Organizing Committee
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.

Image for "Global Arithmetic Dynamics"
Global Arithmetic: Image courtesy of M. Zieve

Confirmed Speakers & Participants

Talks will be presented virtually or in-person as indicated in the schedule below.

  • Speaker
  • Poster Presenter
  • Attendee
  • Virtual Attendee

Workshop Schedule

Monday, March 19, 2012
TimeEventLocationMaterials
8:30 - 8:55am EDTRegistration & Welcome Coffee11th Floor Collaborative Space 
8:55 - 9:00am EDTWelcome - Jill Pipher, Director, ICERM11th Floor Lecture Hall 
9:00 - 9:50am EDTThe Morton-Silverman conjecture, and extending self-maps to projective space - Bjorn Poonen, MIT11th Floor Lecture Hall 
9:55 - 10:20am EDTCoffee/Tea Break11th Floor Collaborative Space 
10:20 - 11:00am EDTExceptional linear subvarieties for the d^th power map - Bianca Viray, Brown University11th Floor Lecture Hall 
11:05 - 11:55am EDTA generalization of the Nagell-Lutz theorem - Aaron Levin, Michigan State University11th Floor Lecture Hall 
12:00 - 1:45pm EDTBreak for Lunch and Free Time  
1:45 - 2:35pm EDTEquidistribution and the Dynamical Uniform Boundedness Conjecture - Rob Benedetto, Amherst College11th Floor Lecture Hall
2:40 - 3:30pm EDTA Uniform Boundedness Theorem for Polynomials - Michael Zieve, University of Michigan11th Floor Lecture Hall
3:35 - 4:00pm EDTCoffee/Tea Break11th Floor Collaborative Space 
4:00 - 5:00pm EDTCritically finite endomorphisms of projective space - Sarah Koch, Harvard University11th Floor Lecture Hall 
Tuesday, March 20, 2012
TimeEventLocationMaterials
9:00 - 9:50am EDTAdelic intersection problems in arithmetic geometry and dynamics - Felipe Voloch, University of Texas at Austin11th Floor Lecture Hall 
9:55 - 10:20am EDTCoffee/Tea Break11th Floor Collaborative Space 
10:20 - 11:00am EDTSimple and Critically Good Reduction For Endomorphisms of P1 - Jung Kyu Canci, University of Basel11th Floor Lecture Hall 
11:05 - 11:55am EDTUsing Sage for Mathematical Research - William Stein, University of Washington11th Floor Lecture Hall 
11:55 - 12:00pm EDTGroup Photo11th Floor Lecture Hall 
12:00 - 1:45pm EDTBreak for Lunch & Free Time  
1:45 - 5:00pm EDTProblem Session and Collaborative Work/Coffee & Tea Break11th Floor Collaborative Space 
7:00 - 10:00pm EDTHackathon10th Floor Collaborative Space 
Wednesday, March 21, 2012
TimeEventLocationMaterials
9:00 - 9:50am EDTDynamical exponential and transcendence - Shou-Wu Zhang, Princeton University11th Floor Lecture Hall 
9:55 - 10:20am EDTCoffee/Tea Break11th Floor Collaborative Space 
10:20 - 11:00am EDTGood reduction and computational applications - Benjamin Hutz, CUNY Graduate Center11th Floor Lecture Hall 
11:05 - 11:55am EDTIteration 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 - 1:45pm EDTBreak for Lunch & Free Time  
1:45 - 5:00pm EDTProgress Reports and Collaborative Work/Coffee & Tea Break11th Floor Collaborative Space 
7:00 - 9:00pm EDTPoster Session and Dessert Reception11th Floor Lecture Hall 
Thursday, March 22, 2012
TimeEventLocationMaterials
9:00 - 9:50am EDTTBA - Tom Scanlon, University of California, Berkeley11th Floor Lecture Hall 
9:55 - 10:20am EDTCoffee/Tea Break11th Floor Collaborative Space 
10:20 - 11:10am EDTAlgebraic Dynamics Difference Fields and Model Theory - Zoe Chatzidakis, Université de Paris VII (Denis Diderot)11th Floor Lecture Hall 
11:15 - 11:55am EDTA property of integral division points on curves and its dynamical system analogue - Su-Ion Ih, University of Colorado11th Floor Lecture Hall 
11:55 - 12:00pm EDTSurvey Distribution, please return to 11th floor reception desk  
12:00 - 1:45pm EDTBreak for Lunch & Free Time  
1:45 - 2:35pm EDTOn the (failure of) equidistribution of periodic points - Juan Rivera-Letelier, Pontificia Universidad Catolica de Chile11th Floor Lecture Hall 
2:35 - 5:00pm EDTProgress Reports and Collaborative Work/Coffee & Tea Break11th Floor Collaborative Space 
7:00 - 10:00pm EDTHackathon10th Floor Collaborative Space 
Friday, March 23, 2012
TimeEventLocationMaterials
9:00 - 10:20am EDTMorning of Collaborations and Coffee/Tea Break11th Floor Collaborative Space 
10:20 - 11:10am EDTSpecial orbifolds: arithmetic and hyperbolic aspects - Frederic Campana, Université de Nancy I (Henri Poincaré)11th Floor Lecture Hall 
11:15 - 12:05pm EDTWhich bilinear recurrences give theta sequences - Noam Elkies, Harvard University11th Floor Lecture Hall
12:10 - 2:00pm EDTBreak for Lunch & Free Time  
2:00 - 3:00pm EDTWrapup Problem Session and Progress Reports 11th Floor Lecture Hall 

