Title: Kähler-Einsten metric, K-stability and moduli of Fano varieties
Speaker: Chenyang Xu, Princeton Univeristy, 11/13
Abstract: A complex variety with a positive first Chern class is called a Fano variety. The question of whether a Fano variety has a Kähler-Einstein metric has been a major topic in complex geometry since the 1980s. The Yau-Tian-Donaldson Conjecture predicts the existence of such a metric is equivalent to an algebraic condition called K-stability. In the last decade, algebraic geometry, or more specifically higher dimensional geometry has played a surprising role in advancing our understanding of K-stability, which leads to the solution of the Yau-Tian-Donaldson Conjecture for all Fano varieties.
Title: Tropical Algebra
Speaker: Kalina Mincheva, Tulane Unversity (11/06)
Abstract: Tropical geometry provides a new set of purely combinatorial tools, which has been used to approach classical problems. In tropical geometry most algebraic computations are done on the classical side - using the algebra of the original variety. The theory developed so far has explored the geometric aspect of tropical varieties as opposed to the underlying (semiring) algebra and there are still many commutative algebra tools and notions without a tropical analogue. In the recent years, there has been a lot of effort dedicated to developing the necessary tools for commutative algebra using different frameworks, among which prime congruences, tropical ideals, tropical schemes. These approaches allows for the exploration of the properties of tropicalized spaces without tying them up to the original varieties and working with geometric structures inherently defined in characteristic one (that is, additively idempotent) semifields. In this talk we explore the relationship between tropical ideals and congruences and what they remember about the geometry of a tropical variety. We will describe the 'coordinate semiring', what information it carries, and if time permits, we will talk about integral closure and some examples of normalization in tropical geometry.