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Organizing Committee
University of Illinois
University of Warwick
University of Michigan, Ann Arbor
Harvard University
Brown University
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
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.
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Workshop Schedule
Monday, April 16, 2012
8:30 - 9:20 AM EDT
Registration & Welcome Coffee
9:20 - 9:30 AM EDT
Welcome - Jeff Brock, Deputy Director, ICERM
Location: 11th Floor Lecture Hall
10:30 - 11:00 AM EDT
Coffee/Tea Break
Location: 11th Floor Collaborative Space
12:00 - 2:30 PM EDT
Break for Lunch
3:30 - 4:00 PM EDT
Coffee/Tea Break
Location: 11th Floor Collaborative Space
5:00 - 6:30 PM EDT
Welcome Reception
Location: 11th Floor Collaborative Space
Tuesday, April 17, 2012
Tutorial Week Schedule
Thursday, April 12, 2012
10:30 - 11:50 AM EDT
Postcritically Finite Rational Maps and their Deformations - Jeremy Kahn, Brown University
Location: 11th Floor Lecture Hall
1:30 - 2:30 PM EDT
Applications and Examples of using a new program `Dynamics Explorer' to study the dynamics of complex mappings - Suzanne Boyd, University of Wisconsin and Brian Boyd
Location: 11th Floor Lecture Hall
2:30 - 2:45 PM EDT
Coffee/Tea Break
Location: 11th Floor Collaborative Space
2:45 - 3:45 PM EDT
Applications and Examples of using a new program `Dynamics Explorer' to study the dynamics of complex mappings - Suzanne Boyd, University of Wisconsin and Brian Boyd
Location: 11th Floor Lecture Hall
4:00 - 4:30 PM EDT
Sir John Ball "Smooth topology-preserving approximations of rough domains" Reception
Location: 11th Floor Collaborative Space
4:30 - 5:30 PM EDT
Special Colloquium-Smooth topology-preserving approximations of rough domains - Sir John Ball, University of Oxford
Location: 11th Floor Lecture Hall
Lecture Videos
Nikita Selinger
Jacobs University
April 19, 2012 • 12:00 AM
Xavier Buff
Universite de Toulouse III (Paul Sabatier)
April 16, 2012 • 12:00 AM
Araceli Bonifant
University of Rhode Island
April 17, 2012 • 12:00 AM
Joseph Silverman
Brown University
April 17, 2012 • 12:00 AM
Romain Dujardin
Ecole Polytechnique
April 16, 2012 • 12:00 AM
Eva Uhre
Stony Brook University
April 18, 2012 • 12:00 AM
Adam Epstein
University of Warwick
April 19, 2012 • 12:00 AM
