Let M be a finite module of dimension d over a Noetherian local ring (R,m). The set{`(M/IM)/e(I,M)}, where I varies over m-primary ideals, is bounded below by (1/d!)e(R/textannM). If ˆ M is equidimensional, this set is bounded above by a constant depending only on M. The lower bound extends an inequality of Lech and the upper bound answers a question of Stru¨ckrad-Vogel.

Abstract for (3) and (4): Ma formulated a weakened generalised Lech’s conjecture and proved it for a class of rings known as numerically Roberts rings, in equal characteristic p > 0. Using these results, combined with results on Hilbert-Kunz multiplicities, he proved the Lech’s conjecture for 3-dimensional Gorenstein rings of equal characteristic p > 0. In the first part of the talk, we define numerically Roberts rings and prove a few results required for proving the main result, which will be proved in the second part of the talk.

The following two results will be considered. Let (A,M) ⊂ (R,N) be a local flat homomorphism with A a regular local ring such that A contains its residue field. Let I be an ideal in A. Then eR/IR ≥ eR, where e denotes the multiplicity. Ma has stated four conjectures related to Lech’s Conjecture. We will discuss the relationships between these conjectures. If time permits, I will indicate how we can understand C.P. Ramanujam’s geometric interpretation of multiplicity in a more intuitive manner.

This is a talk in Boij-Soderberg theory, which involves the study of Betti cones over quotients of polynomial rings. These were introduced by Boij-Soderberg in 2008, and explored further by Eisenbud-Schreyer in 2009. I will give a quick introduction to this theory and the main problems. Finally, I will point out how of a result of mine (joint with Rajiv Kumar) on the construction of pure modules over Cohen-Macaulay rings follows immediately from the work of Eisenbud-Schreyer using Noether Normalization, and the Auslander-Buchsbaum Formula (which are two important results the students proved in the reading course). I will try to make the talk as self-contained as possible. All are welcome.

Let $q$ be a positive integer and let $(a,q)=1$ be a given residue class. Let $p(a,q)$ denote the least prime congruent to $a \mod{q}$. Linnik's theorem tells us that there is a constant $L>0$, such that the $p(a,q) \ll q^L$. The best known value today is $L = 5.18$. A conjecture of Erdos asks if there exist primes $p_1$ and $p_2$, both less than $q$, such that $p_1 p_2 \equiv a \mod{q}$. Recently, Ramar\'{e} and Walker proved that for all $q \geq 2$, there are primes $p_1, p_2, p_3$, each less than $q^{16/3}$, such that $p_1 p_2 p_3 \equiv a \mod{q}$. Their proof combines additive combinatorics with sieve theoretic techniques. We sketch the ideas involved in their proof and talk about a joint work with Olivier Ramar\'{e}, where we refine this method and obtain an improved exponent of $q$.

Let X be a smooth scheme of finite type over C, and let X' be the corresponding complex analytic variety. Grothendieck proved that the complex cohomologies H^q(X') can be calculated as the hypercohomologyes of the algebraic de Rham complex on X. We will present Grothendieck's proof.

The spaces SL(2, R) and SL(2,R)/SL(2, Z) look quite difficult to visualize at first glance. In this talk, we shall see that these two spaces are actually homeomorphic to some nice-looking spaces.