The Robinson-Schensted-Knuth (RSK) correspondence is a bijection from the set of matrices with non-negative integer entries onto the set of pairs of semistandard Young tableaux (SSYT) of the same shape. SSYT can be expressed as integral Gelfand-Tsetlin patterns. We will show how Viennot's light-and-shadows algorithm for computing the RSK correspondence can be extended from matrices with non-negative integer entries to matrices with non-negative real entries, giving rise to real Gelfand-Tsetlin patterns. This real version of the RSK correspondence is piecewise-linear. Indeed, interesting combinatorial problems count lattice points in polyhedra, and interesting bijections are induced by volume-preserving piecewise-linear maps.

We shall discuss the results of Marina Ratner on unipotent flows, and the techniques involved.

A tricolored-sum-free set in F_2^n is a collection of triples {(a_i,b_i,c_i)}_{I=1..m} such that a) for each I, a_i+b_i+c_i=0 b) If a_i+b_j+c_k = 0, then I=j=k. The notion of a tricoloured-sum-free set generalizes the notion of a capset to F_2^n. The basic question here is: How large can a tricolored-sum-free set be? We will see the following two (recent) results. i) Kleinberg's upper bound of 6\binom{n}{n/3} for a tricolored-sum-free set. This in conjunction with a previous result of his establishing a lower bound of \binom{n {n/3}2^{-\sqrt{16n/3}} gives almost asymptotically tight results. ii) Ben Green (in 2005) proved the following BOOLEAN REMOVAL LEMMA: Given \epsilon>0 there exists \delta depending only on epsilon such that the following holds: Write N=2^n. If X,Y,Z are subsets of F_2^n if by deleting \epsilon N elements from X,Y, Z altogether, one can eliminate all arithmetic triangles (triples (x,y,z) with \in X,y\in Y,z\in Z such that x+y+z=0) then there are at most \delta N^2 arithmetic triangles in (X,Y,Z). Green's proof establishes a bound for1/(\delta) which is a tower of 2s of length poly(1/\epsilon). We will look at a recent result of Fox and Lovasz (junior) who obtained an almost tight bound for this delta-epsilon dependence with \delta =O(\epsilon^{O(1)}).

We will give some basic definitions and take a few examples of motives. The reference is A. J. Scholl's article (1991, Seattle).

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.

This will be a continuation of the overview from the last week. Some details will be briefly recalled from the last time, for continuity and the benefit of new audience if any.

In a land-mark paper in 1956, J. Milnor showed that there are non standard differential structures on the 7-dimensional sphere. Six years later along with Kervaire, he introduced an abelian group structure on the set of equivalence classes of smooth structures on spheres of all dimension and determined these groups in several cases. We shall present some of the salient features of this work. This is the second talk on this topic.

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.