# 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

**Office Hours:**

Tuesdays 13:00 – 15:00

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 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 and photographs of fruit fly wings for developmental morphological studies.

### Selected Grants

R2 [Reciprocal Relationships]: Mentorships to Strengthen and Sustain STEM Teachers awarded by National Science Foundation (Co-Principal Investigator). 2020 to 2025

HDR TRIPODS: Innovations in Data Science: Integrating Stochastic Modeling, Data Representation, and Algorithms awarded by National Science Foundation (Senior Investigator). 2019 to 2022

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

### 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)

Miller, E. “Graded greenlees-may duality and the cech hull.” *Local Cohomology and Its Applications*, 2001, pp. 233–53.

Katthän, L., et al. “When is a Polynomial Ideal Binomial After an Ambient Automorphism?” *Foundations of Computational Mathematics*, vol. 19, no. 6, Dec. 2019, pp. 1363–85. *Scopus*, doi:10.1007/s10208-018-9405-0.
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Berenstein, A., et al. “Andrei Zelevinsky, 1953–2013.” *Advances in Mathematics*, vol. 300, Sept. 2016, pp. 1–4. *Scopus*, doi:10.1016/j.aim.2016.06.006.
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Kahle, T., et al. “Irreducible decomposition of binomial ideals.” *Compositio Mathematica*, vol. 152, no. 6, June 2016, pp. 1319–32. *Scopus*, doi:10.1112/S0010437X16007272.
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Bendich, P., et al. “Persistent homology analysis of brain artery trees.” *Annals of Applied Statistics*, vol. 10, no. 1, 2016, pp. 198–218.
Open Access Copy

Miller, E. “Fruit flies and moduli: Interactions between biology and mathematics.” *Notices of the American Mathematical Society*, vol. 62, no. 10, Jan. 2015, pp. 1178–84. *Scopus*, doi:10.1090/noti1290.
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Zamaere, C. B., et al. “Systems of parameters and holonomicity of A-hypergeometric systems.” *Pacific Journal of Mathematics*, vol. 276, no. 2, Jan. 2015, pp. 281–86. *Scopus*, doi:10.2140/pjm.2015.276.281.
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Huckemann, S., et al. “Sticky central limit theorems at isolated hyperbolic planar singularities.” *Electronic Journal of Probability*, vol. 20, Jan. 2015. *Scopus*, doi:10.1214/EJP.v20-3887.
Full Text Open Access Copy

Miller, E., et al. “Polyhedral computational geometry for averaging metric phylogenetic trees.” *Advances in Applied Mathematics*, vol. 68, Jan. 2015, pp. 51–91. *Scopus*, doi:10.1016/j.aam.2015.04.002.
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Gopalkrishnan, M., et al. “A geometric approach to the global attractor conjecture.” *Siam Journal on Applied Dynamical Systems*, vol. 13, no. 2, Jan. 2014, pp. 758–97. *Scopus*, doi:10.1137/130928170.
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Skwerer, S., et al. “Tree-oriented analysis of brain artery structure.” *Journal of Mathematical Imaging and Vision*, vol. 50, no. 1, Jan. 2014, pp. 126–43. *Scopus*, doi:10.1007/s10851-013-0473-0.
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## Pages

Miller, E. “Theory and applications of lattice point methods for binomial ideals.” *Combinatorial Aspects of Commutative Algebra and Algebraic Geometry: The Abel Symposium 2009*, 2011, pp. 99–154. *Scopus*, doi:10.1007/978-3-642-19492-4_8.
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Miller, Ezra. “Topological Cohen-Macaulay criteria for monomial ideals.” *Combinatorial Aspects of Commutative Algebra*, edited by V. Ene and E. Miller, vol. 502, AMER MATHEMATICAL SOC, 2009, pp. 137–55.

Miller, E., and B. Sturmfels. “Monomial ideals and planar graphs.” *Lecture Notes in Computer Science (Including Subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*, vol. 1719, 1 Jan. 1999, pp. 19–28. *Scopus*, doi:10.1007/3-540-46796-3_3.
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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 »