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Global Arithmetic Dynamics (March 19-23, 2012)


Organizing Committee

Global Arithmetic

[Global Arithmetic: Image courtesy of M. Zieve]
Description

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:Pn→Pn is a morphism of degree d defined over K, then the number of K-rational preperiodic points of f is bounded by a constant depending only on D, n, and d. It is a vast generalization of the Mazur-Merel 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 Manin-Mumford and Mordell-Lang, 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 (xi) be a of sequence of algebraic points whose f-canonical heights go to zero. Under suitable hypotheses, it is known that the Galois conjugates of the xi 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: Local-Global Problems. There are many local-global principles in arithmetic geometry, such as those related to the Hasse and Brauer-Manin obstructions. The workshop will explore analogous local-global principles for dynamical systems, including especially the distribution of orbits modulo primes and/or in p-adic or complex neighborhoods.

  • Domenico Aiello
    (University of Massachusetts, Amherst)
  • Ekaterina Amerik*
    (Université de Paris XI, Paris-Sud)
  • Jacqueline Anderson
    (Brown University)
  • Matthieu Arfeux
    (Université de Toulouse III, Paul Sabatier)
  • Cecile Armana
    (University of Münster)
  • Fabrizio Barroero
    (Technische Universitat Graz)
  • Rob Benedetto*
    (Amherst College)
  • Anupam Bhatnagar
    (City University of New York)
  • Fedor Bogomolov*
    (Courant Institute of Mathematical Sciences)
  • Araceli Bonifant
    (University of Rhode Island)
  • Andrew Bridy
    (University of Wisconsin, Madison)
  • Xavier Buff
    (Université de Toulouse III, Paul Sabatier)
  • Frederic Campana*
    (Universite de Nancy I)
  • Jung Kyu Canci*
    (University of Basel)
  • Zoe Chatzidakis*
    (Université de Paris VII, Denis Diderot)
  • Laura DeMarco
    (University of Illinois at Chicago)
  • John Doyle
    (University of Georgia)
  • Dzmitry Dudko
    (University of Goettingen)
  • Noam Elkies*
    (Harvard University)
  • Adam Epstein
    (University of Warwick)
  • Xander Faber
    (University of Hawaii)
  • Paul Fili
    (University of Rochester)
  • Tanya Firsova
    (Stony Brook University)
  • Derek Garton
    (University of Wisconsin, Madison)
  • William Gignac
    (University of Michigan)
  • Mikhail Hlushchanka
    (Jacobs University Bremen)
  • Liang-Chung Hsia
    (National Central University)
  • Hsiu Lien Huang
    (Academia Sinica)
  • Benjamin Hutz*
    (City University of New York)
  • Sui-ion Ih
    (University of Colorado)
  • Patrick Malte Ingram
    (University of Waterloo)
  • Jeremy Kahn
    (Brown University)
  • Shu Kawaguchi
    (Osaka City University)
  • Sarah Koch*
    (Harvard University)
  • Holly Krieger
    (University of Illinois at Chicago)
  • Par Kurlberg
    (Kungliga Tekniska Hogskolan)
  • ChongGyu Lee
    (University of Illinois at Chicago)
  • Aaron Levin*
    (Michigan State University)
  • Alon Levy
    (ICERM)
  • Huaibin Li
    (Pontificia Universidad Catolica de Chile)
  • Jan-Li Lin
    (Indiana University)
  • Karl-Olof Lindahl
    (Linnaeus University)
  • Sijun Liu
    (University of Michigan)
  • Mahdi Majidi-Zolbanin
    (City College, CUNY)
  • Michelle Manes
    (University of Hawaii)
  • Alice Medvedev
    (University of California, Berkeley)
  • Ricardo Menares
    (Pontificia Universidad Catolica de Chile)
  • John Milnor
    (Stony Brook University)
  • Khoa Dang Nguyen
    (University of California, Berkeley)
  • Andrew Obus
    (Max-Planck-Institut fur Mathematik)
  • Alina Ostafe
    (Macquarie University)
  • Frank James Palladino
    (University of Rhode Island)
  • Donghoon Park
    (Korea Advanced Institute of Science and Technology)
  • Jorge Alejandro Pineiro
    (Bronx Community College, CUNY)
  • Bjorn Poonen*
    (MIT)
  • Lukas Pottmeyer
    (University of Tubingen)
  • Juan Rivera-Letelier
    (Pontificia Universidad Catolica de Chile)
  • John Anthony Roberts
    (University of New South Wales)
  • Robert Rumely
    (University of Georgia)
  • Thomas Scanlon*
    (University of California, Berkeley)
  • Zachary Louis Scherr
    (University of Michigan)
  • Dierk Schleicher
    (Jacobs University)
  • Thomas Sharland
    (University of Warwick)
  • Nessim Sibony
    (Universite de Paris XI)
  • Joseph Silverman
    (Brown University)
  • Martin Sombra
    (ICREA and University of Barcelona)
  • William Stein*
    (University of Washington)
  • Michael Stoll*
    (Universität Bayreuth)
  • Lucien Szpiro
    (City University of New York)
  • Bianca Thompson
    (University of Hawaii)
  • Adam David Towsley
    (University of Rochester)
  • Thomas Tucker
    (University of Rochester)
  • Bianca Viray
    (Brown University)
  • Paul Vojta
    (University of California, Berkeley)
  • Felipe Voloch*
    (University of Texas)
  • Chi-Hao Wang
    (National Central University)
  • Xiaoguang Wang
    (Chinese Academy of Sciences)
  • Siman Wong
    (University of Massachusetts, Amherst)
  • Kazuhiko Yamaki
    (Kyoto University)
  • Shou-Wu Zhang*
    (Princeton University)
  • Michael Zieve
    (University of Michigan)
DatesTitleOrganizers
March 19-23, 2012Global Arithmetic DynamicsXander Faber, Michelle Manes, Lucien Szpiro, Thomas Tucker, Michael Zieve (Chair)

