14 November, 2022
Title: Variational Inference: Ideas and Advancements
Speaker: Mr. Ananyapam De, 5th Year BS-MS, IISER Kolkata
Abstract: Intractable posteriors are a common hurdle in Bayesian inference and Probabilistic Machine Learning. Although MCMC methods are widely used to tackle this, they are notoriously slow. Variational inference is a technique which helps us circumvent this problem. I will introduce the problem of intractable posteriors, discuss the core idea of variational inference, optimization of the ELBO and the developments in this technique which make it more powerful and scalable.
17 October, 2022
Title: Polynomial convexity and convexification of closure of strongly pseudoconvex domain
Speaker: Mr. Sanjoy Chatterjee, PhD Student, IISER Kolkata
Abstract: We will show that the closure of a bounded pseudoconvex domain, which is spirallike with respect to a globally asymptotic stable holomorphic vector field and whose boundary is transversal to that vector field is polynomially convex. Next we will provide a necessary and sufficient condition on univalent function on strongly convex domain for embedding it into a filtering $L^{d}$-Loewner chain.
10 October, 2022
Title: Uniform algebras generated by holomorphic and close-to-pluriharmonic functions in several complex variables
Speaker: Mr. Golam Mostafa Mondal, PhD Student, IISER Kolkata
Abstract: We will start by stating a well-known theorem due to Stone and Weier- strass in the context of uniform algebra. As an application, we will first discuss some known approximation results in one and several complex variables for which we need polynomial convexity, which is a fundamental notion in the theory of uniform approximation. An overview of these two topics will be given, and one new result will be presented. In particular, we will provide a result in which the presence of an analytic disc is the only obstruction for polynomial approximation.
26 September, 2022
Title: On the Hartman-Grobman Theorem
Speaker: Mr. Debanjit Mondal, PhD Student, IISER Kolkata
Abstract: According to Hartman-Grobman Theorem, near a hyperbolic equilibrium point $x_0$ the nonlinear system $\dot{x} = f(x)$ and linear system $\dot{x}= Ax$, where $A=Df(x_0)$ has the same qualitative structure, i.e. 1st and 2nd system are locally topologically conjugate. In this talk we will see a proof of this theorem and using this theorem we will draw phase diagram of a nonlinear system.
19 September, 2022
Title: On the Goldbach Conjecture
Speaker: Mr. Habibur Rahaman, PhD Student, IISER Kolkata
Abstract: One of the famous unsolved problems in number theory is 'The Goldbach Conjecture'. Though it is unsolved yet, the best result in this direction is Chen's theorem, which says that every sufficiently large even number can be written as a sum of a prime and an almost prime. In this talk, we discuss what is a general sieve and we will see the proof of Chen's theorem.
12 September, 2022
Title: Classification of representation of $\mathfrak{sl}(2,\mathbb{C})$
Speaker: Mr. Jitender Sharma, PhD Student, IISER Kolkata
Abstract: In this talk we will classify all the representation of $\mathfrak{sl}(2,\mathbb{C})$. We will see that all the representation comes from the Lie group $SU(2)$, group of special unitary matrices of order $2$.
29 August, 2022
Title: Biorthogonal family construction and its application to controllability
Speaker: Mr. Manish Kumar, PhD Student, IISER Kolkata
Abstract: We all know about the orthogonal family from our basic linear algebra course but have you heard of Biorthogonal family? The center of this talk will be to define it, and construct one such family for a particular family of complex exponential functions, along with its need to solve a moment problem and hence a control problem for a system.
22 August, 2022
Title: On The Fermat's Last Theorem Modulo a Prime
Speaker: Mr. Rajiv Mishra, PhD Student, IISER Kolkata
Abstract: Fermat's "last theorem" states that for any natural number $n\geq 3$, the equation $x^n+y^n=z^n$ does not have any solution in the natural numbers. Schur tried to prove the Fermat's last theorem by showing that the equation $x^n+y^n=z^n$ does not have any solution if we do arithmetic modulo any prime. But he found that this is not true, in fact, for any natural number $n$, the equation $x^n+y^n=z^n(\text{mod }p)$ has a solution in natural numbers for any sufficiently large prime number $p$. We shall see how graph theoretic tools are used to solve this number theory problem.