Ezra Miller
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 Ktheory 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; massaction kinetics of chemical reactions; computational geometry, symbolic computation, and combinatorial game theory; and geometric statistics of data sampled from highly nonEuclidean 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.
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 Ktheory 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; massaction kinetics of chemical reactions; computational geometry, symbolic computation, and combinatorial game theory; and geometric statistics of data sampled from highly nonEuclidean 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.
Current Appointments & Affiliations
 Professor of Mathematics, Mathematics, Trinity College of Arts & Sciences 2009
 Professor in the Department of Statistical Science, Statistical Science, Trinity College of Arts & Sciences 2015
Contact Information
 209 Physics Bldg, 120 Science Drive, Durham, NC 27708
 Box 90320, Durham, NC 277080320
 (919) 6602846
 CV (Curriculum Vitae)
 Talks, seminars, and lecture series
 http://math.duke.edu/~ezra
 Background

Education, Training, & Certifications
 Ph.D., University of California at Berkeley 2000
 B.S., Brown University 1995
 Recognition

In the News

APR 24, 2015 Duke Research Blog 
MAR 20, 2014 Fox8 News 
MAR 17, 2014 News & Observer 
MAR 12, 2014 
JAN 24, 2014 
JAN 22, 2014 Los Angeles Times


Awards & Honors
 Fellow. American Mathematical Society. 2013
 Fellow of the American Mathematical Society. American Mathematical Society. September 2012
 Faculty Early Career Development (CAREER) Program. National Science Foundation. 2009
 Faculty Early Career Development (CAREER) Program. National Science Foundation. 2005
 Expertise

Subject Headings
 Research

Selected Grants
 Algebraic and Geometric Methods In Data Analysis awarded by National Science Foundation 2017  2020
 Integrative Middle School STEM Teacher Preparation: A Collaborative Capacity Building Project at Duke University awarded by National Science Foundation 2014  2017
 Combinatorics in geometry awarded by National Science Foundation 2010  2016
 Conference Proposal: Meeting on Combinatorial Commutative Algebra (MOCCA 2014) awarded by National Science Foundation 2014  2015
 CAREER: Discrete Structures in Continuous Contexts awarded by National Science Foundation 2009  2011
 Publications & Artistic Works

Selected Publications

Conference Papers
 Miller, E. (2011). Theory and Applications of Lattice Point Methods for Binomial Ideals. Full Text Link to Item
 Miller, E. (2009). Topological CohenMacaulay criteria for monomial ideals. Link to Item

