##### 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`self-maps of

`P`has a natural structure as an affine variety. The dynamical moduli space

^{n}`M`is the quotient of

_{d}^{n}`Rat`by the conjugation action of the group

_{d}^{n}`PGL`. Problems to be investigated include the geometry of

_{n+1}`M`, the distribution of special maps such as post-critically finite maps in

_{d}^{n}`M`, dynamical modular curves associated to one-parameter families of maps with a marked point of period

_{d}^{n}`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

- Jackie Anderson
- Matthieu Arfeux
- Cecile Armana
- Matthieu Astorg
- Fabrizio Barroero
- Rob Benedetto
- Anupam Bhatnagar
- Paul Blanchard
- Araceli Bonifant
- Joshua Bowman
- Andrew Bridy
- Xavier Buff
- Arnaud Chéritat
- Mark Comerford
- Daniel Cuzzocreo
- Diana Davis
- Laura DeMarco
- Romain Dujardin
- Adam Epstein
- Timo Erkama
- Derek Garton
- Jonah Gaster
- William Gignac
- Mikhail Hlushchanka
- Wei Ho
- Liang-Chung Hsia
- John Hubbard
- Benjamin Hutz
- Patrick Ingram
- Rafe Jones
- Jeremy Kahn
- Linda Keen
- Jan Kiwi
- Sarah Koch
- Janne Kool
- Robert Kozma
- ChongGyu Lee
- Tan Lei
- Alon Levy
- Huaibin Li
- Jan-Li Lin
- Kathryn Lindsey
- Jinsong Liu
- Khudoyor Mamayusupov
- Michelle Manes
- John Milnor
- Khoa Nguyen
- Frank Palladino
- Kevin Pilgrim
- Jorge Pineiro
- Bjorn Poonen
- Remus Radu
- Paul Reschke
- Juan Rivera-Letelier
- Bastien Rossetti
- Robert Rumely
- Zachary Scherr
- Dierk Schleicher
- Nikita Selinger
- Thomas Sharland
- Joseph Silverman
- Brian Stout
- Scott Sutherland
- Lucien Szpiro
- Raluca Tanase
- Michael Tepper
- Bianca Thompson
- Thomas Tucker
- Eva Uhre
- Bianca Viray
- Paul Vojta
- Xiaoguang Wang
- Chi-Hao Wang
- Hexi Ye
- Ilies Zidane
- Michael Zieve

#### 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 topology-preserving approximations of rough domains" Reception | 11th Floor Collaborative Space | |

4:30 - 5:30 | Special Colloquium-Smooth topology-preserving 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}.

_{d}

^{n}

It is known that `M _{2}^{1}` is isomorphic to the affine plane and that

`M`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

_{d}^{1}`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 post-critically finite maps in the moduli space `M _{d}^{1}` in both the complex and the p-adic topologies.

##### Problem 3: Dynamical Modular Curves.

A one-parameter family of maps, for example `f _{c}(z)=z^{2}+c`, with marked points or orbits of order

`N`, yields dynamical modular curves

`X`and

_{0}(N)`X`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

_{1}(N)`N`. The arithmetic properties of

`X`and

_{0}(N)`X`are closely related to the uniform boundedness conjecture for the families that they parameterize.

_{1}(N)##### 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 non-archimedean methods to answer classical questions about the boundary of dynamical moduli spaces over the complex numbers.