Participation is by invitation only. All participants must register.
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.
8:30 - 9:30 PM Dinner at La Salle
7:30 - 9:45 AM Breakfast at La Salle 10:00 - 11:00 AM Charles Favre | Heights of Polynomial Dynamical Pairs 11:00-11:30 AM Break 11:30 - 12:30 PM Joseph Silverman | Dynamical Degrees and Arithmetic Degrees: History, Conjectures and Future Directions 12:30 - 1:30 PM Lunch at La Salle 1:30 - 4:30 PM Discussion & Recreation* 4:30- 5:00 PM Tea 5:00 - 6:00 PM Holly Krieger | The Arakelov-Zhang Pairing and Dynamical Heights 6:00 - 6:15 PM Break 6:15 - 7:15 PM Yohsuke Matsuzawa | Heights and Endomorphisms on Projective Varieties 8:00 - 9:30 PM Dinner: Kaminstüberl Fondue Restaurant
7:30 - 9:45 AM Breakfast at La Salle 10:00 - 11:00 AM Curtis McMullen | Billiards and the Arithmetic of Non-Arithmetic Groups 11:00-11:30 AM Break 11:30 - 12:30 PM Simion Filip | Dynamics on K3s: Berkovich and Tropical Versions 12:30 - 1:30 PM Lunch at La Salle 1:30 - 4:30 PM Discussion & Recreation* 4:30- 5:00 PM Tea 5:00 - 6:00 PM Junyi Xie | The Betti Foliation, the Canonical Height and the Geometric Bogomolov Conjecture 6:00 - 6:15 PM Break 6:15 - 7:15 PM Problem Session 8:00 - 9:30 PM Dinner
7:30 - 9:30 AM Breakfast 9:45 - 2:00 PM Guided Hike to Partnach Gorge 2:00 - 3:00 PM Lunch at Ganesha Restaurant in the Retreat 3:00 - 4:30 PM Recreation & Discussion 4:30 - 5:00 PM Tea 5:00 - 6:00 PM John Lesieutre | Numerical Dimension Revisited 6:00 - 6:15 PM Break 6:15 - 7:15 PM Nicole Looper | Uniform Boundedness, Equidistribution and the Arithmetic of Dynamically Small Points 7:30 - 8:15 PM Dinner 8:30 - 9:30 PM Concert: Messiaen: Quatuor pour la fin du temps
7:30 - 9:45 AM Breakfast at La Salle 10:00 - 11:00 AM Shou-Wu Zhang | Admissible Pairing of Algebraic Cycles 11:00 - 11:30 AM Break 11:30 - 12:30 PM Jason Bell | Some Dynamical Problems Motivated by Questions in Noncommutative Algebra 12:30 - 1:30 PM Lunch at La Salle 1:30 - 4:30 PM Discussion & Recreation* 4:30- 5:00 PM Tea 5:00 - 6:00 PM Ad Hoc Talk 6:00 - 6:15 PM Break 6:15 - 7:15 PM Problem Session 8:00 - 9:30 PM Dinner
7:30 - 9:45 AM Breakfast at La Salle 10:00 - 11:00 AM De-Qi Zhang | Equivariant Minimal Model Program with a View Towards Algebraic and Arithmetic Dynamics 11:00-11:30 AM Break 11:30 - 12:30 PM Keiji Oguiso | Inertia Groups, Decomposition Groups and Smooth Projective Varieties with Nonfinite Generated Automorphism Groups 12:30 - 1:30 PM Lunch at La Salle 1:30 - 4:30 PM Discussion & Recreation* 4:30- 5:00 PM Tea 5:00 - 6:00 PM Ad Hoc Talk 6:00 - 6:15 PM Break 6:15 - 7:15 PM Ad Hoc Talk 8:00 - 9:30 PM Dinner: Ganesha Restaurant in the Retreat
SESSIONS Pavilion located at the Schloss Elmau Retreat MEALS La Salle unless otherwise noted TEA & DISCUSSION Pavilion located at the Schloss Elmau Retreat EXCURSION Meet in Schloss Elmau Lobby SATURDAY DEPARTURE Meet in Schloss Elmau Lobby
*Participants may explore the hotel property and its surrounding areas as well as engage in informal discussion with other participants.
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.
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.
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.
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.
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.
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.
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.
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.
Matthew Baker Georgia Institute of Technology Jason Bell University of Waterloo Nguyen-Bac Dang Stony Brook University Laura DeMarco Northwestern University Jeff Diller University of Notre Dame Charles Favre École Polytechnique Simion Filip Institute for Advanced Study and Clay Mathematics Institute Patrick Ingram York University Mattias Jonsson University of Michigan Shu Kawaguchi Doshisha University Holly Krieger University of Cambridge John Lesieutre Pennsylvania State University Nicole Looper University of Cambridge Yohsuke Matsuzawa University of Tokyo Myrto Mavraki Northwestern University Curtis McMullen Harvard University Keiji Oguiso University of Tokyo Matthew Satriano University of Waterloo Joseph Silverman Brown University Tom Tucker University of Rochester Junyi Xie Université de Rennes Shou Wu Zhang Princeton University De-Qi Zhang National University of Singapore
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.
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.
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.
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 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.
Schloss Elmau Luxury Spa & Cultural Hideaway
In Elmau 2, 82493 Krün
Tel: +49 (0) 8823 18-0
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.
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.
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.
Reimbursement and Travel Policy
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.
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.
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.