Organizing Committee
Abstract

This workshop will bring together dynamicists, number theorists, and algebraic geometers to study the geometry and arithmetic of dynamical moduli spaces. The set Ratdn of rational degree d self-maps of Pn has a natural structure as an affine variety. The dynamical moduli space Mdn is the quotient of Ratdn by the conjugation action of the group PGLn+1. Problems to be investigated include the geometry of Mdn, the distribution of special maps such as post-critically finite maps in Mdn, dynamical modular curves associated to one-parameter families of maps with a marked point of period N, and degeneration of families of maps and the associated points on the boundary of moduli space. A tutorial session will be held the week before this workshop.

Global Arithmetic: Image courtesy of M. Zieve

Confirmed Speakers & Participants

Workshop Schedule

Monday, April 16, 2012
TimeEventLocationMaterials
8:30 - 9:20Registration & Welcome Coffee  
9:20 - 9:30Welcome - Jeff Brock, Deputy Director, ICERM11th Floor Lecture Hall 
9:30 - 10:20Ramification avoidance and a dynamical Mordell-Lang conjecture - Tom Tucker, University of Rochester11th Floor Lecture Hall
10:30 - 11:00Coffee/Tea Break11th Floor Collaborative Space 
11:00 - 11:50Bers, Brown and Lyapunov - Romain Dujardin, Ecole Polytechnique11th Floor Lecture Hall
12:00 - 2:30Break for Lunch  
2:30 - 3:20Measure-theoretic rigidity for Mumford curves - Janne Kool, Universiteit Utrecht11th Floor Lecture Hall
3:30 - 4:00Coffee/Tea Break11th Floor Collaborative Space 
4:00 - 4:50Admissible Covers and Rescaling Limits (after Matthieu Arfeux) - Xavier Buff, Universite de Toulouse III (Paul Sabatier)11th Floor Lecture Hall
5:00 - 6:30Welcome Reception11th Floor Collaborative Space  
Tuesday, April 17, 2012
TimeEventLocationMaterials
9:30 - 10:20Variations on a Theme-Fields of Definition, Fields of Moduli, Automorphisms, and Twists - Michelle Manes, University of Hawaii11th Floor Lecture Hall
10:30 - 11:00Coffee/Tea Break11th Floor Collaborative Space 
11:00 - 11:50Arithmetic Properties of Dominant Self-Maps of P^N - Joseph H. Silverman, Brown University11th Floor Lecture Hall
12:00 - 2:30Break for Lunch  
2:30 - 3:20Puiseux series dynamics and leading monomials of escape regions - Jan Kiwi, Pontificia Universidad Catolica de Chile11th Floor Lecture Hall
3:30 - 4:00Coffee/Tea Break11th Floor Collaborative Space 
4:00 - 4:50Antipode Preserving Cubic Rational Maps - Araceli Bonifant, University of Rhode Island11th Floor Lecture Hall
Wednesday, April 18, 2012
TimeEventLocationMaterials
10:00 - 10:50On Mordell-Lang in algebraic groups of unipotent rank 1 - Paul Vojta, University of California, Berkeley11th Floor Lecture Hall
11:00 - 11:30Coffee/Tea Break11th Floor Collaborative Space 
11:30 - 12:20Limits of polynomial-like quadratic rational maps - Eva Uhre, Stony Brook University11th Floor Lecture Hall
12:30 - 12:40Group Photo11th Floor Lecture Hall 
12:40 - 7:00Free afternoon for collaborations  
7:00 - 9:00Poster Session and Dessert Reception11th Floor Lecture Hall and Collaborative Space 
Thursday, April 19, 2012
TimeEventLocationMaterials
9:30 - 10:20A proof of Pilgrim's conjecture - Nikita Selinger, Jacobs University11th Floor Lecture Hall
10:30 - 11:00Coffee/Tea Break11th Floor Collaborative Space 
11:00 - 11:50The Space of Morphisms on a Projective Variety- Construction and Applications - Alon Levy, ICERM11th Floor Lecture Hall
12:00 - 2:30Break for Lunch  
2:30 - 3:20Mating Discontinuity - Adam Epstein, University of Warwick11th Floor Lecture Hall
3:30 - 3:35Survey Distribution11th Floor Lecture Hall 
3:35 - 4:00Coffee/Tea Break11th Floor Collaborative Space 
4:00 - 4:50Discussions and Collaborations11th Floor Collaborative Space 
Friday, April 20, 2012
TimeEventLocationMaterials
9:30 - 10:20Quadratic Rational Maps- the Totally Marked Moduli Space - John Milnor, Stony Brook University11th Floor Lecture Hall
10:30 - 11:00Coffee/Tea Break11th Floor Collaborative Space 
11:00 - 11:50Rational maps with prescribed critical values - Tan Lei, University d'Angers11th Floor Lecture Hall
12:00 - 2:30Break for Lunch  
2:30 - 3:20Resultant and conductor for self maps of the projective line - Lucien Szpiro, City University of New York (CUNY)11th Floor Lecture Hall
3:30 - 4:00Coffee/Tea Break11th Floor Collaborative Space 
4:00 - 4:50Parabolic blowups and limits of dynamical systems - John Hubbard, Cornell University11th Floor Lecture Hall 

