Listed in this page are some of the talks that I have presented and will be presenting at IISER Kolkata (for the time being).
- Tangent Bundle, Learning Seminar on Morse Theory, IISER Kolkata, February, 2018.
- Tangent Spaces, Learning Seminar on Morse Theory, IISER Kolkata, February, 2018.
- Topological and Smooth Manfolds, Learning Seminar on Morse Theory, IISER Kolkata, January, 2018.
- On Sard's Theorem, IISER Kolkata, October 2017.
- On Einstein-Gauss-Bonnet gravity, IISER Kolkata, April 2017.
- Tautalogical Bundles over Grassmannian, IISER Kolkata, April 2017.
- Grassmannian as a metric space, IISER Kolkata, December 2016.
I talked about pushforwards, definition of tangent bundle; discussed natural topology on it, charts and smooth transition maps, Hausdorff property and concluded that $TM$ is a smooth manifold of dimension $2\times$dim$(M)$. I also discussed tangent bundle of $\mathbb{R}^n$ $(T\mathbb{R}^n\cong \mathbb{R}^n\times \mathbb{R}^n)$ and tangent bundle of $S^1$ $(TS^1\cong S^1\times \mathbb{R}^n)$. While giving the talk, sir told us to compute $T(S^1\times S^2)$.
I gave three definitions of tangent spaces: equivalence class of curves, derivation of smooth real-valued functions and germs. Then, I showed the equivalence between these definitions, tangent spaces as a vector space of dimension n(=dimension of manifold), $T_p(\mathbb{R}^n)\cong \mathbb{R}^n$, $T_pS^1 =$ set of all vectors perpendicular to the radius.
I gave talk on topological manifolds, smooth structure, smooth manifolds, examples ($\mathbb{R}^n$, $S^n$), smooth atlas, maximal atlas. While giving the talk, I came to know a computation of transition functions of $S^n$ just from Euclidean Geometry.
This was a talk as a part of my discussion with Dr. Somnath Basu. Sard's Theorem tells that set of all critical values of a smooth function on a manifold has Lebesgue Measure zero. Prior to this talk, I gave few talks on Inverse and Implicit Function Theorem.
This talk was my final master's thesis presentation at IISER Kolkata. Gauss-Bonnet gravity is a modification of Einstein's general relativity where a scalar quantity is added to the Lagragian of Einstein-Hilbert action. The scalar quantity is called the Gauss-Bonnet term (this is a special case of Lovelock gravity). The interesting part of this gravitational theory is that the equations of motions remains second order differential equations as oppose to other $f(R)$ or Lovelock gravities. I talked about existence of Schwarzchild analogue and TOV analogue solutions, Birkhoff's Theorem for this gravity and the existence of a Global Monopole solution in pure-Gauss-Bonnet gravity.
I talked about grassmannian being a smooth manifold of dimension $n(n-k)$ and the tautalogical vector bundle of rank $k$, denoted as ${\gamma}^{k}_{n}$ over it. I topologized grassmannian as quotient space of Steifel Manifold (set of all $k$-frames in $\mathbb{R}^n$). While giving the talk, I came to that Gram-Schimdt process is a retract (abstract property of Gram-Schimdt).
Grassmannian is the set of all $k$-dimensional subspaces of $\mathbb{R}^n$, denoted as $G_k(\mathbb{R}^n)$ or $G_{k,n}$. For $n=1$, it is nothing but the real projective space $(\mathbb{R}P^n)$. In the talk, I showed an existence of a metric in Grassmannian and verified that the space is compact and path-connected (therefore, connected). The metric is given by $$d(L,L^\prime)=\text{sup}_{x\in L\cap S^{n-1}}\text{inf}_{y\in L^\prime}d_E(x,y)=\text{sup}_{x\in L\cap S^{n-1}}d(x,L^\prime)\;\;\; \forall L,L^\prime\in G_{k,n}$$.
