Lecture 1: (Commutative Algebra/Algebraic Geometry seminar)

Speaker : Nitin Nitsure

Title: Zariski's Main Theorem

Time, Day, Date, Venue : Monday, 6th Jan, 11:30 am, Ramanujan Hall

Abstract : This lecture will discuss the algebraic formulation, and sketch
its proof by Peskine.

Speaker: Peter Sarnak, IAS Princeton 

Date & Time: Monday, 06 January 2025, 4 pm

Venue: Seminar Room 2, Ground Floor, VMCC

Title: Bass Note Spectra of Locally Uniform Geometries

Abstract: We formulate and report on the problem of the bass note spectrum of an invariant operator as one varies over locally uniform geometries. In the Euclidean setting this recasts classical problems of Mahler from the geometry of numbers in a new light. For certain operators, homogeneous dynamics can be used decisively. In the non-Euclidean setting of hyperbolic manifolds we review some recent developments using the conformal bootstrap method and random covers to study the bass note spectra. We highlight the theme and impact of rigidity.

Lecture 3 : (Commutative Algebra/Algebraic Geometry seminar)

Speaker: Nitin Nitsure

Time, Day, Date, Venue : Wednesday, 8th Jan, 11:30 am, Ramanujan Hall

Title: Zariski's Main Theorem (Lecture 2)

In this lecture we will discuss the scheme-theoretic generalizations by
Grothendieck, and outline their deductions from the algebraic formulation.
 

Lecture 4: Statistics and Probability Seminar

Time, Day, Date, Venue :  Wednesday, 8th Jan, 3:00-4:00 pm, Ramanujan Hall

Speaker: Samiran Ghosh

Affiliation: UT Health Houston

Title: The intervention is efficacious, but will it be acceptable in a
real-world setting?” Welcome to the world of Implementation Science

Abstract : "Intervention science is a multidisciplinary field that focuses
on the design, implementation, and evaluation of interventions aimed at
improving health, social, or behavioral outcomes. These interventions can
include drugs, devices, and even policy-level or economic interventions.
While research studies continue to demonstrate successful interventions
that improve healthcare, their dissemination remains relatively slow.
Multiple studies show a significant gap between the adoption of effective
interventions in practice, which, if addressed, could result in a decrease
in human suffering, reduced mortality, and ultimately, an improvement in
public health.

In this talk, we will explore the science of implementation science,
particularly through the lens of randomized controlled trials (RCTs). We
will address the challenges of implementing and disseminating
interventions, illustrating key points with practical examples. Hybrid
effectiveness-implementation studies provide a promising approach by
simultaneously evaluating both the effectiveness of interventions and the
processes of their implementation in real-world settings. Throughout this
discussion, we will highlight both the challenges and successes of
implementation efforts across diverse contexts. By emphasizing the
importance of effective implementation, we can ensure that health
interventions deliver their intended impact, ultimately improving health
outcomes and advancing public health objectives."

Speaker: Vedansh Arya (University of Jyväskylä)
Date and Time: Wednesday, January 8, 2025, at 4:15 PM
Venue: Ramanujan Hall

Title: Harnack Estimate for Non-Homogeneous Parabolic Equations

Abstract: In this talk, we present a scale-invariant Harnack inequality
for certain non-homogeneous parabolic equations within a suitable
intrinsic geometry influenced by the nonlinearity. This result, in
particular, establishes Hölder continuity. Additionally, we discuss a
Harnack-type estimate on a global scale that provides a quantitative
formulation of the strong minimum principle. The talk is based on joint
work with Vesa Julin.

Speaker Bio:  Vedansh Arya has been a postdoctoral researcher in the
Department of Mathematics and Statistics at the University of Jyväskylä,
Finland, since 2022. He completed his Ph.D. in Mathematics at the TIFR
Centre for Applicable Mathematics, Bangalore, in the same year. His
research interests span regularity theory for elliptic and parabolic PDEs,
unique continuation problems, free boundary problems, and mean curvature
flow.

Lecture 4 (Commutative Algebra/Algebraic Geometry seminar)

Speaker: R.V Gurjar

Time, Day,Date,Venue: Thursday 9th Jan, 11:30 am in Ramanujan Hall

Title: 2 new proofs of Zariski's Main theorem

Abstract: This lecture will supplement the lecture series by Prof.
Nitsure. In this talk Prof. Gurjar will discuss two new proof of Zariski's
Main Theorem, substantially different from the proof of Peskine discussed
by Nitsure in his first lecture.

