Advisor: Dr. Jeffrey Ovall

-Worked on the MS Thesis project 'Finite Element Method for Elliptic Partial Differential Equations' under the guidance of Dr. Ambuj Pandey and Dr. Sombuddha Bhattacharyya.-Studied the weak formulation of elliptic second-order PDEs and the existence and uniqueness of solutions in Sobolev spaces.-Using the Galerkin Method, we devised a computational framework for approximating the weak solutions on finite-dimensional subspaces of the Sobolev space formed by dividing the domain into a finite number of elements. -Implemented FEM in MATLAB and C++ to solve various PDEs and analyzed the error magnitudes as computed by the method.-Extended our study to the Helmholtz equation, where the error magnitudes depend on the wave number of the equation.

-Worked on the project 'Applied Differential Equations for Scientific Research' under Dr. Cristian Rios. -Studied functional analytic concepts such as the Riesz Representation Theorem and the Lax-Milgram Theorem to answer the questions related to the existence of solutions to linear elliptic equations. -Analyzed the existence of weak solutions to the equations by studying the Sobelov Spaces and their properties, which generalize to the classical existence under appropriately smooth boundary conditions-Studied the prototypes: Laplace, Heat and Wave Equation and explored the Galerkin Approximation and the Finite Element Method at the end.

-Worked on the project 'Introduction to Knot Theory' under Dr. Dheeraj Kulkarni.-Focused on the embedding of knots and links in the 3-sphere, the classification of orientable surfaces, analysis of link diagrams and invariants, and link factorization. -Learned about Seifert surfaces and Seifert Matrix, Alexander and Conway's polynomials associated with the knots.