Algebraic, Complex and Arithmetic Dynamics

Date & Time


Location

Schloss Elmau
In Elmau 2, 82493 Krün Germany

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Registration Closed
Invitation Only

Participation is by invitation only. All participants must register.

Organizers:
Laura DeMarco, Northwestern University
Mattias Jonsson, University of Michigan

Organized by Laura DeMarco (Northwestern University) and Mattias Jonsson (University of Michigan), the goal of this symposium is to formulate a vision for future developments in complex, algebraic, and arithmetic dynamics. One concrete objective is to produce high quality expository material presenting the methods, recent results, and goals of these active research areas. Another is to identify key problems and directions of research connected with the study of height functions and complexity growth in algebraic dynamical systems.

Discussion topics include:

  • Degree growth of rational maps.
  • The growth of heights along orbits.
  • Height functions on parameter spaces.

 

  • Agendaplus--large

    SUNDAY

    8:30 - 9:30 PMDinner at La Salle

    MONDAY

    7:30 - 9:45 AMBreakfast at La Salle
    10:00 - 11:00 AMCharles Favre | Heights of Polynomial Dynamical Pairs
    11:00-11:30 AMBreak
    11:30 - 12:30 PMJoseph Silverman | Dynamical Degrees and Arithmetic Degrees: History, Conjectures and Future Directions
    12:30 - 1:30 PMLunch at La Salle
    1:30 - 4:30 PMDiscussion & Recreation*
    4:30- 5:00 PMTea
    5:00 - 6:00 PMHolly Krieger | The Arakelov-Zhang Pairing and Dynamical Heights
    6:00 - 6:15 PMBreak
    6:15 - 7:15 PMYohsuke Matsuzawa | Heights and Endomorphisms on Projective Varieties
    8:00 - 9:30 PMDinner: Kaminstüberl Fondue Restaurant

    TUESDAY

    7:30 - 9:45 AMBreakfast at La Salle
    10:00 - 11:00 AMCurtis McMullen | Billiards and the Arithmetic of Non-Arithmetic Groups
    11:00-11:30 AMBreak
    11:30 - 12:30 PMSimion Filip | Dynamics on K3s: Berkovich and Tropical Versions
    12:30 - 1:30 PMLunch at La Salle
    1:30 - 4:30 PMDiscussion & Recreation*
    4:30- 5:00 PMTea
    5:00 - 6:00 PMJunyi Xie | The Betti Foliation, the Canonical Height and the Geometric Bogomolov Conjecture
    6:00 - 6:15 PMBreak
    6:15 - 7:15 PMProblem Session
    8:00 - 9:30 PMDinner

    WEDNESDAY

    7:30 - 9:30 AMBreakfast
    9:45 - 2:00 PMGuided Hike to Partnach Gorge
    2:00 - 3:00 PMLunch at Ganesha Restaurant in the Retreat
    3:00 - 4:30 PMRecreation & Discussion
    4:30 - 5:00 PMTea
    5:00 - 6:00 PMJohn Lesieutre | Numerical Dimension Revisited
    6:00 - 6:15 PMBreak
    6:15 - 7:15 PMNicole Looper | Uniform Boundedness, Equidistribution and the Arithmetic of Dynamically Small Points
    7:30 - 8:15 PMDinner
    8:30 - 9:30 PMConcert: Messiaen: Quatuor pour la fin du temps

    THURSDAY

    7:30 - 9:45 AMBreakfast at La Salle
    10:00 - 11:00 AMShou-Wu Zhang | Admissible Pairing of Algebraic Cycles
    11:00 - 11:30 AMBreak
    11:30 - 12:30 PMJason Bell | Some Dynamical Problems Motivated by Questions in Noncommutative Algebra
    12:30 - 1:30 PMLunch at La Salle
    1:30 - 4:30 PMDiscussion & Recreation*
    4:30- 5:00 PMTea
    5:00 - 6:00 PMAd Hoc Talk
    6:00 - 6:15 PMBreak
    6:15 - 7:15 PMProblem Session
    8:00 - 9:30 PMDinner

