Dynamics on Projective Spaces
TIME AND VENUE
- Time
- Tuesdays, 14:00–17:00 (Beijing time, UTC+8), starting
- Venue
- Room 123, South Math Building
- Organizer
- Tianle Yang · 10211510098@stu.ecnu.edu.cn
INTRODUCTION
In Fall 2026, I will organize a seminar on dynamics on projective spaces. We will explore recent developments connecting complex dynamics, arithmetic dynamics, and algebraic geometry.
In the AI era, attention is the most valuable resource. I appreciate the time and attention you bring to this seminar, and I hope to make the talks engaging and the discussions worthwhile.
Lecture notes will be posted on this page once the draft is ready. You are welcome to read ahead, explore questions with AI tools, and bring them to the seminar.
AUDIENCE AND BACKGROUND
The seminar is intended for graduate students and researchers interested in dynamics on projective spaces. Helpful background includes basic algebraic geometry, as covered in Hartshorne's Algebraic Geometry and Lazarsfeld's Positivity in Algebraic Geometry I. A basic knowledge of measures and currents in complex geometry, as well as height functions and arithmetic geometry, will also be helpful. Familiarity with arithmetic or complex dynamics will be helpful.
SCHEDULE, NOTES, AND VIDEOS
Scroll horizontally to see all columns, including video links.
| DATE | TOPIC | LECTURE NOTES |
|---|---|---|
| Review of moduli problems and hilbert schemes | subsection 1 and 2 of section 1 | |
| (planned) | Moduli spaces and automorphisms of dynamical systems on projective spaces | subsection 3 and 4 of section 1 |
| (planned) | Review of complex geometry: forms, currents, and hermitian metrics | To be posted |
PLANNED TOPICS
Topics may include:
- Moduli spaces of dynamical systems on projective spaces
- Equilibrium measures, canonical heights, and bifurcation
- Equidistribution theorem of Galois orbits of small height points
- Special/exceptional maps
- Unlikely intersections in dynamics
- Rigidity in dynamics
- Dynamical uniform boundedness
- Valuation theory and the valuative tree for surfaces
- Singularities of ramification divisors
- Other possible related topics
REFERENCES
Possible references include the following books and papers:
- Laura DeMarco. "Critical Orbits and Arithmetic Equidistribution"
- Laura DeMarco. "Dynamical Moduli Spaces and Elliptic Curves"
- Laura DeMarco. "Dynamics of Rational Maps: Lyapunov Exponents, Bifurcations, and Capacity"
- Charles Favre and Mattias Jonsson. "The Valuative Tree"
- T. Gauthier, Johan Taflin, and Gabriel Vigny. "Sparsity of Postcritically Finite Maps of Pk and Beyond: A Complex Analytic Approach"
- Patrick Ingram, Rohini Ramadas, and Joseph Silverman. "Post-Critically Finite Maps on Pn for n ≥ 2 Are Sparse"
- Zhuchao Ji, Jiarui Song, and Junyi Xie. "Exceptional Maps and Abelian Points in Backward Orbits"
- Zhuchao Ji and Junyi Xie. "DAO for Curves"
- Zhuchao Ji and Junyi Xie. "Genus and Gonality of Small Curves, Dynamical Uniform Boundedness, and Bifurcation"
- Zhuchao Ji and Junyi Xie. "The Multiplier Spectrum Morphism Is Generically Injective"
- Alon Levy. "The Space of Morphisms on Projective Space"
- Yujie Luo and Sheng Meng. "Log Calabi-Yau Structure for Endomorphisms on Pn"
- Sheng Meng. "Log Calabi-Yau Structure of Projective Threefolds Admitting Polarized Endomorphisms"
- J. H. Silverman. "Moduli Spaces and Arithmetic Dynamics"
- Xinyi Yuan and Shou-Wu Zhang. "Adelic Line Bundles on Quasi-Projective Varieties"
- Robin Zhang. "Dynamical Uniform Boundedness for Unicritical Polynomials and Elliptic Curves of Large Rank"