Complex Analysis Seminar City University of New York Graduate School and University Center Mathematics Program

The Complex Analysis Seminar is organized by Professors Linda Keen and Fred Gardiner.

The seminar meets weekly during the semester and covers topics in complex analysis including (but not exclusively) Teichmüller theory, Kleinian groups, hyperbolic geometry and complex dynamics.

The seminar time is now  Fridays 1:45-3:15  at the CUNY Graduate Center in Room 4419. For up-to-date schedule information, see the CUNY Graduate Center seminar bulletin.  Dates of lectures by local speakers may change to accommodate outside speakers.  Students are encouraged also to attend  the Complex Analysis Student Seminar, Fridays 10:30-12:00 in room 4419;
 

  Schedule for Fall  2004  -   Fridays at 1:45     Room 4419

• Sep 10:   Speaker: Fred Gardiner,     Title: "A hyperelliptic Riemann surface for the full horse shoe map"
  
  Abstract: Analysis of the full horseshoe map leads to the following topics in complex analysis:
  1. Koebe uniformization.
  2. extremal length.
  3. Teichmuller disks
  4. Weierstrass factorization of entire functions.
  5. transverse measured foliations.
  6. quadratic differentials.
  7. Bloch domains and contraction.
  8. hyperelliptic Riemann surfaces.
  9. pseudo-Anosov mappings.
  We present the example and discuss as many of these topics as time allows.

• Sep 17:    Speaker:  No Meeting     University Closed

• Sep 24:   Speaker:  No Meeting    University Closed

• Oct 1:   Speaker:  Aaron Wooton, Univ of Arizona     Title: Finding Explicit Defining Polynomials for Cyclic Prime Covers of the Riemann Sphere
  
  Abstract: Historically, compact Riemann surfaces were defined as the domain of definition of irreducible complex polynomials in two variables. For many current problems however, it is now common to regard compact Riemann surfaces using uniformization as the quotient of a simply connected surface. Both descriptions have their advantages and disadvantages, yet presently there is no known method to extract one description from the other. We shall discuss a specific case where one can move between these two different descriptions. Specifically we shall describe a method which produces explicit defining polynomials for $p$-gonal surfaces - compact Riemann surfaces which admit a cyclic prime cover of the Riemann sphere for a given prime $p$. Time permitting, we shall also discuss the uniqueness of the equations derived.

• Oct 8:   Speaker:  Laura De Marco, Univ of Chicago   Title: Iteration at the boundary of moduli space
  
  Abstract:The space of rational maps, Rat(d), is the set of all holomorphic self-maps of the Riemann sphere of degree d>1, with the topology of uniform convergence. The group of Mobius transformations acts on Rat(d) by conjugation and the quotient is the moduli space M(d). We study limiting dynamics at the boundary of M(d).

• Oct 15:   Speaker:  Nik Lakic, CUNY Lehman    Title: "Accumulation points for iterated function systems on non-relatively compact subdomains"

• Oct 22:   Speaker:  Eric Bedford, Univ of Indiana    Title: "Degree growth of rational mappings in higher dimension"

• Oct 29:   Speaker:  Jun Hu, CUNY Brooklyn    Title: "Dynamics of a one-parameter family of cubic polynomials"

• Nov 5:   Speaker:  Ege Fujikawa, Tokyo Inst of Tech., Japan    Title: "Dynamics of quasiconformal mapping class groups acting on asymptotic Teichmuller spaces"

• Nov 12:   Speaker:  Sudeb Mitra, Queens    Title: "Quasiconformal trivializations of holomorphic motions".
  Abstract: For a closed subset of the Riemann sphere, its Teichmuller space is the universal parameter space of holomorphic motions of that set. In this talk we will define "quasiconformal trivializations" of holomorphic motions, and discuss some of their basic properties. The main goal is to give a new characterization of quasiconformal trivializations in terms of holomorphic maps into the universal parameter space of the given holomorphic motion. Quasiconformal trivializations have found some interesting applications to problems in geometric function theory, and they show some surprising connections with Teichmuller theory.

• Nov 19:   Speaker:  Greg Markowsky, Grad Center    Title: "Efficient quasiconformal mappings."

• Dec 3:   Speaker:  Bob Devaney, Boston University    Title: McMullen Domains and Their Environs
  Abstract: In this talk we describe the parameter plane for the families of rational maps z^n + C / z^d. It is known that, when n,d > 2, these families have the property that there is an open, simply connected region about 0 in the parameter plane (the McMullen domain) having the property that any map corresponding to such a parameter has Julia set that is a Cantor set of simple closed curves. We will describe the structure of the parameter plane just outside this domain. We show that there are infinitely many special curves converging on the boundary of the McMullen domain. The jth curve contains the centers of (n-2)n^j +1 baby Mandelbrot sets as well as the same number of centers of Sierpinski holes. The latter are regions where the Julia set is a Sierpinski curve.

• Feb 4:   Speaker:  Meiyu Su, LIU    Title: TBA

• Feb 11:   Speaker:  Ser Tan Peow, Natl Univ Singapore    Title: "Generalized Markoff Maps and McShane's Identity"
  Abstract: We prove that a generalized version of McShane's identity holds for certain two generator subgroups of SL(2,C) satisfying certain simple conditions first given by Bowditch called the Bowditch Q-conditions, and also give variations which apply to incomplete hyperbolic structures on punctured torus bundles over the circle, and the complete hyperbolic three manifolds obtained by hyperbolic Dehn surgery on these manifolds.