Ezra Miller, Professor of Mathematics

Education:

Ph.D., University of California - Berkeley, 2000

ScB, Brown University (Math), 1995

AB, Brown University (Music), 1995

B.S., Brown University, 1995

Contact Info:

209 Physics Bldg, 120 Science Drive, Durham, NC 27708

(919) 660-2846

ezra@math.duke.edu

http://math.duke.edu/~ezra

Office Hours:

Office hours: Monday, 16:15 – 17:15 in Physics 209 or outside
                       Thursday, 13:15 – 14:30 in Physics 209 or outside

Research Interests: Geometry, algebra, combinatorics, algorithms, probability, statistics, biology, neuroscience and other applications

Professor Miller's research centers around problems in geometry, algebra, topology, combinatorics, and computation originating in mathematics and the sciences, including biology, chemistry, computer science, medical imaging, and statistics.

The techniques range, for example, from abstract algebraic geometry of 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 homology; from functorial perspectives on homological algebra in the derived category to constructions of complexes based on combinatorics of cell decompositions; or from central limit theorems via geodesic contraction on 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. Current datasets under consideration include MRI images of blood vessels in human brains and mouse lungs, vein structures in fruit fly wings for developmental morphological studies, and fMRI time-course images of human brains for classification of schizophrenia patients.

Comments:

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; Lie 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 lungs, 3D folded protein structures, and photographs of fruit fly wings for developmental morphological studies.

• Math 501.01, Algebraic structures i Synopsis

  Physics 119, TuTh 01:25 PM-02:40 PM

Representative Publications   (More Publications)

1. Miller, E. "Fruit flies and moduli: Interactions between biology and mathematics." Notices of the American Mathematical Society 62.10 (November, 2015): 1178-1184. [doi]
2. Bendich, P; Marron, JS; Miller, E; Pieloch, A; Skwerer, S. "Persistent homology analysis of brain artery trees." Annals of Applied Statistics 10.1 (2016): 198-218. [arXiv:1411.6652], [1411.6652v1], [doi]  [abs]
3. Huckemann, S; Mattingly, JC; Miller, E; Nolen, J. "Sticky central limit theorems at isolated hyperbolic planar singularities." Electronic Journal of Probability 20 (2015): 1-34. [repository], [doi]  [abs]
4. with Bernd Sturmfels. Combinatorial commutative algebra. Graduate Texts in Mathematics227 Springer-Verlag, 2005. xiv+417 pp. [MR2006d:13001]
5. with Matusevich, LF; Miller, E; Walther, U. "Homological methods for hypergeometric families." Journal of the American Mathematical Society 18.4 (October, 2005): 919-941. [MR2007d:13027], [math.AG/0406383], [doi]
6. with Miller, E; Pak, I. "Metric combinatorics of convex polyhedra: Cut loci and nonoverlapping unfoldings." Discrete & Computational Geometry 39.1-3 (January, 2008): 339-388. [MR2008m:52027], [math.MG/0312253], [doi]  [abs]
7. with Knutson, A; Miller, E; Shimozono, M. "Four positive formulae for type A quiver polynomials." Inventiones Mathematicae 166.2 (November, 2006): 229-325. [MR2007k:14098], [math.AG/0308142], [doi]  [abs]
8. with Gopalkrishnan, M; Shiu, A. "A projection argument for differential inclusions, with applications to persistence of mass-action kinetics." Sigma (Symmetry, Integrability, and Geometry: Methods and Applications) 9 (2012). (paper 025, 25 pages) [math.DS/1208.0874], [DOI:10.3842/SIGMA.2013.025], [doi]  [abs]
9. Miller, E; Kahle, T; O'Neill, C. "Irreducible decomposition of binomial ideals." Compositio Mathematica 152.6 (June, 2016): 15 pages. [arXiv:1503.02607], [1503.02607], [doi]  [abs]
10. with Anderson, D; Griffeth, S; Miller, E. "Positivity and Kleiman transversality in equivariant K-theory of homogeneous spaces." Journal of the European Mathematical Society 13.1 (January, 2011): 57-84. [math.AG/0808.2785], [DOI:10.4171/JEMS/244], [doi]  [abs]
11. with Guo, A; Miller, E. "Lattice point methods for combinatorial games." Advances in Applied Mathematics 46.1-4 (January, 2011): 363-378. ([math.CO/0908.3473] [math.CO/1105.5420]) [repository], [doi]  [abs]
12. Miller, E. "The Alexander duality functors and local duality with monomial support." Journal of Algebra 231.1 (September, 2000): 180-234. [MR2001k:13028], [pdf], [doi]  [abs]