    FRIDAY

    7:30 - 9:45 AMBreakfast at La Salle
    10:00 - 11:00 AMDe-Qi Zhang | Equivariant Minimal Model Program with a View Towards Algebraic and Arithmetic Dynamics
    11:00-11:30 AMBreak
    11:30 - 12:30 PMKeiji Oguiso | Inertia Groups, Decomposition Groups and Smooth Projective Varieties with Nonfinite Generated Automorphism Groups
    12:30 - 1:30 PMLunch at La Salle
    1:30 - 4:30 PMDiscussion & Recreation*
    4:30- 5:00 PMTea
    5:00 - 6:00 PMAd Hoc Talk
    6:00 - 6:15 PMBreak
    6:15 - 7:15 PMAd Hoc Talk
    8:00 - 9:30 PMDinner: Ganesha Restaurant in the Retreat

    LOCATIONS

    SESSIONSPavilion located at the Schloss Elmau Retreat
    MEALSLa Salle unless otherwise noted
    TEA & DISCUSSIONPavilion located at the Schloss Elmau Retreat
    EXCURSION Meet in Schloss Elmau Lobby
    SATURDAY DEPARTUREMeet in Schloss Elmau Lobby

    *Participants may explore the hotel property and its surrounding areas as well as engage in informal discussion with other participants.

  • Abstractsplus--large

    Jason Bell
    University of Waterloo

    Some Dynamical Problems Motivated by Questions in Noncommutative Algebra

    Bell will give an overview of some of the dynamical questions that arise when studying the representation theory of algebras in noncommutative projective geometry, highlighting some of the results already obtained in this direction.

     

    Holly Krieger
    University of Cambridge

    The Arakelov-Zhang Pairing and Dynamical Heights

    For any two rational maps of the Riemann sphere with algebraic coefficients, the Arakelov-Zhang pairing of their canonical heights provides an arithmetic measure of the dynamical distance between the two maps. Krieger will discuss how this pairing can be used, together with quantitative equidistribution, to provide bounds on points of small height for both maps, as done in recent joint work with DeMarco and Ye. Krieger will highlight some of the many open questions about the behavior of this pairing in moduli.

     

    John Lesieutre
    Pennsylvania State University

    Numerical Dimension Revisited

    The Iitaka dimension of a line bundle \(D\) on a projective variety \(X\) is the dimension of the image of the rational map given by \(|mD|\) for large and divisible \(m\). The Iitaka dimension is not a numerical invariant of \(D\), and there are several approaches to constructing a “numerical dimension,” which should be an analogous invariant depending only on the numerical class of \(D\). Lesieutre will discuss some divisors of dynamical origin whose behavior with respect to these invariants is pathological and which provide counterexamples to some conjectures from birational geometry. The examples hinge on a sort of dynamical positivity property, which also arises in arithmetic contexts. Lesieutre will then pose some related problems about degree growth on varieties with large groups of pseudoautomorphisms.

     

    Nicole Looper
    University of Cambridge

    Uniform Boundedness, Equidistribution and the Arithmetic of Dynamically Small Points

    In this talk, Looper will discuss a uniform boundedness theorem for unicritical, along with the relevant tools from, Diophantine geometry. Looper will also discuss connections to other results concerning points of small canonical height relative to polynomials.

     

    Keiji Oguiso
    University of Tokyo

    Inertia Groups, Decomposition Groups and Smooth Projective Varieties with Nonfinite Generated Automorphism Groups

    The so-called ‘Coble problem’ concerning complexities of the decomposition group and inertia group of the special smooth rational curve on a classic complex Coble surface is a long-standing problem which is still open. In this talk, Oguiso will attempt to negatively answer another long-standing problem — the finite generation problem of the automorphism group of a smooth projective variety of any dimension \(\ge 2\) over an algebraically closed field, under the assumption that the base field is not an algebraic closure of a finite field or not of characteristic \(2\). In our construction, the decomposition group of a smooth rational curve of some special K3 surface, which is closely related to classical Coble surfaces, and the arithmetic of the base field play essential and delicate roles, as Oguiso will explain. It also turns out that the inertia group in our construction has rich complex dynamics. This talk is partly based on Oguiso’s joint work with Coung-Tien Dinh and is also much inspired by the works of Lesieutre and Dolgachev.

