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

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

Let X be a smooth projective variety contained in CP^n. We say X is nondegenerate if it is not contained in any proper hyperplane. Given a variety of dimension m in CP^n, we intersect it with m many hyperplanes and get bunch of points. This number is independent of the choice of the hyperplanes and is defined to be the degree of the variety. A set of k points in CP^n is said to be in general position, if every subset of n+1 points spans all of CP^n. We will prove the general position theorem which states that given an irreducible nondegenerate curve C in CP^n ( where n is \geq 3) of degree d, a general hyperplane meets C in d points which are in general position. Using this we will show any nondegenerate variety X in CP^n , degree \geq 1+codim(X).

In this talk, we will continue the discussion on the control of wave equation from where we stopped at the previous lecture. The control of wave equation using Hilbert uniqueness method will be discussed.

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

abstract of the talk.

The Minimal Model Program (MMP) or the Mori Program is one of the fundamental tools of birational classifications and the construction of Moduli Spaces of higher dimensional varieties. In this talk I will explain the main ideas of MMP and its various applications. Finally I will talk about the recent progress of MMP in positive characteristic.

In this talk we will tell what the purpose of an error-correcting code is, and we will in particular study linear codes. We will relate some of the most important properties of such codes with those of another class of mathematical objects, namely matroids. These are objects that arise in a natural way, either from undirected (multi)graphs, or, as the name indicates, from matrices. Furthermore, if time permits, we will sketch briefly how algebraic geometry over finite fields enter the picture when defining and producing codes with good properties.

In this talk, we will look at three different ways to calculate the local cohomology modules and some of the properties of these modules. No pre-requisites are required for the talk, so all are welcome!

Homogenization is a branch of science where we try to understand microscopic structures via a macroscopic medium. Hence, it has applications in various branches of science and engineering. This study is basically developed from material science in the creation of composite materials though the contemporary applications are much far and wide. It is a process of understanding the microscopic behavior of an in-homogeneous medium via a homogenized medium. Mathematically, it is a kind of asymptotic analysis. We are interested in the asymptotic behavior of elliptic boundary value problems posed in domains with highly oscillating boundary. In fact, we will consider different types of unfolding operators to study many types of oscillating boundary domains with various model problems posed in them. We will also see some interesting optimal boundary control problem posed in such domains. In total, the presentation will start from the asymptotic behavior of Laplacian in a simple rectangular oscillating boundary domains to the future possibilities of the shapes of oscillations in the boundary keeping in mind the mathematical issues arise in topology optimization.