Complex Analysis and Dynamics Seminar
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Fall 2014 Schedule
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Sep 5: Max Fortier Bourque (Graduate Center of CUNY)
Moving one Riemann Surface Inside Another
In this talk, I will sketch a proof of the following parametric homotopy
principle: If two holomorphic embeddings between Riemann surfaces are
homotopic, then they are isotopic through holomorphic embeddings. The proof
uses a generalization of Teichmuller's theorem for quasiconformal
embeddings, the geometry of quadratic differentials, and Gardiner's
variational formula for extremal length.
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Sep 12: Feng Luo (Rutgers University)
A Dilogarithm Identity on the Moduli Space of Curves
We establish an identity for closed hyperbolic surfaces whose
terms depend on the dilogarithms of the lengths of simple closed geodesics
in all 3-holed spheres and 1-holed tori in the surface. This is joint work
with Ser Peow Tan.
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Sep 19: Yunhui Wu (Rice University)
Translation Lengths of Parabolic Isometries of CAT(0) Spaces and
its Application to the Geometry and Topology of Hadamard Manifolds
A CAT(0) space is a complete path-metric space with a certain inequality property.
In this talk, we will discuss the translation length of parabolic isometries of CAT(0) spaces.
As an application, we will connect several open problems and conjectures on Hadamard
manifolds and moduli spaces of closed surfaces.
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Sep 26: No seminar
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Oct 3: No seminar
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Oct 10: Athanase Papadopoulos (Université de Strasbourg)
A Survey on Thurtson's Metric
I will survey some classical and some new results on Thurston's metric on Teichmuller space,
and present some generalizations and open questions.
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Oct 17: Nikita Selinger (Stony Brook University)
Classification of Thurston Maps with Parabolic Orbifolds
In a joint work with M. Yampolsky, we give a classification of Thurston maps
with parabolic orbifolds based on our previous results on characterization of canonical
Thurston obstructions. The obtained results yield a partial solution to the problem of
algorithmically checking combinatorial equivalence of two Thurston maps.
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Oct 31: Yaar Solomon (Stony Brook University)
Separated nets in ${\mathbb R}^d$ with a bounded displacement to ${\mathbb Z}^d$
Given a separated net $Y$ in ${\mathbb R}^d$, we study the question of whether
$Y$ can be mapped injectively on ${\mathbb Z}^d$ with every point moving at most
$M$ (for some constant $M$). This notion gives rise to an equivalence
relation on the set of separated nets, called bounded displacement (BD)
equivalence, which is more delicate than the biLipschitz equivalence
relation. There is a correspondence between separated nets and tilings
of ${\mathbb R}^d$, and nets that correspond to periodic tilings are clearly BD
to ${\mathbb Z}^d$. We answer the above question for substitution tilings, which
form a nice class of tilings that are often non-periodic. All the
relevant background and definitions will be given in the talk.
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Nov 7: Ara Basmajian (Hunter College and Graduate Center of CUNY)
Geometric Structures on Infinite Type Surfaces
While the geometric theory of finite type surfaces is well developed, the study of
hyperbolic geometric structures on infinite type surfaces (that is, infinitely generated fundamental group) is in its infancy. In this talk we first describe some of the known results about the geometry and topology of such surfaces and then consider surfaces $X$ constructed by gluing pairs of pants along their cuffs. When finitely many pants are used the geometric completion of $X$ is well understood (namely, attach a hyperbolic funnel to each boundary geodesic of $X$). In joint work with Dragomir Saric, we consider the case with infinitely many pairs of pants and give criteria for the resulting hyperbolic structure to be geometrically complete. These constructions lead to new phenomena for the deformation theory of such surfaces.
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Nov 14: Rodrigo Trevino (Courant Institute, New York University)
Flat Surfaces, Bratteli Diagrams and Adic Transformations
I will survey some recent developments in the theory of flat
surfaces of finite area and translation flows, including both compact
and (infinite genus) non-compact surfaces. In particular, I will
concentrate on a new point of view based on a joint paper with K.
