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
- algebraic varieties, polyhedra, manifolds

**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. 1178-1184, ISSN 0002-9920 [doi] - Bendich, P; Marron, JS; Miller, E; Pieloch, A; Skwerer, S,
*Persistent Homology Analysis of Brain Artery Trees.*, The Annals of Applied Statistics, vol. 10 no. 1 (January, Accepted, 2016), pp. 198-218 [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, Springer-Verlag, New York, ISBN 0-387-22356-8 [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. 919-941, ISSN 0894-0347 [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. 1-3 (2008), pp. 339-388, ISSN 0179-5376 [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. 229-325, ISSN 0020-9910 [MR2007k:14098], [math.AG/0308142], [doi] [abs] - 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] - 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] - 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] - 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] - 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]

- Miller, E,