We cordially invite you to the 1st Math Symposium of Infosys Young Investigators: an in-House faculty symposium of the Department of Mathematics, IISc, on Wednesday, 6th November, 2019. In it, the most recent cohort of Infosys Young Investigators in IISc Mathematics will present snapshots of their research funded by the Infosys Foundation.

Date: 6th November, 2019 (Wednesday)

Venue: Lecture Hall-1, Department of Mathematics

Time      Speaker      Title
2:00 pm – 2:30 pm      Subhojoy Gupta      Meromorphic geometric structures on surfaces
2:35 pm – 3:05 pm      Vamsi Pritham Pingali      Interpolation or the lack of thereof from affine hypersurfaces;
a vector bundle version of the Monge-Ampere equation
3:05 pm – 3:25 pm        Tea
3:25 pm – 3:55 pm      Apoorva Khare      Polymath-14 - Groups with norms; Distance matrices and Zariski density
5:00 pm        High Tea

Each lecture will be of 30 minutes, with a 5 minute break for Q&A and change of speaker.

### Abstracts

#### Lecture 1 ​

Speaker: Subhojoy Gupta

Title: ​ Meromorphic geometric structures on surfaces

Abstract: I shall present results from two projects of mine that were supported by the Infosys Foundation. Both concern geometric structures on a punctured Riemann surface X, that are associated with holomorphic quadratic differentials on X via certain differential equations.

The first concerns projective structures, which are determined by quadratic differentials via the Schwarzian differential equation. If we fix the orders of the poles at the punctures, the space of such meromorphic projective structures admits a monodromy map to the space of surface-group representations to PSL(2,C).

I shall discuss a recent result characterizing the image of the monodromy map in the case the poles have order at most two. This is an analogue of a theorem of Gallo-Kapovich-Marden for closed surfaces, and clarifies a remark of Poincaré.

The second concerns solutions of the non-linear PDE satisfied by harmonic maps from X to a hyperbolic surface of the same topological type. In this case, a holomorphic quadratic differential is obtained as the Hopf differential of the harmonic map. In the case that X is a closed surface, this defines a homeomorphism between the Teichmüller space of X and the space of holomorphic quadratic differentials on X. This was work of M. Wolf, and independently N. Hitchin, in the mid-1980s. I shall discuss the analogue of this theorem in the case X has punctures, with the assumption that the orders of the poles are all greater than two.

#### Lecture 2​

Speaker: Vamsi Pritham Pingali

Title: Interpolation or the lack of thereof from affine hypersurfaces; a vector bundle version of the Monge-Ampere equation

Abstract: There are two themes of my research funded by the Infosys Foundation. I shall present a typical representative of each theme.

1) PDE arising from Differential Geometry and Physics : The Monge–Ampere equation is a well-studied PDE in complex geometry and its solvability has ramifications in various sub-areas. Inspired by its success, in a preprint, I introduced a vector bundle version of it, and proved a Kobayashi–Hitchin correspondence (essentially, the PDE has a solution if and only if some condition from algebraic geometry is met) in a special case, namely, for some equivariant rank-2 vortex bundles.

2) Analytic studies in Algebraic Geometry : A natural question arising from applied mathematics is, “When can one extend finite-energy analytic functions from subsets of $\mathbb{C}^n$ to all of space whilst preserving the finite-energy condition ?” In a joint work with D. Varolin, we discuss examples and counterexamples of such subsets arising as zeroes of polynomials.

#### Lecture 3​

Speaker: Apoorva Khare

Title: ​ Polymath-14 - Groups with norms; Distance matrices and Zariski density

Abstract: I shall present details of two disparate projects that received funding from the Infosys Foundation. We first discuss the Polymath-14 project, which arose out of a discussion literally made possible by Infosys funding! In this project, we show that a group is abelian and torsion-free if and only if it admits a “norm”, or equivalently a homogeneous length function. This question was motivated by probability, connects algebra, geometry, and analysis, was solved in five days on a blog, and used a computer. (Joint work as D.H.J. Polymath, with Tobias Fritz, Siddhartha Gadgil, Pace Nielsen, Lior Silberman, and Terence Tao.)

Next, I discuss recent joint work with Projesh Nath Choudhury, in which we study distance matrices of trees. We propose a model that subsumes all previous variants to date (starting with Graham, Pollak, and Lovasz). In this model, we compute the determinant, cofactor-sum, and inverse of the distance matrix (and its minors), subsuming prior results, and answering an open question of Bapat et al. The proofs use Zariski density, as our results hold over all unital commutative rings.

Contact: +91 (80) 2293 2711, +91 (80) 2293 2265 ;     E-mail: chair.math[at]iisc[dot]ac[dot]in
Last updated: 08 Aug 2022