Mathematics Projects

PhD thesis is Spectral multiplicity for random operators with projection valued randomness.

Master's project thesis is on the Schauder Regularity of Poisson's Equation, this gives a bound on solution of the Poisson's equation and consequently on a large class of elliptic partial differential equations. The bound is not only on the function itself but also on its derivative.

Reports and Projects done in IMSc till now:

Unimodular Extension of Unimodular Vector in a Polynomial Ring: It is basically the first step of Serre's conjecture. It is a version of Quillen-Suslin theorem.
Brownian motion: Existence and Regularity: Here the mathematical definition and proof of existence of Brownian motion is given. The Holder regularity of individual paths of Brownian motion is also proved. This is a measure theoretic approach to Brownian motion.
Hilbert's Third Problem and Scissor Congruence: Here an homological proof of Jessen-Sydler theorem is given and provides an answer to Hilbert's Third problem. Here the main approach is by homology theory of groups and spectral sequences. This is one of the direct application of homological algebra.

Project done in Mathematics during BS/MS program are:

Peter-Weyl's Theorem, this theorem is about existence of "Fourier coefficients" for elements in L²(G), where G is a (finite) group. This is one of the basic starting point of group representation theory and is connection between harmonic analysis and representation theory.
CS-decomposition of a Unitary matrix , this decomposition of matrix is similar to decomposition of complex numbers in sine and cosines. The main goal of this decomposition is to get smaller block of square matrices for future simplification of some other problem.
Localization and Nullstellensatz , this is the starting point of algebraic geometry. Simply speaking what it answers is the relation between ideals of k[x₁,x₂,...,x_n] and A_kⁿ which is affine space. This may not look profound but it gives relation between geometry and algebra.