I am broadly interested in Differential Geometry and Algebraic Topology. I also interested in Algebraic Geometry and Category Theory. In future, I want to explore the mathematical formalisms of physical theories.
I have worked with Dr. Somnath Basu in my 5th year on Grassmannian (set of all $k$-dimensional subspaces of $\mathbb{R}^n$, $G_{k,n})$. Where, we have shown that Grassmannian is a metric space. The induced topology on grassmannian has some nice topological properties: compact and path-connected. The detailed report on grassmannian is given here. In the next term, I worked on the smooth manifold structure and tautalogical bundle over Grassmannian.
I have done my Master's Thesis under Prof. Narayan Banerjee. We worked on Gauss-Bonnet gravity. Einstein-Gauss-Bonnet gravitational theory is a modification of Einstein’s general relativity, where we added a scalar term to the Lagrangian of Einstein-Hilbert action. The beauty of this theory is that it gives the second order differential equation $(G_{\mu\nu}+\alpha H_{\mu\nu}=0)$ in metric like, Einstein’s field equations. Motivated by this fact, we tried investigate if this theory has the same kind of solutions to GR. We showed that there exists a Schwarzchild like solution in this theory and Birkhoff’s theorem also holds. To our surprise, we got the Global Monopole metric as a solution to the field equations consisting only the Gauss-Bonnet terms $(H_{\mu\nu}=0)$. Generally, global monopole occurs due to the breaking of global SO(3) symmetry. We also showed that there exist the Reisnner-Nordstrom analogue and TOV analogue to the Einstein-Gauss-Bonnet gravity in five dimensions. We aim to investigate the geometry of our problem to understand occurance of the global monopole metric.
