Global Arithmetic Dynamics (March 1923, 2012)
 Xander Faber
(University of Hawaii)  Michelle Manes
(University of Hawaii)  Lucien Szpiro
(City University of New York)  Thomas Tucker
(University of Rochester)  Michael Zieve
(University of Michigan), Chair)
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:P^{n}→P^{n} is a morphism of degree d defined over K, then the number of Krational preperiodic points of f is bounded by a constant depending only on D, n, and d. It is a vast generalization of the MazurMerel 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 ManinMumford and MordellLang, 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 (x_{i}) be a of sequence of algebraic points whose fcanonical heights go to zero. Under suitable hypotheses, it is known that the Galois conjugates of the x_{i} 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: LocalGlobal Problems. There are many localglobal principles in arithmetic geometry, such as those related to the Hasse and BrauerManin obstructions. The workshop will explore analogous localglobal principles for dynamical systems, including especially the distribution of orbits modulo primes and/or in padic or complex neighborhoods.



Dates  Title  Organizers 

March 1923, 2012  Global Arithmetic Dynamics  Xander Faber, Michelle Manes, Lucien Szpiro, Thomas Tucker, Michael Zieve (Chair) 
Monday  March 12, 2012  

Time  Description  Speaker  Location  Abstracts  Slides 
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  Description  Speaker  Location  Abstracts  Slides 
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  Description  Speaker  Location  Abstracts  Slides 
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  Description  Speaker  Location  Abstracts  Slides 
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  Modeltheoretic properties for algebraic dynamics, via ACFA  Alice Medvedev, UC Berkeley  11th Floor Lecture Hall 
Friday  March 16, 2012  

Time  Description  Speaker  Location  Abstracts  Slides 
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 RiveraLetelier, Pontificia Universidad Catolica de Chile  11th Floor Lecture Hall  
3:30  4:00  Coffee/Tea Break  11th Floor Collaborative Space 