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.