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.

Image for "Moduli Spaces Associated to Dynamical Systems"
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, April 16, 2012
TimeEventLocationMaterials
8:30 - 9:20am EDTRegistration & Welcome Coffee  
9:20 - 9:30am EDTWelcome - Jeff Brock, Deputy Director, ICERM11th Floor Lecture Hall 
9:30 - 10:20am EDTRamification avoidance and a dynamical Mordell-Lang conjecture - Tom Tucker, University of Rochester11th Floor Lecture Hall
10:30 - 11:00am EDTCoffee/Tea Break11th Floor Collaborative Space 
11:00 - 11:50am EDTBers, Brown and Lyapunov - Romain Dujardin, Ecole Polytechnique11th Floor Lecture Hall
12:00 - 2:30pm EDTBreak for Lunch  
2:30 - 3:20pm EDTMeasure-theoretic rigidity for Mumford curves - Janne Kool, Universiteit Utrecht11th Floor Lecture Hall
3:30 - 4:00pm EDTCoffee/Tea Break11th Floor Collaborative Space 
4:00 - 4:50pm EDTAdmissible Covers and Rescaling Limits (after Matthieu Arfeux) - Xavier Buff, Universite de Toulouse III (Paul Sabatier)11th Floor Lecture Hall
5:00 - 6:30pm EDTWelcome Reception11th Floor Collaborative Space  
Tuesday, April 17, 2012
TimeEventLocationMaterials
9:30 - 10:20am EDTVariations on a Theme-Fields of Definition, Fields of Moduli, Automorphisms, and Twists - Michelle Manes, University of Hawaii11th Floor Lecture Hall
10:30 - 11:00am EDTCoffee/Tea Break11th Floor Collaborative Space 
11:00 - 11:50am EDTArithmetic Properties of Dominant Self-Maps of P^N - Joseph H. Silverman, Brown University11th Floor Lecture Hall
12:00 - 2:30pm EDTBreak for Lunch  
2:30 - 3:20pm EDTPuiseux series dynamics and leading monomials of escape regions - Jan Kiwi, Pontificia Universidad Catolica de Chile11th Floor Lecture Hall
3:30 - 4:00pm EDTCoffee/Tea Break11th Floor Collaborative Space 
4:00 - 4:50pm EDTAntipode Preserving Cubic Rational Maps - Araceli Bonifant, University of Rhode Island11th Floor Lecture Hall
Wednesday, April 18, 2012
TimeEventLocationMaterials
10:00 - 10:50am EDTOn Mordell-Lang in algebraic groups of unipotent rank 1 - Paul Vojta, University of California, Berkeley11th Floor Lecture Hall
11:00 - 11:30am EDTCoffee/Tea Break11th Floor Collaborative Space 
11:30 - 12:20pm EDTLimits of polynomial-like quadratic rational maps - Eva Uhre, Stony Brook University11th Floor Lecture Hall
12:30 - 12:40pm EDTGroup Photo11th Floor Lecture Hall 
12:40 - 7:00pm EDTFree afternoon for collaborations  
7:00 - 9:00pm EDTPoster Session and Dessert Reception11th Floor Lecture Hall and Collaborative Space 
Thursday, April 19, 2012
TimeEventLocationMaterials
9:30 - 10:20am EDTA proof of Pilgrim's conjecture - Nikita Selinger, Jacobs University11th Floor Lecture Hall
10:30 - 11:00am EDTCoffee/Tea Break11th Floor Collaborative Space 
11:00 - 11:50am EDTThe Space of Morphisms on a Projective Variety- Construction and Applications - Alon Levy, ICERM11th Floor Lecture Hall
12:00 - 2:30pm EDTBreak for Lunch  
2:30 - 3:20pm EDTMating Discontinuity - Adam Epstein, University of Warwick11th Floor Lecture Hall
3:30 - 3:35pm EDTSurvey Distribution11th Floor Lecture Hall 
3:35 - 4:00pm EDTCoffee/Tea Break11th Floor Collaborative Space 
4:00 - 4:50pm EDTDiscussions and Collaborations11th Floor Collaborative Space 
Friday, April 20, 2012
TimeEventLocationMaterials
9:30 - 10:20am EDTQuadratic Rational Maps- the Totally Marked Moduli Space - John Milnor, Stony Brook University11th Floor Lecture Hall
10:30 - 11:00am EDTCoffee/Tea Break11th Floor Collaborative Space 
11:00 - 11:50am EDTRational maps with prescribed critical values - Tan Lei, University d'Angers11th Floor Lecture Hall
12:00 - 2:30pm EDTBreak for Lunch  
2:30 - 3:20pm EDTResultant and conductor for self maps of the projective line - Lucien Szpiro, City University of New York (CUNY)11th Floor Lecture Hall
3:30 - 4:00pm EDTCoffee/Tea Break11th Floor Collaborative Space 
4:00 - 4:50pm EDTParabolic blowups and limits of dynamical systems - John Hubbard, Cornell University11th Floor Lecture Hall 

