In this talk we shall summarize results on the structure and representation theory of semisimple algebraic groups. This will to prepare ground for the subsequent talks on the Borel-Weil-Bott theorem which explains how (loosely speaking) representations of semisimple algebraic groups can be obtained as sheaf cohomology groups associated to certain line bundles.

In this talk, we study the basics of defining ideal of the Rees algebra of Ideal I and what makes the ideal to be of linear type. Further, we prove that ideals generated by a regular sequences are of linear type.

A result of D.N. Bernstein proved in the late seventies gives an upper bound on the number of common solutions of n multivariate Laurent polynomials in n indeterminates in terms of the mixed volumes of their Newton polytopes. This bound refines the classical Bezout's bound. Bernstein's Theorem has several proofs using techniques from numerical analysis, intersection theory and tori varieties. B. Teissier proved the theorem using intersection theory. A proof using theory of toric varieties can be found in the book by W. Fulton on the same subject. In this talk, I will outline an algebraic proof similar to the standard proof of Bezout's Theorem. This proof, found in collaboration with N.V. Trung, uses basic results about Hilbert functions of multigraded algebras first proved by van der Waerden.

We will explain how to attach an L-function to a motive, what the critical points of this L-function are, and Deligne's conjectures on the values of the L-function at critical points.

In this talk, we study the basics of defining ideal of the Rees algebra of Ideal I and what makes the ideal to be of linear type. Further, we prove that ideals generated by a regular sequences are of linear type.

For a finite abelian group G with |G|= n, the arithmetical invariant EA(G) is defined to be the least integer k such that any sequence S with length k of elements in G has a A weighted zero-sum subsequence of length n. When A={1}, it is the Erdos-Ginzburg-Ziv constant and is denoted by E(G). Similarly, the Davenport Constant DA(G) is defined to be the least integer k such that any sequence S with length k of elements in G has a non-empty A weighted zero-sum subsequence. For certain sets A, we already know some general bounds for these weighted constants corresponding to the cyclic group Z_n. We try to find out bounds for these combinatorial invariants for random A. We got few results in this connection. In this talk I would like to present those results and discuss about an extremal problem related to the cardinality of A

In this talk, we study the basics of defining ideal of the Rees algebra of Ideal I and what makes the ideal to be of linear type. Further, we prove that ideals generated by a regular sequences are of linear type.

We will talk about a result of M. Katz and W. Messing, which says the following. From the Riemann hypothesis and the hard Lefschetz theorem in l-adic cohomology, the corresponding facts for any Weil cohomology follow.

We will prove a purely numerical criterion due to M. Artin to test the rationality of a surface singularity. In practice this is the criterion which is used when a rational surface singularity is being considered.