


Lecture series on algebraic stacks
Monday, 21 August, 11:30 am

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, Bombay1968', 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 MoretBailly. 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 SpanierWhitehead 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

Venue : Room 215
Host: Tony Puthenpurakal
Speaker: Tony Puthenpurakal
Affiliation: IIT Bombay
Title: Triangulated and Derived Categories2
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 BirkhoffJames orthogonality and its connection with dilation Abstract: BirkhoffJames 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.