Video Links
Description Speaker
Video
Abstract
Slides
A Uniform Boundedness Theorem for Polynomials Michael Zieve, University of Michigan
Video
PDF
PDF
A generalization of the Nagell-Lutz theorem Aaron Levin, Michigan State University
Video
PDF
Adelic intersection problems in arithmetic geometry and dynamics Felipe Voloch, University of Texas at Austin
Video
PDF
Algebraic dynamics, difference fields and model theory Zoe Chatzidakis, Centre National de la Recherche Scientifique (CNRS)-Paris 7
Video
PDF
Critically finite endomorphisms of projective space Sarah Koch, Harvard University
Video
PDF
Dynamical exponential and transcendence Shou-Wu Zhang, Princeton University
Video
PDF
Equidistribution and the Dynamical Uniform Boundedness Conjecture Rob Benedetto, Amherst College
Video
PDF
PDF
Exceptional linear subvarieties for the d^th power map Bianca Viray, Brown University
Video
PDF
Good reduction and computational applications Benjamin Hutz, CUNY Graduate Center
Video
PDF
Iteration of algebraic points under a rational self-map Ekaterina Amerik, Higher School of Economics (Moscow) and Universite Paris-Sud (Orsay)
Video
PDF
On the (failure of) equidistribution of periodic points Juan Rivera-Letelier, Pontificia Universidad Catolica de Chile
Video
PDF
Simple and Critically Good Reduction for Endomorphisms of P1 Jung Kyu Canci, University of Basel
Video
PDF
Special orbifolds-arithmetic and hyperbolic aspects Frederic Campana, Universite Nancy 1
Video
PDF
The Morton-Silverman conjecture, and extending self-maps to projective space Bjorn Poonen, Massachusetts Institute of Technology
Video
PDF
Using Sage for Mathematical Research William Stein, University of Washington
Video
PDF

MondayMarch 12, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
2:30 - 3:30Fields with several automorphismsAlice Medvedev, UC Berkeley11th Floor Lecture Hall
3:30 - 4:00Coffee/Tea Break11th Floor Collaborative Space
4:00 - 5:00The dynamics of birational maps over finite fields and their signaturesJohn A. G. Roberts, The University of New South Wales, Sydney, Australia11th Floor Lecture Hall

TuesdayMarch 13, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
10:30 - 11:50Postcritically Finite Rational Maps and their DeformationsJeremy Kahn, Brown University11th Floor Lecture Hall
3:30 - 4:00Coffee/Tea Break11th Floor Collaborative Space
4:00 - 5:00Professional Development: Ethics in research I11th Floor Lecture Hall

WednesdayMarch 14, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
3:14 - 4:00Coffee/Tea Break and Pi Day Treats!11th Floor Collaborative Space
4:00 - 5:00Postdoc/Grad Student Seminar11th Floor Lecture Hall

ThursdayMarch 15, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
10:30 - 11:50Postcritically Finite Rational Maps and their DeformationsJeremy Kahn, Brown University11th Floor Lecture Hall
1:30 - 2:30Introduction to Height FunctionsYu Yasufuku, Nihon University11th Floor Lecture Hall
2:30 - 3:00Coffee/Tea Break11th Floor Collaborative Space
3:00 - 4:00Model-theoretic properties for algebraic dynamics, via ACFAAlice Medvedev, UC Berkeley11th Floor Lecture Hall

FridayMarch 16, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
11:00 - 12:00Computational Working Group/Seminar11th Floor Lecture Hall
2:30 - 3:30Equidistribution of small points on $\mathbb{P}^1$Juan Rivera-Letelier, Pontificia Universidad Catolica de Chile11th Floor Lecture Hall
3:30 - 4:00Coffee/Tea Break11th Floor Collaborative Space

