Moduli Spaces Associated to Dynamical Systems
(April 16-20, 2012)


Organizing Committee
Description

Complex.pAdic

 

[Global Arithmetic: Image courtesy of M. Zieve]

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.

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.

  • Jacqueline Anderson
    (Brown University)
  • Matthieu Arfeux
    (Université de Toulouse III (Paul Sabatier))
  • Cecile Armana
    (Westfälische Wilhelms-Universität Münster)
  • Matthieu Astorg
    (Université de Toulouse III (Paul Sabatier))
  • Fabrizio Barroero
    (Technische Universität Graz)
  • Rob Benedetto
    (Amherst College)
  • Anupam Bhatnagar
    (City University of New York (CUNY))
  • Paul Blanchard
    (Boston University)
  • Araceli Bonifant *
    (University of Rhode Island)
  • Joshua Bowman
    (Stony Brook University)
  • Andrew Bridy
    (University of Wisconsin)
  • Xavier Buff *
    (Université de Toulouse III (Paul Sabatier))
  • Arnaud Cheritat
    (Université de Toulouse III (Paul Sabatier))
  • Mark Comerford
    (University of Rhode Island)
  • Daniel Cuzzocreo
    (Boston University)
  • Diana Davis
    (Brown University)
  • Laura DeMarco
    (University of Illinois)
  • Romain Dujardin *
    (École Polytechnique)
  • Adam Epstein
    (University of Warwick)
  • Timo Erkama
    (University of Eastern Finland)
  • Derek Garton
    (University of Wisconsin)
  • Jonah Gaster
    (University of Illinois)
  • William Gignac
    (University of Michigan)
  • Mikhail Hlushchanka
    (Jacobs University)
  • Wei Ho
    (Columbia University)
  • Liang-Chung Hsia
    (National Taiwan Normal University)
  • John Hubbard *
    (Cornell University)
  • Benjamin Hutz
    (City University of New York (CUNY))
  • Patrick Ingram
    (Colorado State University)
  • Rafe Jones
    (College of the Holy Cross)
  • Jeremy Kahn
    (Brown University)
  • Linda Keen
    (Herbert H. Lehman College, CUNY)
  • Jan Kiwi *
    (Pontificia Universidad Catolica de Chile)
  • Sarah Koch
    (Harvard University)
  • Janne Kool*
    (Universiteit Utrecht)
  • Robert Kozma
    (Stony Brook University)
  • ChongGyu Lee
    (University of Illinois)
  • Tan Lei *
    (Université d'Angers)
  • Alon Levy *
    (Institute for Computational and Experimental Research in Mathematics (ICERM))
  • Huaibin Li
    (Pontificia Universidad Catolica de Chile)
  • Jan-Li Lin
    (Indiana University)
  • Kathryn Lindsey
    (Cornell University)
  • Jinsong Liu
    (Chinese Academy of Sciences)
  • Khudoyor Mamayusupov
    (Jacobs University)
  • Michelle Manes *
    (University of Hawaii)
  • John Milnor *
    (SUNY)
  • Khoa Nguyen
    (University of California, Berkeley)
  • Frank Palladino
    (University of Rhode Island)
  • Kevin Pilgrim
    (Indiana University)
  • Jorge Pineiro
    (Bronx Community College, CUNY)
  • Bjorn Poonen
    (Massachusetts Institute of Technology)
  • Remus Radu
    (Cornell University)
  • Paul Reschke
    (University of Illinois)
  • Juan Rivera-Letelier
    (Pontificia Universidad Catolica de Chile)
  • Bastien Rossetti
    (Université de Toulouse III (Paul Sabatier))
  • Robert Rumely
    (University of Georgia)
  • Zachary Scherr
    (University of Michigan)
  • Dierk Schleicher
    (Jacobs University)
  • Nikita Selinger*
    (Jacobs University)
  • Thomas Sharland
    (University of Warwick)
  • Joseph Silverman *
    (Brown University)
  • Brian Stout
    (City University of New York (CUNY))
  • Scott Sutherland
    (Stony Brook University)
  • Lucien Szpiro *
    (City University of New York (CUNY))
  • Raluca Tanase
    (Cornell University)
  • Michael Tepper
    (Pennsylvania State University, Ogontz Campus)
  • Bianca Thompson
    (University of Hawaii at Manoa)
  • Thomas Tucker *
    (University of Rochester)
  • Eva Uhre*
    (Stony Brook University)
  • Bianca Viray
    (Brown University)
  • Paul Vojta *
    (University of California, Berkeley)
  • Chi-Hao Wang
    (National Central University)
  • Xiaoguang Wang
    (Chinese Academy of Sciences)
  • Hexi Ye
    (University of Illinois)
  • Ilies Zidane
    (Université de Toulouse III (Paul Sabatier))
  • Michael Zieve
    (University of Michigan)

