Mathematics Colloquium

================

Speaker: Prof. Agnid Banerjee, TIFR-CAM

Date and time: 2 September 4-5 pm

Venue: Ramanujan Hall

Title: Space like a strong unique continuation for some fractional parabolic equations

Abstract: I will talk about some recent work on space like a strong unique continuation for fractional heat type equations. This is based on a joint work with Vedansh Arya, Donatella Danielli and Nicola Garofalo. I will also try to report on some subsequent works of mine on this subject which are also in part joint with Venky Krishnan and Soumen Senapati. 

Note: The colloquium is addressed to a general audience.

Commutative algebra seminar

Tuesday, 30 August 2022, 4.00 pm--5.30 pm

Venue: Ramanujan Hall

Speaker: J. K. Verma

Title: Fibers of blowing ups, symbolic Rees algebras, and set-theoretic complete intersections

Abstract: In this series of three talks, I will present classical results of Cowsik-Nori about fibers of blowing ups, criteria for Noetherian property of symbolic Rees algebras, due to Huneke, Goto-Nishida, and its relation with set-theoretic complete intersections. If time permits, I will show some concrete examples of set-theoretic complete intersection ideals in polynomial rings such as monomial space curves, Fermat ideals, and certain ideals of hyperplane arrangements.

Mathematics Colloquium

Venue: Ramanujan Hall, Mathematics Department, Room 214

Date and time: Wednesday, 7 September 2022 at 4 pm

Speaker: Dr. Akashdeep Dey, Princeton University

Title: A comparison of the Almgren-Pitts and the Allen-Cahn min-max theory

Abstract: Min-max theory for the area functional was developed by Almgren
and Pitts to construct closed minimal hypersurfaces in arbitrary closed
Riemannian manifolds. There is an alternate PDE-based approach to the
construction of minimal hypersurfaces. This approach is based on the study
of the limiting behavior of solutions to the Allen-Cahn equation. In my
talk, I will briefly describe the Almgren-Pitts min-max theory and the
Allen-Cahn's min-max theory and discuss the question of to what extent these
two theories agree.

Note: The Mathematics Colloquia are accessible to the general audience.

Mathematics Colloquium

Speaker: Prof. Ravi Raghunathan

Date: Friday, 9 September 2022

Duration: 4.00-5.00 pm

Venue: Ramanujan Hall, Mathematics Department, Room 214

Title: Sphere packing, the Uncertainty Principle, $E_8$ and Fourier 
interpolation on the real line

Abstract: The talk will focus on two theorems of M. Viazovska who was 
awarded the Fields medal in Helsinki earlier this year. The first 
theorem resolved the problem of sphere packing in eight dimensions, 
while the closely related second theorem (proved jointly with D. 
Radchenko) establishes a new "Fourier interpolation" result for even 
Schwartz functions on the real line.

The first half of the talk should be completely accessible to anyone 
with a first course in linear algebra and multivariable calculus (MA 
109, MA 111, MA 106), and thus to most undergraduate students at IIT. 
The second half will require some familiarity with basic complex 
analysis (MA 205).

Commutative algebra seminar

Date: Tuesday, 13 September 2022

Time: 3.30-5.00 pm

Venue: Ramanujan Hall

Speaker: J. K. Verma

Title: Noetherian symbolic Rees algebras-III

Abstract: We shall give examples of ideals whose symbolic Rees algebra is Noetherian. These  are the

(1) ideals of space curves of multiplicity 3,

(2) ideals of a finite set of points in projective space,

(3) height one prime ideals in two-dimensional normal local domains.

We shall also discuss necessary and sufficient conditions for the symbolic Rees algebra of a prime ideal to be Noetherian in terms of set-theoretic complete intersections.

Mathematics Colloquium

Date and time: 14 September 2022 at 4 pm

Venue: Ramanujan Hall (Room 213, II Floor)

Title:  Boundary null-controllability of 1d linearized compressible 
Navier-Stokes
System by one control force.