MondayMarch 19, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
8:30 - 8:55Registration & Welcome Coffee11th Floor Collaborative Space
8:55 - 9:00WelcomeJill Pipher, Director, ICERM11th Floor Lecture Hall
9:00 - 9:50The Morton-Silverman conjecture, and extending self-maps to projective spaceBjorn Poonen, MIT11th Floor Lecture Hall
9:55 - 10:20Coffee/Tea Break11th Floor Collaborative Space
10:20 - 11:00Exceptional linear subvarieties for the d^th power mapBianca Viray, Brown University11th Floor Lecture Hall
PDF
11:05 - 11:55A generalization of the Nagell-Lutz theoremAaron Levin, Michigan State University11th Floor Lecture Hall
PDF
12:00 - 1:45Break for Lunch and Free Time
1:45 - 2:35Equidistribution and the Dynamical Uniform Boundedness ConjectureRob Benedetto, Amherst College11th Floor Lecture Hall
PDF
2:40 - 3:30A Uniform Boundedness Theorem for PolynomialsMichael Zieve, University of Michigan11th Floor Lecture Hall
PDF
3:35 - 4:00Coffee/Tea Break11th Floor Collaborative Space
4:00 - 5:00Critically finite endomorphisms of projective spaceSarah Koch, Harvard University11th Floor Lecture Hall
PDF

TuesdayMarch 20, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
9:00 - 9:50Adelic intersection problems in arithmetic geometry and dynamicsFelipe Voloch, University of Texas at Austin11th Floor Lecture Hall
PDF
9:55 - 10:20Coffee/Tea Break11th Floor Collaborative Space
10:20 - 11:00Simple and Critically Good Reduction For Endomorphisms of P1 Jung Kyu Canci, University of Basel11th Floor Lecture Hall
11:05 - 11:55Using Sage for Mathematical ResearchWilliam Stein, University of Washington11th Floor Lecture Hall
PDF
11:55 - 12:00Group Photo11th Floor Lecture Hall
12:00 - 1:45Break for Lunch & Free Time
1:45 - 5:00Problem Session and Collaborative Work/Coffee & Tea Break11th Floor Collaborative Space
7:00 - 10:00Hackathon10th Floor Collaborative Space

WednesdayMarch 21, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
9:00 - 9:50Dynamical exponential and transcendenceShou-Wu Zhang, Princeton University11th Floor Lecture Hall
PDF
9:55 - 10:20Coffee/Tea Break11th Floor Collaborative Space
10:20 - 11:00Good reduction and computational applicationsBenjamin Hutz, CUNY Graduate Center11th Floor Lecture Hall
PDF
11:05 - 11:55Iteration of algebraic points under a rational self-mapEkaterina Amerik, Higher School of Economics (Moscow) and Universite Paris-Sud (Orsay)11th Floor Lecture Hall
PDF
12:00 - 1:45Break for Lunch & Free Time
1:45 - 5:00Progress Reports and Collaborative Work/Coffee & Tea Break11th Floor Collaborative Space
7:00 - 9:00Poster Session and Dessert Reception11th Floor Lecture Hall

ThursdayMarch 22, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
9:00 - 9:50TBATom Scanlon, University of California, Berkeley11th Floor Lecture Hall
9:55 - 10:20Coffee/Tea Break11th Floor Collaborative Space
10:20 - 11:10Algebraic Dynamics Difference Fields and Model TheoryZoe Chatzidakis, Universit� de Paris VII (Denis Diderot)11th Floor Lecture Hall
11:15 - 11:55A property of integral division points on curves and its dynamical system analogueSu-Ion Ih, University of Colorado11th Floor Lecture Hall
PDF
11:55 - 12:00Survey Distribution, please return to 11th floor reception desk
12:00 - 1:45Break for Lunch & Free Time
1:45 - 2:35On the (failure of) equidistribution of periodic pointsJuan Rivera-Letelier, Pontificia Universidad Catolica de Chile11th Floor Lecture Hall
PDF
2:35 - 5:00Progress Reports and Collaborative Work/Coffee & Tea Break11th Floor Collaborative Space
7:00 - 10:00Hackathon10th Floor Collaborative Space

FridayMarch 23, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
9:00 - 10:20Morning of Collaborations and Coffee/Tea Break11th Floor Collaborative Space
10:20 - 11:10Special orbifolds: arithmetic and hyperbolic aspectsFrederic Campana, Universit� de Nancy I (Henri Poincar�)11th Floor Lecture Hall
11:15 - 12:05Which bilinear recurrences give theta sequencesNoam Elkies, Harvard University11th Floor Lecture Hall
PDF
12:10 - 2:00Break for Lunch & Free Time
2:00 - 3:00Wrapup Problem Session and Progress Reports 11th Floor Lecture Hall