2:30pm» Jishnu Ray : University of British Columbia, Vancouver- 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
- 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.