Complex Analysis and Dynamics Seminar
Department of Mathematics

Past seminars:
Fall 2006,
Spring 2007

Fall 2014: 
Sep 5: Max Fortier Bourque (Graduate Center of CUNY)
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.

Sep 12: Feng Luo (Rutgers University)
We establish an identity for closed hyperbolic surfaces whose
terms depend on the dilogarithms of the lengths of simple closed geodesics
in all 3holed spheres and 1holed tori in the surface. This is joint work
with Ser Peow Tan.

Sep 19: Yunhui Wu (Rice University)
A CAT(0) space is a complete pathmetric 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.

Sep 26: No seminar

Oct 3: No seminar

Oct 10: Athanase Papadopoulos (Université de Strasbourg)
I will survey some classical and some new results on Thurston's metric on Teichmuller space,
and present some generalizations and open questions.

Oct 17: Nikita Selinger (Stony Brook University)
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.

Oct 24: No seminar because of the
AhlforsBers Colloquium at Yale University

Oct 31: Yaar Solomon (Stony Brook University)
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 nonperiodic. All the
relevant background and definitions will be given in the talk.

Nov 7: Ara Basmajian (Hunter College and Graduate Center of CUNY)
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.

Nov 14: Rodrigo Trevino (Courant Institute, New York University)
I will survey some recent developments in the theory of flat
surfaces of finite area and translation flows, including both compact
and (infinite genus) noncompact 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 nontechnical and full of examples.

Nov 21: Steven Frankel (Yale University)
A flow is called quasigeodesic if each flowline is uniformly efficient at measuring distances on the large scale.
In a hyperbolic 3manifold, quasigeodesic flows are exactly the ones that one can study "from infinity." We will illustrate how
the 3dimensional dynamics of a quasigeodesic flow is reflected in a simpler 1dimensional 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 wellknown CannonThurston theorem.

Nov 28: No seminar

Dec 5: Nessim Sibony (University of ParisSud, Orsay)
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 Riemannsurfaces in ${\mathbb P}^2$. This is joint work with T.C. Dinh.

Dec 12: Marian Gidea (Yeshiva University)

Feb 6: Igor Rivin (Temple University)
Feb 20: Jayadev Athreya (University of Illinois at UrbanaChampaign) and Tanya Firsova (Kansas State University)
Feb 27: John Loftin (Rutgers University)
Mar 27: Ferran Valdez (UNAM, Mexico)