Abstracts of Talks

Bodil Branner (DTU) : Polynomial Vector Fields in one complex variable in

Abstract: Vector fields ξP = P(z) d/dz in for which P is a monic centered polynomial are classified by means of a combinatorial invariant C ( ξP ) and an analytic invariant A( ξP ). The combinatorial invariant describes the topology and the analytic invariant the geometry. Giving admissible combinatorial and analytic data sets, we construct through surgery a unique monic centered polynomial vector field realizing the given invariants. This result is joint work with Kealey Dias. It generalizes the pioneering work by Douady, Estrada, and Sentenac, who proved the result for the structurally stable polynomial vector fields. We end by describing the result of Kealey Dias on enumerating the combinatorial classes.

Xavier Buff (IMT) : Transversality for Herman Rings (joint work with Adam Epstein)

Abstract: Assume f is a rational having a Herman ring of period p with Brjuno rotation number α. We show that the locus of rational maps near f which have a Herman ring of period p and rotation number α close to the Herman ring of f is a smooth ₵-analytic submanifold of codimension 1.

In addition, if f has several cycles of Herman rings with Brjuno rotation numbers, the corresponding ₵-analytic submanifolds intersect transversally at f.

Laura DeMarco (UIC): Polynomial dynamics:  conjugacy classes and shift automorphisms

In a beautiful article published in 1991, Blanchard, Devaney, and Keen proved that the fundamental group of the shift locus (in the space of degree d polynomials) surjects onto the automorphism group of the 1-sided shift (on d symbols).  In this talk, I aim to describe how my recent work with Kevin Pilgrim on the moduli space of polynomials relates to the results of Blanchard-Devaney-Keen.

Clifford Earle (Cornell) : Uniqueness properties of analytic families of Riemann surfaces

Abstract: In our joint work on augmented Teichmüller and moduli spaces, Albert Marden and I construct analytic families of Riemann surfaces over certain quotients of classical Teichmüller spaces. Our analysis of these families led us to some uniqueness theorems for more general families. These will be the topic of this talk.

Núria Fagella (Universitat de Barcelona) : A separation Theorem for entire transcendental functions

Abstract: (This is joint work with Anna Benini.) Let f be an entire transcendental map in class B and of finite order (or a finite composition of finite order functions).  For this class of maps we prove a separation theorem analogous to the Goldberg-Milnor-Kiwi separation theorem for polynomials. Such results have several applications, as for example  that there cannot be any Cremer  periodic point in the boundary of a periodic Siegel disk.

Jeremy Kahn (Stony Brook): Counting Surfaces in Closed Hyperbolic 3-Manifolds

Abstract: Given a closed hyperbolic three-manifold M, let c (g , M) denote the number of genus g subgroups of π1(M), and let ĉ (g , M) denote
1/g(c (g , M) )1/g  . We show that for g large ĉ (g , M) is bounded by constants above and below, and conjecture the ĉ (g , M) converges to a universal constant divided by the volume of M. This is joint work with Vladimir Markovic.

Mikhail Lyubich (Stony Brook) : A priori bounds for some quadratic polynomials.

Abstract: We will survey recent results, joint with Jeremy Kahn, on a priori bounds for some infinitely renormalizable quadratic maps.

Howard Masur (University of Chicago):
The Weil-Petersson geodesic flow on the moduli space of Riemann surfaces.

Abstract:  There are several interesting mapping class group invariant metrics on Teichmüller space that descend to metrics on the quotient moduli space. One of these is the Weil-Petersson metric. It is an incomplete Kahler metric of negative sectional curvature. The curvature is not bounded away from 0 or negative infinity. In this talk I will give background on this metric and discuss the following theorem which is joint with Keith Burns and Amie Wilkinson.

Theorem:  The Weil-Petersson geodesic flow is ergodic on moduli space.

John Milnor (Stony Brook): Cubic Maps, work with Bonifant and Kiwi

  Abstract: Discussion of the geometry asociated with cubic polynomial maps with a periodic critical point. The talk will make use of Kiwi's analytic version of the Branner-Hubbard puzzle.

Yair Minsky (Yale) : Dynamics of Out (Fn) on Character Varieties
Abstract: Automorphisms of the free group Fn act by precomposition on representations of Fn into a group G, and this descends to an action of Out (Fn) on conjugacy classes or on characters. When G = PSL(2,) this action is closely related to the hyperbolic geometry of the corresponding groups, but has some surprising features. The action is properly discontinuous on a set strictly larger than the characters of the Schottky groups (the "primitive-stable" set), and is minimal and ergodic on the set of "redundant" characters. There are many primitive-stable representations whose image groups are lattices, but the relation between primitive-stability and the topology and geometry of these lattices is elusive.
Some of this is joint work with T. Gelander and Y. Moriah.

Caroline Series (University of Warwick) : Top terms of trace polynomials in Kra's plumbing construction

Kra's plumbing construction manufactures a surface S by `plumbing' together a suitable family of triply punctured spheres. This gives a natural pants decomposition of S, together with a projective  structure for which the associated holonomy representation ρ depends  on the `plumbing parameters' τ. In particular Trace ρ(γ), for γ in the fundamental group of S, is a polynomial in the τ. Simple curves on S can be described in terms of their Dehn-Thurston coordinates relative to the pants decomposition. After explaining the construction, we show that if γ is simple there is a remarkably easy formula relating the coefficients of the top terms of ρ(γ) and its Dehn-Thurston coordinates. The formula generalizes ones previously obtained by Keen, Parker and Series for the once and twice punctured torus. The proof involves a rather interesting result on matrix products. This is joint work with Sara Maloni.

Mitsuhiko Shishikura (Kyoto University) : Smoothness of hairs for some transcendental entire functions

Abstract: For complex exponential functions, Devaney and Krych have shown that there are "hairs" which consist of escaping points. Viana showed that these hairs are smooth (infinitely differentiable).  In a joint work with M. Kisaka, we try to extend this result to a larger class of entire functions. If the time permits, we will also discuss a possible application to the Julia sets of Cremer polynomials.

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