows.

Abstract:

R.G. Swan and J. Towber showed that if (a

2

, b, c) is a unimodular row

over any commutative ring R then it can be completed to an invertible

matrix over R. This was strikingly generalised by A.A. Suslin who showed

that if (a

r!

0 , a1, . . . , ar) is a unimodular row over R then it can be com-

pleted to an invertible matrix. As a consequence A.A. Suslin proceeds to

conclude that if 1

r! ∈ R, then a unimodular row v(X) ∈ Umr+1(R[X])

of degree one, with v(0) = (1, 0, . . . , 0), is completable to an invertible

matrix. Then he asked

(Sr(R)): Let R be a local ring such that r! ∈ GL1(R), and let p =

(f0(X), . . . , fr(X)) ∈ Umr+1(R[X]) with p(0) = e1(= (1, 0, . . . , 0)). Is it

possible to embed the row p in an invertible matrix?

Due to Suslin, one knows answer to this question when r = d + 1,

without the assumption r! ∈ GL1(R). In 1988, Ravi Rao answered this

question in the case when r = d.

In this talk we will discuss about the Suslin’s question Sr(R) when r =

d − 1. We will also discuss about two important ingredients; “homotopy

and commutativity principle” and “absence of torsion in Umd+1(R[X])

Ed+1(R[X]) ”,

to answer Suslin’s question in the case when r = d − 1, where d is the

dimension of the ring.

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

Speaker: R V Gurjar.

Affiliation: IIT Bombay.

Date and Time: Monday 19 August, 3:30 pm - 5:00 pm.

Venue: Room 215, Department of Mathematics.

Title: Ramification in Commutative Algebra and Algebraic Geometry.

Abstract: We will consider mainly the following situation. Let R,S be

complete normal local domains over an alg. closed field k of char. 0 such

that S is integral over R. Our aim is to describe three ideals in S; I_N,

I_D, I_K (Noether, Dedekind, Kahler differents resp.) each of which

capture the ramified prime ideals in S over R. In general these three

ideals are not equal. An important special case when all are equal is when

S is flat over R. We will prove many of these statements.

The case when there is a finite group G of k-automorphisms of S such that

R is the ring of invariants is already very interesting. Then many nice

results are proved.

These include works of Auslander-Buchsbaum, Chevalley-Shephard-Todd,

Balwant Singh, L. Avramov, P. Roberts, P. Griffith, P. Samuel,....

I will try to discuss all these results.

I believe that these results and ideas involved in them will be very

valuable to students and faculty both.

Prerequisites. Basic knowledge of Commutative Algebra and language of

Algebraic Geometry (no sheaf theory!). I will

- Time:
- 4:00pm - 5:00pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- Mathematics Colloquium
Speaker: Karthik Adimurthi.
Affiliation: TIFR CAM, Bangalore.
Date and Time: Monday 19 August, 4:00 pm - 5:00 pm.
Venue: Ramanujan Hall, Department of Mathematics.
Title: Uniform boundedness and Lipschitz estimates for quasilinear
parabolic equations.
Abstract: In this talk, we will discuss some well known regularity issues
concerning equations of the form $u_t - div |\nabla u|^{p-2} \nabla u = 0$
for $1
2$) and the singular case ($p<2$) separately. Moreover in several instances, the estimates are not even stable as $p\rightarrow 2$. In this talk, I shall discuss two regularity estimates and give an overview on how to obtain uniform $L^{\infty}$ and $C^{0,1}$ estimates in the full range $\frac{2N}{N+2}

- Time:
- 2:45pm - 3:45pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- Mathematics Colloquium

Speaker: Rahul Gupta.

Affiliation: University of Freiburg, Germany.

Date and Time: Wednesday 21 August, 2:45 pm - 3:45 pm.

Venue: Ramanujan Hall, Department of Mathematics.

Title: Milnor $K$-theory and Chow groups.

Abstract: In this talk, we shall define Milnor $K$-groups and Chow groups.

We study various properties of these and also theorems relating both

groups. In particular, we talk about Bloch's formula and Totaro's map.

Towards the end, I shall state my results in this direction, which are

joint work with Prof A. Krishna.

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

Speaker: Debraj Chakrabarti.

Affiliation: Central Michigan University, USA.

Date and Time: Wednesday 21 August, 4:00 pm - 5:00 pm.

