##### Organizing Committee

- Xander Faber
*University of Hawaii* - Michelle Manes
*Mathematics Department* - Lucien Szpiro
*City University of New York (CUNY)* - Thomas Tucker
*University of Rochester* - Michael Zieve
*University of Michigan*

##### Abstract

This workshop will examine global arithmetic dynamics from the perspectives of number theory, algebraic geometry, and model theory. It will introduce aspects of this topic to a larger audience, and clarify connections between different perspectives. In addition, there will be extensive discussion periods in which participants can collaborate on theoretical and computational aspects of the topic.

#### Confirmed Speakers & Participants

- Domenico Aiello
- Ekaterina Amerik
- Jackie Anderson
- Matthieu Arfeux
- Cecile Armana
- Fabrizio Barroero
- Rob Benedetto
- Anupam Bhatnagar
- Araceli Bonifant
- Andrew Bridy
- Xavier Buff
- Frederic Campana
- Jung Kyu Canci
- Zoe Chatzidakis
- Laura DeMarco
- John Doyle
- Dzmitry Dudko
- Noam Elkies
- Adam Epstein
- Xander Faber
- Paul Fili
- Tanya Firsova
- Derek Garton
- William Gignac
- Chad Gratton
- Wade Hindes
- Liang-Chung Hsia
- Hsiu Lien Huang
- Benjamin Hutz
- Su-ion Ih
- Patrick Ingram
- Jeremy Kahn
- Sarah Koch
- Holly Krieger
- Par Kurlberg
- ChongGyu Lee
- Benjamin LeVeque
- Aaron Levin
- Alon Levy
- Huaibin Li
- Jan-Li Lin
- Karl-Olof Lindahl
- Sijun Liu
- Michelle Manes
- Alice Medvedev
- Ricardo Menares
- John Milnor
- Igor Minevich
- Khoa Nguyen
- Andrew Obus
- Alina Ostafe
- Frank Palladino
- Donghoon Park
- Jorge Pineiro
- Bjorn Poonen
- Lukas Pottmeyer
- Jamie Rahr
- Juan Rivera-Letelier
- John Roberts
- Robert Rumely
- Thomas Scanlon
- Zachary Scherr
- Dierk Schleicher
- Thomas Sharland
- Joseph Silverman
- Martin Sombra
- William Stein
- Michael Stoll
- Brian Stout
- Lucien Szpiro
- Michael Tepper
- Lubos Thoma
- Bianca Thompson
- Adam Towsley
- Thomas Tucker
- Bianca Viray
- Paul Vojta
- Jose Voloch
- Chi-Hao Wang
- Xiaoguang Wang
- Lloyd West
- Siman Wong
- Elizabeth Wulcan
- Kazuhiko Yamaki
- Yu Yasufuku
- Tianjun Ye
- Shou-Wu Zhang
- Liang Zhao
- Michael Zieve

#### Workshop Schedule

##### Monday, March 19, 2012

##### Tuesday, March 20, 2012

Time | Event | Location | Materials |
---|---|---|---|

9:00 - 9:50 | Adelic intersection problems in arithmetic geometry and dynamics - Felipe Voloch, University of Texas at Austin | 11th Floor Lecture Hall | |

9:55 - 10:20 | Coffee/Tea Break | 11th Floor Collaborative Space | |

10:20 - 11:00 | Simple and Critically Good Reduction For Endomorphisms of P1 - Jung Kyu Canci, University of Basel | 11th Floor Lecture Hall | |

11:05 - 11:55 | Using Sage for Mathematical Research - William Stein, University of Washington | 11th Floor Lecture Hall | |

11:55 - 12:00 | Group Photo | 11th Floor Lecture Hall | |

12:00 - 1:45 | Break for Lunch & Free Time | ||

1:45 - 5:00 | Problem Session and Collaborative Work/Coffee & Tea Break | 11th Floor Collaborative Space | |

7:00 - 10:00 | Hackathon | 10th Floor Collaborative Space |

##### Wednesday, March 21, 2012

Time | Event | Location | Materials |
---|---|---|---|

9:00 - 9:50 | Dynamical exponential and transcendence - Shou-Wu Zhang, Princeton University | 11th Floor Lecture Hall | |

9:55 - 10:20 | Coffee/Tea Break | 11th Floor Collaborative Space | |

10:20 - 11:00 | Good reduction and computational applications - Benjamin Hutz, CUNY Graduate Center | 11th Floor Lecture Hall | |

11:05 - 11:55 | Iteration of algebraic points under a rational self-map - Ekaterina Amerik, Higher School of Economics (Moscow) and Universite Paris-Sud (Orsay) | 11th Floor Lecture Hall | |

12:00 - 1:45 | Break for Lunch & Free Time | ||

1:45 - 5:00 | Progress Reports and Collaborative Work/Coffee & Tea Break | 11th Floor Collaborative Space | |

7:00 - 9:00 | Poster Session and Dessert Reception | 11th Floor Lecture Hall |

##### Thursday, March 22, 2012

Time | Event | Location | Materials |
---|---|---|---|

9:00 - 9:50 | TBA - Tom Scanlon, University of California, Berkeley | 11th Floor Lecture Hall | |

