# Ezra Miller

- Professor of Mathematics
- Professor in the Department of Statistical Science (Secondary)

**External address:**209 Physics Bldg, 120 Science Drive, Durham, NC 27708

**Internal office address:**Box 90320, Durham, NC 27708-0320

**Phone:**(919) 660-2846

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

Algebraic and geometric methods in data analysis awarded by <a href=https://scholars.duke.edu/display/insnationalsciencefoundation>National Science Foundation</a> (2017 to 2020)

Integrative Middle School STEM Teacher Preparation: A Collaborative Capacity Building Project at Duke University awarded by <a href=https://scholars.duke.edu/display/insnationalsciencefoundation>National Science Foundation</a> (2014 to 2017)

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)

Knutson, A, Miller, E, and Shimozono, M. "Four positive formulae for type A quiver polynomials." *Inventiones Mathematicae* 166.2 (2006): 229-325.
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Miller, E, and Reiner, V. "Stanley's simplicial poset conjecture, after M. Masuda." *Communications in Algebra* 34.3 (2006): 1049-1053.
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Matusevich, LF, and Miller, E. "Combinatorics of rank jumps in simplicial hypergeometric systems." *Proceedings of the American Mathematical Society* 134.5 (2006): 1375-1381.
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Matusevich, LF, Miller, E, and Walther, U. "Homological methods for hypergeometric families." *Journal of the American Mathematical Society* 18.4 (2005): 919-941.
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Miller, E. "Alternating formulas for K-theoretic quiver polynomials." *Duke Mathematical Journal* 128.1 (2005): 1-17.
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Kogan, M, and Miller, E. "Toric degeneration of Schubert varieties and Gelfand-Tsetlin polytopes." *Advances in Mathematics* 193.1 (2005): 1-17.
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Knutson, A, and Miller, E. "Gröbner geometry of Schubert polynomials." *Annals of Mathematics* 161.3 (2005): 1245-1318.
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Miller, E, and Reiner, V. "Reciprocal domains and Cohen-Macaulay d-complexes in ℝd." *Electronic Journal of Combinatorics* 11.2 N (2005): 1-9.

Helm, D, and Miller, E. "Algorithms for graded injective resolutions and local cohomology over semigroup rings." *Journal of Symbolic Computation* 39.3-4 SPEC. ISS. (2005): 373-395.
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Victor Reiner, . "Reciprocal domains and Cohen-Macaulay d-complexes in R^d." *Electronic Journal of Combinatorics* 11.2 (2004): Note 1, 9 pp.-. (Academic Article)