Past projects
Laplacian on Riemannian manifolds (Spring semester 2018)
In the spring semester of 2018, I studied the Laplacian, and more generally elliptic partial differential operators acting on Riemannian manifolds. I looked at the Hodge theorem, and also studied how the eigenvalues of the Laplacian affect the geometry of the manifold, and vice versa. Here is a summary of some of things I studied over the semester.
Homotopy Theory and \(L^2\)-invariants (Summer 2017)
In the summer of 2017, I was a summer student at the University of Münster, and I took courses in Homotopy Theory and Homological Algebra. I also participated in a seminar on \(L^2\)-invariants.
In particular, I worked out the \(L^2\)-Betti numbers of certain amenable spaces, and read a few other results. I also learnt a fair amount of homotopy theory, and wrote up a few of the more technical results.
Fourier Analysis (Summer 2016)
In the summer of 2016, I did a reading project on Fourier Analysis and its applications under Prof. Manjunath Krishnapur.
I read a proof for Weyl's equidistribution theorem for quadratic polynomials with irrational coefficients, and I wrote up the following expository article on the proof of the theorem. I also read a proof of Roth's theorem which states that a sufficiently large susbet of the natural numbers will contain three term arithmetic progression. Here's the expository article I wrote up after understanding the proof.
ProvingGround (Summer 2015)
In the summer of 2015, I worked with Prof. Siddhartha Gadgil on his automated theorem prover ProvingGround.
I also learnt a little bit of combinatorial group theory on my own to understand the Andrews Curtis conjucture, which naturally led to algebraic topology