Dr. Philip Mummert

Associate Professor of Practice
Former Assistant Head
Mathematics Department
Purdue University
West Lafayette, Indiana

Research

Research Statement

I'm interested in discrete dynamical systems (iteration theory) of several complex variables.

The basis for my research involving dynamical systems stems from the following line of inquiry: what effect does the initial state have on the long-term behavior of the system, and do small changes in the iterative procedure itself lead to radically different dynamics? The most well-known examples come from complex dynamics; iterating a simple quadratic function in the complex plane quickly gives rise to exotic looking fractals that are the locus of chaos, points stuck in the middle of a tug-of-war between the infinite and the finite. I have tried to understand a higher-dimensional analogue: complex Hénon maps are fundamental to research in two-variable dynamics as a simple family of invertible functions that exhibit a wide range of dynamical phenomena. Primary questions concern the transition from order to chaos, structural stability, and developing and understanding relevant computer algorithms. What are simple models and classifications for the dynamics? How do the dynamics and the Julia set vary with the parameters of the system? What computer models and algorithms can calculate the relevant data and draw informative illustrations? Questions like these have motivated my research.

My interests have broadened to more general iteration questions and research topics that can readily include undergraduates. These experiences have led to two joint publications, involvement in several conferences, and students who are better prepared for their future pursuits. I also enjoy learning about complex analysis, real analysis, mathematical physics, discrete math, computer algorithms, and financial mathematics.

Dissertation Horseshoes, solenoids, and holomorphic motions for Hénon maps
Advisors Dr. Gregery Buzzard and Dr. John Smillie

Awards

Distinguished Lecture in the Taylor University School of Natural and Applied Sciences, 2012
Research Fellowship in Complex Analysis in Several Variables at the Institut Mittag-Leffler, Djursholm, Sweden, Spring 2008

Past undergraduate research projects

Complex and Symbolic Dynamics, with Sarah Stoops, Anna Durham, and Paije Smith, Fall 2014 and Spring 2015.
Voronoi Vertex Iteration, with Justin Southworth and Bill Solyst, Summer 2012.
Austrian Solitaire with Rachel DeMeo and Daniel Kasper, Summer 2010.
Discrete Complex Analysis with Joseph Seaborn, Summer 2009.
Evolution of finite one-dimensional Cellular Automata with Joseph Seaborn, Summer 2008.
Numerical computation of zeroes of holomorphic functions with S. Bell and A. Jenkins, Purdue REU Summer 2006.

Publications

"Austrian Solitaire," submitted (2023); available on arXiv
"Differentiating Iteratively," Amer. Math. Monthly 120-6 (2013): 545. (with K. Constantine).
"The Evolution of Finite 1-Dimensional Cellular Automata updated with k-Rules," J. of Cell. Autom. 3-3 (2011): 505--515 (with J. Seaborn, A. Churchill, and M. Bardzell).
"A Complex Finite Calculus," Involve 3-3 (2010): 273--287 (with J. Seaborn).
"Holomorphic Shadowing for Hénon Maps," Nonlinearity 21 (2008): 2887--2898.
"The Solenoid and Holomorphic Motions for Hénon Maps," Indiana Univ. Math. J. 56(2007): 2739--2762.