Lecture series on algebraic stacks

Monday, 21 August, 11:30 am

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Venue: Ramanujan Hall

Host: Sudarshan Gurjar

Speaker : Nitin Nitsure

Affiliation: TIFR, Mumbai (retd)

Title: Algebraic Spaces

Abstract:  In his talk `The Implicit Function Theorem in Algebraic Geometry', delivered in the legendary conference `Algebraic Geometry, Bombay-1968', Michael Artin expanded the category of spaces used in algebraic geometry by introducing `algebraic spaces', which was a step beyond the schemes of Grothendieck. In the 55 years that have passed since then, algebraic spaces have become an important part of Algebraic Geometry. In particular, the theory of algebraic stacks makes foundational use of them.

In this lecture, we will introduce algebraic spaces, just assuming that the audience is familiar with schemes at the level of the first few sections of Chapter 2 of Hartshorne's textbook Algebraic Geometry'. After giving some motivation and examples, some basic properties of algebraic spaces will be outlined.

This lecture is the beginning of a series of about 12 talks titled `An introduction to Algebraic Stacks'. The aim of this series is to cover the first half of the textbook `Champs Algebriques' by Laumon and Moret-Bailly. This knowledge is essential in order to follow a whole lot of modern research in both arithmetic and geometry.

Topology and Related Topics Seminar 

Tuesday, 22 August 2023, 2.15 -3:30  pm

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Venue: Ramanujan Hall, Department of Mathematics

Host:  Rekha Santhanam

Speaker: Priyanka Magar

Affiliation: IIT Bombay

Title: A brief introduction to the stable homotopy category

Abstract:

 The talk will begin with some stable phenomenon that guides us to the Spanier-Whitehead category, which has its own interesting properties. Next, we'll explore how the Brown representability theorem brings in the idea of Spectra and, as a consequence, gives rise to the stable homotopy category.

Commutative Algebra Seminar

Tuesday, August 22, 3:30 pm

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Venue : Room 215

Host: Tony Puthenpurakal

Speaker: Tony Puthenpurakal

Affiliation: IIT Bombay

Title: Triangulated and Derived Categories-2

Abstract: We continue our introduction to Triangulated categories

Algebraic Groups seminar

Tuesday, 22 August 2023, 4 pm

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Venue: Ramanujan Hall

Host: Shripad Garge

Speaker: Chayan Karmakar

Affiliation: IIT Bombay

Title: Derivations and Tangent Spaces - II

Abstract: We study derivations with the aim of introducing Lie algebras for algebraic groups.

Combinatorics, Commutative Algebra and Algebraic Geometry (CACAAG) Seminar

Wednesday, 23 August, 5:30 pm

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Host: Madhusudan Manjunath

Venue: Ramanujan Hall.

Speaker: Madhusudan Manjunath

Title: The Chow Ring of a simplicial toric variety.

Abstract: We will give a combinatorial presentation of the Chow ring of a simplicial toric variety and discuss its applications. 

Annual progress seminar

Date:  24th August, 2023

Time:  10:00 AM -- 11:00 AM

Venue: Room 215

Host: Manoj Keshari

Speaker: Sai Krishna

Title: An algebraic characterization of the affine three space in 
arbitrary characteristic

Abstract:  We give an algebraic characterization of the affine three 
space over an algebraically closed field of arbitrary characteristic. We 
will look at a possible application of this characterization to 
reformulate the question of whether  certain ring is isomorphic to the 
polynomial ring in three variables over an algebraically closed field of 
positive characteristic.

Annual Progress Seminar

Time:  02:00 PM -- 02:35 PM

Host: Sourav Pal

Speaker: Priyanka Aroda

Title: Contractions and their dilations

Abstract:   A contraction is an operator on a Hilbert space which has norm
atmost 1. We study Sz.-Nagy's dilation theorem which models contractions
as a part of nicer operators on larger spaces. We also discuss a more
direct and explicit construction of a unitary dilation of a contraction,
which is due to Schaffer. We study an analogue to Sz.-Nagy's dilation
theorem, due to Ando, for a pair of commuting contractions.

Topology and Related Topics Seminar 

Thursday, 24 August 2023, 2.30 -3:45  pm

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Venue: Room 215

Host:  Rekha Santhanam

Speaker: Soumya Deb Samanta 

Affiliation: IIT Bombay

Title: An Introduction to Riemannian Manifolds

Abstract: In this talk, I will give a brief introduction to the different aspects of Riemannian geometry: Riemannian metrics,  affine connections, covariant derivatives, geodesics, curvature, Jacobi fields, and variation formulas. This talk will serve as a prelude to my next talk which will involve some topological aspects of Riemannian Manifolds.

 Analysis Seminar

Time:  02:40 PM -- 03:30 PM

Host: Sourav Pal

Speaker: Saikat Roy

Title: Applications of Birkhoff-James orthogonality and its connection
with dilation

Abstract: Birkhoff-James orthogonality is a generalized notion of
orthogonality that
extends the inner product orthogonality to the setting of Banach spaces.
In this talk, we present some of its applications in geometry of Banach
spaces and its relation with norming properties of some subspaces of
Banach space. We also consider operator orthogonality and its connection
with the unitary and isometric dilations. We study the case when unitary
(isometric) dilations of two contractions preserve the orthogonality
relation between the basic contractions.