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)

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
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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
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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
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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
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9:55 - 10:20Coffee/Tea Break11th Floor Collaborative Space
10:20 - 11:00Good reduction and computational applicationsBenjamin Hutz, CUNY Graduate Center11th Floor Lecture Hall
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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
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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, Université 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
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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
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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, Université de Nancy I (Henri Poincaré)11th Floor Lecture Hall
11:15 - 12:05Which bilinear recurrences give theta sequencesNoam Elkies, Harvard University11th Floor Lecture Hall
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12:10 - 2:00Break for Lunch & Free Time
2:00 - 3:00Wrapup Problem Session and Progress Reports 11th Floor Lecture Hall