Speaker:  Shirshendu Chowdhury, Department of Mathematics and 
Statistics, IISER Kolkata,

Abstract: In the first part of the talk, we introduce the concept of 
controllability of Differential Equations. Then we give some examples in 
finite (ODE) and infinite dimensional(PDE) contexts. We recall the 
controllability results of the Transport and Heat equation.

In the second part of the talk, we consider compressible Navier-Stokes 
equations in one dimension, linearized around a constant steady state  
$(Q_0, V_0 ) $, with $Q_ 0 > 0, V 0 >0 $.  It is a Coupled system of 
transport and heat type equations.  We study the boundary 
null-controllability of this linearized system in the interval $(0,1)$ 
when a Dirichlet control function is acting either only on the density 
or only on the velocity component at one end of the interval.

We obtain null controllability using one boundary control in the space 
${H}^s_{per}(0,1)\times L^2(0,1)$ for any $s>\frac{1}{2}$ provided the 
time $T>1$, where ${H}_{per}^s(0,1)$ denotes the Sobolev space of 
periodic functions. The proof is based on spectral analysis and on 
solving a mixed parabolic-hyperbolic moments problem and a parabolic 
hyperbolic joint Ingham-type inequality.

This is a recent joint work (https://arxiv.org/abs/2204.02375 [1], 2022) 
with Kuntal Bhandari, Rajib Dutta and Jiten Kumbhakar.

Note: Mathematics Colloquia are accessible to the general audience

Links:
------
[1] https://arxiv.org/abs/2204.02375

Dear all,

We will have *Basudev Pattanayak* speaking in our RTAG seminar from 2pm to
3:30pm on Thursday.


Here are the necessary details for his talk:

Time: Thursday, 15 September, 2:00 – 3:30 pm.

Venue : Ramanujan Hall, Department of mathematics.

Title: A Visit to the Local Langlands Conjecture

Abstract: In this series of talks, we first recall some important results
of class field theory. Then we will discuss the representation theory of
p-adic groups. Here we will discuss the Hecke algebra attached to
Bushnell-Kutzko types. With little basic setup, later we will state the
local Langlands Conjecture and its enhancement. For some special cases, we
will discuss their proofs.

We have now a dedicated website where one can find the notes and resources
from the past meets and announcements of the upcoming meetings:

https://sites.google.com/view/rtag/

Please join us!
 

Mathematics Colloquium

Time and Date: 4-5 pm Wednesday, 21 September 2022

Speaker: Prof. Saikat Mazumdar, IIT Bombay

Title: Searching conformally for metrics with constant curvature.

Abstract: I will start by surveying the Yamabe problem, which asks to find a (conformal) metric with constant scalar curvature on a compact Riemannian manifold. This amounts to solving a nonlinear PDE involving the Laplacian. The solution to the Yamabe problem highlighted the role played by local and global geometry of the manifold and the unexpected connection to the positive mass theorem of general relativity.

I will first discuss the case of compact surfaces, and introduce some tools and techniques from Calculus of Variations, Nonlinear Analysis on Manifolds, and PDEs. In the remaining time, I will discuss the higher-order version of the Yamabe problem: “Given a compact Riemannian manifold, does there exists a conformal metric with constant Q-curvature”?

Note: The speakers make an effort to make Mathematics Colloquia accessible to the general audience.

We will have *Basudev Pattanayak* speaking in our RTAG seminar from 2pm to
3:30pm on Thursday.


Here are the necessary details for his talk:

Time: Thursday, 22 September, 2:00 – 3:30 pm.

Venue : Ramanujan Hall, Department of mathematics.

Title: A Visit to the Local Langlands Conjecture - 2

Abstract: In this series of talks, we first recall some important results
of class field theory. Then we will discuss the representation theory of
p-adic groups. Here we will discuss the Hecke algebra attached to
Bushnell-Kutzko types. With little basic setup, later we will state the
local Langlands Conjecture and its enhancement. For some special cases, we
will discuss their proofs.



We have now a dedicated website where one can find the notes and resources
from the past meets and announcements of the upcoming meetings:

https://sites.google.com/view/rtag/

Please join us!

