- 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.
Fellowships, Supported Research, & Other Grants
Combinatorics in geometry and algebra with applications to the natural sciences awarded by National Science Foundation (2010 to 2016)
CAREER: Discrete structures in continuous contexts awarded by National Science Foundation (2005 to 2010)
Miller, E. "Affine stratifications from finite misère quotients." Journal of Algebraic Combinatorics 37.1 (2013): 1-9. Full Text
Hotz, T, Huckemann, S, Le, H, Marron, JS, Mattingly, JC, Miller, E, Nolen, J, Owen, M, Patrangenaru, V, and Skwerer, S. "Sticky central limit theorems on open books." The Annals of Applied Probability 23 (2013): 2238-2258. (Academic Article) Full Text Open Access Copy
Gopalkrishnan, M, and Shiu, A. "A projection argument for differential inclusions, with applications to persistence of mass-action kinetics (Accepted)." SIGMA (Symmetry, Integrability, and Geometry: Methods and Applications) (2012). (Academic Article)
Guo, A, and Miller, E. "Erratum: Lattice point methods for combinatorial games (Advances in Applied Mathematics (2011) 46:1 (363-378))." Advances in Applied Mathematics 48.1 (2012): 269-271. Full Text
Anderson, D, Griffeth, S, and Miller, E. "Positivity and Kleiman transversality in equivariant K-theory of homogeneous spaces." Journal of the European Mathematical Society (2011): 57-84. Full Text
Miller, E, Novik, I, and Swartz, E. "Face rings of simplicial complexes with singularities." Mathematische Annalen 351.4 (2011): 857-875. Full Text
Dickenstein, A, Matusevich, LF, and Miller, E. "Combinatorics of binomial primary decomposition." Mathematische Zeitschrift 264.4 (April 2010): 745-763. Full Text
Dickenstein, A, Matusevich, LF, and Miller, E. "Binomial D-modules." Duke Mathematical Journal 151.3 (January 1, 2010): 1-13.