Mathematics

Embeddings and analogies between right-angled Artin groups and mapping class groups

Elizabeth Field (UIUC Math)

Abstract: During this talk, we will discuss the relationships which arise between right-angled Artin groups and mapping class groups of surfaces. We will begin by exploring when a right-angled Artin group can be embedded into the mapping class group of a surface and conclude by discussing various analogies between these two types of groups. In particular, we will see how the acylindrical action of a right-angled Artin group on its extension graph leads us to a classification of the elements of a right-angled Artin group which is analogous to the Nielson-Thurston classification of the elements of a mapping class group. This talk will assume no prior knowledge of either mapping class groups or of right-angled Artin groups.

Towards a Model Theory for Logarithmic Transseries

Allen Gehret   [email] (UIUC Math)

Abstract: For the past few years I have been working on a project to show that the (ordered valued differential) field of logarithmic transseries, $\mathbb{T}_{\log}$, has a good model theory (model completeness, QE, etc). The first part of this project was to show that the asymptotic couple (=value group + additional structure induced by the derivation) has a good model theory. After completing this first part, for the past year I have turned my attention to the field $\mathbb{T}_{\log}$ itself. This project is very similar to the recent results of Aschenbrenner, van der Hoeven and van den Dries in showing that the (ordered valued differential) field of logarithmic-exponential transseries, $\mathbb{T}$, has a good model theory. In this talk I will describe recent progress in the direction of proving model completeness for this structure, as well as the general strategy moving forward. I will also draw parallels between the two fields $\mathbb{T}$ and $\mathbb{T}_{\log}$ to illustrate the obstructions in $\mathbb{T}_{\log}$ that are not present in $\mathbb{T}$.