Venue: Ramanujan Hall, Department of Mathematics.

Title: The flat-earth society: conformal mapping from Claudius Ptolemy to

Louis Nirenberg.

Abstract: The problem of constructing flat representations of spherical

surfaces arises naturally in geography and astronomy while making maps. We

look at a mathematical formulation of this problem using the notion of

conformal mapping, and discuss its relation with complex analysis. After

reviewing the contributions of Gauss, Riemann, and Poincaré to this

problem, we end with some glimpses of 20th century developments. This will

be an expository talk accessible to undergraduate and postgraduate

students.

- Time:
- 11:30am - 12:30pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- Number theory seminar.

Speaker: Aprameyo Pal.

Affiliation: University of Duisburg-Essen, Germany.

Date and Time: Thursday 22 August, 11:30 am - 12:30 pm.

Venue: Ramanujan Hall, Department of Mathematics.

Title: A central value formula of degree 6 complex L-series and arithmetic

applications.

Abstract: We prove an explicit central value formula for a family of

complex L-series of degree 6 for GL2 × GL3 which arise as factors of

certain Garret--Rankin triple product L-series associated with modular

forms. Our result generalizes a previous formula of Ichino involving

Saito--Kurokawa lifts, and as an application, we prove Deligne's

conjecture about the algebraicity of the central values of the considered

L-series up to the relevant periods. I would also include some other

arithmetic applications towards subconvexity problem, construction of

associated p-adic L function etc. This is joint work with Carlos de Vera

Piquero.

- Time:
- 2:00pm - 3:00pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- Combinatorics seminar.

Speaker: Deepanshu Kush.

Affiliation: IIT Bombay.

Date and Time: Tuesday 22August, 2:00 pm - 3:00 pm.

Venue: Ramanujan Hall, Department of Mathematics.

Title: Normalized Matching Property in Random & Pseudorandom Graphs.

Abstract: Normalized Matching Property (NMP) is a simple and natural

generalization of the famous Hall's marriage condition for bipartite

graphs, to the setting when the sizes of the two vertex classes are

distinct. It is a well-studied notion in the context of graded posets and

several well-known ones are known to have it (for instance the boolean

lattice or the poset of subspaces of a finite dimensional vector space).

However, in this talk, we will consider NMP with a 'random twist': if for

every possible edge in a bipartite graph, we toss a coin in order to

decide if we keep it or not, how biased must the coin be to expect to have

NMP in the graph with high probability? We shall arrive at a sharp

threshold for this event. Next, what can we say about explicit graphs that

are known to behave 'random-like'? One of the earliest notions of a

pseudorandom graph was given by Thomason in the 80s. We shall prove an

'almost' vertex decomposition theorem: every Thomason pseudorandom

bipartite graph admits - except for a negligible portion of its vertex set

- a partition of its vertex set into trees that have NMP and which arise

organically through the Euclidean GCD algorithm.

- Time:
- 3:45pm - 4:45pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- Dr. P. V. Sukhatme Memorial Lecture.

Speaker: Rajeeva Karandikar.

Affiliation: Chennai Mathematical Institute.

Date and Time: Thursday 22 August, 3:45 pm.

Venue: Ramanujan Hall, Department of Mathematics.

Title: On Connections between Partial Differential Equations and Diffusion

Processes.

Abstract: In this talk we will describe connections between second order

partial differential equations and Markov processes associated with them.

This connection had been an active area of research for several decades.

The talk is aimed at Analysts and does not assume familiarity with

probability theory.

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

Speaker: Anuj Jakhar.

Affiliation: Institute of Mathematical Sciences, Chennai.

Date and Time: Friday 23 August, 4:30 pm - 5:30 pm.

Venue: Ramanujan Hall, Department of Mathematics.

Title: On discriminant and integral basis of pure number fields.

Abstract:

By a pure number field we mean an algebraic number field of the type Q(

√n a)

where the polynomial x

n − a with integer coefficients is irreducible over the field

Q of rationals. In this talk our aim is to provide a formula for the discriminant

of pure number fields K = Q(

√n a) where for each prime p dividing n, p does not

divide the gcd of a and vp(a); vp(a) stands for the highest power of p dividing a.

We also describe explicitly an integral basis of such fields. This takes care of all

pure fields K = Q(

√n a), where either a, n are coprime or a is squarefree.

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

Speaker: R V Gurjar.

