Complex Analysis and Dynamics Seminar

Department of Mathematics
Graduate Center of CUNY

Fridays 2:00 - 3:00 pm
Room 5417
Organizers: Jun Hu and Saeed Zakeri


Past seminars:

Fall 2006, Spring 2007, Fall 2007 Spring 2008, Fall 2008, Spring 2009, and Fall 2009


Spring 2010:

Jan. 29: Ara Basmajian (Hunter College and Graduate Center of CUNY)
Universal Length Bounds for Non-Simple Closed Geodesics

We consider the relationship between the length of a closed geodesic on a hyperbolic surface and its self-intersection number. In the 1993 paper The stable neighborhood theorem and lengths of closed geodesics it was shown that a closed geodesic with intersection number k has length bounded from below by a constant Mk that goes to infinity with k. The Mk are universal constants in that they only depend on k and not on any particular hyperbolic structure. In this talk we show that the Mk grow like Log k. Our techniques are elementary and involve using a quantitative version of the Margulis lemma.


Feb. 5: Sarah Koch (Harvard University)
Böttcher Coordinates in Cm

A well-known theorem of Böttcher asserts that an analytic germ f:(C,0) → (C,0) which has a superattracting fixed point at 0, more precisely of the form f(z) = a zk + o(zk) for some non-zero a, is analytically conjugate to the power map za zk by an analytic germ (C,0) → (C,0) which is tangent to the identity at 0. In this talk, we generalize this result and give a Böttcher criterion for analytic maps in several complex variables.


Feb. 12: No meeting


Feb. 19: Trevor Clark (Toronto/Stony Brook University)
Regular or Stochastic Dynamics in Higher Degree Families of Unimodal Maps

About twenty years ago, Palis conjectured that typical dynamical systems should possess good statistical properties. Through the work of Avila, Lyubich, de Melo and Moreira, this has been proven for unimodal maps with a non-degenerate critical point. I will show how to remove the condition on the critical point in analytic families of unimodal maps; along the way proving that the hybrid classes in the space of unimodal maps yield a lamination near all but countably many maps in the family. The essential difference in the higher degree case is the presence of non-renormalizable maps without "decay of geometry." The key to their study is the use of a generalized renormalization operator, which has much in common with the usual renormalization operator.


Feb. 26: Seminar cancelled due to heavy snow


Mar. 5: Reza Chamanara (Brooklyn College of CUNY)
Rigidity of Inversive Distance Disk Patterns

The study of configurations of disks with prescribed combinatorial and geometric patterns is an important part of combinatorial geometry. The most well-known example is the theory of circle packing with its numerous applications in discrete analytic function theory. The basis of the theory of circle packing is the so called Koebe's theorem which states that given any triangulation K of the sphere S2, there is a circle packing PK in S2 with the tangency pattern prescribed by K. Moreover, the disks in PK have mutually disjoint interiors and PK is unique up to application of a Mobius map. Theorems by Andreev and Rivin on the existence and uniqueness of convex compact or convex ideal hyperbolic polyhedra can be interpreted as theorems about configurations of disks with prescribed patterns of intersection. I will discuss the possible extensions of such results to patterns of pairwise disjoint disks on the sphere. In particular, I will show how such patterns define a weighted planar graph. Moreover, I will explain why the weighted graph determines the disk pattern up to application of a Mobius map.


Mar. 12: Robert Devaney (Boston University)
Dynamics of Family of Maps zn + C/zn: Why the Case n = 2 is Crazy

In this talk we describe some of the interesting dynamics associated to the family of complex maps zn + C/zn, where C is a complex parameter and n > 1. It turns out that the case n = 2 is very different from the case n > 2. For example, when n > 2 there is a "McMullen domain" in the parameter space which is surrounded by infinitely many "Mandelpinski" necklaces, but there is no such structure when n = 2. Also, when n = 2, as C tends to 0, the Julia sets of these maps converge to the closed unit disk, but this never happens when n > 2. So the dynamical behavior of these maps near the parameter C = 0 is much more complicated when n = 2.


Mar. 19: Gabino Gonzalez-Diez (UAM, Spain)
On Beauville Surfaces and the Genus of the Curves Arising in Their Construction

A Beauville surface is a 2-dimensional compact complex manifold of the form S = (C1 x C2) / G, where C1 and C2 are compact Riemann surfaces of hyperbolic type and G is a finite group acting freely on the product C1 x C2 in such a way that each of the factors is preserved by the action and, moreover, the quotient Ci / G is an orbifold of genus zero with three cone points. Beauville surfaces were introduced by Catanese following an initial example of Beauville in which C1 = C2 is the Fermat curve of genus 6 with the equation X05+X15+X25 = 0 and G is isomorphic to Z5 x Z5.
In this talk, I shall discuss questions such as which groups G and which genera g1, g2 of C1, C2 can arise in the construction of Beauville surfaces. In particular, I will show that g1 and g2 have to be at least 6 and that if g1 = g2 = 6 then S agrees with (one of the two) Beauville examples. The proof of this fact will rely on methods belonging to the theory of Fuchsian groups and Riemann surfaces.


Mar. 26: Xiao Jun Huang (Rutgers University)
TBA




Apr. 2: No Meeting


Apr. 9: Jian Song (Rutgers University)
TBA




Apr. 16: Moira Chas (Stony Brook University)
TBA




Apr. 23: Melkana Brakalova (Fordham University)
TBA




Apr. 30: Shou-Wu Zhang (Columbia University)
Calabi Theorem and Algebraic Dynamics

In this talk, I will first state a theorem of Calabi for both complex and p-adic manifolds and show its applications to dynamical systems on complex varieties in terms of preperiodic points. Then I will discuss dynamical Manin-Mumford conjectures with both supporting examples and counterexamples. This is a report on joint works with Xinyi Yuan, and with Ghioca and Tucker respectively.


May 7: Clifford Earle (Cornell University)
TBA




May 14: Sudeb Mitra (Queens College of CUNY)
TBA