Algebraic Groups Seminar: Tuesday, 11/04/2023, 4 pm

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Host: Shripad M. Garge : Venue: 114

Speaker: Dibyendu Biswas, Affiliation: IIT Bombay

Title: Commutative linear algebraic groups II

Abstract: We initiate the study of tori in this lecture.

Number Theory Seminar : Wednesday: 12/04/2023, 2.30 pm

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Host: Preeti Raman : Venue: Room 216

Speaker: Parul Gupta : Affiliation: IISER, Pune

Title: A ruled residue theorem for function fields of elliptic curves

Abstract: Let E be a field and v be a valuation on E. The ruled residue theorem, proved by J. Ohm in 1983, characterizes residue field extensions for valuations on a rational function field E(X) in one variable. It states that for any extension of v from E to E(X), the corresponding field extension at the level of residue fields is either algebraic or ruled, i.e., it is a rational function field in one variable over a finite extension of the residue field of E. In this talk, we will discuss residue field extensions for valuation extensions of v to the function field F of an elliptic curve over E. In this case, we show that there exists at most one extension of v to F such that the corresponding residue field extension is neither algebraic nor ruled. We also describe the cases where such an extension occurs. This is joint work with K. J. Becher and S. Mishra.

Algebraic Geometry seminar

Wednesday, 12th April 2023 at 4 pm

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Host: Madhusudan Manjunath 

Venue: Room 215

Speaker: Saurav Bhaumik, IIT Bombay

Title: The Hard Lefschetz Theorem

Abstract: We will state and prove the hard Lefschetz theorem.

Ph.D. Defence Seminar

Thursday, 13th April 2023, 11 am

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Venue:  Ramanujan Hall

Host: Debanjana Mitra

Speaker: Sakil Ahamed

Title:  Control Problems for Certain Linear Coupled Systems.

Algebraic Geometry seminar: Thursday, 13 April at 11.30 am.

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Host: Sudarshan Gurjar : Venue: Ramanujan Hall

Speaker: Nitin Nitsure : Affiliation: TIFR (retd)

Title: Derived Categories and derived functors: a first look-2

Abstract: The category of abelian groups, and that of modules over a ring is part of common knowledge. These are examples of Abelian categories.  More advanced examples include sheaves of abelian groups on a topological space, quasi-coherent sheaves on a scheme, etc. Functors such as Hom, tensor product or global sections on these categories are not exact but have higher derived functors, such as Ext, Tor, and sheaf cohomology, which give long exact sequences that are basic tools in both theory and computations. A more evolved version of abelian categories and derived functors are the so-called `derived categories' made from them, which support a corresponding version of derived functors. These are indispensable in various topics in mathematics, such as duality theory, deformation theory, D-modules, etc. At first sight, this subject appears to be forbiddingly technical. These two lectures are a `first look' at the subject. We will focus on conceptual points and simple examples so that the students can go on to read the more complete technical accounts in the literature with
greater ease.

Commutative Algebra Seminar: Thursday, 13/04/2023, 4 pm

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Venue: Ramanujan Hall : Host: Tony J. Puthenpurakal

Speaker: Prof. Ananthnarayan Hariharan : Affiliation: IIT Bombay

Title: Boij-Soderberg Theory over Standard-Graded Rings

Abstract: The Boij-Soderberg conjectures, which appeared in 2008, signified a paradigm shift in the study of Betti numbers of graded modules over polynomial rings. In this talk, we identify questions arising out of these conjectures, that are relevant over standard graded k-algebras, and some attempts to resolve them.