Dr. Jason Mireles-James

Education

• Ph.D. Mathematics. University of Texas at Austin. December 2009

Research Interests

• Computational Topology in Dynamical Systems: Conley Index Theory, Set Oriented Numerics, computational aspects of Morse/Floer theory, and Applications.
• A Posteriori Analysis and Numerics in Dynamical Systems: Parameterization of invariant manifolds with rigorous error bounds. Applications to computer assisted proof of connecting dynamics for diffeomorphisms, ordinary and partial differential equations.
• KAM Theory: existence and numerical approximation of invariant manifolds in spite of small divisors.

Research Description

• The computer as the mathematician's "laboratory" for studying global dynamics of nonlinear systems. I'm especially interested in the computation and visualization of smooth invariant manifolds.
• Reformulation of qualitative questions about nonlinear dynamical systems into quantitative functional equations, and numerical methods for studying these functional equations.
• Rigorous numerical methods and computer assisted proof in analysis, especially constructive a-posteriori existence (or "shadowing") theorems for invariant manifolds, connecting dynamics, and chaotic motions.

Recent Publications

• Parameterization of invariant manifolds for periodic orbits (II): a-posteriori analysis and computer assisted error bounds. With Roberto Castelli and J.P. Lessard (Submitted).

•  Fourier-Taylor Parameterization of Unstable Manifolds for Parabolic Partial Differential Equations: Formalism, Implementation, and Rigorous Validation with Christian Reinhardt (Submitted). 

• Parameterization method for unstable manifolds of delay differential equations. with Chris Groothedde (submitted). 

• Validated computation of heteroclinic sets. with Maciej Capinski (submitted). 

• High-order parameterization of stable/unstable manifolds for long periodic orbits of maps.  with Jorge Gonzalez (submitted).number 2

Scholarly Activities

• Seminar:  Professor Erik Lundberg and I organize the Analysis and Applications Seminar in the Department of Mathematical Sciences at FAU. Here is a link to the seminar page:
  • Hompage/Schedule (Fall 2016)

Personal Website

http://brain2.math.fau.edu/~jmirelesjames/index.html