Organizing Committee
 Laura DeMarco
University of Illinois  Adam Epstein
University of Warwick  Sarah Koch
University of Michigan, Ann Arbor  Curtis McMullen
Harvard University  Joseph Silverman
Brown University
Abstract
This workshop will bring together dynamicists, number theorists, and algebraic geometers to study the geometry and arithmetic of dynamical moduli spaces. The set Rat_{d}^{n} of rational degree d selfmaps of P^{n} has a natural structure as an affine variety. The dynamical moduli space M_{d}^{n} is the quotient of Rat_{d}^{n} by the conjugation action of the group PGL_{n+1}. Problems to be investigated include the geometry of M_{d}^{n}, the distribution of special maps such as postcritically finite maps in M_{d}^{n}, dynamical modular curves associated to oneparameter families of maps with a marked point of period N, and degeneration of families of maps and the associated points on the boundary of moduli space. A tutorial session will be held the week before this workshop.
Confirmed Speakers & Participants
 Speaker
 Poster Presenter
 Attendee

Jackie Anderson
Bridgewater State University

Matthieu Arfeux
Universite de Toulouse III (Paul Sabatier)

Cecile Armana
Westfälische WilhelmsUniversität Münster

Matthieu Astorg
Universite de Toulouse III (Paul Sabatier)

Fabrizio Barroero
Technische Universität Graz

Rob Benedetto
Amherst College

Anupam Bhatnagar
City University of New York (CUNY)

Paul Blanchard
Boston University

Araceli Bonifant
University of Rhode Island

Joshua Bowman
Smith College

Andrew Bridy
University of Wisconsin

Xavier Buff
Universite de Toulouse III (Paul Sabatier)

Arnaud Chéritat
Institut de Mathématiques de Toulouse

Mark Comerford
University of Rhode Island

Daniel Cuzzocreo
Boston University

Diana Davis
Northwestern University

Laura DeMarco
University of Illinois

Romain Dujardin
Ecole Polytechnique

Adam Epstein
University of Warwick

Timo Erkama
University of Eastern Finland

Derek Garton
University of Wisconsin

Jonah Gaster
University of Illinois

William Gignac
University of Michigan

Mikhail Hlushchanka
Jacobs University

Wei Ho
Columbia University

LiangChung Hsia
National Taiwan Normal University

John Hubbard
Cornell University

Benjamin Hutz
Saint Louis University

Patrick Ingram
York University

Rafe Jones
College of the Holy Cross

Jeremy Kahn
Brown University

Linda Keen
Herbert H. Lehman College, CUNY

Jan Kiwi
Pontificia Universidad Catolica de Chile

Sarah Koch
University of Michigan, Ann Arbor

Janne Kool
Universiteit Utrecht

Robert Kozma
Stony Brook University

ChongGyu Lee
University of Illinois

Tan Lei
Universite d'Angers

Alon Levy
Institute for Computational and Experimental Research in Mathematics (ICERM)

Huaibin Li
Pontificia Universidad Catolica de Chile

JanLi Lin
Indiana University

Kathryn Lindsey
University of Chicago

Jinsong Liu
Chinese Academy of Sciences

Khudoyor Mamayusupov
Jacobs University

Michelle Manes
Mathematics Department

John Milnor
SUNY

Khoa Nguyen
University of California, Berkeley

Frank Palladino
University of Rhode Island

Kevin Pilgrim
Indiana University

Jorge Pineiro
Bronx Community College, CUNY

Bjorn Poonen
MIT

Remus Radu
Stony Brook University

Paul Reschke
University of Illinois at UrbanaChampaign

Juan RiveraLetelier
Pontificia Universidad Catolica de Chile

Bastien Rossetti
Universite de Toulouse III (Paul Sabatier)

Robert Rumely
University of Georgia

Zachary Scherr
University of Michigan

Dierk Schleicher
Jacobs University

Nikita Selinger
Jacobs University

Thomas Sharland
University of Warwick

Joseph Silverman
Brown University

Brian Stout
City University of New York (CUNY)