9:55 - 10:20 | Coffee/Tea Break | 11th Floor Collaborative Space | |

10:20 - 11:10 | Algebraic Dynamics Difference Fields and Model Theory - Zoe Chatzidakis, Université de Paris VII (Denis Diderot) | 11th Floor Lecture Hall | |

11:15 - 11:55 | A property of integral division points on curves and its dynamical system analogue - Su-Ion Ih, University of Colorado | 11th Floor Lecture Hall | |

11:55 - 12:00 | Survey Distribution, please return to 11th floor reception desk | ||

12:00 - 1:45 | Break for Lunch & Free Time | ||

1:45 - 2:35 | On the (failure of) equidistribution of periodic points - Juan Rivera-Letelier, Pontificia Universidad Catolica de Chile | 11th Floor Lecture Hall | |

2:35 - 5:00 | Progress Reports and Collaborative Work/Coffee & Tea Break | 11th Floor Collaborative Space | |

7:00 - 10:00 | Hackathon | 10th Floor Collaborative Space |

##### Friday, March 23, 2012

#### Tutorial Week Schedule

##### Monday, March 12, 2012

Time | Event | Location | Materials |
---|---|---|---|

2:30 - 3:30 | Fields with several automorphisms - Alice Medvedev, UC Berkeley | 11th Floor Lecture Hall | |

3:30 - 4:00 | Coffee/Tea Break | 11th Floor Collaborative Space | |

4:00 - 5:00 | The dynamics of birational maps over finite fields and their signatures - John A. G. Roberts, The University of New South Wales, Sydney, Australia | 11th Floor Lecture Hall |

##### Tuesday, March 13, 2012

Time | Event | Location | Materials |
---|---|---|---|

10:30 - 11:50 | Postcritically Finite Rational Maps and their Deformations - Jeremy Kahn, Brown University | 11th Floor Lecture Hall | |

3:30 - 4:00 | Coffee/Tea Break | 11th Floor Collaborative Space | |

4:00 - 5:00 | Professional Development: Ethics in research I | 11th Floor Lecture Hall |

##### Wednesday, March 14, 2012

Time | Event | Location | Materials |
---|---|---|---|

3:14 - 4:00 | Coffee/Tea Break and Pi Day Treats! | 11th Floor Collaborative Space | |

4:00 - 5:00 | Postdoc/Grad Student Seminar | 11th Floor Lecture Hall |

##### Thursday, March 15, 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 | Introduction to Height Functions - Yu Yasufuku, Nihon University | 11th Floor Lecture Hall | |

2:30 - 3:00 | Coffee/Tea Break | 11th Floor Collaborative Space | |

3:00 - 4:00 | Model-theoretic properties for algebraic dynamics, via ACFA - Alice Medvedev, UC Berkeley | 11th Floor Lecture Hall |

##### Friday, March 16, 2012

Time | Event | Location | Materials |
---|---|---|---|

11:00 - 12:00 | Computational Working Group/Seminar | 11th Floor Lecture Hall | |

2:30 - 3:30 | Equidistribution of small points on $\mathbb{P}^1$ - Juan Rivera-Letelier, Pontificia Universidad Catolica de Chile | 11th Floor Lecture Hall | |

3:30 - 4:00 | Coffee/Tea Break | 11th Floor Collaborative Space |

#### Problems

##### Problem 1: The Uniform Boundedness Conjecture.

This fundamental conjecture in arithmetic dynamics says that for given positive integers D, n, and d with d>1, if K/Q is a number field of degree D and if f:Pn→Pn is a morphism of degree d defined over K, then the number of K-rational preperiodic points of f is bounded by a constant depending only on D, n, and d. It is a vast generalization of the Mazur-Merel theorem on uniform boundedness of torsion points on elliptic curves.

##### Problem 2: Dynamical Intersection Theorems.

Two fundamental arithmetic results for abelian varieties are theorems of Raynaud and Faltings, orginally formulated as conjectures by Manin-Mumford and Mordell-Lang, respectively. There are a number of dynamical analogues of these conjectures, which roughly say that the orbit of a point should intersect a subvariety only finitely often unless the orbit of the entire subvariety has special properties. Only special cases of the dynamical conjectures have been proven. We expect these problems to be a major focus of the workshop.

##### Problem 3: Global Applications of Equidistribution Theorems.

Let (xi) be a of sequence of algebraic points whose f-canonical heights go to zero. Under suitable hypotheses, it is known that the Galois conjugates of the xi are equidistributed with respect to the complex and Berkovich invariant measures. A focus of the program will be on global arithmetic applications of this and other similar equidistribution theorems.

##### Problem 4: Arithmetic Dynamics Over Function Fields.

Many theorems in arithmetic geometry were first proven in the easier setting of function fields. A theme for the workshop will be the analogy for arithmetic dynamics between number fields and function fields.

##### Problem 5: Local-Global Problems.

There are many local-global principles in arithmetic geometry, such as those related to the Hasse and Brauer-Manin obstructions. The workshop will explore analogous local-global principles for dynamical systems, including especially the distribution of orbits modulo primes and/or in p-adic or complex neighborhoods.