Lucas Schuermann is a rising sophomore at Columbia University in New York City, majoring in applied mathematics and computer science. Lucas hopes to pursue a PhD in mathematics or computer science leading to developing new technologies in emerging sectors, especially artificial intelligence and machine learning. Lucas previously worked in high school with a fellow student and Dr. Kimball Milton of the University of Oklahoma's Department of Physics investigating theoretical torques arising from the quantum mechanical Casimir effect in systems with inhomogeneous dielectric regions. Lucas also collaborated with Dr. Michael Jablonski of the University of Oklahoma's Department of Mathematics on a numerical algorithm to classify unimodular Einstein Lie groups with applications to resolving the Alekseevskii conjecture. Lucas has been working this year in Columbia's Bionet lab on the Neurokernel project, an attempt to emulate the entire brain of the fruit fly Drosophila melanogaster on massively parallel computing systems. Over the summer of 2015, Lucas is additionally working in Columbia's Computer Graphics Group as an NSF fellow on research in the field of discrete differential geometry. He is looking forward to learning from both Stephen Wolfram and the other incredible faculty and participants of the Summer School.
Project: Functionals of Cellular Automaton Rules
We model the collection of possible neighborhood configurations in an elementary cellular automaton as a discrete manifold over which the transition function or rule takes on Boolean values. We find the Dirichlet energy of different rules through the Boolean differential calculus by summing over all possible neighborhood configurations. We also perform spectral analysis of different elementary cellular automaton rules using the Walsh–Hadamard transform. The same analysis can be applied to transition functions that act on an entire causal network of cells, with cells in the first layer as the input and cells in the last layer as the output. We compare this measure of cellular automaton behavior with existing ones, including entropy-based measures and Lyapunov exponent estimators. We obtain a method for gradual rule modification that preserves the Dirichlet energy, with applications to enhancing genetic algorithms that attempt to find the right rule for performing a computation. Finally, we discuss possible extensions of this approach and directions for future research.
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