Gotzmann's regularity theorem establishes a bound on Castelnuovo-Mumford regularity using a binomial representation (the Macaulay representation) of the Hilbert polynomial of a standard graded algebra. Gotzmann's persistence theorem shows that once the Hilbert function of a homogeneous ideal achieves minimal growth then it grows minimally for ever. We start with a proof of Eisenbud-Goto's theorem to establish regularity in terms of graded Betti numbers. Then we discuss Gotzmann's theorems in the language of commutative algebra.

We will define the moduli of Higgs bundles and the Hitchin fibration, which is a morphism from the moduli of Higgs bundles to an affine space. Then we will describe the general fibre in terms of the spectral cover.

We are interested in understanding cuspidal support of smooth representations of p-adic reductive group via understanding the restriction of a smooth representation to a maximal compact subgroup. This is motivated by arithmetic applications via local Langlands correspondence. We will explain the case of general linear groups and classical groups.

A needle dropped on a plain with parallel and uniformly separated rulings, crosses exactly one of them, with finite probability. le Comte de Buffon, a French nobleman, posed the interesting problem of precisely determining this probability in 1777. This value, with much surprise, involves the number pi. E. Barbier in 1860 rigorously proved that the probability of a short needle of length l crossing exactly one line is 2l/\pi d if the uniform separation, d, between the lines is greater than l. In this seminar, the needle problem and Barbier’s proof are discussed. In addition to this, we attempt a classroom demonstration of needle problem and verify whether the ascertains are in agreement with the theoretical results

Consider a Noetherian ring R and an ideal I of R. Ratliff asked a question that what happens to Ass(R/I^n) as n gets large ? He was able to answer that question for the integral closure of I. Meanwhile Brodmann answered the original question, and proved that the set Ass(R/I^n) stabilizes for large n. We will discuss the proof of stability of Ass(R/I^n). We will also give an example to show that the sequence is not monotone. The aim of this series of talks to present the first chapter of S. McAdam, Asymptotic prime divisors, Lecture Notes in Mathematics 1023, Springer-Verlag, Berlin, 1983.

Gotzmann's regularity theorem establishes a bound on Castelnuovo-Mumford regularity using a binomial representation (the Macaulay representation) of the Hilbert polynomial of a standard graded algebra. Gotzmann's persistence theorem shows that once the Hilbert function of a homogeneous ideal achieves minimal growth then it grows minimally for ever. We start with a proof of Eisenbud-Goto's theorem to establish regularity in terms of graded Betti numbers. Then we discuss Gotzmann's theorems in the language of commutative algebra.

In this second talk of the series, I will continue the discussion on Higgs bundles with focus on some of the moduli aspects of the theory.

The conjecture of Erdos-Heilbronn (1964) states the following: Suppose G is a a finite group and and we have a G-sequence (g_1,g_2,...,g_l) of pairwise distinct g_i where l>2|G|^{1/2}, there is a subsequence (g_{i_1},g_{i_2},..,g_{i_t}) (for some t) such that \prod_{j} g_{i_j} = 1. The conjecture is open in its fullness, but has been settled (up to a constant) in some special cases of groups. We will see the proof of the E-H conjecture for cyclic groups, by Szemeredi.

Consider a Noetherian ring R and an ideal I of R. Ratliff asked a question that what happens to Ass(R/I^n) as n gets large ? He was able to answer that question for the integral closure of I. Meanwhile Brodmann answered the original question, and proved that the set Ass(R/I^n) stabilizes for large n. We will discuss the proof of stability of Ass(R/I^n). We will also give an example to show that the sequence is not monotone. The aim of this series of talk to present the first chapter of S. McAdam, Asymptotic prime divisors, Lecture Notes in Mathematics 1023, Springer-Verlag, Berlin, 1983.

Consider a graded ideal in the polynomial ring in several variables. We shall discuss criterion for the graded ideal and its power to have linear resolution. Then we focus our attention to study linear resolution of monomial ideals. Monomial ideals are the bridge between commutative algebra and the combinatorics. Monomial ideals are also significant because they appear as initial ideals of arbitrary ideals. Since many properties of an initial ideal are inherited by its original ideal, one often adopt this strategy to decipher properties of general ideals. The first talk is meant for covering the preliminary results on resolution and regularity of monomial ideal. The aim of this series of talk is to present the result in ArXiv:1709.05055 .