We will continue with the material in Chapter 1 in Cassels and Frohlich.

Let S be a Noetherian ring, and R = S/I. It is always possible to construct a differential graded algebra (DG-algebra) resolution of R over S due to a result of Tate. If R is the residue field of S, then Gulliksen proved that such a DG-algebra resolution is minimal. We shall discuss the construction of the Tate resolution in our talk.

Huneke and Itoh independently proved a celebrated result on integral closure of powers of an ideal generated by a regular sequence. As a consequence of this theorem, one can find the Hilbert-Samuel polynomial of the integral closure filtration of I if the normal reduction number is at most 2. We prove Hong and Ulrich's version of the intersection theorem.

We will define special singularities of algebraic or analytic varieties called rational singularities introduced my M. Artin. After discussing some equivalent criterion for rationality we will give many naturally occuring examples. Next, we will describe the results of Artin in dimension 2. Important results due to Brieskorn, Lipman, Mumford, Tjurina will be mentioned. If time permits some results by Spivakovsky, Le dung Trang-M. Tosun, Gurjar-Wagh will be mentioned. No proofs wil be given.

We shall describe recent progress on the question of writing down the reductions of certain local Galois representations. We shall focus on the case of half integral slopes (especially slope 3/2) where the behaviour of the reduction is both more complicated and more interesting. Our proof uses the mod p Local Langlands Correspondence to reduce the problem to computing the reductions of certain locally algebraic representations of GL_2 of the p-adics on certain functions on the underlying tree

We will continue with the material in Chapter 1 in Cassels and Frohlich.

This talk will be introduction to syzygies: basic theorems, examples and some early applications. This is the first seminar on this topic.

I will give introduce the basic ideas in homotopy theory, along the way state some classical theorems and if time permits some recent results. The talk will be expository and will have few or no proofs possibly.

Given a closed smooth Riemannian manifold M, the Laplace operator is known to possess a discrete spectrum of eigenvalues going to infinity. We are interested in the properties of the nodal sets and nodal domains of corresponding eigenfunctions in the high energy (semiclassical) limit. We focus on some recent results on the size of nodal domains and tubular neighbourhoods of nodal sets of such high energy eigenfunctions (joint work with Bogdan Georgiev).

Given a finite set of points P in R^2 and a finite family of lines L in R^2, an incidence is a pair (p,l), where p\in P, l\in L and p is a point in l. The Szemeredi-Trotter Theorem states that the number of incidences is atmost a constant multiple of (|L||P|)^{2/3} + |L| + |P|. We give a proof by Tao, which uses the method of cell partitions.