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.

In these talks, we will see relations between Hilbert series of a module and its graded Betti numbers. This gives relations between the graded Betti numbers of a modules which are known as Herzog-Kuhl equations. As an application, we show that the property of R being Cohen-Macaulay is characterized by the existence of a pure Cohen-Macaulay R-module of finite projective dimension.

In this lecture, we will understand the statement of the local Langlands correspondence in the Archimedean case. This lecture will be based on the article available here https://www.math.stonybrook.ed u/~aknapp/pdf-files/motives.pdf

In this second lecture we will continue with the material in Chapter 1 in Cassels and Frohlich.

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.

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.

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.

In these talks, we will see relations between Hilbert series of a module and its graded Betti numbers. This gives relations between the graded Betti numbers of a modules which are known as Herzog-Kuhl equations. As an application, we show that the property of R being Cohen-Macaulay is characterized by the existence of a pure Cohen-Macaulay R-module of finite projective dimension.