Tutorial Week Schedule

Thursday, April 12, 2012
TimeEventLocationMaterials
10:30 - 11:50Postcritically Finite Rational Maps and their Deformations - Jeremy Kahn, Brown University11th Floor Lecture Hall 
1:30 - 2:30Applications and Examples of using a new program `Dynamics Explorer' to study the dynamics of complex mappings - Suzanne Boyd, University of Wisconsin and Brian Boyd11th Floor Lecture Hall 
2:30 - 2:45Coffee/Tea Break11th Floor Collaborative Space 
2:45 - 3:45Applications and Examples of using a new program `Dynamics Explorer' to study the dynamics of complex mappings - Suzanne Boyd, University of Wisconsin and Brian Boyd11th Floor Lecture Hall 
4:00 - 4:30Sir John Ball "Smooth topology-preserving approximations of rough domains" Reception11th Floor Collaborative Space 
4:30 - 5:30Special Colloquium-Smooth topology-preserving approximations of rough domains - Sir John Ball, University of Oxford11th Floor Lecture Hall 
Friday, April 13, 2012
TimeEventLocationMaterials
10:00 - 11:00Moduli spaces: drawing pictures and doing computations I - Sarah Koch, Harvard University and Xavier Buff, Université de Toulouse III (Paul Sabatier)11th Floor Lecture Hall 
11:30 - 12:30Moduli spaces: drawing pictures and doing computations II - Sarah Koch, Harvard University and Xavier Buff, Université de Toulouse III (Paul Sabatier)11th Floor Lecture Hall 
2:30 - 3:30Computational Working Group11th Floor Lecture Hall 
3:30 - 4:00Coffee/Tea Break11th Floor Collaborative Space 

Problems

Problem 1: The Geometry of Mdn.

It is known that M21 is isomorphic to the affine plane and that Md1 is a rational variety, but many fundamental questions remain. A major goal of the workshop will be to study the geometry of Mdn and the associated moduli spaces in which one adds level structure, for example by adding a marked point of period N or a marked finite orbit of order N. A motivating question is whether the resulting varieties are of general type if N is sufficiently large.

Problem 2: Distribution of Special Points.

An example of the type of problem to be considered is the distribution of post-critically finite maps in the moduli space Md1 in both the complex and the p-adic topologies.

Problem 3: Dynamical Modular Curves.

A one-parameter family of maps, for example fc(z)=z2+c, with marked points or orbits of order N, yields dynamical modular curves X0(N) and X1(N) that are analogous to classical modular curves. A good deal is known about the geometry of these curves, but little has been proven about their arithmetic except for some small values of N. The arithmetic properties of X0(N) and X1(N) are closely related to the uniform boundedness conjecture for the families that they parameterize.

Problem 4: The Boundary of Moduli Space.

The boundary of a moduli space and a natural method for completing the space are of fundamental importance in understanding the underlying objects and their degenerations. Recent work of Kiwi has used Berkovich space dynamics over Laurent series fields to analyze degenerations of complex dynamical systems. A goal of the workshop is to exploit these non-archimedean methods to answer classical questions about the boundary of dynamical moduli spaces over the complex numbers.

Associated Semester Workshops

Complex and Arithmetic Dynamics
Complex and p-adic Dynamics
Global Arithmetic Dynamics

Lecture Videos

Mating Discontinuity

Adam Epstein
University of Warwick
April 19, 2012

Antipode Preserving Cubic Rational Maps

Araceli Bonifant
University of Rhode Island
April 17, 2012

Bers, Brown and Lyapunov

Romain Dujardin
Ecole Polytechnique
April 16, 2012