Abstract. Using resolution of singularity we will give a geometric proof of Teissier's Conjecture about the multiplicity of the product of two ideals. A stronger form of this proved in Jugal Verma's lecture will also be proved. We will also give a short proof of C.P.Ramanujam's geometric interpretation of the multiplicity of a local ring.

Abstract: Let C be an algebraic curve defined over a finite field with q elements. The Hasse-Weil bound gives an upper bound on the number of rational points on C. An algebraic curve is called maximal if this upper bound is attained. On of the most important examples of a maximal curve is the Hermitian curve, which can be defined by the equation x^q+x=y^(q+1) over the field GF(q^2) with q^2 elements. It has genus q(q-1)/2 and it is not hard to show that any maximal curve over GF(q^2) has genus at most q(q-1)/2. One of the main open problems in this area is to classify (the genera of) all maximal curves for a given finite field GF(q^2). In a recent work together with Maria Montanucci, a new family of maximal curves was discovered. In this talk I will give an introduction to the topic as well as present this new family of curves.

Abstract : We discuss analogues of the theorem "A CW complex with even dimensional cells has free homology" for G-spaces. For the cyclic group of order p, the "freeness theorem" was first proved by Lewis, and subsequently the result was improved by Ferland -Lewis. We generalize this result to other cyclic groups (in joint work with Surojit Ghosh).

Abstract: We will present basic facts about the analytic spread of an ideal and then present the Rees-Sharp proof of Teissier's conjecture for 2-dimensional local rings.

Abstract: I'd prove that the Picard group of a tropical curve of genus g is a real torus of dimension g.