Tutorial Week Schedule

Thursday, April 12, 2012
TimeEventLocationMaterials
10:30 - 11:50am EDTPostcritically Finite Rational Maps and their Deformations - Jeremy Kahn, Brown University11th Floor Lecture Hall 
1:30 - 2:30pm EDTApplications 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 - 2:45pm EDTCoffee/Tea Break11th Floor Collaborative Space 
2:45 - 3:45pm EDTApplications 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 - 4:30pm EDTSir John Ball "Smooth topology-preserving approximations of rough domains" Reception11th Floor Collaborative Space 
4:30 - 5:30pm EDTSpecial Colloquium-Smooth topology-preserving approximations of rough domains - Sir John Ball, University of Oxford11th Floor Lecture Hall 
Friday, April 13, 2012
TimeEventLocationMaterials
10:00 - 11:00am EDTModuli 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 - 12:30pm EDTModuli 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 - 3:30pm EDTComputational Working Group11th Floor Lecture Hall 
3:30 - 4:00pm EDTCoffee/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
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
Global Arithmetic Dynamics
Image for "Global Arithmetic Dynamics"
Mar 19 - 23, 2012
Semester Workshop

Lecture Videos

Rational maps with prescribed critical values

Tan Lei
Universite d'Angers
April 20, 2012
Moduli Spaces Associated to Dynamical Systems
Play     Abstract     Slides

Quadratic Rational Maps- the Totally Marked Moduli Space

John Milnor
SUNY
April 20, 2012
Moduli Spaces Associated to Dynamical Systems
Play     Abstract     Slides

Mating Discontinuity

Adam Epstein
University of Warwick
April 19, 2012
Moduli Spaces Associated to Dynamical Systems
Play     Abstract

The Space of Morphisms on a Projective Variety- Construction and Applications

Alon Levy
Institute for Computational and Experimental Research in Mathematics (ICERM)
April 19, 2012
Moduli Spaces Associated to Dynamical Systems
Play     Abstract

A proof of Pilgrim's conjecture

Nikita Selinger
Jacobs University
April 19, 2012
Moduli Spaces Associated to Dynamical Systems
Play     Abstract

Limits of polynomial-like quadratic rational maps

Eva Uhre
Stony Brook University
April 18, 2012
Moduli Spaces Associated to Dynamical Systems
Play     Abstract

On Mordell-Lang in algebraic groups of unipotent rank 1

Paul Vojta
University of California, Berkeley
April 18, 2012
Moduli Spaces Associated to Dynamical Systems
Play     Abstract     Slides

Antipode Preserving Cubic Rational Maps

Araceli Bonifant
University of Rhode Island
April 17, 2012
Moduli Spaces Associated to Dynamical Systems
Play     Abstract

Puiseux series dynamics and leading monomials of escape regions

Jan Kiwi
Pontificia Universidad Catolica de Chile
April 17, 2012
Moduli Spaces Associated to Dynamical Systems
Play     Abstract     Slides

Arithmetic Properties of Dominant Self-Maps of P^N

Joseph Silverman
Brown University
April 17, 2012
Moduli Spaces Associated to Dynamical Systems
Play     Abstract

Variations on a Theme-Fields of Definition, Fields of Moduli, Automorphisms, and Twists

Michelle Manes
Mathematics Department
April 17, 2012
Moduli Spaces Associated to Dynamical Systems
Play     Abstract     Slides

Admissible Covers and Rescaling Limits (after Matthieu Arfeux)

Xavier Buff
Universite de Toulouse III (Paul Sabatier)
April 16, 2012
Moduli Spaces Associated to Dynamical Systems
Play     Abstract     Slides

Bers, Brown and Lyapunov

Romain Dujardin
Ecole Polytechnique
April 16, 2012
Moduli Spaces Associated to Dynamical Systems
Play     Abstract     Slides

Ramification avoidance and a dynamical Mordell-Lang conjecture

Thomas Tucker
University of Rochester
April 16, 2012
Moduli Spaces Associated to Dynamical Systems
Play     Abstract