Algebraic groups Seminar

Monday, 3 April at 4 pm

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Venue: 114, Mathematics Department

Host: Shripad Garge

Speaker: Dibyendu Biswas

Affiliation: IIT Bombay

Title: Commutative linear algebraic groups I

Abstract: We begin the study of commutative linear algebraic groups

Algebraic Geometry seminar

Wednesday, 5 April, 2023, 11.30 am

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Venue: Ramanujan Hall

Host: Sudarshan Gurjar

Speaker: Nitin Nitsure

Former Affiliation: TIFR, Mumbai

Title: An introduction to derived categories

Geometry and topology Seminar

Thursday, 6 April, 2023 at 11.30 am

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Venue: Ramanujan Hall

Host: Rekha Santhanam

Speaker: Kashyap Rajeevsarathy

Affiliation: IISER, Bhopal

Title: Generating the liftable mapping class groups of regular cyclic covers

Abstract: Let $\mathrm{Mod}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $g \geq 1$. We show that the liftable mapping class group $\mathrm{LMod}_k(S_g)$ of the $k$-sheeted regular cyclic cover of $S_g$ is self-normalizing in $\mathrm{Mod}(S_g)$ and that $\mathrm{LMod}_k(S_g)$ is maximal in $\mathrm{Mod}(S_g)$ when $k$ is prime. Moreover, we establish the existence of a normal series of $\mathrm{LMod}_k(S_g)$ that generalizes a well-known normal series of congruence subgroups in $\mathrm{SL}(2,\mathbb{Z})$. Furthermore, we give an explicit finite generating set for $\mathrm{LMod}_k{S_g)$ for $g \geq 3$ and $k \geq 2$, and when $(g,k) = (2,2)$. As an application, we provide a finite generating set for the liftable mapping class group of the infinite-sheeted regular cyclic covering of $S_g$ for $g \geq 3$ by the infinite ladder surface.

Commutative algebra seminar

Thursday, 6 April 2023 at 4 pm

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 Venue: Ramanujan Hall

Host: Tony Puthenpurakal

Speaker: H. Ananthnarayan

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 from these conjectures, that are relevant over standard graded k-algebras, and some attempts to resolve them.

IPDF extension seminar

Date: 6 April, 5 pm

Venue: Ramanujan Hall

Host: J. K. Verma

Speaker: Suprajo Das, IIT Bombay

Title: Toward Computing the Epsilon Multiplicity

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.

Ph.D. Defence Seminar

Monday, 17 April, 2023 at 11 am

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Venue: Ramanujan Hall

Host: Sandip Singh

Speaker: Shashank Vikram Singh

Affiliation: IIT Bombay

Title: Arithmeticity and Thinness of Some Hypergeometric Groups

Number Theory Seminar

Monday, 17/04/2023 at 4 pm

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Venue: Ramanujan Hall

Host: Sandip Singh

Speaker: C. S. Rajan

Affiliation: Ashoka University

Title: Classical Branching Laws

Abstract: We consider the classical branching laws; for example, that of understanding the decomposition of the highest weight representation of GL(n) to that of GL(n-1), or restriction from odd orthogonal to even orthogonal and symplectic branching laws. We outline an approach via the Weyl character formula. This is joint work with Sagar Srivastava.

Algebraic Geometry Seminar

Tuesday, 18 April, 2023 at 11.30 am

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Venue: Ramanujan Hall

Host: Sudarshan Gurjar

Speaker: Nitin Nitsure

Affiliation: TIFR (Retd)

Title: The cotangent complex

IPDF Seminar

Tuesday, 18 April 2023, 2.30 pm

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Venue: Ramanujan Hall

Host: Rekha Santhanam

Speaker: Navnath Daundkar

Affiliation: IIT Bombay

Title: Topological Complexity of fibre bundles, lower bounds on higher topological complexity and its Applications

Abstract:  For a path-connected space, Farber introduced a numerical homotopy invariant called topological complexity. Farber gave an additive upper bound on the topological complexity of product spaces. In the first part of this talk, we show that fibre bundles have such an upper bound under certain conditions. As an application, we obtain a tight upper bound on the topological complexity of higher-dimensional Klein bottles. This is based on the joint work with Prof. Soumen Sarkar. In the next part of this talk, we study a higher analogue of topological complexity. Here we improve the usual cohomological lower bound by introducing the concept of higher TC-weights. Then we use this to show that nth topological complexity of Seifert fibre manifolds is either 3n or 3n+1. This is based on joint work with Prof. Rekha Santhanam and Soumyadip Thandar.

