Mapping classes of surfaces of finite type have been classified by Nielsen and Thurston. For surfaces of infinite type (e.g. surfaces of infinite genus), no such classification is known. I will talk about the difficulties that arise when trying to generalize the Nielsen-Thurston classification to infinite-type surfaces and present a first result in this direction, concerning maps which - loosely speaking - do not show any pseudo-Anosov behavior. Joint work with Mladen Bestvina and Jing Tao.[-]

Consider an orientable surface S with negative Euler characteristic, a minimal set of generators of the fundamental group of S, and a hyperbolic metric on S. Each unbased homotopy class C of closed oriented curves on S determines three numbers: the word length (that is, the minimal number of letters needed to express C as a cyclic word in the generators and their inverses), the minimal geometric self-intersection number, and finally the geometric length. Also, the set of free homotopy classes of closed directed curves on S (as a set) is the vector space basis of a Lie algebra discovered by Goldman. This Lie algebra is closely related to the intersection structure of curves on S. These three numbers, as well as the Goldman Lie bracket of two classes, can be explicitly computed (or approximated) using a computer. We will discuss the algorithms to compute or approximate these numbers, and how these computer experiments led to counterexamples to existing conjectures, formulations of new conjectures and (sometimes) subsequent theorems.This talk means to be accessible to mathematically young people.These results are joint work with different collaborators; mainly Arpan Kabiraj, Steven Lalley and Rachel Zhang.[-]

Given a finite-type surface, there are two important objects naturally associated to it. The first is the mapping class group and the second is the curve graph, which the mapping class group acts on via isometries. This action is well understood and has been extremely useful in understanding the algebraic and geometric properties of mapping class groups. For instance, the elements acting loxodromically on the curve graph and precisely the pseudo-Anosov homeomorphisms. In this talk I'll discuss recent joint work with Carolyn Abbott and Nicholas Miller as well as a project with Sam Taylor regarding infinite-type mapping classes that act as loxodromic isometries on graphs associated to infinite-type surfaces. The aim of these projects is to work towards a Nielsen-Thurston type classification of mapping classes for infinite-type surfaces to understand which homeomorphisms are the generalizations of pseudo-Anosovs is in this setting.[-]