- Professor of Mathematics
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.
Knutson, A, Miller, E, and Shimozono, M. "Four positive formulae for type A quiver polynomials." Inventiones Mathematicae 166.2 (2006): 229-325. Full Text
Miller, E, and Reiner, V. "Stanley's simplicial poset conjecture, after M. Masuda." Communications in Algebra 34.3 (2006): 1049-1053. Full Text
Matusevich, LF, and Miller, E. "Combinatorics of rank jumps in simplicial hypergeometric systems." Proceedings of the American Mathematical Society 134.5 (2006): 1375-1381. Full Text
Matusevich, LF, Miller, E, and Walther, U. "Homological methods for hypergeometric families." Journal of the American Mathematical Society 18.4 (2005): 919-941. Full Text
Miller, E. "Alternating formulas for K-theoretic quiver polynomials." Duke Mathematical Journal 128.1 (2005): 1-17. Full Text
Kogan, M, and Miller, E. "Toric degeneration of Schubert varieties and Gelfand-Tsetlin polytopes." Advances in Mathematics 193.1 (2005): 1-17. Full Text
Miller, E, and Reiner, V. "Reciprocal domains and Cohen-Macaulay d-complexes in ℝd." Electronic Journal of Combinatorics 11.2 N (2005): 1-9.
Helm, D, and Miller, E. "Algorithms for graded injective resolutions and local cohomology over semigroup rings." Journal of Symbolic Computation 39.3-4 SPEC. ISS. (2005): 373-395. Full Text
Victor Reiner, . "Reciprocal domains and Cohen-Macaulay d-complexes in R^d." Electronic Journal of Combinatorics 11.2 (2004): Note 1, 9 pp.-. (Academic Article)