Lindsey, where we develop a close connection of Bratteli diagrams and
flat surfaces. I will also state a criterion for unique ergodicity in
the spirit of Masur's criterion which holds in this very general
setting and which implies Masur's criterion in moduli spaces of
(compact) flat surfaces. No knowledge of anything will be assumed, and
the talk will non-technical and full of examples.
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Nov 21: Steven Frankel (Yale University)
Quasigeodesic Flows and Dynamics at Infinity
A flow is called quasigeodesic if each flowline is uniformly efficient at measuring distances on the large scale.
In a hyperbolic 3-manifold, quasigeodesic flows are exactly the ones that one can study "from infinity." We will illustrate how
the 3-dimensional dynamics of a quasigeodesic flow is reflected in a simpler 1-dimensional discrete dynamical system at infinity: the universal circle. This is a topological circle, equipped with an action of the fundamental group, that lies at the edge of the orbit space of the flow. We will see that one can find closed orbits in a flow by looking at the action on the universal circle. We will also show that the universal circle provides a generalization of the well-known Cannon-Thurston theorem.
The universal circle for a quasigeodesic flow (due to Calegari) has a relative for a pseudo-Anosov flow (Calegari-Dunfield). The content of this talk completes a large part of Calegari's program to show that every quasigeodesic flow on a closed hyperbolic manifold can be deformed, keeping it quasigeodesic, to a pseudo-Anosov flow.
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Nov 28: No seminar
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Dec 5: Nessim Sibony (University of Paris-Sud, Orsay)
Nevanlinna's Theory and Holomorphic Dynamics
I will discuss some analogies between the second main theorem in Nevanlinna's theory and results in holomorphic dynamics.
The two main examples will be equidistribution results for endomorphisms of ${\mathbb P}^k$ and equidistribution results
for singular foliations by Riemann-surfaces in ${\mathbb P}^2$. This is joint work with T.C. Dinh.
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Dec 12: The seminar will feature two talks:
1:50-2:50 Marian Gidea (Yeshiva University)
Recent Progress in the Arnold Diffusion Problem
In 1964, V.I. Arnold conjectured that integrable Hamiltonian systems
subjected to typical small perturbations always have some
trajectories that travel a significant distance in the phase space.
The problem has been later on classified into an a priori unstable
case, when the unperturbed Hamiltonian depends on action-angle and
hyperbolic variables, and an a priori stable case, when the
unperturbed Hamiltonian depends on action-angle variables only. We
will present a geometric/topological approach to this problem that
relies on two ingredients: the existence of hyperbolic invariant
manifolds, and Poincare recurrence. We will discuss applications to
both the a priori unstable and the a priori stable case.
2:55-3:55 Edson Vargas (University of Sao Paulo)
Invariant Measures for Critical Coverings of the Circle
We study ergodic properties of a critical double covering of the
circle, say $f$. This is a smooth double covering of the circle which has only
one critical point, which we assume to be of finite order $> 1$. Examples of
these maps are the Arnold maps $f_b$, induced by
$$x \mapsto b + 2x + \frac{1}{\pi} \sin(2 \pi x).$$
We assume that $f$ is topologically conjugate to the double covering $L_2$, induced
by $x \mapsto 2x$. Although Lebesgue measure on the circle is invariant by $L_2$,
we prove that it may happen that f has no absolutely continuous invariant
measure (acim). One cause of this kind of behavior is a strong recurrence of the
critical point. We can study this from a combinatorial point of view and, as a
consequence, we get that there is an uncountable set of parameters $b$ such that
the critical covering $f_b$ has no acim. These type of results were obtained before
in the context of unimodal maps by H. Bruin, J. Guckenheimer, F. Hofbauer,
S. Johnson, G.Keller, T. Nowicki, S. van Strien and others. In the critical
covering case there is no dynamical symmetry around the critical point and
this cause some new combinatorial difficulties which need to be understood.
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