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.[-]

The shear coordinate is a countable coordinate system to describe increasing self-maps of the unit circle, which is furthermore invariant under modular transformations. Characterizations of circle homeomorphism and quasisymmetric homeomorphisms were obtained by D. Šarić. We are interested in characterizing Weil-Petersson circle homeomorphisms using shears. This class of homeomorphisms arises from the Kähler geometry on the universal Teichmüller space.
For this, we introduce diamond shear which is the minimal combination of shears producing WP homeomorphisms. Diamond shears are closely related to the log-Lambda length introduced by R. Penner, which can be viewed as a renormalized length of an infinite geodesic. We obtain sharp results comparing the class of circle homeomorphisms with square summable diamond shears with the Weil-Petersson class and Hölder classes. We also express the Weil-Petersson metric tensor and symplectic form in terms of infinitesimal shears and diamond shears.
This talk is based on joint work with Dragomir Šarić and Catherine Wolfram. See https://arxiv.org/abs/2211.11497.[-]

In this talk I will discuss the existence of complete extremal metrics on the complement of simple normal crossings divisors in compact Kähler manifolds, and stability of pairs, in the toric case. Using constructions of Legendre and Apostolov-Calderbank-Gauduchon, we completely characterize when this holds for Hirzebruch surfaces. In particular, our results show that relative stability of a pair and the existence of extremal Poincaré type/cusp metrics do not coincide. However, stability is equivalent to the existence of a complete extremal metric on the complement of the divisor in our examples. It is the Poincaré type condition on the asymptotics of the extremal metric that fails in general.
This is joint work with Vestislav Apostolov and Hugues Auvray.[-]

In this talk I will consider the problem of counting the number of pairs (z,w) of saddle connections on a translation surface whose holonomy vectors have bounded virtual area. That is, we fix a positive A and require that the absolute value of the cross product of the holonomy vectors of z and w is bounded by A. One motivation is the result of Smillie-Weiss that for a lattice surface there is a constant A such that if z and w have virtual area bounded by A then they are parallel. We show that for any A there is a constant $c_A$ such that for almost every translation surface the number of pairs with virtual area bounded by A and of length at most R is asymptotic to $c_AR^2$ as R goes to infinity. This is joint work with Jayadev Athreya and Samantha Fairchild.[-]

Several important problems in complex dynamics are centered around the local connectivity of Julia sets of polynomials and of the Mandelbrot set. Importantly, when the Julia set of a polynomial is locally connected, the topological dynamics ofthe map can be completely described as a quotient of a power map on the circle.Local connectivity of the Julia set is less significant for transcendental entire functions. Nevertheless, by restricting to a class of transcendental entire functions, known as docile functions, we obtain a similar concept by describing the topological dynamics as a quotient of a simpler disjoint-type map. We will discuss the notion ofdocile functions, as well as some of their properties. This is joint work with Lasse Rempe.[-]