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
Ktheory 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; massaction kinetics of chemical
reactions; computational geometry, symbolic
computation, and combinatorial game theory; and
geometric statistics of data sampled from highly
nonEuclidean 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 timecourse 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 nonEuclidean geometric datasets
 commutative algebra and representation theory
 algorithms in algebra and geometry
 combinatorics of games, simpicial complexes, trees
 Representative Publications
 Miller, E, Fruit Flies and Moduli: Interactions between Biology and Mathematics,
Notices of the American Mathematical Society, vol. 62 no. 10
(November, 2015),
pp. 11781184, ISSN 00029920 [doi]
 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
(Accepted, 2016),
pp. 198218 [arXiv:1411.6652], [1411.6652v1], [doi] [abs]
 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]
 with Bernd Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics, vol. 227
(2005),
pp. xiv+417, SpringerVerlag, New York, ISBN 0387223568 [MR2006d:13001]
 with Matusevich, LF; Miller, E; Walther, U, Homological methods for hypergeometric families,
Journal of the American Mathematical Society, vol. 18 no. 4
(2005),
pp. 919941, ISSN 08940347 [MR2007d:13027], [math.AG/0406383], [doi]
 with Miller, E; Pak, I, Metric combinatorics of convex polyhedra: Cut loci and nonoverlapping unfoldings,
Discrete & Computational Geometry, vol. 39 no. 13
(2008),
pp. 339388, ISSN 01795376 [MR2008m:52027], [math.MG/0312253], [doi] [abs]
 with Knutson, A; Miller, E; Shimozono, M, Four positive formulae for type A quiver polynomials,
Inventiones Mathematicae, vol. 166 no. 2
(2006),
pp. 229325, ISSN 00209910 [MR2007k:14098], [math.AG/0308142], [doi] [abs]
 with Gopalkrishnan, M; Shiu, A, A projection argument for differential inclusions, with applications to persistence of massaction 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]
 Kahle, T; Miller, E; O’Neill, C, Irreducible decomposition of binomial ideals,
Compositio Mathematica, vol. 152 no. 06
(June, Accepted, 2016),
pp. 13191332 [arXiv:1503.02607], [1503.02607], [doi] [abs]
 with Anderson, D; Griffeth, S; Miller, E, Positivity and Kleiman transversality in equivariant Ktheory of homogeneous spaces,
Journal of the European Mathematical Society, vol. 13 no. 1
(2011),
pp. 5784, ISSN 14359855 [math.AG/0808.2785], [DOI:10.4171/JEMS/244], [doi] [abs]
 with Guo, A; Miller, E, Lattice point methods for combinatorial games,
Advances in Applied Mathematics, vol. 46 no. 14
(2011),
pp. 363378, ISSN 01968858 ([math.CO/0908.3473] [math.CO/1105.5420].) [repository], [doi] [abs]
 Miller, E, The Alexander duality functors and local duality with monomial support,
Journal of Algebra, vol. 231 no. 1
(2000),
pp. 180234 [MR2001k:13028], [pdf] [abs]
