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.

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.

We will start with some general results about compact complex manifolds of dimension 2 (including non-algebraic ones) like intersection theory, Hodge Index Theorem, Riemann-Roch Theorem. The goal is to outline the classification of minimal smooth projective surfaces, and describe the main properties of surfaces in each class. Due to time constraints almost no proofs will be given.

We will start with some general results about compact complex manifolds of dimension 2 (including non-algebraic ones) like intersection theory, Hodge Index Theorem, Riemann-Roch Theorem. The goal is to outline the classification of minimal smooth projective surfaces, and describe the main properties of surfaces in each class. Due to time constraints almost no proofs will be given.

It is well known that the prime numbers are equidistributed in arithmetic progressions. Such a phenomenon is also observed more generally for a class of multiplicative functions. We derive some variants of such results and give a few applications. We also discuss an interesting application that relates to the Chowla conjecture on correlations of the Mobius function, and show its relevance to the twin prime conjecture.

This is the first in a series of lectures on class field theory. We begin with Chapter 1 in Cassels and Frohlich.

A famous conjecture of Sato and Tate (now a celebrated theorem of Taylor et al) predicts that the normalised p-th Fourier coefficients of a non-CM Hecke eigenform follow the semicircle distribution as we vary the primes p. In 1997, Serre obtained a distribution law for the vertical analogue of the Sato-Tate family, where one fixes a prime p and considers the family of p-th coefficients of Hecke eigenforms. In this talk, we address a situation in which we vary the primes as well as families of Hecke eigenforms. In 2006, Nagoshi obtained distribution measures for Fourier coefficients of Hecke eigenforms in these families. We consider another quantity, namely the number of primes p for which the p-th Fourier coefficient of a Hecke eigenform lies in a fixed interval I. On averaging over families of Hecke eigenforms, we obtain a conditional central limit theorem for this quantity. This is joint work with Kaneenika Sinha.