     

    Junyi Xie
    Université de Rennes

    The Betti Foliation, the Canonical Height and the Geometric Bogomolov Conjecture

    With Cantat, Habegger and Gao, Xie will prove the geometric Bogomolov conjecture over a function field of characteristic zero. This generalizes recent work of Habegger and Gao, who proved the geometric Bogomolov conjecture over a function field of a curve of characteristic zero.

     

    De-Qi Zhang
    National University of Singapore

    Equivariant Minimal Model Program with a View Toward Algebraic and Arithmetic Dynamics

    Zhang will elaborate the notion of ‘int-amplified’ endomorphism f of a normal projective variety X, a property weaker than ‘polarized’ yet preserved by products. Zhang will show that the existence of such a single f guarantees that every Minimal Model Program (MMP) is equivariant w.r.t. a finite-index submonoid of the whole monoid SEnd(X) of all surjective endomorphisms of X. Applications of the equivariant MMP are discussed: Kawaguchi-Silverman conjecture on the equivalence of arithmetic and dynamic degrees, and characterization of a subvariety with Zariski dense periodic points. Some parts are based on joint work with Cascini and Meng.

     

    Shou-Wu Zhang
    Princeton University

    Admissible Pairing of Algebraic Cycles

    For a smooth and projective variety X over a global field of dimension n with an adelic polarization, Zhang proposes canonical local and global height pairings for two cycles Y , Z of pure codimension p, q satisfying p + q = n + 1. Zhang will discuss some applications to algebraic dynamical systems and Shimura varieties.

  • Participantsplus--large
    Matthew BakerGeorgia Institute of Technology
    Jason Bell University of Waterloo
    Nguyen-Bac DangStony Brook University
    Laura DeMarcoNorthwestern University
    Jeff DillerUniversity of Notre Dame
    Charles FavreÉcole Polytechnique
    Simion FilipInstitute for Advanced Study and Clay Mathematics Institute
    Patrick IngramYork University
    Mattias JonssonUniversity of Michigan
    Shu KawaguchiDoshisha University
    Holly KriegerUniversity of Cambridge
    John LesieutrePennsylvania State University
    Nicole LooperUniversity of Cambridge
    Yohsuke MatsuzawaUniversity of Tokyo
    Myrto MavrakiNorthwestern University
    Curtis McMullenHarvard University
    Keiji OguisoUniversity of Tokyo
    Matthew SatrianoUniversity of Waterloo
    Joseph SilvermanBrown University
    Tom TuckerUniversity of Rochester
    Junyi XieUniversité de Rennes
    Shou Wu ZhangPrinceton University
    De-Qi ZhangNational University of Singapore
  • AVplus--large

    Participants will have access to a projector and screen for computer-based talks and blackboards for those who prefer to give board-based talks.

    High-speed Internet access is available.

  • Wednesday Excursionplus--large

    On Wednesday symposium activities are shortened for a guided hiking tour from Schloss-Elmau through the Partnach Gorge. The hike difficulty is moderate and will take approximately 3 hours.

    If you will participate in the hike, hiking boots or shoes appropriate for traversing rocky/wet terrain, warm clothing (e.g. a sweater), sun protection (e.g. light cap) and waterproofs (e.g. raincoat or umbrella) are strongly recommended.

    A small satchel containing a light snack and water is provided, but feel free to bring your own backpack in which to carry a camera and other items that may be useful to you.

  • Travelplus--large

    Air

    The foundation will coordinate and purchase air and train travel to the symposium. Travel specifications can be provided by clicking the registration link above.

