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

UC San Diego, December 6, 2015, 9:30am-3:30pm

AP&M 6402 (Halkin room)

Schedule

9:30am-10am Breakfast
10am-10:50am Daniel Smith Formal schemes and vanishing

Abstract: We will start with an overview of formal schemes and (some of) their applications. We will then use the "different" nature of the structure sheaf of a formal scheme to transition to a discussion of sheaf cohomology. This discussion will eventually focus on vanishing theorems, where we will state some known and desired results.
11am-11:50am Susan Elle Dimension 5 Ore extensions and AS-regular algebras

Abstract: We examine Ore extensions, which are non-commutative polynomial rings with a very nice multiplication, and the possible relations that can define them in dimension 5. We then look more generally at Artin-Schelter regular algebras which are bigraded and ask how the possible defining relations change.
12pm-1:30pm Lunch break
1:30pm-2:20pm Zonglin Jiang An Introduction to Rigid Analytic Geometry

Abstract: Rigid analytic geometry is the geometry associated with non-archimedean fields. As schemes are defined as locally spectrum of some commutative rings, rigid analytic spaces are defined as locally "spectrum" of Tate algebras. In this talk, I'll explain what Tate algebras and their spectrum are, then discuss the globalization, coherent sheaves, cohomology and so on.
2:30pm-3:20pm Perry Strahl The Picard Group of the Moduli Space of Genus 0 Stable Quotients

Abstract: The moduli space of stable quotients is an alternative compactification to Kontsevich's moduli space of stable maps. In genus 0, both moduli spaces contain an open locus whose closed points correspond to tuples

(ℙ1, p1, ..., pm, f: ℙ1→G(r, n))

where pi are distinct points on ℙ1 and f is a morphism of fixed degree d to the Grassmannian, together with a stability condition. The difference between the two moduli spaces outside this locus is the moduli space of stable quotients allows the morphism to degenerate to a rational map to the Grassmannian, and the moduli space of stable maps allows more irreducible components in the domain curve. In this talk we will discuss the relationship between the Picard groups of the two moduli spaces.