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.