- Professor of Mathematics
Professor Miller's research centers around problems in geometry, algebra, topology, combinatorics, statistics, probability, and computation originating in mathematics and the sciences, including biology, chemistry, computer science, and medical imaging.
The techniques range, for example, from abstract algebraic geometry or commutative algebra of ideals and varieties to concrete metric or discrete geometry of polyhedral spaces; from deep topological constructions such as equivariant K-theory and stratified Morse theory to elementary simplicial and persistent homology; from functorial perspectives on homological algebra in the derived category to specific constructions of complexes based on combinatorics of cell decompositions; from geodesic contraction applied to central limit theorems for samples from stratified spaces to dynamics of explicit polynomial vector fields on polyhedra.
Beyond motivations from within mathematics, the sources of these problems lie in, for example, graphs and trees in evolutionary biology and medical imaging; mass-action kinetics of chemical reactions; computational geometry, symbolic computation, and combinatorial game theory; and geometric statistics of data sampled from highly non-Euclidean spaces. Examples of datasets under consideration include MRI images of blood vessels in human brains, vein structures in fruit fly wings for developmental morphological studies, and weather data.
Algebraic and Geometric Methods In Data Analysis awarded by National Science Foundation (Principal Investigator). 2017 to 2020
Integrative Middle School STEM Teacher Preparation: A Collaborative Capacity Building Project at Duke University awarded by National Science Foundation (Co Investigator). 2014 to 2017
Combinatorics in geometry awarded by National Science Foundation (Principal Investigator). 2010 to 2016
Conference Proposal: Meeting on Combinatorial Commutative Algebra (MOCCA 2014) awarded by National Science Foundation (Principal Investigator). 2014 to 2015
CAREER: Discrete Structures in Continuous Contexts awarded by National Science Foundation (Principal Investigator). 2009 to 2011
Bendich, P, Marron, JS, Miller, E, Pieloch, A, and Skwerer, S. "Persistent homology analysis of brain artery trees." Annals of Applied Statistics 10.1 (2016): 198-218. Open Access Copy
Miller, E. "Fruit Flies and Moduli: Interactions between Biology and Mathematics." Notices of the American Mathematical Society 62.10 (November 1, 2015): 1178-1184. Full Text
Miller, E, Owen, M, and Provan, JS. "Polyhedral computational geometry for averaging metric phylogenetic trees." Advances in Applied Mathematics 68 (July 2015): 51-91. Full Text
Huckemann, S, Mattingly, J, Miller, E, and Nolen, J. "Sticky central limit theorems at isolated hyperbolic planar singularities." Electronic Journal of Probability 20.0 (2015). Full Text Open Access Copy
Berkesch Zamaere, C, Griffeth, S, and Miller, E. "Systems of parameters and holonomicity of A -hypergeometric systems." Pacific Journal of Mathematics 276.2 (2015): 281-286. Full Text
Skwerer, S, Bullitt, E, Huckemann, S, Miller, E, Oguz, I, Owen, M, Patrangenaru, V, Provan, S, and Marron, JS. "Tree-oriented analysis of brain artery structure." Journal of Mathematical Imaging and Vision 50.1 (January 1, 2014): 126-143. Full Text
Gopalkrishnan, M, Miller, E, and Shiu, A. "A Geometric Approach to the Global Attractor Conjecture." SIAM Journal on Applied Dynamical Systems 13.2 (January 2014): 758-797. Full Text
Miller, E. "Topological Cohen-Macaulay criteria for monomial ideals." 2009.
Possibilities for using geometry and topology to analyze statistical problems in biology raise a host of novel questions in geometry, probability, algebra, and combinatorics that demonstrate the power of biology to influence the future of pure... read more »