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Moduli Spaces and Dynamical Systems
(tentative schedule of talks)
- Introduction to moduli spaces associated to dynamical systems
Dynamical moduli spaces
- Definitions from dynamics
- Moduli spaces: what they are and why they're useful
- Moduli spaces in dynamical systems
- The Uniform boundedness conjecture
Dynamical modular curves
- Introduction to geometric invariant theory (GIT)
- Construction of moduli spaces M_d^n using GIT
- M_2^1 = A^2
- Rationality of M_d^1
Applications and further topics
- Definition and construction of dynamical modular curves for x^2+c
- Irreducibility and genus formulas
- Rational points on dynamical modular curves
- Other dynamical modular curves
Field of moduli and field of definition
- Canonical heights over function fields
- Arithmetic properties of post-critically finite maps
- Specialization theorems
- Definitions and examples
- Tools for determining when FOM = FOD
- Application to rational points on moduli spaces
Background Reading and References
- Laura DeMarco: Dynamical moduli spaces (and complex analysis)
- Part I. Building blocks and bifurcations
- Part II. Boundaries and barycenters
- Adam Epstein: Transversality and Holomorphic Dynamics
- Michelle Manes: Level Structures on Dynamical Moduli Spaces
- Michael Tepper: Isotriviality and M_d^n
- Graduate Student Reports (30 minutes each)
- Anupam Bhatnagar: Points of Canonical Height Zero on Projective Varieties
- Joey Lee: Height Estimates for Dominant Morphisms on Projective Varieties
- Alon Levy: Semistable Reduction for Dynamical Systems
- Silverman, Joseph H. The Arithmetic of Dynamical Systems.
Graduate Texts in Mathematics, 241. Springer, New York, 2007.
(Chapter 4 is particularly germane to our theme.)
- Levy, Alon. The Space of Morphisms on Projective Space.
To appear in Acta Arithmetica. (arXiv:0903.1318)
- Manes, Michelle. Moduli spaces for families of rational maps on P^1.
J. Number Theory. 129, 2009, 1623--1663. (arXiv:0902.1813)
- Morton, Patrick. On certain algebraic curves related to polynomial maps.
Compositio Math. 103, 1996, 319--350.
- Silverman, Joseph H. The space of rational maps on P^1.
Duke Math. J. 94, 1998, 41--77. (arXiv:math/9609212)
- Silverman, Joseph H., The field of definition for dynamical systems on P^1.
Compositio Math. 98, 1995, 269--304.