# Complex Analysis and Dynamics Seminar

## Fall 2017 Schedule

#### Sep 8: Vincent Delecroix (LaBRI, France) Counting Triangulations: From Elementary Combinatorics to the Geometries of Moduli Spaces of Complex Curves

How many triangulations of the sphere are there with n vertices and so that all vertices have degree not larger than $6$? It appears that there is a formula for these numbers due to P. Engel and P. Smillie in terms of the function $\sigma_9(n)$ (= sum of the $9$-th power of the divisors of the integer $n$). Even more interesting is that this sequence has a polynomial asymptotic whose leading coefficient is a Thurston (or Deligne-Mostow) volume of the moduli space $\mathcal{M}_{0,12}$ of configurations of points on the sphere. In this talk I will discuss several relations between combinatorial problems of graphs on surfaces (such as triangulations and meanders) and the geometries of moduli spaces of curves (intersection theory, Thurston volumes and Masur-Veech volumes).
Part of the material from this talk comes from a collaboration with E. Goujard, P. Zograf and A. Zorich.

#### Oct 6: Jun Hu (Brooklyn College and Graduate Center of CUNY) Dynamics of Regularly Ramified Rational Maps in Some One-Parameter Families

$\newcommand{\EC}{\widehat{\mathbb{C}}}$ Let $\EC$ be the Riemann sphere. A rational map $f:\EC \rightarrow \EC$ is said to be regularly ramified if for every point $q\in \EC$, all pre-images of $q$ under $f$ have equal indices (meaning that $f$ has same local degrees at all pre-images of $q$). Up to conjugacy by a Mobius transformation, any regularly ramified rational map $f$ can be written as a quotient map of a finite Kleinian group post-composed by a Mobius transformation, and $f$ can have only two or three critical values. In this paper, we classify the Julia sets of such maps in some one-parameter families $f_{\lambda }$, where $\lambda$ is a complex parameter. The maps in these families have a common super attracting fixed point of order $=2$ or $>2$. We show they have classifications similar to the classifications of the Julia sets of maps in the families $f_n^{c}(z)=z^n+c/z^n$, where $n$ is a positive integer $=2$ or $>2$ and $c$ is a complex number. A new type of Julia set is also presented, which has not appeared in the literature and which are called exploded McMullen necklaces. We first prove that none of the maps in these families can have Hermann rings in their Fatou sets. Then we prove: if the super attracting fixed point of $f_{\lambda }$ has order greater than $2$, then the Julia set $J(f_{\lambda })$ is either connected, a Cantor set, or a McMullen necklace (either exploded or not); if the super attracting fixed point has order equal to $2$, then $J(f_{\lambda })$ is either connected or a Cantor set. This is a joint work with Oleg Muzician and Yingqing Xiao.

#### Oct 13: Steven Bradlow (University of Illinois at Urbana-Champaign) Exotic Components of Surface Group Representation Varieties, and Their Higgs Bundle Avatars

Moduli spaces of Higgs bundles on a Riemann surface correspond to representation varieties for the surface fundamental group. For representations into complex semisimple Lie groups, the components of these spaces are labeled by obvious topological invariants. This is no longer true if one restricts to real forms of the complex groups. Factors other than the obvious invariants lead to the existence of extra `exotic' components which can have special significance. Formerly, all known instances of such exotic components were attributable to one of two distinct mechanisms. Recent Higgs bundle results for the groups $SO(p,q)$ shed new light on this dichotomy and reveal new examples outside the scope of the two known mechanisms. This talk will survey what is known about the exotic components and, without assuming familiarity with Higgs bundles, describe the new $SO(p,q)$ results.

#### Oct 20: Stephen Preston (Brooklyn College and Graduate Center of CUNY) Global Existence and Blowup for Geodesics in Universal Teichmuller Spaces

I will discuss the Riemannian approach to universal Teichmuller space and the universal Teichmuller curve, inspired by Tromba and Yamada, together with the Weil-Petersson metric. The universal Teichmuller space can be viewed as coming from the space of hyperbolic metrics on the upper half-plane modulo diffeomorphisms that fix the boundary, and we end up with the Weil-Petersson metric on the diffeomorphism group of the real line. Meanwhile the universal Teichmuller curve, as described by Teo, has a natural metric called the Velling-Kirillov metric. I will discuss how one can view this as coming from the space of Euclidean metrics on the upper half-plane.
Both situations yield a right-invariant metric on the diffeomorphism group of the reals. Geodesics on such groups can be written as an Euler-Arnold equation on the Lie algebra, and this equation may be studied using PDE techniques. In particular I will show that all initially-smooth geodesics on the universal Teichmuller curve end in finite time, while all initially-smooth geodesics in the universal Teichmuller space remain smooth for all time. This is joint work with my student, Pearce Washabaugh.

#### Nov 3: Felipe Ramírez (Wesleyan University) Counterexamples and Questions in Inhomogeneous Approximation

Khintchine’s Theorem (1924) states that almost all (respectively, almost no) real numbers can be approximated by rationals at a given rate, provided that the rate is monotonic and corresponds to a divergent (resp. convergent) series. In 1941, Duffin and Schaeffer showed by way of example that the monotonicity condition cannot be removed. They formulated their famous and resistant Duffin—Schaeffer Conjecture in response to this example. I will discuss an analogue of this situation for inhomogeneous approximations. From the point of view of dynamics, this talk is about toral translations.

#### Nov 17: Chenxi Wu (Rutgers University, New Brunswick) An Upper Bound on the Asymptotic Translation Length on the Curve Complex

A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will give an introduction on prior works on the study of this asymptotic translation length, and present an improved upper bound for the asymptotic translation length for pseudo-Anosov maps inside a fibered cone, which generalizes the previous result on sequences with small translation length on curve graphs by Kin and Shin. This is a joint work by Hyungryul Baik and Hyunshik Shik.

#### Dec 1: Martin Bridgeman (Boston College) Schwarzian Derivatives, Projective Structures, and the Weil-Petersson Gradient Flow for Renormalized Volume

We consider complex projective structures and their associated locally convex pleated surface. We relate their geometry in terms of the $L^2$ and $L^{\infty}$ norms the Schwarzian derivative. We show that these give a unifying approach that generalizes a number of well-known results for convex cocompact hyperbolic structures including bounds on the Lipschitz constant for the retract and the length of the bending lamination. We then use these bounds to study the Weil-Petersson gradient flow of renormalized volume on the space CC(N) of convex cocompact hyperbolic structures on a compact manifold N with incompressible boundary. This leads to a proof of the conjecture that the renormalized volume has infimum given by one-half the simplicial volume of DN, the double of N. Joint work with Jeffrey Brock and Kenneth Bromberg.

#### Dec 8: Zeno Huang (College of Staten Island and Graduate Center of CUNY) On Weil-Petersson Geometry on Universal Teichmuller Space

I will describe Hilbert structure on the universal Teichmuller space where the Weil-Petersson metric is well defined (via Takhatajan-Teo). It has been shown that the WP metric is negatively curved and Einstein. Instead of curvature tensor, we consider its curvature operator and study some fundamental properties of this operator. In particular we show the operator is non-positive, bounded, and noncompact. This is based on joint work with Y. Wu.