- Professor of Mathematics
- Professor in the Department of Statistical Science (Secondary)
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 <a href=https://scholars.duke.edu/display/insnationalsciencefoundation>National Science Foundation</a> (2010 to 2016)
CAREER: Discrete structures in continuous contexts awarded by <a href=https://scholars.duke.edu/display/insnationalsciencefoundation>National Science Foundation</a> (2005 to 2010)
Dickenstein, A, Matusevich, LF, and Miller, E. "Binomial D-modules." Duke Mathematical Journal 151.3 (January 1, 2010): 1-13.
Ezra Miller, . "Binomial D-modules." Duke Mathematical Journal 151.3 (2010): 385-429. (Academic Article)
Knutson, A, Miller, E, and Yong, A. "Gröbner geometry of vertex decompositions and of flagged tableaux." Journal fur die Reine und Angewandte Mathematik 630 (2009): 1-31. Full Text
Jow, SY, and Miller, E. "Multiplier ideals of sums via cellular resolutions." Mathematical Research Letters 15.2-3 (March 1, 2008): 359-373.
Miller, E. "What is.. a toric variety?." Notices of the American Mathematical Society 55.5 (2008): 586-587.
Shin Yao Jow, . "Multiplier ideals of sums via cellular resolutions." Mathematical Research Letters 15.2 (2008): 359-373. (Academic Article)
Miller, E, and Pak, I. "Metric combinatorics of convex polyhedra: Cut loci and nonoverlapping unfoldings." Discrete and Computational Geometry 39.1-3 (2008): 339-388. Full Text
Ezra, M, and Speyer, DE. "A kleiman-bertini theorem for sheaf tensor products." Journal of Algebraic Geometry 17.2 (2008): 335-340.
Jow, S-Y, and Miller, E. "Multiplier ideals of sums via cellular resolutions." Mathematical Research Letters 15.2-3 (2008): 359-373.