Georgia Institute of Technology College of Sciences

This is joint work with Pakwut Jiradilok. Let X be a smooth, proper curve of genus 3 over a complete and algebraically closed nonarchimedean field. We say X is a K_4-curve if the nonarchimedean skeleton G of X is a metric K_4, i.e. a complete graph on 4 vertices.We prove that X is a K_4-curve if and only if X has an embedding in p^2 whose tropicalization has a strong deformation retract to a metric K_4. We then use such an embedding to show that the 28 odd theta characteristics of X are sent to the seven odd theta characteristics of g in seven groups of four. We give an example of the 28 bitangents of a honeycomb plane quartic, computed over the field C{{t}}, which shows that in general the 4 bitangents in a given group need not have the same tropicalizations.

Recent work by J. and N. Giansiracusa, myself, and O. Lorscheid suggests that the tropical geometry of a toric variety $X$, or more generally of a logarithmic scheme $X$, can be formalized as a "Berkovich analytification" of a scheme over the field $\mathbb{F}_1$ with one element that is canonically associated to $X$.The goal of this talk is to introduce the theory of Artin fans, originally due to D. Abramovich and J. Wise, which can be used to lift rather unwieldy $\mathbb{F}_1$-geometric objects to the more familiar realm of algebraic stacks. Artin fans are \'etale locally isomorphic to quotient stacks of toric varieties by their big tori and their glueing data has a completely combinatorial description in terms of Kato fans.I am going to explain how to use the ideas surrounding the notion of Artin fans to study tropicalization maps associated to toric varieties and logarithmic schemes. Surprisingly these techniques allow us to give a reinterpretation of Tevelev's theory of tropical compactifications that can be generalized to compactifications of subvarieties in logarithmically smooth compactifcations of smooth varieties. For example, we can introduce definitions of tropical pairs and schoen varieties in terms of Artin fans that are equivalent to Tevelev's notions.

This is joint work with Rob Kirby. Trisections are to 4-manifolds as Heegaard splittings are to 3-manifolds; a Heegaard splitting splits a 3-manifolds into 2 pieces each of which looks like a regular neighborhood of a bouquet of circles in R^3 (a handlebody), while a trisection splits a 4-manifold into 3 pieces of each of which looks like a regular neighborhood of a bouquet of circles in R^4. All closed, oriented 4-manifolds (resp. 3-manifolds) have trisections (resp. Heegaard splittings), and for a fixed manifold these are unique up to a natural stabilization operation. The striking parallels between the two dimensions suggest a plethora of interesting open questions, and I hope to present as many of these as I can.

The talk will survey the main definitions and properties of patchy vector fields and patchy feedbacks, with applications to asymptotic feedback stabilization and nearly optimal feedback control design. Stability properties for discontinuous ODEs and robustness of patchy feedbacks will also be discussed.