# 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.

### Selected Grants

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)

"Graded greenlees-may duality and the cech hull." *Local Cohomology and its Applications.* January 1, 2001. 233-253.

Berenstein, A, Braverman, M, Miller, E, Retakh, V, and Weitsman, J. "Andrei Zelevinsky, 1953–2013." *Advances in Mathematics* 300 (September 2016): 1-4.
Full Text

Kahle, T, Miller, E, and O’Neill, C. "Irreducible decomposition of binomial ideals." *Compositio Mathematica* 152.06 (June 2016): 1319-1332.
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Bendich, P, Marron, JS, Miller, E, Pieloch, A, and Skwerer, S. "Persistent Homology Analysis of Brain Artery Trees." *The Annals of Applied Statistics* 10.1 (January 2016): 198-218.
Full Text 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.
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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

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.
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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.
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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* 23.6 (December 2013): 2238-2258.
Full Text Open Access Copy

Gopalkrishnan, M. "A Projection Argument for Differential Inclusions, with Applications to Persistence of Mass-Action Kinetics." *Symmetry, Integrability and Geometry: Methods and Applications* (March 26, 2013).
Full Text

## Pages

Miller, E. "Theory and Applications of Lattice Point Methods for Binomial Ideals.": Springer Berlin Heidelberg, 2011. Full Text

Miller, E. "Topological Cohen-Macaulay criteria for monomial ideals." 2009.

Miller, E, and Sturmfels, B. "Monomial Ideals and Planar Graphs." (1999): 19-28. Full Text

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 »