Consider the following three properties of a general group G: (1) Algebra: G is abelian and torsion-free. (2) Analysis: G is a metric space that admits a "norm", namely, a translation-invariant metric d(.,.) satisfying: d(1,g^n) = |n| d(1,g) for all g in G and integers n. (3) Geometry: G admits a length function with "saturated" subadditivity for equal arguments: l(g^2) = 2 l(g) for all g in G.

Fluid-structure interaction problems have been an important area of research in recent years. Such systems occur, for instance, in aerodynamics (flow around an aircraft), medicine (blood flow in vessels), zoology (swimming of aquatic animals). The mathematical study of these problems rises several challenges, the main one being due to the fact that the domain filled by the fluid is one of the unknowns of the problem. In this talk, we present some recent advances in the mathematical analysis of some particulate flows. We show that a variety of such system admits a global in time unique solution for small initial data in the $L^p$ - $L^q$ framework.

We discuss on feedback stabilizability of the infinite dimensional system via the solution of Riccati equation. We touch upon some interesting aspects of an optimal control problem. At the end, if possible, we shall discuss on some interesting open problems in this direction.

Branching random walk arises naturally in mathematical biology, statistical physics and probability theory. Roughly speaking, it models a system of growing particles or organisms that invades an environment in a systematic fashion. Two famous statistical physicists (Eric Brunet and Bernard Derrida) made conjectures about the long run configurations of positions of particles in a branching random walk, and asked an open question in their seminal work in 2011. Their question was answered positively by Maillard (2013), and the conjectures were mathematically proved recently by Madaule (2017) under certain conditions. In this talk, we shall concentrate on the PhD thesis of Ayan Bhattacharya, who verified Brunet-Derrida conjectures outside the Maillard-Madaule setup. If time permits, some other recent related work will also be discussed. This talk will be based on joint work with Ayan Bhattcharya and Rajat Subhra Hazra. The papers are available in https://arxiv.org/abs/1411.5646 and https://arxiv.org/abs/1601.01656.

The celebrated Helton-Howe trace formula for hyponormal operators is derived as a consequence of Krein's trace formula . In many situation trace is a 'special' non-commutative integral of 'operator-functions ' and the said formula relates the non-commutative integral with the 'volume Lebegue integral' ( in the usual commutative sense ) .

Prof. Chaudhuri has not given the abstract to keep the matter a little mysterious!!

We continue the discussion of the asymptotic behavior of a C_0 semigroup. Then, the different notions of controllability of a linear system in infinite dimension will be introduced in an abstract set up. For example, we study the heat equation in details.

In the previous talk, we saw that given an ideal I in a polynomial ring R, the monomials not in the initial ideal of I form a basis of R/I. In this talk, we will see further examples of how to use the initial ideal of I to get more information about I (or R/I).

Holomorphic vector bundles on compact Riemann surfaces and its other avatar, namely algebraic vector bundles on smooth, complex projective curves has a long and rich history with important contributions by the Indian Algebraic Geometry school. I will introduce this subject with special focus on bundles which come from representations of the fundamental group of the surface. Much of the talk will be accessible to Ph.D students.

Attached.