Algebraic Geometry Seminar
Tuesday, 30  January 2024, 11:30 am

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Host: Sudarshan Gurjar

Venue: Ramanujan Hall

Speaker: Nitin Nitsure
Affiliation: Bhaskaracharya Pratishthana, Pune

Title: Separated and proper morphisms of schemes

Abstract:  The lecture will begin with closed and universally closed morphisms of schemes. We will prove, following B Poonen, that any scheme that is universally closed over Spec k (where k is any field) is quasi-compact. This will be followed by some basics of quasi-compact morphisms, preparing the way to study separated and proper morphisms via specialization of points. The lecture will be easily accessible to students that have looked at section 4 of Chapter 2 of Hartshorne's “Algebraic Geometry”

Analysis Seminar

Tuesday, 30  Jan. 2 pm

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Venue: Ramanujan hall

Host: Sourav Pal

Speaker: Himadri Halder

Affiliation: IIT Bombay

Title: Bohr's theorem for the polydisc and the symmetrized polydisc

Abstract: In this talk, we will discuss Bohr's theorem for the family of holomorphic functions in the polydisc and symmetrized polydisc. We obtain several new estimates of the multidimensional Bohr radius for polydisc. Moreover, we will also study Bohr's zone for the symmetrized polydisc.  In addition, we will answer certain open questions related to the exact values of multidimensional Bohr radius by using the concept of arithmetic Bohr radius for vector-valued holomorphic functions defined in complete Reinhardt domains in \mathbb{C}^n. Our discussion will include several interesting earlier results as well as our recent works on Bohr's theorem problem.

Analysis Seminar

Tuesday, 30  Jan. 2:30 pm

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Venue: Ramanujan hall

Host: Sourav Pal

Speaker: Swapan Jana

Affiliation: IIT Bombay

Title: Positive definite functions and a generalized Naimark theorem

Abstract: We study the operator-valued positive definite functions on a group and characterize them in terms of positive block matrices. We present a generalization of a theorem of Naimark which states that the positive definite functions and unitary representations are in one-to-one correspondence. We discuss an alternative proof of Brehmer's positivity for doubly commuting operators. We classify all commuting unitary representations over a finite group and present a structure theorem. In the end, we show that the power of a positive-definite function and a unitary representation need not be a positive definite function and a unitary representation respectively. We also present a characterization of such a class of functions.

Analysis Seminar

Tuesday, 30  Jan. 3 pm

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Venue: Ramanujan hall

Host: Sourav Pal

Speaker: Nitin Tomar

Affiliation: IIT Bombay

Title: Operators associated with the pentablock and their relations with biball and symmetrized bidisc

Abstract: We discuss properties of commuting operator triples for which the closed pentablock is a spectral set. An explicit dilation for such a triple is obtained. Also, we show the interactions of such an operator triple with the operator pairs associated with the biball and the symmetrized bidisc.

Commutative Algebra Seminar

Tuesday 30 Jan 2024, 4.00-5:30 pm

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Venue: Room 215

Host: Tony J. Puthenpurakal

Speaker : Samarendra Sahoo

Affiliation: IIT Bombay

Title: The Eisenbud operator and complexity of modules

Abstract: The Eisenbud Operator is an important tool for studying the behavior of infinite free resolution of a module. In this talk, we discuss Eisenbud Operator and its properties as well as how the Betti numbers are related to the complexity of modules.

Algebraic Groups seminar
Tuesday, 30 January 2024, 4 pm -- 5.30 pm
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Venue: Ramanujan Hall
Host: Shripad M. Garge

Speaker: Shripad M. Garge
Affiliation: IIT Bombay

Title: Finite morphisms and normal varieties

Abstract: We introduce the notion of normal varieties and prove the following version of Zariski's main theorem: A bijective and birational morphism of irreducible varieties, \phi: X \to Y, is an isomorphism if Y is normal. 

Mathematics Colloquium

Wednesday, 31 Jan. 4 pm

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Venue: Ramanujan Hall

Host: Prachi Mahajan

Speaker: Debraj Chakrabarti

Affiliation: Central Michigan University, Mt. Pleasant, MI

Title: Convexity conditions and complex analysis

Abstract: The global geometric properties of a space affect the properties of functions that can be defined on it: for example, on a connected topological space a locally constant function is constant. In complex analysis,  complex convexity properties such as pseudo convexity play a crucial role in the construction and properties of holomorphic functions and other analytic objects. These conditions become very important in several complex variables since in domains in the plane the required convexity conditions are automatically satisfied. 

