## 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.

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, Université 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
11:55 - 12:00Survey Distribution, please return to 11th floor reception desk
12:00 - 1:45Break for Lunch & Free Time
1:45 - 2:35On the (failure of) equidistribution of periodic pointsJuan Rivera-Letelier, Pontificia Universidad Catolica de Chile11th Floor Lecture Hall
2:35 - 5:00Progress Reports and Collaborative Work/Coffee & Tea Break11th Floor Collaborative Space
7:00 - 10:00Hackathon10th Floor Collaborative Space

FridayMarch 23, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
9:00 - 10:20Morning of Collaborations and Coffee/Tea Break11th Floor Collaborative Space
10:20 - 11:10Special orbifolds: arithmetic and hyperbolic aspectsFrederic Campana, Université de Nancy I (Henri Poincaré)11th Floor Lecture Hall
11:15 - 12:05Which bilinear recurrences give theta sequencesNoam Elkies, Harvard University11th Floor Lecture Hall
12:10 - 2:00Break for Lunch & Free Time
2:00 - 3:00Wrapup Problem Session and Progress Reports 11th Floor Lecture Hall