    Participants arriving via plane will arrive via Munich International Airport (MUC) Airport.

    Train

    The closet train station to Schloss Elmau is in Klais. Those traveling by train should book their tickets directly and request reimbursement at the conclusion of the meeting. Should you require booking assistance, please contact Meghan Fazzi.

    All participants are required to arrive on Sunday prior to the meeting’s start and attend the entire week. No accommodations will be made for partial participation.

    Passports

    Passports are required.

    Local Ground Transportation

    Arrival at Munich International Airport

    The foundation will arrange for your transfer from MUC airport to Schloss Elmau via a private car. After claiming your luggage at baggage claim you will proceed through customs and to the exit where a uniformed driver holding a Simons Symposia sign will greet you.

    Travel time from Munich to Schloss Elmau is 90 minutes.

    Arrival at Klais Train Station

    The foundation will arrange for your transfer from Klais train station to Schloss Elmau via a hotel shuttle.

    Travel time from Klais to Elmau is approximately 10 minutes.

    Departure from Schloss Elmau

    The day prior to your departure you will receive a departure letter advising on your departure time to the airport or train station via private car. Please meet your driver in the hotel lobby ten minutes prior to your noted departure time. If for any reason your flight departure time changes, please visit the front desk to update hotel staff.

  • Hotelplus--large

    Schloss Elmau Luxury Spa & Cultural Hideaway

    In Elmau 2, 82493 Krün
    Germany
    Tel: +49 (0) 8823 18-0
    www.schloss-elmau.de

    Check-in time is at 4:30 PM
    Check-out time is 11:30 AM

    Additional Night at Elmau

    The meeting concludes on Friday evening. Participants have the option to depart on Saturday following the conclusion of the meeting or depart on Sunday at no additional cost. Please note your desired departure date when registering.

    Dress Code

    Business casual clothing should be worn during the symposium.

    The weather can change very quickly; please bring cold-weather clothing appropriate for spring in the mountains.

  • Mealsplus--large

    Sunday and Saturday: Meals on arrival and departure days (including while traveling) are reimbursable by the foundation. Your meals on these days are not to exceed $115 USD.

    Monday through Friday: All meals occurring during the meeting week will be covered directly by the Simons Foundation. Participants are expected to be present for all foundation-hosted meals.

    Breakfast is served starting at 7:30 AM at La Salle restaurant. Food at La Salle is served buffet-style and is open to all hotel guests; tables are reserved for all symposium participants so that the group may dine together.

    Lunch is served daily at La Salle.

    Dinner is hosted around the hotel property at one of the restaurants. Locations are noted on the agenda.

  • Contactsplus--large

    Registration and Travel Assistance

    Christophe Vergnol, Protravel
    chrisvgroup@protravelinc.com
    (973) 334-9300 ext. 10

    Meeting Questions and Assistance

    Meghan Fazzi
    Senior Executive Assistant, Simons Foundation
    mfazzi@simonsfoundation.org
    (212) 524-6080

  • Reimbursement and Travel Policyplus--large

    Reimbursement

    Any expenses not directly paid for by the Simons Foundation are subject to reimbursement based on the travel policy. Please review the policy and complete when submitting your expenses. Receipts are required for any expenses over $50 USD and are due within THIRTY DAYS (30) after the conclusion of the symposium. All expenses are reconciled through the foundation’s online reimbursement platform. Additional information regarding reimbursement is sent via email on the final day of the symposium.

    Should you have any questions, please contact Meghan Fazzi.

    Guest Policy

    It is the foundation’s policy to discourage spouses, families or others from joining participants during the meeting. Participants who choose to invite guests to come before or stay on after the meeting do so at their own expense. The Simons Foundation will not cover expenses associated with such extensions.

    Additional Nights

    Should you decide to extend your stay, Schloss Elmau will honor the conference rate of €337.50/night for up to two nights. Any nights beyond those associated with the conference are not reimbursable by the foundation.

    Please note on your registration form if you wish to extend your stay.

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