Scott Sutherland
Stony Brook University

Lucien Szpiro
City University of New York (CUNY)

Raluca Tanase
Cornell University

Michael Tepper
Pennsylvania State University, Ogontz Campus

Bianca Thompson
University of Hawaii at Manoa

Thomas Tucker
University of Rochester

Eva Uhre
Stony Brook University

Bianca Viray
University of Washington

Paul Vojta
University of California, Berkeley

Xiaoguang Wang
Chinese Academy of Sciences

ChiHao Wang
National Central University

Hexi Ye
University of Illinois

Ilies Zidane
Universite de Toulouse III (Paul Sabatier)

Michael Zieve
University of Michigan
Workshop Schedule
Monday, April 16, 2012
Tuesday, April 17, 2012
Wednesday, April 18, 2012
Thursday, April 19, 2012
Friday, April 20, 2012
Tutorial Week Schedule
Thursday, April 12, 2012
Time  Event  Location  Materials 

10:30  11:50  Postcritically Finite Rational Maps and their Deformations  Jeremy Kahn, Brown University  11th Floor Lecture Hall  
1:30  2:30  Applications and Examples of using a new program `Dynamics Explorer' to study the dynamics of complex mappings  Suzanne Boyd, University of Wisconsin and Brian Boyd  11th Floor Lecture Hall  
2:30  2:45  Coffee/Tea Break  11th Floor Collaborative Space  
2:45  3:45  Applications and Examples of using a new program `Dynamics Explorer' to study the dynamics of complex mappings  Suzanne Boyd, University of Wisconsin and Brian Boyd  11th Floor Lecture Hall  
4:00  4:30  Sir John Ball "Smooth topologypreserving approximations of rough domains" Reception  11th Floor Collaborative Space  
4:30  5:30  Special ColloquiumSmooth topologypreserving approximations of rough domains  Sir John Ball, University of Oxford  11th Floor Lecture Hall 
Friday, April 13, 2012
Time  Event  Location  Materials 

10:00  11:00  Moduli spaces: drawing pictures and doing computations I  Sarah Koch, Harvard University and Xavier Buff, Université de Toulouse III (Paul Sabatier)  11th Floor Lecture Hall  
11:30  12:30  Moduli spaces: drawing pictures and doing computations II  Sarah Koch, Harvard University and Xavier Buff, Université de Toulouse III (Paul Sabatier)  11th Floor Lecture Hall  
2:30  3:30  Computational Working Group  11th Floor Lecture Hall  
3:30  4:00  Coffee/Tea Break  11th Floor Collaborative Space 
Problems
Problem 1: The Geometry of M_{d}^{n}.
It is known that M_{2}^{1} is isomorphic to the affine plane and that M_{d}^{1} is a rational variety, but many fundamental questions remain. A major goal of the workshop will be to study the geometry of Mdn and the associated moduli spaces in which one adds level structure, for example by adding a marked point of period N or a marked finite orbit of order N. A motivating question is whether the resulting varieties are of general type if N is sufficiently large.
Problem 2: Distribution of Special Points.
An example of the type of problem to be considered is the distribution of postcritically finite maps in the moduli space M_{d}^{1} in both the complex and the padic topologies.
Problem 3: Dynamical Modular Curves.
A oneparameter family of maps, for example f_{c}(z)=z^{2}+c, with marked points or orbits of order N, yields dynamical modular curves X_{0}(N) and X_{1}(N) that are analogous to classical modular curves. A good deal is known about the geometry of these curves, but little has been proven about their arithmetic except for some small values of N. The arithmetic properties of X_{0}(N) and X_{1}(N) are closely related to the uniform boundedness conjecture for the families that they parameterize.
Problem 4: The Boundary of Moduli Space.
The boundary of a moduli space and a natural method for completing the space are of fundamental importance in understanding the underlying objects and their degenerations. Recent work of Kiwi has used Berkovich space dynamics over Laurent series fields to analyze degenerations of complex dynamical systems. A goal of the workshop is to exploit these nonarchimedean methods to answer classical questions about the boundary of dynamical moduli spaces over the complex numbers.