Research Projects and reading courses
We offer short research projects and reading courses on advanced topics to students from Pakistan with no research experience. Interested students can contact us, and we will try to find a volunteer supervisor. If you are interested in any available topics below, please write to us at themathetf@gmail.com (attach your CV, recent transcript, and motivation for the chosen topic). Please note the title of the project in the subject of your email.
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
Mathematical Modelling:
Parameter estimation in ODE Models using Markov chain Monte Carlo (MCMC)
Evolutionary dynamics on graphs
Kurtz's limit theorems: Deterministic ODEs as limit of underlying stochastic processes
A general modelling framework for analysing multiplayers games in networks
Comparative analysis of epidemic models
Population interactions and coexistence
Network-based disease spread
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
Scientific Computing:
Numerical solution of one-dimensional Burgers equation
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