## Research Areas

The Department aims to promote close collaboration between various mathematical disciplines and with other applied areas. The areas of current research are:

### Algebra and Number Theory

• #### Algebraic Geometry and Noncommutative Geometry (Abhishek Banerjee, Amalendu Krishna)

Schemes over symmetric monoidial categories, bivariant Chow theory. Hochschild and cyclic cohomology. Algebraic cycles, algebraic K-theory, $A^1$-homotopy theory of varieties, arithmetic geometry, and $K$-theory of algebraic stacks.

• #### Representation Theory (Apoorva Khare, Pooja Singla, R. Venkatesh)

Representations of: finite groups and finite dimensional associative algebras; p-adic groups and other linear groups with entries from local rings; finitely generated discrete groups. Representations of Lie algebras: semisimple, Kac–Moody, Borcherds, and current algebras; quantum groups; algebras with triangular decomposition.

• #### Number Theory (Soumya Das, Shaunak Deo, Mahesh Kakde, Radhika Ganapathy)

Automorphic forms (and their $p$-adic families) and representations, mod $p$ modular and Hilbert modular forms and their Hecke algebras, Galois representations, L-functions, Iwasawa theory, Arithmetic Geometry, Langlands programme, Analytic Number theory, Algebraic K-theory.

### Analysis

• #### Calculus of Variations (Swarnendu Sil)

Existence and regularity/singularity of minimizers and/or critical points of possibly nonconvex, noncoercive functionals, weak continuity and weak lower semicontinuity, compensated compactness and concentration compactness, differential inclusions.

• #### Differential Equations (Thirupathi Gudi, A.K. Nandakumaran, Swarnendu Sil)

Homogenization of partial differential equations, controllability, viscosity solutions. Numerical methods for partial differential equations. Existence and regularity of nonlinear elliptic partial differential equations and systems.

Hilbert modules, multivariable operator theory, complex geometry.

• #### Harmonic Analysis (E. K. Narayanan, S. Thangavelu)

Analysis on the Heisenberg group and generalisations such as H-type groups, analysis on symmetric spaces of non-compact type and on semisimple Lie groups, spectral multipliers of Laplcians and sub-Laplacians on these spaces, integral geometry on homogeneous spaces and relations with complex analysis.

• #### Positivity and Matrix Analysis (Apoorva Khare)

Positive matrices/kernels and functions operating on them, (total) positivity preservation, structured matrices and kernels.

• #### Several Complex Variables (Gautam Bharali, Purvi Gupta, Vamsi Pritham Pingali, Kaushal Verma)

Holomorphic mappings, holomorphic interpolation, Ohsawa-Takegoshi type extension theorems. Invariant metrics: estimates, metric geometry of hyperbolic domains. Domains in $\mathbb C^n$: convexity and finite-type conditions, optimal and random polyhedral approximations, integral representations and holomorphic function spaces. Convexity properties of real submanifolds in complex spaces. Pluripotential theory and holomorphic dynamical systems.

### Discrete Mathematics

• #### Combinatorics (Arvind Ayyer, Basudeb Datta, Srikanth K. Iyer, Apoorva Khare, R. Venkatesh)

Algebraic and enumerative combinatorics, random geometric graphs, graph limits, symmetric functions, Schur polynomials, determinantal identities; Coxeter groups, root systems, structure theory of Borcherds–Kac–Moody algebras and connections to algebraic graph theory, lattice polytopes and polyhedra; Combinatorial aspects of simplicial complexes, Tessellation and tiling problems.

### Geometry and Topology

• #### Algebraic and Combinatorial Topology (Basudeb Datta)

Combinatorial manifolds, PL-manifolds, minimal triangulation of manifolds, triangulation of spheres and projective planes with few vertices, pseudomanifolds with small excess, equivelar polyhedral maps.

• #### Differential Geometry and Geometric Analysis (Ved Datar, Subhojoy Gupta, Vamsi Pritham Pingali, Harish Seshadri, Swarnendu Sil)

Manifolds of positive curvature, Einstein manifolds, conformal geometry, Teichmüller theory, Kähler geometry, complex Monge–Ampere type equations, special metrics on vector bundles, and degenerations of canonical metrics and connections. Higher dimensional Gauge theory, Yang–Mills equation. Harmonic, $p$-harmonic and polyharmonic maps.

• #### Low Dimensional Topology (Siddhartha Gadgil, Subhojoy Gupta)

Topology of three-manifolds and smooth four-manifolds, hyperbolic geometry, geometric group theory, Heegaard Floer theory and its relations to geometric topology.

### Logic and Foundations of Mathematics

• #### Automated Theorem Proving (Siddhartha Gadgil)

Homotopy type theory and its applications to automated theorem proving.

### Mathematical Physics

• #### Theoretical Physics (Arvind Ayyer, Vamsi Pritham Pingali)

Statistical mechanics, exactly solvable models, partial differential equations arising from string theory and general relativity.

• #### Nonlinear Dynamics (Govindan Rangarajan)

Couded dynamical systems, Synchronization, Turing patterns, applications of Lie algebraic methods to nonlinear Hamiltonian systems, fractal dimensional analysis, generalized replicator dynamics.

### Probability and its Applications

• #### Probability theory and Stochastic Processes (Arvind Ayyer, Mrinal K. Ghosh, Srikanth K. Iyer, Manjunath Krishnapur, Sanchayan Sen)

Random matrix theory, zeroes of analytic functions. Diffusion and related topics: first passage time problems for anomalous diffusion, measure-valued diffusion, branching processes. Stochastic dynamic games, stability and control of stochastic systems, applications to manufacturing systems. Stochastic differential equations. Stochastic geometry, random geometric graphs.

• #### Time Series Analysis (Srikanth K. Iyer, Govindan Rangarajan)

Time series and long-memory processes. Application of time series analysis techniques to neuroscience, especially to brain-machine interface; applications to geophysics.

• #### Mathematical Finance (Mrinal K. Ghosh, Srikanth K. Iyer, Govindan Rangarajan)

Option pricing, portfolio optimization, interest rate models, credit risk models.

Contact: +91 (80) 2293 2711, +91 (80) 2293 2265 ;     E-mail: chair.math[at]iisc[dot]ac[dot]in
Last updated: 20 Mar 2023