Tutorial Week Schedule

Monday, March 12, 2012
TimeEventLocationMaterials
2:30 - 3:30pm EDTFields with several automorphisms - Alice Medvedev, UC Berkeley11th Floor Lecture Hall 
3:30 - 4:00pm EDTCoffee/Tea Break11th Floor Collaborative Space 
4:00 - 5:00pm EDTThe dynamics of birational maps over finite fields and their signatures - John A. G. Roberts, The University of New South Wales, Sydney, Australia11th Floor Lecture Hall 
Tuesday, March 13, 2012
TimeEventLocationMaterials
10:30 - 11:50am EDTPostcritically Finite Rational Maps and their Deformations - Jeremy Kahn, Brown University11th Floor Lecture Hall 
3:30 - 4:00pm EDTCoffee/Tea Break11th Floor Collaborative Space 
4:00 - 5:00pm EDTProfessional Development: Ethics in research I11th Floor Lecture Hall 
Wednesday, March 14, 2012
TimeEventLocationMaterials
3:14 - 4:00pm EDTCoffee/Tea Break and Pi Day Treats!11th Floor Collaborative Space 
4:00 - 5:00pm EDTPostdoc/Grad Student Seminar11th Floor Lecture Hall 
Thursday, March 15, 2012
TimeEventLocationMaterials
10:30 - 11:50am EDTPostcritically Finite Rational Maps and their Deformations - Jeremy Kahn, Brown University11th Floor Lecture Hall 
1:30 - 2:30pm EDTIntroduction to Height Functions - Yu Yasufuku, Nihon University11th Floor Lecture Hall 
2:30 - 3:00pm EDTCoffee/Tea Break11th Floor Collaborative Space 
3:00 - 4:00pm EDTModel-theoretic properties for algebraic dynamics, via ACFA - Alice Medvedev, UC Berkeley11th Floor Lecture Hall 
Friday, March 16, 2012
TimeEventLocationMaterials
11:00 - 12:00pm EDTComputational Working Group/Seminar11th Floor Lecture Hall 
2:30 - 3:30pm EDTEquidistribution of small points on $\mathbb{P}^1$ - Juan Rivera-Letelier, Pontificia Universidad Catolica de Chile11th Floor Lecture Hall 
3:30 - 4:00pm EDTCoffee/Tea Break11th 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.

Associated Semester Workshops

Complex and Arithmetic Dynamics
Image for "Complex and Arithmetic Dynamics"
Jan 30 - May 4, 2012
Semester Program
Complex and p-adic Dynamics
Image for "Complex and p-adic Dynamics"
Feb 13 - 17, 2012
Semester Workshop
Moduli Spaces Associated to Dynamical Systems
Image for "Moduli Spaces Associated to Dynamical Systems"
Apr 16 - 20, 2012
Semester Workshop

Lecture Videos

Special orbifolds-arithmetic and hyperbolic aspects

Frederic Campana
Université de Nancy I (Henri Poincaré)
March 23, 2012
Global Arithmetic Dynamics
Play

On the (failure of) equidistribution of periodic points

Juan Rivera-Letelier
Pontificia Universidad Catolica de Chile
March 22, 2012
Global Arithmetic Dynamics
Play

Algebraic dynamics, difference fields and model theory

Zoe Chatzidakis
Centre National de la Recherche Scientifique (CNRS)
March 22, 2012
Global Arithmetic Dynamics
Play

Iteration of algebraic points under a rational self-map

Ekaterina Amerik
Universite de Paris XI (Paris-Sud)
March 21, 2012
Global Arithmetic Dynamics
Play

Good reduction and computational applications

Benjamin Hutz
Saint Louis University
March 21, 2012
Global Arithmetic Dynamics
Play

Dynamical exponential and transcendence

Shou-Wu Zhang
Princeton University
March 21, 2012
Global Arithmetic Dynamics
Play

Using Sage for Mathematical Research

William Stein
University of Washington
March 20, 2012
Global Arithmetic Dynamics
Play

Simple and Critically Good Reduction for Endomorphisms of P1

Jung Kyu Canci
Universität Basel
March 20, 2012
Global Arithmetic Dynamics
Play

Adelic intersection problems in arithmetic geometry and dynamics

Jose Voloch
University of Texas at Austin
March 20, 2012
Global Arithmetic Dynamics
Play

Critically finite endomorphisms of projective space

Sarah Koch
University of Michigan, Ann Arbor
March 19, 2012
Global Arithmetic Dynamics
Play

A Uniform Boundedness Theorem for Polynomials

Michael Zieve
University of Michigan
March 19, 2012
Global Arithmetic Dynamics
Play     Slides

Equidistribution and the Dynamical Uniform Boundedness Conjecture

Rob Benedetto
Amherst College
March 19, 2012
Global Arithmetic Dynamics
Play

A generalization of the Nagell-Lutz theorem

Aaron Levin
Michigan State University
March 19, 2012
Global Arithmetic Dynamics
Play

Exceptional linear subvarieties for the d^th power map

Bianca Viray
University of Washington
March 19, 2012
Global Arithmetic Dynamics
Play

The Morton-Silverman conjecture, and extending self-maps to projective space

Bjorn Poonen
MIT
March 19, 2012
Global Arithmetic Dynamics
Play