We shall begin with the homotopy invariance property of K-theory. After reviewing monoids and monoid algebras, we present some results which are monoid version of the homotopy invariance property in K-theory. This answers a question of Gubeladze. Next, we will discuss the monoid version of Weibel's vanishing conjecture and some results in this direction. Finally, we will talk about the homology stability for groups. Here we present a result which improves homology stability for symplectic groups. If the time permits, some application of the homology stability will be given to the hermitian K-theory.

In this talk, we will mainly discuss on the control of wave equation. At the first part of the talk, we will give an overview of the control of wave equation and mention some important results in this direction. Then in the second part, the control of wave equation using Hilbert uniqueness method will be discussed.

In this talk, we attempt to give a brief introduction to the theory of Hypocoercivity which has become an indispensable tool in the study of relaxation to equilibrium states for mathematical models arising in statistical physics. The essential ideas behind this theory will be motivated via simple examples. The role of certain functional inequalities while deriving explicit rates of convergence will be made precise during this talk. This talk concludes by addressing a certain degenerate kinetic Fokker-Planck equation. Incidentally, the study of the trend to equilibrium for this degenerate model finds link to the acclaimed Geometric Condition from the theory of control for wave propagation.

Selmer group is an important object of study in number theory. We will discuss a twisting result in the setting of so called "non-commutative" Iwasawa theory. We will further use this to deduce a duality result for certain Selmer groups. (This talk is based on joint works with T. Ochiai, G. Zabradi and S. Shekhar.

The aim of the talk is to give a brief survey on the Wilf's conjecture, and to present a commutative algebra formulation of it. We will verify Wilf's conjecture in some cases. A numerical semigroup $S$ is a subset of the nonnegative integers $N$ that is closed under addition, contains 0, and has finite complement in $N$. The Frobenius number $F$ of numerical semigroup $S$ is the largest integer not in $S$. Let $d$ be the minimal number of generators of $S$ and $n$ be the number of representable integers in the interval $[0, F]$. Wilf's conjecture states that $F +1 \leq n d$.

See Attachment.

The second Strang lemma gives an error estimate for linear problems written in variational formulation, such as elliptic equations. It covers both conforming and non-conforming methods, it is widely spread in the finite element community, and usually considered as the starting point of any convergence analysis. For all its potency, it has a number of limitations which prevents its direct application to other popular methods, such as dG methods, Virtual Element Methods, Hybrid High Order schemes, Mimetic Methods, etc. Ad-hoc adaptations can be found for some of these methods, but no general `second Strang lemma' has been developed so far in a framework that covers all these schemes, and others, at once. In this talk, I will present a `third Strang lemma' that is applicable to any discretisation of linear variational problems. The main idea to develop a framework that goes beyond FEM and covers schemes written in a fully discrete form is to estimate, in a discrete energy norm, the difference between the solution to the scheme and some interpolant of the continuous solution. I will show that this third Strang lemma is much simpler to prove, and use, than the second Strang lemma. It also enables us to define a clear notion of consistency, including for schemes for which such a notion was not clearly defined so far, and for which the Lax principle `stability + consistency implies convergences' holds. I will also extend the analysis to the Aubin-Nitsche trick, presenting a generalisation of this trick that covers fully discrete schemes and provides improved error estimates in a weaker norm than the discrete energy norm. We will see that the terms to estimate when applying this Aubin-Nitsche trick are extremely similar to those appearing when applying the third Strang lemma; work done in the latter case can therefore be re-invested when looking for improved estimates in a weaker norm. I will conclude by briefly presenting applications of the third Strang lemma and the abstract Aubin-Nitsche trick to discontinuous Galerkin and Finite Volume methods.

A file with the title and abstract of his talk is attached

"In this talk, we present a class of new higher order WENO spatial approximation schemes to solve the hyperbolic conservation laws along with the use of total variation diminishing (TVD)/strong stability preserving Runge-Kutta (SSPRK) temporial derivative approximation techniques. The objective of these developments aim for the improvements, in obtaining higher resolution and efficiency of the solution which retains the desired order of accuracy in smooth regions and in the presence of critical points. These improvements have been achieved by mainly focusing on the construction of new smoothness indicators which plays a key role in the spatial derivative approximation of flux function via the nonlinear weights in WENO algorithm. With these new measurements, higher-order WENO schemes viz fifth and seventh-order WENO schemes have been constructed and subsequently imposed a sufficient condition on the parameters in the weight functions which recovers the optimal order for smooth regions of solution that includes the critical points. Numerical results show that these new schemes achieve optimal-order of accuracy. These schemes also show the advantage of resolving the sharper results for shock waves, contact discontinuities and the regions that contain high-frequency waves."

We will discuss Zariski's theory of complete modules and their relation to base points of linear systems.