The Selmer group of an elliptic curve over a number field encodes several arithmetic data of the curve providing a p-adic approach to the Birch and Swinnerton Dyer, connecting it with the p-adic Lfunction via the Iwasawa main conjecture. Under suitable extensions of the number field, the dual Selmer becomes a module over the Iwasawa algebra of a certain compact p-adic Lie group over Z_p (the ring of padic integers), which is nothing but a completed group algebra. The structure theorem of GL(2) Iwasawa theory by Coates, Schneider and Sujatha (C-S-S) then connects the dual Selmer with the “reflexive ideals” in the Iwasawa algebra. We will give an explicit ring-theoretic presentation, by generators and relations, of such Iwasawa algebras and sketch its implications to the structure theorem of C-S-S. Furthermore, such an explicit presentation of Iwasawa algebras can be obtained for a much wider class of p-adic Lie groups viz. pro- p uniform groups and the pro-p Iwahori of GL(n,Z_p). If we have time, alongside Iwasawa theoretic results, we will state results (joint with Christophe Cornut) constructing Galois representations with big image in reductive groups and thus prove the Inverse Galois problem for p-adic Lie extensions using the notion of “p-rational” number fields.

R.G. Swan and J. Towber showed that if (a2, b, c) is a unimodular row over any commutative ring R then it can be completed to an invertible matrix over R. This was strikingly generalised by A.A. Suslin who showed that if (a r! 0 , a1, . . . , ar) is a unimodular row over R then it can be com- pleted to an invertible matrix. As a consequence A.A. Suslin proceeds to conclude that if 1 r! ∈ R, then a unimodular row v(X) ∈ Umr+1(R[X]) of degree one, with v(0) = (1, 0, . . . , 0), is completable to an invertible matrix. Then he asked (Sr(R)): Let R be a local ring such that r! ∈ GL1(R), and let p = (f0(X), . . . , fr(X)) ∈ Umr+1(R[X]) with p(0) = e1(= (1, 0, . . . , 0)). Is it possible to embed the row p in an invertible matrix? Due to Suslin, one knows answer to this question when r = d + 1, without the assumption r! ∈ GL1(R). In 1988, Ravi Rao answered this question in the case when r = d. In this talk we will discuss about the Suslin’s question Sr(R) when r = d − 1. We will also discuss about two important ingredients; “homotopy and commutativity principle” and “absence of torsion in Umd+1(R[X]) Ed+1(R[X]) ”, to answer Suslin’s question in the case when r = d − 1, where d is the dimension of the ring.

Given one uniform(0,1) random variable we show that one can generate a sequence of iid uniform r.v. and give some applications.

In this work we study a stochastic three-dimensional Landau-Lifschitz-Gilbert equation perturbed by pure jump noise in the Marcus canonical form. We show existence of weak martingale solutions taking values in a two-dimensional sphere $S^2$ and discuss certain regularity results. The construction of the solution is based on the classical Faedo-Galerkin approximation, the compactness method and the Jakubowski version of the Skorokhod Theorem for nonmetric spaces. This is a joint work with Zdzislaw Brzezniak (University of York) and has been published in Commun. Math. Phys. (2019), https://doi.org/10.1007/s00220-019-03359-x.

This work is concerned about some optimal control problems associated to the evolution of two isothermal, incompressible, immiscible fluids in a two-dimensional bounded domain. The Cahn-Hilliard-Navier-Stokes model consists of a Navier亡tokes equation governing the fluid velocity field coupled with a convective Cahn蓬illiard equation for the relative concentration of one of the fluids. A distributed optimal control problem is formulated as the minimization of a cost functional subject to the controlled nonlocal Cahn-Hilliard-Navier-Stokes equations. We establish the first-order necessary conditions of optimality by proving the Pontryagin maximum principle for optimal control of such system via the seminal Ekeland variational principle. The optimal control is characterized using the adjoint variable. We also study another control problem which is similar to that of data assimilation problems in meteorology of obtaining unknown initial data using optimal control techniques when the underlying system is same as above.

In this talk, we will revisit some of the known bounds on the number of rational points on hypersurfaces of a given degree defined over a finite field. We will recall a conjecture proposed by Homma and Kim towards a tight upper bound on the number of rational points on a nonsingular hyperface contained in an even dimensional projective space over a finite field. Finally, we will present a recent work towards proving the above mentioned conjecture for nonsingular threefolds contained in a four-dimensional projective space.

Nematic liquid crystals are quintessential examples of soft matter, intermediate in character between solids and liquids, with long-range orientational order. We model spatio-temporal pattern formation for nematic liquid crystals on two-dimensional polygonal geometries, which are relevant for applications. We work within the powerful continuum Landau-de Gennes theory for nematic liquid crystals. We illustrate the complex solution landscapes on square domains as a function of the square size, temperature and boundary conditions, reporting a novel Well Order Reconstruction Solution on nnao-scale geometries. We discuss generalizations to arbitrary 2D polygons, using symmetry-based and variational techniques to study stable patterns in distinguished asymptotic limits. We conclude by reviewing recent work on stabilization of interior vortices by magneto-nematic coupling in ferronematics, which leads to new possibilities for magneto-mechanical effects in nematic-based materials. This is joint work with researchers in Peking University, Shanghai Jiao Tong, IIT Delhi, IIT Bombay, Illinois Technological University and University of Verona.

The following questions arise quite naturally from what we see around us. Why are soap bubbles that float in air approximately spherical? Why does a herd of reindeer form a round shape when attacked by wolves? Of all geometric objects having a certain property, which ones have the greatest area or volume; and of all objects having a certain property, which ones have the least perimeter or surface area? These problems have been stimulating much mathematical thought. Mathematicians have been trying to answer such questions and this has led to a branch of mathematical analysis known as “shape optimisation problems”. A typical shape optimisation problem is, as the name suggests, to find a shape which is optimal in the sense that it minimises a certain cost functional while satisfying given constraints. Isoperimetric problems form a special class of shape optimisation problems. A typical isoperimetric problem is to enclose a given area with a shortest possible curve. In many cases, the functional being minimised depends on solution/s of a given partial differential equation defined on a variable domain. The plan is to give a glimpse of a few shape optimization problems we have worked on.

Our goal is to give an outline of the proof of the prime number theorem. Let π(x) be the prime-counting function that gives the number of primes less than or equal to x. The prime number theorem then states that π(x) is asymptotically equal to x/log x. The proof involves application of the methods of complex analysis to the study of the real valued function π(x).

In this seminar we shall discuss the analogue of Quillen-Suslin's local-global principle for the transvection subgroups of the full automorphism groups, and its application to generalise results in classical K-theory from the free modules to the projective modules.