After introducing classical complex convexity conditions, we discuss partial convexity conditions, which lead to the vanishing of Dolbeault cohomology in specified degrees. At the end, we will introduce some new geometric conditions that allow us to prove the vanishing of Dolbeault cohomology in domains in complex manifolds.

This is ongoing joint work with Phil Harrington and Andy Raich of the University of Arkansas.

We will try to make the talk accessible to a wide audience.

Analysis Seminar
Thursday,  1 February 2024, 3.30 pm
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Venue: meet.google.com/fxo-yvfh-avr
Host: Santanu Dey

Speaker: Repana Devendra
Affiliation: IIT Madras

Title: C*-extreme points of unital entanglement breaking maps on matrix algebras.

Abstract: In this talk, we discuss various characterizations of C*-extreme points of the C*-convex set of unital entanglement breaking (EB-) maps on matrix algebras.  A key contribution is made through the establishment of a Radon-Nikodym type theorem for a specific class of EB-maps, offering a comprehensive understanding of these points. Additionally, a noncommutative counterpart to the Krein-Milman theorem is derived for
C*-convexity within the set of unital EB- maps. Lastly, we highlight the significant connection between C*-extreme points and the notion of degradability in the context of quantum information theory.

Commutative algebra seminar

IPDF talk
Friday, 2nd Feb, 3:30-4:30 pm

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Host: Manoj Keshari

Venue: https://meet.google.com/gyb-jfbn-oiu?authuser=0

Speaker: Parnashree Ghosh

Affiliation: ISI Kolkata

Title: Applications of exponential maps to the epimorphism and Zariski
cancellation problem

Abstract: In the first part, we will discuss the Epimorphism Problem and also discuss the famous Abhyankar-Sathaye Epimorphism Conjecture. We will introduce ``Generalised Asanuma varieties" (GAV) of higher dimensions \geq 3 and see some necessary and sufficient conditions for these varieties to be isomorphic to the affine space. We see that this characterization immediately yields a family of higher dimensional hyperplanes satisfying the Abhyankar-Sathaye Conjecture.

In the second part, we see some necessary conditions for two GAVs to be isomorphic and also describe automorphisms of a certain subfamily of GAV. These results show that for each d \geq 3, there is a family of infinitely many pairwise non-isomorphic rings which are counterexamples to the Zariski Cancellation Problem for dimension d in positive characteristic. We further give a complete description of two important invariants called Makar-Limanov and Derksen invariants of a certain subfamily of GAV.

This talk is based on a joint work with Neena Gupta.

Algebraic Groups seminar

Friday, 2 February, 4.00-5.15

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Venue: Room 105

Host: Dipendra Prasad

Speakers: Dibyendu Biswas, Chayan Karmakar, Mohammed Saad and Deep Makadiya

Affiliation: IIT Bombay

Title: Regular elements in semi-simple algebraic groups

Abstract: We will discuss a variety of topics in Algebraic groups through reading some of the papers that have become classics in the subject. The first few seminars will be on the paper by ROBERT STEINBERG, Regular elements of semi-simple algebraic groups Publications mathématiques de l'I.H.É.S., tome 25 (1965), p. 49-80

Analysis seminar

IPDF talk

Friday, 2nd Feb. 4 pm - 5 pm 

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Venue: Online, link TBA

Host: Sanjay Pusti

Speaker: Jitendra Kumar Senapati 

Affiliation : BITS Pilani, Goa

Title: Restriction theorem for the Fourier-Dunkl transform and its applications to Strichartz inequalities

Abstract: We define the Fourier-Dunkl transform, which generalizes the Fourier transform. We prove Strichartz’s restriction theorem for the Fourier-Dunkl transform for certain surfaces, namely, cone, paraboloid, sphere, and hyperboloid, and its generalization to the family of orthonormal functions. Finally, as an application of these restriction theorems, we establish versions of Strichartz estimates for orthonormal families of initial data associated with the Schrodinger propagator, wave propagator, and Klein-Gordon propagator for the case of the Dunkl Laplacian. This restriction theorem generalizes Stein-Tomas and Strichartz’s restrictions theorems in special cases. This is a joint work with B. Pradeep, S. S. Mondal, and H. Mejjaoli.