ThursdayApril 12, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
10:30 - 11:50Postcritically Finite Rational Maps and their DeformationsJeremy 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 mappingsSuzanne 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 mappingsSuzanne 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 domainsSir John Ball, University of Oxford11th Floor Lecture Hall

FridayApril 13, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
10:00 - 11:00Moduli spaces: drawing pictures and doing computations ISarah 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 IISarah 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

MondayApril 16, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
8:30 - 9:20Registration & Welcome Coffee
9:20 - 9:30WelcomeJeff Brock, Deputy Director, ICERM11th Floor Lecture Hall
9:30 - 10:20Ramification avoidance and a dynamical Mordell-Lang conjectureTom Tucker, University of Rochester11th Floor Lecture Hall
PDF
10:30 - 11:00Coffee/Tea Break11th Floor Collaborative Space
11:00 - 11:50Bers, Brown and LyapunovRomain Dujardin, Ecole Polytechnique11th Floor Lecture Hall
PDF
PDF
12:00 - 2:30Break for Lunch
2:30 - 3:20Measure-theoretic rigidity for Mumford curvesJanne Kool, Universiteit Utrecht11th Floor Lecture Hall
PDF
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
PDF
PDF
5:00 - 6:30Welcome Reception11th Floor Collaborative Space

TuesdayApril 17, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
9:30 - 10:20Variations on a Theme-Fields of Definition, Fields of Moduli, Automorphisms, and TwistsMichelle Manes, University of Hawaii11th Floor Lecture Hall
PDF
PDF
10:30 - 11:00Coffee/Tea Break11th Floor Collaborative Space
11:00 - 11:50Arithmetic Properties of Dominant Self-Maps of P^NJoseph H. Silverman, Brown University11th Floor Lecture Hall
PDF
12:00 - 2:30Break for Lunch
2:30 - 3:20Puiseux series dynamics and leading monomials of escape regionsJan Kiwi, Pontificia Universidad Catolica de Chile11th Floor Lecture Hall
PDF
PDF
3:30 - 4:00Coffee/Tea Break11th Floor Collaborative Space
4:00 - 4:50Antipode Preserving Cubic Rational MapsAraceli Bonifant, University of Rhode Island11th Floor Lecture Hall
PDF

WednesdayApril 18, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
10:00 - 10:50On Mordell-Lang in algebraic groups of unipotent rank 1Paul Vojta, University of California, Berkeley11th Floor Lecture Hall
PDF
PDF
11:00 - 11:30Coffee/Tea Break11th Floor Collaborative Space
11:30 - 12:20Limits of polynomial-like quadratic rational mapsEva Uhre, Stony Brook University11th Floor Lecture Hall
PDF
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

ThursdayApril 19, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
9:30 - 10:20A proof of Pilgrim's conjectureNikita Selinger, Jacobs University11th Floor Lecture Hall
PDF
10:30 - 11:00Coffee/Tea Break11th Floor Collaborative Space
11:00 - 11:50The Space of Morphisms on a Projective Variety- Construction and ApplicationsAlon Levy, ICERM11th Floor Lecture Hall
PDF
12:00 - 2:30Break for Lunch
2:30 - 3:20Mating DiscontinuityAdam Epstein, University of Warwick11th Floor Lecture Hall
PDF
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

FridayApril 20, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
9:30 - 10:20Quadratic Rational Maps- the Totally Marked Moduli SpaceJohn Milnor, Stony Brook University11th Floor Lecture Hall
PDF
PDF
10:30 - 11:00Coffee/Tea Break11th Floor Collaborative Space
11:00 - 11:50Rational maps with prescribed critical valuesTan Lei, University d'Angers11th Floor Lecture Hall
PDF
PDF
12:00 - 2:30Break for Lunch
2:30 - 3:20Resultant and conductor for self maps of the projective lineLucien Szpiro, City University of New York (CUNY)11th Floor Lecture Hall
PDF
3:30 - 4:00Coffee/Tea Break11th Floor Collaborative Space
4:00 - 4:50Parabolic blowups and limits of dynamical systemsJohn Hubbard, Cornell University11th Floor Lecture Hall