I will motivate the subject of (Newton-) Okounkov bodies from the perspective of intersection theory by presenting  the statement of Theorem 4.9 in Kaveh and Khovanskii's 2012 Annals paper. Time-permitting, I will discuss examples: a notable special case of this result is Kushnirenko's theorem on the degree of a projectively-embedded toric variety.

We extend the asymptotic Samuel function of an ideal to a filtration and study some properties of the function. We look at the relation between integral closure of filtrations and the asymptotic Samuel function. We further study the notion of projective equivalence of filtrations. The talk follows the pre-print ‘The Asymptotic Samuel Function of a Filtration’ by Dale Cutkosky and Smita Praharaj.

We continue the discussion on projective equivalence of filtrations. We further look at the discrete valued filtrations and show that they have particularly nice properties . We generalize some of the theory of Rees valuations of ideals to these filtrations.

This is the first part of the talk on the Quillen-Suslin theorem, formerly known as Serre's Conjecture. The assertion is that a finitely generated projective module over a polynomial ring over a field is always free. In 1976, Quillen and Suslin independently provided a complete proof of this theorem. Quillen was awarded a fields medal in part because of his proof this conjecture. In this talk we'll discuss a simplification of Suslin's proof given by Vaserstein. 

We will finish the proof of Quillen-Suslin. Please note the non-standard meeting time for the seminar this week.

Infinity categories, as formulated by André Joyal and developed by Jacob Lurie, have become a useful framework for dealing with higher categorical phenomena in different areas, including representation theory and algebraic geometry. The goal of this talk is to introduce just enough background on simplicial homotopy theory to define infinity categories, including the prerequisites to read the first chapter of Lurie's book, Higher Topos Theory. If time allows, we will discuss basic properties and examples of infinity categories.

This is Part II of Benjamin’s talk.

In this talk we will introduce the notion of a Kaplansky devissage and show its key properties. We will use these properties to prove that any projective module is some direct sum of countably generated projective submodules. We then define the direct sum property for modules and show that any countably generated module with the direct sum property is free. We finally show that any projective module over a local ring has the direct sum property letting us conclude that all projective modules over local rings are free.