Abstract: This talk will be based on the following elementary and nice exposition http://www.math.stonybrook.edu/~roblaz/Reprints/Green. Laz.Simple.Pf.Petri.pdf Using some simple facts about projective space, cohomology, cohomology of line bundles on projective space, we shall prove the following theorems: 1. Noether's theorem - Projective normality of the canonical embedding of non-hyperelliptic curves. 2. Petri's -theorem - Let X be a smooth and projective curve of genus g \geq 5. Assume that X carries a line bundle A of degree g-1 with h^0(A)=2. Further assume that both A and \Omega_X\otimes A^* are generated by their global sections. Then the homogeneous ideal of X in its canonical embedding is generated by degree 2 elements.

Abstract: We will work over the finite field F_q, q = p^k. The Reed-Solomon code with parameters (q,d), denoted as RS(q,d), is the linear space of all polynomial functions from F_q to F_q with degree atmost d. The Reed-Muller code with parameters (q,m,d), denoted as RM(q,m,d), is the m-variable analog of RS(q,d), defined to be the linear space of all polynomial functions from F_q^m to F_q with total degree atmost d. A nonempty set N in F_q^m is called a Nikodym set if for every point p in F_q^m, there is a line L passing through p such that all points on L, except possibly p, are contained in N. Using the polynomial method and the code RM(q,m,q-2), we can prove the lower bound |N| >= q^m / m!. We will outline this proof. We will then define a new linear code called the m-lift of RS(q,d), denoted as L_m(RS(q,d)), and show that RM(q,m,d) is a proper subspace of L_m(RS(q,d)). We will use this fact crucially, in a proof very similar to the earlier one, to obtain the improved lower bound |N| >= (1 - o(1)) * q^m, when we fix p and allow q to tend to infinity. This result is due to Guo, Kopparty and Sudan.

Abstract: Let V and V' be irreducible representations of a complex semisimple Lie algebra g with highest weight vectors v and v' of weights m and m' respectively. For w in the Weyl group, let M(m,m',w) denote the cyclic g-submodule of V tensor V' generated by the vector v tensor wv' (where wv' denotes a non-zero vector in V' of weight wm'). It was conjectured by Kostant and proved by Kumar that the irreducible representation V(m,m',w) whose highest weight is the unique dominant Weyl conjugate of m+wm' occurs with multiplicity exactly one in the decomposition of M(m,m',w) into irreducibles. Since M(m,m',w0) equals V tensor V', where w0 denotes the longest element of the Weyl group, it follows from this that V(m,m',w) occurs in the decomposition of V tensor V'. This corollary was conjectured earlier by Parthasarathy, Ranga Rao, and Varadarajan (PRV) and proved by Mathieu independently of Kumar. There's a subsequent proof by Littelmann of the PRV conjecture using his theory of Lakshmibai-Seshadri paths. I will talk about joint work with Mrigendra Kushwaha and Sankaran Viswanath where we consider such a path approach to Kostant's refinement of the PRV.

Abstract: Using the linear complexity on finite sequences over a finite field F, I will define a metric on F^n. We will develop a coding theory using the new metric. I will also give the exact expression of number of finite sequences of length n with a fixed linear complexity l. A possible application is a new cryptosystem similar to the McEliece cryptosystem but with a different metric on the code.

Abstract: This is the first of two lectures where we'll cover Groebner bases of toric ideals. We start with an introduction to toric ideals and then study their Grobener bases. Our main goal will be a theorem of Bernd Sturmfels from 1991 that relates (certain) initial ideals of toric ideals to regular triangulations of an associated point configuration. The lectures are based on Chapters 4 and 8 of the book ``Groebner Bases and Convex Polytopes'' by Strumfels.