# Complex Analysis and Dynamics Seminar

## Spring 2014 Schedule

#### Feb 7: Alistair Fletcher (Northern Illinois University)Poincare Linearizers in Higher Dimensions

It is well known that the behaviour of a holomorphic function near a fixed point is determined by its derivative there. In the case of a repelling fixed point $z_0$, the function can be conjugated to $z \mapsto f'(z_0) z$ and the class of functions which do the conjugating are called Poincare linearizers. We will discuss extending this idea to the setting of quasiregular mappings in higher dimensions, and in particular exploring the dynamics of quasiregular Poincare linearizers.

#### Feb 21: Frederick Gardiner (Brooklyn College and Graduate Center of CUNY)The Quasidisc Cocycle

Outline of the talk with some diagrams

#### Feb 28: Linda Keen (Lehman College and Graduate Center of CUNY)Discreteness and the Hyperbolic Geometry of Hexagons

Deciding if two matrices in PSL$(2,{\mathbb C})$ generate a discrete group is a hard question. The generators determine a hexagon in ${\mathbb H}^3$. In many cases, it is possible to determine from the hexagon if the group discrete. Combining earlier work on enumerating sequences of pairs of generators using Farey sequences of rationals with a study of the geometry of right angled hexagons in ${\mathbb H}^3$, we have a conjecture on how to determine a sequence of rationals and hexagons, with their corresponding pairs of generators, so that either one the generators will converge to the identity, the hexagons will become degenerate and the group is non-discrete, or the hexagons will converge in ${\mathbb H}^3$ and the group is discrete. This talk is based on joint work with Jane Gilman.

#### Mar 7: Andrew Sanders (University of Illinois at Chicago)A New Proof of Bowen's Theorem on Hausdorff Dimension of Quasi-circles

A quasi-Fuchsian group is a discrete group of Mobius transformations of the Riemann sphere which is isomorphic to the fundamental group of a compact surface and acts properly on the complement of a Jordan curve: the limit set. In 1979, Bowen proved a remarkable rigidity theorem on the Hausdorff dimension of the limit set of a quasi-Fuchsian group: it is equal to 1 if and only if the limit set is a round circle. This theorem now has many generalizations. We will present a new proof of Bowen's result as a by-product of a new lower bound on the Hausdorff dimension of the limit set of a quasi-Fuchsian group. This lower bound is in terms of the differential geometric data of an immersed, incompressible minimal surface in the quotient manifold. If time permits, generalizations of this result to other convex-co-compact surface groups will be presented.

#### Mar 14: Gabriele Mondello (University of Rome)On the Cohomological Dimension of the Moduli Space of Riemann Surfaces

The moduli space of compact Riemann surfaces of fixed genus has a finite étale cover which is a complex manifold. Thus it makes sense to speak of its de Rham or cohomology of coherent sheaves. Its de Rham cohomological dimension was determined by Harer in the 1980's. Conjectural vanishing of its coherent cohomology in high degree would shed new light on many vanishing theorems for the tautological classes and for de Rham cohomology. In this talk, we will give an estimate of coherent cohomological dimension of the moduli space of Riemann surfaces, which works in every genus though it is not optimal. In the proof, flat surfaces will come into play.

#### Mar 21: Araceli Bonifant (University of Rhode Island)Fjords in a Parameter Space for Antipode Preserving Cubic Maps

This talk will describe the topological properties of the "fjords" that appear in the parameter space for antipode preserving cubic maps with a critical fixed point.

#### Mar 28: Sergiy Merenkov (University of Illinois at Urbana-Champaign and City College of New York)Quasisymmetric maps between Sierpinski carpet Julia sets

I will discuss recent rigidity results for quasisymmetric maps between Sierpinski carpets that are Julia sets of postcritically finite rational maps. This is joint work with M. Bonk and M. Lyubich.

#### Apr 4: Patrick Hooper (City College and Graduate Center of CUNY)Topologizing the Space of all Translation Surfaces

A translation surface is a surface equipped with an atlas of charts to the plane where the transition functions are translations. We do not insist that the inherited metric on the surface be complete. There is an intimate connection between dynamics on these surfaces (straight-line or geodesic flows), and dynamics on spaces of translation surfaces (Teichmuller flow) through renormalization. These ideas are most developed in the finite genus case, but recent work has shown that the connection persists in infinite genus. Making this relationship concrete requires topologizing spaces of translation surfaces. I will explain how to do this, and discuss some of the connections to dynamics.

#### Apr 11: Giulio Tiozzo (ICERM and Yale University)An Entropic Tour Along the Mandelbrot Set

The notion of topological entropy, arising from information theory, is a fundamental tool to understand the complexity of a dynamical system. When the dynamical system varies in a family, the natural question arises of how the entropy changes with the parameter. Recently, W. Thurston has introduced these ideas in the context of complex dynamics by defining the "core entropy" of a quadratic polynomial as the entropy of a certain forward-invariant subset of the Julia set called the Hubbard tree. As we shall see, the core entropy is a purely topological / combinatorial quantity which nonetheless captures the richness of the fractal structure of the Mandelbrot set. In particular, we shall see how to relate the variation of such a function to the geometry of the Mandelbrot set.

#### Apr 25: Qiongling Li (Rice University)Asymptotics of Certain Families of Higgs Bundles in the Hitchin Component

In this talk, I will first go through Higgs bundles, basic construction of Hitchin components inside the moduli space of Higgs bundles. I will then introduce recent work with Brian Collier on the asymptotic behavior of certain families in Hitchin components. Namely, in the family of Higgs bundles $(\mathcal{E},t{\phi})$, we try to analyze the asymptotic behavior of the corresponding representation $\rho_t$ as $t\rightarrow \infty$ in two special cases.

#### May 2: Kealey Dias (BCC of CUNY)On Parameter Space of Single-variable Complex Polynomial Vector Fields

The space of single-variable complex polynomial vector fields $\Xi_d \simeq \mathbb{C}^{d-1}$ can be decomposed into loci $\mathcal{C}$ in which vector fields share a combinatorial invariant (topological equivalence with a labeling). The main result of this talk aims to prove that such a class is homeomorphic to $\mathbb{H}_+^s \times \mathbb{R}_+^h$, which corresponds to the set of the so-called analytic invariants associated to the class. The construction in the proof of this theorem, which utilizes holomorphic dependence on parameters in the Measurable Riemann Mapping Theorem, will pave the way for understanding a class of bifurcations of these vector fields.

#### May 9: Jon Chaika (University of Utah)The Limit Set in PMF of Some Teichmuller Geodesics

Teichmuller space is topologically an open ball which has numerous compactifications. In joint work with H. Masur and M. Wolf, we show that there are Abelian differentials with minimal but not uniquely ergodic vertical foliations so that their limit set in Thurston's compactification, PMF, a) is a unique point; b) is a line segment; c) an ergodic (but not uniquely ergodic) minimal Abelian differential which has a line segment as its limit set; d) an ergodic (but not uniquely ergodic) minimal Abelian differential which has a unique point as its limit set. These examples arise from Veech's example of minimal and not uniquely ergodic $\mathbb{Z}_2$ skew products of rotations which are related to two tori glued along a slit. Masur proved that the geodesic defined by a quadratic differential with uniquely ergodic vertical foliation has a (unique) limit in PMF and that it was what one would expect. Lenzhen constructed an example of a non-minimal quadratic differential that did not have a limit in PMF (the limit set was a line segment). This talk will focus on some motivating examples.
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