Affiliation: IIT Bombay.

Date and Time: Monday 26 August, 3:30 pm - 5:00 pm.

Venue: Room 215, Department of Mathematics.

Title: Lecture series on Ramification in Commutative Algebra and Algebraic

Geometry.

Abstract: We will consider mainly the following situation. Let R,S be

complete normal local domains over an alg. closed field k of char. 0 such

that S is integral over R. Our aim is to describe three ideals in S; I_N,

I_D, I_K (Noether, Dedekind, Kahler differents resp.) each of which

capture the ramified prime ideals in S over R. In general these three

ideals are not equal. An important special case when all are equal is when

S is flat over R. We will prove many of these statements.

The case when there is a finite group G of k-automorphisms of S such that

R is the ring of invariants is already very interesting. Then many nice

results are proved.

These include works of Auslander-Buchsbaum, Chevalley-Shephard-Todd,

Balwant Singh, L. Avramov, P. Roberts, P. Griffith, P. Samuel....

I will try to discuss all these results.

I believe that these results and ideas involved in them will be very

valuable to students and faculty both.

Prerequisites. Basic knowledge of Commutative Algebra and language of

Algebraic Geometry (no sheaf theory!). I will "throw in" topological

proofs from time to time to make the results intuitively more clear.

- Time:
- 12:05pm - 1:05pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- Probability and Statistics seminar.

Speaker: Vivek Kumar.

Affiliation: IIT Roorkee.

Date and Time: Tuesday 27 August, 12:05 pm - 1:05 pm.

Venue: Ramanujan Hall, Department of Mathematics.

Title: Existence and uniqueness of solutions of generalised stochastic

Burger equation perturbed by Volterra noise.

Abstract:

In this article, we investigate the existence and uniqueness of local mild solutions for the one-dimensional generalized stochastic Burgers equation (GSBE) containing a non-linearity of polynomial type and perturbed by α-regular cylindrical Volterra

process and having Dirichlet boundary conditions. The Banach fixed point theorem (or

contraction mapping principle) is used to obtain the local solvability results. The L∞-

estimate on both time and space for the stochastic convolution involving the α-regular

cylindrical Volterra process is obtained. Further, the existence and uniqueness of global

mild solution of GSBE up to third order nonlinearity is shown.

2010 Mathematics Subject Classification. Primary: 60H15, 60G22; Secondary: 35Q35,

35R60.

Key-words: Stochastic Burgers equation, Volterra process, γ-Radonifying operator,

Stopping time.

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

Speaker: Sandeep Kunnath.

Affiliation: TIFR CAM, Bangalore.

Date and Time: Wednesday 28 August, 4:00 pm - 5:00 pm.

Venue: Ramanujan Hall, Department of Mathematics.

Title: Sharp Inequalities, their extremals and related problems.

Abstract: Inequalities play an important role in the analysis of partial

differential equations. The best constants involved in these equations and

the case equality in these inequalities are of particular interest as they

are connected with many interesting phenomenon in various problems. In

this talk we will discuss some of these inequalities and related problems.

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

Speaker: Guhan Venkat.

Affiliation: Universite Laval, Quebec, Canada.

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

Venue: Ramanujan Hall, Department of Mathematics.

Title: Stark-Heegner cycles for Bianchi modular forms.

Abstract: In his seminal paper in 2001, Henri Darmon proposed a systematic

construction of p-adic points on elliptic curves over the rational

numbers, viz. Stark–Heegner points. In this talk, I will report on the

construction of p-adic cohomology classes/cycles in the

Harris–Soudry–Taylor representation associated to a Bianchi cusp form,

building on the ideas of Henri Darmon and Rotger–Seveso. These local

cohomology classes are conjectured to be the restriction of global

cohomology classes in an appropriate Bloch–Kato Selmer group and have

consequences towards the Bloch–Kato–Beilinson conjecture as well as

Gross–Zagier type results. This is based on a joint work with Chris

Williams (Imperial College London).

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

Speaker: Madhusudan Manjunath.

Affiliation: IIT Bombay.

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

Venue: Ramanujan Hall, Department of Mathematics.

Title: An Introduction to the Geometry of Numbers.

Abstract: We give a gentle introduction to the geometry of numbers. We

start with the classical theory and then treat some of the modern aspects

of this subject. This talk will be accessible to the general audience.