Research Projects and Reading Courses

Interested and dedicated students can contact us to work on any of the following topics:

Mathematical Analysis and Geometric measure theory: 

• A measure-theoretic study of the Central Limit Theorem and the Law of Large Numbers

• Hausdorff dimension of self-similar sets

PDEs and Functional Analysis:

• Theory of function spaces: L^p Space, Sobolev spaces, Schwartz space, distributions

•  Lax–Milgram theorem and its applications to existence of weak solutions for partial differential equations 

• Universality of L^2 and applications to Fourier Series

Calculus of Variations: 

• Direct methods in variational calculus and applications 

Data Science and Mathematical Modelling:

• Parameter estimation using Markov chain Monte Carlo (MCMC)

•  Modelling data of real populations

• Network-based models of epidemics

Scientific Computing:

• Numerical solution of one-dimensional Burgers equation

Theory of ODEs:

• Kurtz's limit theorems: Deterministic ODEs as limit of underlying stochastic processes

Geometry and Topology:

•  (Algebraic topology) Fundamental groups and Singular homology 

• (Differential Geometry) Surfaces in R^3, the Gauss–Bonnet theorem

• (Complex geometry) Riemann Surfaces, the Riemann–Roch theorem

Dynamical Systems and Ergodic Theory: 

• Topological dynamics; Symbolic dynamics.

• Ergodic decomposition of dynamically invariant measures

•  Existence of of invariant measure for a continuous transformation on a compact space.

• Recurrence theorems, Gauss map, and circle rotations.

• Birkhof theorems and its applications.

• Dynamical systems in higher dimensions (Hyperbolic dynamics), hyperbolic Toral Auto-morphisms, horseshoe and solenoid.

Projects in progress:

• Parameter estimation in ODE Models using Markov chain Monte Carlo (MCMC)

• Evolutionary dynamics on graphs

• Classification of closed surfaces

Completed projects: 

• Box and Hausdorff dimension of self-similar sets

• Network-based models of epidemics

• Phenomenological models of epidemics