Graduate Student Workshop in Algebra, Number Theory, and Algebraic Geometry

UC San Diego, March 5, 2016, 9:30am-3:30pm

AP&M 6402 (Halkin room)


9:30am-10am Breakfast
10am-10:50am Robbie Snellman On p-adic Lie groups and applications

Abstract: p-adic Lie groups appear in many branches of mathematics such as number theory, arithmetic geometry, representation theory, etc. In his 1965 paper, Michel Lazard outlined a method for studying p-adic Lie groups via algebraic techniques. An important concept in this study is the notion of a p-valuation on an abstract group G. In this talk, I will give a brief overview of important properties of p-valuable groups along with an application to the 3×3 Heisenberg group H over the p-adic integers. Time permitting, I will sketch how Lazard's techniques allow us to classify certain classes of ideals in the Iwasawa algebra of H, which are invariant under the action of a "contracting monoid".
11am-11:50am Joseph Palmer Integrable systems, fans, and minimal models

Abstract: This talk is meant to be an introduction to a variety of related topics in the sympletic theoretic study of integrable systems. We give special attention to toric and semitoric integrable systems. We discuss invariants of such systems and introduce a novel algebraic method to study toric/semitoric fans. The bulk of this work is joint with Alvaro Pelayo and Daniel M. Kane.
12pm-1:30pm Lunch break
1:30pm-2:20pm Calum Spicer Toric foliations

Abstract: Toric varieties provide interesting and concrete testing bed for questions about varieties in general, indeed many of the subtleties present in higher dimensional geometry already appear in the toric case. We will introduce some basic facts about and examples of toric varieties, as well as their foliated counterparts, and suggest how these relate to the geometry of foliations.
2:30pm-3:20pm Brian Longo Strong approximation, super approximation, and applications

Abstract: We discuss the notion of Strong approximation and how graph theory has led to an effective version: "super approximation." We discuss several applications of super approximation in characteristic zero and how one of these applications, namely the Large Group Sieve method of Lubotzky and Meiri, could possibly carry over to the positive characteristic world.