Speaker: P Amrutha, IISER Thiruvananthapuram

Date & Day: September 22, 2022, Thursday Time: 4.00--5.00 pm

Venue: Room 215

Title: On the partitions and multipartitions not divisible by powers of 2

Abstract:

Given a finite group G and a natural number p, an interesting question one can ask is to count

the number of inequivalent irreducible representations of G whose degree is not divisible by p. This
question originated in a paper by I. G. Macdonald for the case of prime numbers. MacDonald’s
paper was a motivation for the McKay conjecture. The announcement of McKay conjecture in 1971
is the origin of a different kind of counting conjectures of finite groups. Extending Macdonald’s

results to all integers is a much harder problem to study. Motivated by a question from chiral rep-
resentations of the wreath products, we will see a generalization of the above question to composite

numbers of the form 2k and a recursive formula for the groups Sn, An, and (Z/rZ)≀ Sn. Regardless
of the description of the count, even for the smaller integers, a complete characterization of the
irreducibles with a degree not divisible by a given prime number is still missing in the literature.
We will see such characterization for some special cases at the end and further open problems in
this direction. This is joint work with T. Geetha.

Commutative algebra  seminar

Tuesday, 27 September 2022 @3.30 pm.

Venue: Ramanujan Hall

Speaker: Tony Puthenpurakal, IIT Bombay

Title: On coefficient ideals-II

Abstract: Let (A, m) be a Cohen-Macaulay local ring of dimension d ≥ 2 with infinite residue field and let I be an m-primary ideal. Let For 0 ≤ i ≤ d let Ii be the i th-coefficient ideal of I. Also let Ie = Id denote the Ratliff-Rush closure of A. Let G = GI (A) be the associated graded ring of I. We show that if dim H j G+ (G) ∨ ≤ j−1 for 1 ≤ j ≤ i ≤ d−1 then (I n)d−i = Ifn for all n ≥ 1. In particular if G is generalized Cohen-Macaulay then (I n)1 = Ifn for all n ≥ 1. As a consequence we get that if A is an analytically unramified domain with G generalized Cohen-Macaulay, then the S2-ification of the Rees algebra A[It] is L n≥0 Ifn.

Dear all,
 Prof. Nitin Nitsure will give a series of lectures on 'Algebraic Stacks
and Moduli Theory' starting next week. The draft announcement is
attached.
The first talk will be introductory and will be accessible to everyone
with very minimal knowledge of algebraic geometry.

For the benefit of students, this is a very central and important area of
algebraic geometry and Prof. Nitsure is a very good lecturer. Do try and
attend the first lecture in Ramanujan Hall on Tuesday 27th at 5:10 pm. The
first talk will be of 1 hour.

Date 28 September 2022

Time 4-5 pm

Venue: Ramanujan Hall

Speaker: Prof. Eknath Ghate, TIFR, Mumbai

Title: Semi-stable representations as limits of crystalline representations

Abstract: We construct an explicit sequence of crystalline representations
converging to a given irreducible two-dimensional semi-stable
representation of the Galois group of Q_p. The convergence takes place in
the blow-up space of two-dimensional trianguline representations studied
by Colmez and Chenevier. It is connected to a classical formula going back
to Greenberg and Stevens expressing the L-invariant as a logarithmic
derivative.

Our convergence result can be used to compute the reductions of any
irreducible two-dimensional semi-stable representation in terms of the
reductions of certain nearby crystalline representations of exceptional
weight. For instance, using our zig-zag conjecture on the reductions of
crystalline representations of exceptional weights, we recover completely
the work of Breuil-Mezard and Guerberoff-Park on the reductions of
irreducible semi-stable representations of weights at most p+1, at least
on the inertia subgroup. As new cases of the zig-zag conjecture are
proved, we further obtain some new information about the reductions for
small odd weights.

Finally, we use the above ideas to explain away some apparent violations
to local constancy in the weight of the reductions of crystalline
representations of small weight that were noted in our earlier work and
which provided the initial impetus for this work.

This is joint work with Anand Chitrao and Seidai Yasuda.