Research Interests for Ezra Miller

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.

Keywords:
algebra, Algebra, algorithms, applications, biology, Biology, combinatorics, evolution, geometry, Geometry, neuroscience, Probability, statistics, Statistics, topology, Topology
Areas of Interest:

  • algebraic varieties, polyhedra, manifolds
  • applications to biology, computer science, chemistry, physics, statistics
  • analysis of non-Euclidean geometric datasets
  • commutative algebra and representation theory
  • algorithms in algebra and geometry
  • combinatorics of games, simpicial complexes, trees
  • Representative Publications
    1. Miller, E, Fruit Flies and Moduli: Interactions between Biology and Mathematics, Notices of the American Mathematical Society, vol. 62 no. 10 (November, 2015), pp. 1178-1184, ISSN 0002-9920 [doi]
    2. Bendich, P; Marron, JS; Miller, E; Pieloch, A; Skwerer, S, Persistent homology analysis of brain artery trees, Annals of Applied Statistics, vol. 10 no. 1 (January, Accepted, 2016), pp. 19 pages [arXiv:1411.6652], [1411.6652v1[abs]
    3. Huckemann, S; Mattingly, J; Miller, E; Nolen, J, Sticky central limit theorems at isolated hyperbolic planar singularities, Electronic Journal of Probability, vol. 20 (2015) [repository], [doi]
    4. with Bernd Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics, vol. 227 (2005), pp. xiv+417, Springer-Verlag, New York, ISBN 0-387-22356-8 [MR2006d:13001]
    5. with Matusevich, LF; Miller, E; Walther, U, Homological methods for hypergeometric families, The Journal of the American Mathematical Society, vol. 18 no. 4 (2005), pp. 919-941, ISSN 0894-0347 [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, vol. 39 no. 1-3 (2008), pp. 339-388, ISSN 0179-5376 [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, vol. 166 no. 2 (2006), pp. 229-325, ISSN 0020-9910 [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), vol. 9 (2013) (paper 025, 25 pages.) [math.DS/1208.0874], [DOI:10.3842/SIGMA.2013.025[abs]
    9. Kahle, T; Miller, E; O’Neill, C, Irreducible decomposition of binomial ideals, Compositio Mathematica, vol. 152 no. 06 (June, Accepted, 2016), pp. 1319-1332 [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, vol. 13 no. 1 (2011), pp. 57-84, ISSN 1435-9855 [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, vol. 46 no. 1-4 (2011), pp. 363-378, ISSN 0196-8858 ([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, vol. 231 no. 1 (2000), pp. 180-234 [MR2001k:13028], [pdf[abs]