# Ezra Miller

- Professor of Mathematics
- Professor in the Department of Statistical Science (Secondary)

**External address:**209 Physics Bldg, 120 Science Drive, Durham, NC 27708

**Internal office address:**Box 90320, Durham, NC 27708-0320

**Phone:**(919) 660-2846

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.

### Fellowships, Supported Research, & Other Grants

Combinatorics in geometry and algebra with applications to the natural sciences awarded by <a href=https://scholars.duke.edu/display/insnationalsciencefoundation>National Science Foundation</a> (2010 to 2016)

CAREER: Discrete structures in continuous contexts awarded by <a href=https://scholars.duke.edu/display/insnationalsciencefoundation>National Science Foundation</a> (2005 to 2010)

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)

Knutson, A, and Miller, E. "Subword complexes in Coxeter groups." *Advances in Mathematics* 184.1 (2004): 161-176.
Full Text

Miller, E. "Mitosis recursion for coefficients of Schubert polynomials." *Journal of Combinatorial Theory. Series A* 103.2 (2003): 223-235.
Full Text

Helm, D, and Miller, E. "Bass numbers of semigroup-graded local cohomology." *Pacific Journal of Mathematics* 209.1 (2003): 41-66.

Miller, E. "Cohen-Macaulay quotients of normal semioroup rings via irreducible resolutions." *Mathematical Research Letters* 9.1 (2002): 117-128.

Miller, E. "Planar graphs as minimal resolutions of trivariate monomial ideals." *Documenta Mathematica* 7.1 (2002): 43-90.

David Perkinson, . "Eight lectures on monomial ideals." *Queen's papers in pure and applied mathematics* 120 (2001): 3-105. (Academic Article)

Miller, E, Sturmfels, B, and Yanagawa, K. "Generic and cogeneric monomial ideals." *Journal of Symbolic Computation* 29.4-5 (2000): 691-708.

Miller, E. "The Alexander duality functors and local duality with monomial support." *Journal of Algebra* 231.1 (2000): 180-234.

Miller, EN. "Icosahedra constructed from congruent triangles." *Discrete and Computanional Geometry* 24.2-3 (2000): 437-451.