I am a research scholar at Indian Institute of Science Education and Research Kolkata in the Department of Mathematics and Statistics. It is my seventh year of the Integrated PhD program. I have been working under Dr. Somnath Basu. My broad area of research is Algebraic Topology, in paricular homological algebra and homotopy theory. I am also interested in Differential Geometry, Algebraic Geometry.
Department of Mathemetics and Statistics
In this project, we studied the Morse Theory and use it to prove Reeb's theorem, which states that if If $M$ is a compact manifold and $f$ is a $C^2$-function on $M$ with only two critical points, both of which are non-degenerate, then $M$ is homeomorphic to a sphere. We followed the book Morse Theory by John Milnor.
Complex Grassmannian $G_k(\mathbb{C}^n)$ consists of all $k$ dimensional complex subspace in $\mathbb{C}^n$. It is a smooth (complex) manifold of real dimension $2k(n-k)$. Using Morse theory, we studied the cell structure of complex Grassmannian by constructing an explicit Morse function on it.
Here, we again studied the cell structure of complex Grassmannian topologically i.e., without using any smooth structure of complex grassmannian. This process is equally applicable for real Grassmannian too. Finally, we compare this cell structure of complex Grassmannian with that of obtained using Morse theory in project II.
Using the CW-structure of the complex Grassmannian, we first compute the cellular homology of it and then cohomology by PoincarÃ© duality. Finally, compute the cohomology ring structute of it using Young Tableau.
The combined project report can be found here.
The two Trace Theorems in Sobolev spaces were studied. Detailed report can be found in the following link.
Studied the proof of the theorem that any closed oriented surface of genus $g\geq 2$ admits metrics of constant negative curvature. Detailed can be found in the report with following link.
Starting with basic Morse theory, we studied Morse homology and its applications. Notes can be found in the following link.
Starting with the basic category theory, we went to model category. Notes can be found in the following link.
It was a semester (Spring 2023) long reading seminar happened weekly and alternate between the two mentioned topics. On one side, we studied Hochschild (co)homology, their relation with (co)homology of loop space of a simply connected space, the celebrated HKR (Hochschild-Kostant-Rosenberg) Theorem and bit of cyclic homology. The main references were ``Introduction to Homological Algebra" by Charles Weibel, ``An Introduction to Hochschild Cohomology" by Sarah Witherspoon, ``Cyclic Homology" by Jean-Louis Loday.
On the other hand the basics of operads were studied and ended with the recognition principle.