Algebraic Groups seminar

Tuesday, 18 April 2023, 4 pm

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Venue: Room 215

Host: Shreepad Garge

Speaker: Dibyendu Biswas

Affiliation: IIT Bombay

Title: Diagonalizable groups

CACAAG Seminar

Thursday, 20 April 2023, 4 pm

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Venue: Room 215

Host:  Madhusudan Manjunath

Speaker: Saurav Bhaumik

Affiliation: IIT Bombay

Title: The Hard Lefschetz theorem - second talk

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

Geometric Group Theory Learning Seminar

21 April 2023 at 11:30 AM

Host: Rekha Santhanam

Venue:  Ramanujan Hall

Speaker: Radhika Gupta (TIFR)

Affiliation: TIFR, Mumbai

Title:  “Growth of Groups.”

PDE seminar

Monday, April 24, 3:30 pm

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Venue: Ramanujan Hall

Host: Saikat Mazumdar

Speaker: Mitesh Modasiya

Affiliation: IISER Pune

Title: Fine boundary regularity results for fully nonlinear mixed local
nonlocal problems

Abstract: In this talk, I will consider Dirichlet problems for fully nonlinear mixed local-nonlocal non-translation invariant operators. I will start by the discussion on global Lipschitz regularity for such problems. Later I will move on to the fine boundary regularity which will require careful construction of appropriate sub/super solutions and a weak version of Harnack inequality. If time permits, I will talk about an overdetermined problem for linear mixed local-nonlocal operators.

IPDF Seminar

Tuesday, 25 April 2023, 11.30 pm

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Venue: Ramanujan Hall

Host: Sudarshan Gurjar

Speaker: Arusha C

Affiliation:  TIFR, Mumbai

Title: Vector Bundles and Parabolic Bundles on Nodal Curves

Abstract:  A great deal is known about vector bundles and parabolic bundles in the case of smooth curves but a similar study on singular curves has been relatively slow.
Interestingly, the results for irreducible nodal curves are very similar to those for smooth curves; however, the proofs are different and much more difficult as there are torsion free
sheaves on a nodal curve that are not locally free unlike the smooth case. Ramanan proved in the 70s that a universal family (also called a Poincar ́e bundle) exists for the moduli problem vector bundles on smooth curves if and only if the rank and degree are coprime. One of the key elements in the proof is the computation of the Picard group of the moduli space. First, we prove the non-existence of a Poincar ́e bundle for the moduli problem of vector bundles on nodal curves when the degree and rank are not coprime closely following that of Ramanan. When the degree is sufficiently high, the pushforward of a Poincar ́e bundle to the moduli space is a vector bundle, called the Picard bundle. Although the existence of Poincar ́e bundles (hence Picard bundles) depend on the rank and degree being relatively prime, there always exists a Poincar ́e family of projective bundles called the projective Poincar ́e bundle. Similarly, there is a projective Picard bundle. Next, we discuss the stability of these bundles. Finally, we move from vector bundles to parabolic bundles. Mehta-Seshadri theorem gives a one to one correspondence between irreducible unitary representations of the fundamental group of a punctured compact Riemann surface and stable parabolic bundles on the compact Riemann surface with a parabolic structure at the punctures. We prove that such a correspondence does not hold for nodal curves.

Algebraic Groups seminar

Tuesday, 25 April 2023, 4.30 pm

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Venue: Room 105

Host: Shreepad Garge

Speaker: Dibyendu Biswas

Affiliation: IIT Bombay

Title: Study of diagonalizable groups

Algebraic Geometry seminar: Thursday, 26 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.

GGT Seminar

Date and time: 28th April 2023, at 11:30 AM 

Venue: Ramanujan Hall.

Host: Rekha Santhanam

Speaker:  Soumyadeb Samanta

Affiliation:  IIT Bombay

Title: Lamplighter groups

Abstract:  In this talk, we shall discuss about the Lamplighter groups, mostly Lamplighter group L_2, it's group operation and presentation. Then we will move on to give an efficient formula for calculating the word length of elements of L_2 and have a look at the Cayley graph of L_2. I shall end with a short discussion on which contexts Lamplighter groups appear in Mathematics.