CAREER: Dynamics in Several Complex Variables, in Context

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Systems that evolve with time appear at the core of nearly all scientific endeavor, including biology, chemistry, physics, and the social sciences. Given the current state of the system, can one predict the future state? How does this evolution of the state of the system depend on the parameters of the system? Many such dynamical systems are far too complicated for a rigorous study, so one resorts to simpler models, which are hoped to indicate the types of behavior that one should expect experimentally. One venue for such simpler models is the iteration of holomorphic maps, the topic of this project. Completing these projects will provide a deeper understanding of fundamental properties of dynamical systems in several complex variables and it will build connections with other areas of mathematics. The broader impacts of this project are largely educational, focusing on both high school students and Ph.D. students. Funding for this project will help to develop an interest and talent in mathematics among high school students, both local and throughout Indiana. It will help to train at least one supported Ph.D. student at Indiana University-Purdue University and also to train Ph.D. students from all over the United States by means of two workshops on holomorphic dynamics in several variables.The main research goal of this project is to explore several examples where the iteration of a holomorphic self-map of a complex manifold has a connection with another area of mathematics or mathematical physics. Not only does the connection allow for applications in the other area, but the context coming from that area usually gives the mapping extra structure, allowing for a significantly deeper study. Four projects are proposed relating dynamics in several complex variables to Thurston theory, the Ising model from statistical physics, the spectra of certain operators arising from self-similar group actions, and complex time ordinary differential equations (the Painleve equations). The proposed research projects are designed around three principles: (1) exploring connections between two or more different areas of mathematics can lead to surprising new results, (2) dynamical systems having an additional context from another field can be studied significantly more deeply, and (3) a study of concrete examples often leads to statements and proofs of more general theorems. Partnered with the main research goals are several educational goals: (1) leading local high school students on specially tailored research projects, (2) helping to run and improve the Indiana University-Purdue University High School Math Contest, (3) leading Ph.D. students on the main research projects mentioned above, and (4) organizing two workshops on dynamics in several complex variables which have a training component for Ph.D. students.