## Global Arithmetic Dynamics (March 19-23, 2012)

Organizing Committee

[Global Arithmetic: Image courtesy of M. Zieve]
Description

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.

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.

DatesTitleOrganizers
March 19-23, 2012Global Arithmetic DynamicsXander Faber, Michelle Manes, Lucien Szpiro, Thomas Tucker, Michael Zieve (Chair)

Description Speaker
Video
Abstract
Slides
A Uniform Boundedness Theorem for Polynomials Michael Zieve, University of Michigan
A generalization of the Nagell-Lutz theorem Aaron Levin, Michigan State University
Adelic intersection problems in arithmetic geometry and dynamics Felipe Voloch, University of Texas at Austin
Algebraic dynamics, difference fields and model theory Zoe Chatzidakis, Centre National de la Recherche Scientifique (CNRS)-Paris 7
Critically finite endomorphisms of projective space Sarah Koch, Harvard University
Dynamical exponential and transcendence Shou-Wu Zhang, Princeton University
Equidistribution and the Dynamical Uniform Boundedness Conjecture Rob Benedetto, Amherst College
Exceptional linear subvarieties for the d^th power map Bianca Viray, Brown University
Good reduction and computational applications Benjamin Hutz, CUNY Graduate Center
Iteration of algebraic points under a rational self-map Ekaterina Amerik, Higher School of Economics (Moscow) and Universite Paris-Sud (Orsay)
On the (failure of) equidistribution of periodic points Juan Rivera-Letelier, Pontificia Universidad Catolica de Chile
Simple and Critically Good Reduction for Endomorphisms of P1 Jung Kyu Canci, University of Basel
Special orbifolds-arithmetic and hyperbolic aspects Frederic Campana, Universite Nancy 1
The Morton-Silverman conjecture, and extending self-maps to projective space Bjorn Poonen, Massachusetts Institute of Technology
Using Sage for Mathematical Research William Stein, University of Washington

MondayMarch 12, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
2:30 - 3:30Fields with several automorphismsAlice Medvedev, UC Berkeley11th Floor Lecture Hall
3:30 - 4:00Coffee/Tea Break11th Floor Collaborative Space
4:00 - 5:00The dynamics of birational maps over finite fields and their signaturesJohn A. G. Roberts, The University of New South Wales, Sydney, Australia11th Floor Lecture Hall

TuesdayMarch 13, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
10:30 - 11:50Postcritically Finite Rational Maps and their DeformationsJeremy Kahn, Brown University11th Floor Lecture Hall
3:30 - 4:00Coffee/Tea Break11th Floor Collaborative Space
4:00 - 5:00Professional Development: Ethics in research I11th Floor Lecture Hall

WednesdayMarch 14, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
3:14 - 4:00Coffee/Tea Break and Pi Day Treats!11th Floor Collaborative Space
4:00 - 5:00Postdoc/Grad Student Seminar11th Floor Lecture Hall

ThursdayMarch 15, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
10:30 - 11:50Postcritically Finite Rational Maps and their DeformationsJeremy Kahn, Brown University11th Floor Lecture Hall
1:30 - 2:30Introduction to Height FunctionsYu Yasufuku, Nihon University11th Floor Lecture Hall
2:30 - 3:00Coffee/Tea Break11th Floor Collaborative Space
3:00 - 4:00Model-theoretic properties for algebraic dynamics, via ACFAAlice Medvedev, UC Berkeley11th Floor Lecture Hall

FridayMarch 16, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
11:00 - 12:00Computational Working Group/Seminar11th Floor Lecture Hall
2:30 - 3:30Equidistribution of small points on \$\mathbb{P}^1\$Juan Rivera-Letelier, Pontificia Universidad Catolica de Chile11th Floor Lecture Hall
3:30 - 4:00Coffee/Tea Break11th Floor Collaborative Space

MondayMarch 19, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
8:30 - 8:55Registration & Welcome Coffee11th Floor Collaborative Space
8:55 - 9:00WelcomeJill Pipher, Director, ICERM11th Floor Lecture Hall
9:00 - 9:50The Morton-Silverman conjecture, and extending self-maps to projective spaceBjorn Poonen, MIT11th Floor Lecture Hall
9:55 - 10:20Coffee/Tea Break11th Floor Collaborative Space
10:20 - 11:00Exceptional linear subvarieties for the d^th power mapBianca Viray, Brown University11th Floor Lecture Hall
11:05 - 11:55A generalization of the Nagell-Lutz theoremAaron Levin, Michigan State University11th Floor Lecture Hall
12:00 - 1:45Break for Lunch and Free Time
1:45 - 2:35Equidistribution and the Dynamical Uniform Boundedness ConjectureRob Benedetto, Amherst College11th Floor Lecture Hall
2:40 - 3:30A Uniform Boundedness Theorem for PolynomialsMichael Zieve, University of Michigan11th Floor Lecture Hall
3:35 - 4:00Coffee/Tea Break11th Floor Collaborative Space
4:00 - 5:00Critically finite endomorphisms of projective spaceSarah Koch, Harvard University11th Floor Lecture Hall

TuesdayMarch 20, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
9:00 - 9:50Adelic intersection problems in arithmetic geometry and dynamicsFelipe Voloch, University of Texas at Austin11th Floor Lecture Hall
9:55 - 10:20Coffee/Tea Break11th Floor Collaborative Space
10:20 - 11:00Simple and Critically Good Reduction For Endomorphisms of P1 Jung Kyu Canci, University of Basel11th Floor Lecture Hall
11:05 - 11:55Using Sage for Mathematical ResearchWilliam Stein, University of Washington11th Floor Lecture Hall
11:55 - 12:00Group Photo11th Floor Lecture Hall
12:00 - 1:45Break for Lunch & Free Time
1:45 - 5:00Problem Session and Collaborative Work/Coffee & Tea Break11th Floor Collaborative Space
7:00 - 10:00Hackathon10th Floor Collaborative Space

WednesdayMarch 21, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
9:00 - 9:50Dynamical exponential and transcendenceShou-Wu Zhang, Princeton University11th Floor Lecture Hall
9:55 - 10:20Coffee/Tea Break11th Floor Collaborative Space
10:20 - 11:00Good reduction and computational applicationsBenjamin Hutz, CUNY Graduate Center11th Floor Lecture Hall
11:05 - 11:55Iteration of algebraic points under a rational self-mapEkaterina Amerik, Higher School of Economics (Moscow) and Universite Paris-Sud (Orsay)11th Floor Lecture Hall
12:00 - 1:45Break for Lunch & Free Time
1:45 - 5:00Progress Reports and Collaborative Work/Coffee & Tea Break11th Floor Collaborative Space
7:00 - 9:00Poster Session and Dessert Reception11th Floor Lecture Hall

ThursdayMarch 22, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
9:00 - 9:50TBATom Scanlon, University of California, Berkeley11th Floor Lecture Hall
9:55 - 10:20Coffee/Tea Break11th Floor Collaborative Space
10:20 - 11:10Algebraic Dynamics Difference Fields and Model TheoryZoe Chatzidakis, Universit� de Paris VII (Denis Diderot)11th Floor Lecture Hall
11:15 - 11:55A property of integral division points on curves and its dynamical system analogueSu-Ion Ih, University of Colorado11th Floor Lecture Hall