Moduli Spaces Associated to Dynamical Systems
(April 1620, 2012)
 Laura DeMarco, Chair
(University of Illinois at Chicago)  Adam Epstein
(University of Warwick)  Sarah Koch
(Harvard University)  Curtis McMullen
(Harvard University)  Joseph Silverman
(Brown University)
This workshop will bring together dynamicists, number theorists, and algebraic geometers to study the geometry and arithmetic of dynamical moduli spaces. The set Rat_{d}^{n} of rational degree d selfmaps of P^{n} has a natural structure as an affine variety. The dynamical moduli space M_{d}^{n} is the quotient of Rat_{d}^{n} by the conjugation action of the group PGL_{n+1}. Problems to be investigated include the geometry of M_{d}^{n}, the distribution of special maps such as postcritically finite maps in M_{d}^{n}, dynamical modular curves associated to oneparameter families of maps with a marked point of period N, and degeneration of families of maps and the associated points on the boundary of moduli space. A tutorial session will be held the week before this workshop.
Problem 1: The Geometry of M_{d}^{n}. It is known that M_{2}^{1} is isomorphic to the affine plane and that M_{d}^{1} is a rational variety, but many fundamental questions remain. A major goal of the workshop will be to study the geometry of M_{d}^{n} and the associated moduli spaces in which one adds level structure, for example by adding a marked point of period N or a marked finite orbit of order N. A motivating question is whether the resulting varieties are of general type if N is sufficiently large.
Problem 2: Distribution of Special Points. An example of the type of problem to be considered is the distribution of postcritically finite maps in the moduli space M_{d}^{1} in both the complex and the padic topologies.
Problem 3: Dynamical Modular Curves. A oneparameter family of maps, for example f_{c}(z)=z^{2}+c, with marked points or orbits of order N, yields dynamical modular curves X_{0}(N) and X_{1}(N) that are analogous to classical modular curves. A good deal is known about the geometry of these curves, but little has been proven about their arithmetic except for some small values of N. The arithmetic properties of X_{0}(N) and X_{1}(N) are closely related to the uniform boundedness conjecture for the families that they parameterize.
Problem 4: The Boundary of Moduli Space. The boundary of a moduli space and a natural method for completing the space are of fundamental importance in understanding the underlying objects and their degenerations. Recent work of Kiwi has used Berkovich space dynamics over Laurent series fields to analyze degenerations of complex dynamical systems. A goal of the workshop is to exploit these nonarchimedean methods to answer classical questions about the boundary of dynamical moduli spaces over the complex numbers.



Thursday  April 12, 2012  

Time  Description  Speaker  Location  Abstracts  Slides 
10:30  11:50  Postcritically Finite Rational Maps and their Deformations  Jeremy Kahn, Brown University  11th Floor Lecture Hall  
1:30  2:30  Applications and Examples of using a new program `Dynamics Explorer' to study the dynamics of complex mappings  Suzanne Boyd, University of Wisconsin and Brian Boyd  11th Floor Lecture Hall  
2:30  2:45  Coffee/Tea Break  11th Floor Collaborative Space  
2:45  3:45  Applications and Examples of using a new program `Dynamics Explorer' to study the dynamics of complex mappings  Suzanne Boyd, University of Wisconsin and Brian Boyd  11th Floor Lecture Hall  
4:00  4:30  Sir John Ball "Smooth topologypreserving approximations of rough domains" Reception  11th Floor Collaborative Space  
4:30  5:30  Special ColloquiumSmooth topologypreserving approximations of rough domains  Sir John Ball, University of Oxford  11th Floor Lecture Hall 
Friday  April 13, 2012  

Time  Description  Speaker  Location  Abstracts  Slides 
10:00  11:00  Moduli spaces: drawing pictures and doing computations I  Sarah Koch, Harvard University and Xavier Buff, Université de Toulouse III (Paul Sabatier)  11th Floor Lecture Hall  
11:30  12:30  Moduli spaces: drawing pictures and doing computations II  Sarah Koch, Harvard University and Xavier Buff, Université de Toulouse III (Paul Sabatier)  11th Floor Lecture Hall  
2:30  3:30  Computational Working Group  11th Floor Lecture Hall  
3:30  4:00  Coffee/Tea Break  11th Floor Collaborative Space 