Lecture 5 (Commutative Algebra/Algebraic Geometry seminar)

Speaker: Nitin Nitsure

Time, Day, Date, Venue : Friday 10th Jan, 11:30 am, Ramanujan Hall

Title: Zariski's Main Theorem (Lecture  3)

Abstract : This lecture will be devoted to some corollaries and applications.
 

Speaker: Dr. Shubham Rastogi
Date and time: January 13, 11:30 am
Venue: Ramanujan Hall

Title: On *-regular isometric dilations

Abstract: Every contraction on a Hilbert space has an isometric dilation.
And\^o extended this result to the pairs of commuting contractions. S.
Parrott showed that this dilation result does not extend to an $n$-tuple
of commuting contractions, in general for $n\geq 3.$ However, provided
that the $n$-tuple satisfies Brehmer's positivity condition, the dilation
exists. In fact, S. Brehmer proved that an $n$-tuple  of commuting
contractions satisfies Brehmer's positivity if and only if it admits a
minimal $*$-regular isometric dilation.  Moreover, D. Gasper and N. Suciu
showed that the minimal $*$-regular isometric dilation comprises doubly
commuting isometries. In this talk, we shall see an extension of this
result to a sequence of commuting contractions.

An $n$-tuple of doubly commuting pure isometries can be modeled by the
tuple of multiplication by the co-ordinate functions on a vector-valued
Hardy space over the polydisc. A similar result does not hold true for a
sequence of doubly commuting pure isometries. This brings us to the
question of characterizing a sequence that has the sequence of
multiplication by the co-ordinate functions on a vector-valued Hardy space
over the Hilbert multidisc, as its minimal $*$-regular isometric dilation.
We will address this question in the talk. The talk is based on a work in
progress with B. K. Das.

Partial Differential Equations seminar
Speaker: Mikko Salo (University of Jyvaskyla, Finland)
Title: Harmonic functions and their analogues in inverse problems
Time, day and date: 2:00:00 PM - 3:00:00 PM, Tuesday, January 14
Venue: Online mode

Commutative Algebra seminar
Speaker: R, V, Gurjar (TIFR Mumbai (Retd))
Title: Automorphisms of Algebraic Varieties.
Time, day and date: 4:00:00 PM - 5:00:00 PM, Tuesday, January 14
Venue: Room 215

Abstract

In the first lecture we will discuss general results about diffeomorphisms (resp. biholomorphisms) of compact differentiable (resp. compact complex) manifolds, and general algebraic varieties.
In the second talk we will indicate a proof of an important result.
Let X be a smooth projective variety of general type. Then the automorphism group of X is finite.
This applies to compact Riemann surfaces of genus > 1. Using this I will indicate the proof of Hurwitz's theorem:
Theorem. Let C be a compact Riemann surface of genus g > 1. Then Aut(C) has order at most 84(g-1). This bound is best possible.

Colloquium:
Speaker: Suman Kumar Sahoo (Department of Mathematics, IIT Bombay)
Title: Propagation of Singularities and Inverse Problems
Time, day and date: 4:00:00 PM - 5:30:00 PM, Wednesday, January 15
Venue: Ramanujan Hall

Abstracts:

The propagation of singularities in partial differential equations describes how the singularities of solutions evolve over time or space. In this talk, we explore how these singularities can be leveraged to recover geometric information about the coefficients of certain PDEs, such as the scattering relation.

Speaker: Beth Romano (King's College, London, U. K.)
Title: An introduction to the local Langlands correspondence
Time, day and date: 4:00:00 PM, Monday, January 20
Venue: Ramanujan Hall

Abstracts:
The local Langlands correspondence conjectures links between two seemingly unrelated areas of mathematics: the Galois theory of local fields and the theory of complex Lie groups. In this talk, we'll use examples to introduce some of the objects of interest from both number theory and geometry. We'll then use these examples to illuminate some of the beautiful interactions between these different areas.

Commutative Algebra seminar
Speaker: R. V. Gurjar (TIFR (retd))
Title: Automorphisms of algebraic varieties II
Time, day and date: 4:00:00 PM, Tuesday, January 21
Venue: Ramanujan Hall

Abstract: 

Commutative Algebra seminar
In the first lecture we will discuss general results about diffeomorphisms (resp. biholomorphisms) of compact differentiable (resp. compact complex) manifolds, and general algebraic varieties.
In the second talk we will indicate a proof of an important result.
Theorem. Let X be a smooth projective variety of general type. Then the automorphism group of X is finite.
This applies to compact Riemann surfaces of genus > 1. Using this I will indicate the proof of Hurwitz's theorem:
Theorem. Let C be a compact Riemann surface of genus g > 1. Then Aut(C) has order at most 84(g-1). This bound is best possible.