Projects
Abelian branched Covers and Symplectcic geography problem
This was my MS thesis project under supervision of Prof. Mainak Poddar and Prof.Doug Park. I built my prerequisites in four manifold theory and Seiberg-Witten theory from the books - "Four manifolds and Kirby calculus" and "Notes on Seiberg Witten theory" by Liviu I. Nicolaescu. I also built my background in symplectic topology by reading Salamon-McDuff's book on the same and learning various surgeries in the symplectic category. Right now we are working on a project to construct exotic symplectic simply coneected four manifolds. The idea of using branched covering techniques in algbraic geometry to construct symplectic exotic manifolds is previously explored in the following paper. The idea if to first construct complex surface and prove it is Lefschtez fibration to show it is symplectic manifold. Then using different surgeries, making it simply connected with minimal changes in signature. Then this manifold is exotic follwos from Taubes theorem on seiberg witten invariants of symplectic four manifolds. We improved the asymptotic formula mentioned in the previous paper. My thesis can be found Here
h-cobordism Theorem
This was a semester project of FALL 2021 done under the supervision of Prof. Mainak Poddar and Prof.Doug Park. Initially, this project was aimed at the study of four-manifolds. Later we got curious about why doesn't h-cobordism theorem hold in dimension four. Hence I studied a detailed outline of the proof following the original paper and a nice outline in the book on four-manifolds by Scorpan. We studied the nuances which fail in dimension 4, how the h-cobordism theorem implies topological Poincare conjecture in high dimensions why does the smooth Poincare conjecture possibly fail.
I received an "O" grade for outstanding performance. For details of the project - report
Algebraic and non -algebraic Manifolds
This mini-project was done as a part of the algebraic geometry course in FALL 2021. I studied the relation between classical variety theory, scheme theory. In the end, I examined complex non-singular algebraic varieties and their association with the category of complex manifolds. Details of the project can be found here.
Symplectic Classes
This was a semester project in SPRING 2020 under Dr. Steven Spallone. The main aim of this project was to understand the theory of symplectic classes and their relationship with the well-known chern and Stiefel Whitney classes. The project was based on Prof. Peter May's work on the same. We took a new approach of assuming the axiomatic theory of Chern classes and proving the uniqueness and existence of the symplectic classes. We derived a new proof of the cohomology rings of the Sp(n) groups by using dual induction. In this way, previously known results can be proved by avoiding the spectral sequences.We had to use spectral sequences in the computation of the classifying spaces of these groups. I learned fiber bundle theory, obstruction theory, and various spectral sequences during this project. In the end, we carefully proved with a counterexample that the integer cohomology classes satisfying axioms of the Steenrod Squares do not exist. Becuase of this, there is no equivalent of Wu's formula in integral cohomology for symplectic classes. The project report can be found here.
Study of Lie Groupoids
This project was done as a part of Tifr-VSRP 2020 under Prof.S.K.Roushon. In this project, we followed various papers of I.Moerdijk on the theory of Lie Groupoids and Orbifolds. The focus was on the study of the category of Lie Groupoids. I studied how the well-known results in algebraic topology, foliation theory, and some differential topology results can be generalized using the theory of Lie Groupoids. The project report can be found here.