Journal Articles
 Berenstein, A, Braverman, M, Miller, E, Retakh, V, & Weitsman, J. (2016, September). Andrei Zelevinsky, 1953–2013. Advances in Mathematics , 300 , 14. Full Text
 Kahle, T, Miller, E, & O’Neill, C. (2016, June). Irreducible decomposition of binomial ideals. Compositio Mathematica , 152 (06), 13191332. Full Text
 Bendich, P, Marron, JS, Miller, E, Pieloch, A, & Skwerer, S. (2016, January). Persistent Homology Analysis of Brain Artery Trees. The annals of applied statistics , 10 (1), 198218. Open Access Copy
 Miller, E. (2015, November 1). Fruit Flies and Moduli: Interactions between Biology and Mathematics. Notices of the American Mathematical Society , 62 (10), 11781184. Full Text
 Miller, E, Owen, M, & Provan, JS. (2015, July). Polyhedral computational geometry for averaging metric phylogenetic trees. Advances in Applied Mathematics , 68 , 5191. Full Text
 Berkesch Zamaere, C, Griffeth, S, & Miller, E. (2015). Systems of parameters and holonomicity of A hypergeometric systems. Pacific Journal of Mathematics , 276 (2), 281286. Full Text
 Huckemann, S, Mattingly, J, Miller, E, & Nolen, J. (2015). Sticky central limit theorems at isolated hyperbolic planar singularities. Electronic Journal of Probability , 20 (0). Full Text Open Access Copy
 Gopalkrishnan, M, Miller, E, & Shiu, A. (2014, January). A Geometric Approach to the Global Attractor Conjecture. SIAM Journal on Applied Dynamical Systems , 13 (2), 758797. Full Text
 Skwerer, S, Bullitt, E, Huckemann, S, Miller, E, Oguz, I, Owen, M, Patrangenaru, V, Provan, S, & Marron, JS. (2014, January 1). Treeoriented analysis of brain artery structure. Journal of Mathematical Imaging and Vision , 50 (1), 126143. Full Text
 Guo, A, & Miller, E. (2013). Algorithms for Lattice Games. International Journal of Game Theory , 42 (4), 777788. Full Text
 Hotz, T, Huckemann, S, Le, H, Marron, JS, Mattingly, JC, Miller, E, Nolen, J, Owen, M, Patrangenaru, V, & Skwerer, S. (2013). Sticky central limit theorems on open books. The Annals of Applied Probability , 23 , 22382258. (Academic Article) Full Text Open Access Copy
 Miller, E. (2013). Affine stratifications from finite misère quotients. Journal of Algebraic Combinatorics , 37 (1), 19. Full Text
 Gopalkrishnan, M, & Shiu, A. (2012). A projection argument for differential inclusions, with applications to persistence of massaction kinetics (Accepted). SIGMA (Symmetry, Integrability, and Geometry: Methods and Applications) . (Academic Article)
 Guo, A, & Miller, E. (2012). Erratum: Lattice point methods for combinatorial games (Advances in Applied Mathematics (2011) 46:1 (363378)). Advances in Applied Mathematics , 48 (1), 269271. Full Text
 Thomas Kahle, . (2012). Decompositions of commutative monoid congruences and binomial ideals (Accepted). Algebra & Number Theory . (Academic Article)
 Anderson, D, Griffeth, S, & Miller, E. (2011). Positivity and Kleiman transversality in equivariant Ktheory of homogeneous spaces. Journal of the European Mathematical Society , 13 (1), 5784. Full Text
 Guo, A, & Miller, E. (2011). Lattice point methods for combinatorial games. Advances in Applied Mathematics , 46 (14), 363378. Full Text Open Access Copy
 Miller, E, Novik, I, & Swartz, E. (2011). Face rings of simplicial complexes with singularities. Mathematische Annalen , 351 (4), 857875. Full Text
 Dickenstein, A, Matusevich, LF, & Miller, E. (2010). Combinatorics of binomial primary decomposition. Mathematische Zeitschrift , 264 (4), 745763. Full Text
 Dickenstein, A, Matusevich, LF, & Miller, E. (2010, January 1). Binomial Dmodules. Duke Mathematical Journal , 151 (3), 113.
 Ezra Miller, . (2010). Binomial Dmodules. Duke Mathematical Journal , 151 (3), 385429. (Academic Article) Link to Item
 Knutson, A, Miller, E, & Yong, A. (2009). Gröbner geometry of vertex decompositions and of flagged tableaux. Journal fur die Reine und Angewandte Mathematik , (630), 131. Full Text
 Jow, SY, & Miller, E. (2008, March 1). Multiplier ideals of sums via cellular resolutions. Mathematical Research Letters , 15 (23), 359373.
 Ezra, M, & Speyer, DE. (2008). A kleimanbertini theorem for sheaf tensor products. Journal of Algebraic Geometry , 17 (2), 335340.
 Jow, SY, & Miller, E. (2008). Multiplier ideals of sums via cellular resolutions. Mathematical Research Letters , 15 (23), 359373.
 Knutson, A, Miller, E, & Yong, A. (2008). Tableau complexes. Israel Journal of Mathematics , 163 , 317343. Full Text
 Miller, E, & Pak, I. (2008). Metric combinatorics of convex polyhedra: Cut loci and nonoverlapping unfoldings. Discrete and Computational Geometry , 39 (13), 339388. Full Text
 Miller, E. (2008). What is.. a toric variety?. Notices of the American Mathematical Society , 55 (5), 586587.
 Shin Yao Jow, . (2008). Multiplier ideals of sums via cellular resolutions. Mathematical Research Letters , 15 (2), 359373. (Academic Article)
 Ning Jia, . (2007). Duality of antidiagonals and pipe dreams. Séminaire Lotharingien de Combinatoire , 58 . (Academic Article)
 Knutson, A, Miller, E, & Shimozono, M. (2006). Four positive formulae for type A quiver polynomials. Inventiones Mathematicae , 166 (2), 229325. Full Text
 Matusevich, LF, & Miller, E. (2006). Combinatorics of rank jumps in simplicial hypergeometric systems. Proceedings of the American Mathematical Society , 134 (5), 13751381. Full Text
 Miller, E, & Reiner, V. (2006). Stanley's simplicial poset conjecture, after M. Masuda. Communications in Algebra , 34 (3), 10491053. Full Text
 Helm, D, & Miller, E. (2005). Algorithms for graded injective resolutions and local cohomology over semigroup rings. Journal of Symbolic Computation , 39 (34 SPEC. ISS.), 373395. Full Text
 Knutson, A, & Miller, E. (2005). Gröbner geometry of Schubert polynomials. Annals of Mathematics , 161 (3), 12451318. Full Text
 Kogan, M, & Miller, E. (2005). Toric degeneration of Schubert varieties and GelfandTsetlin polytopes. Advances in Mathematics , 193 (1), 117. Full Text
 Matusevich, LF, Miller, E, & Walther, U. (2005). Homological methods for hypergeometric families. Journal of the American Mathematical Society , 18 (4), 919941. Full Text
 Miller, E, & Reiner, V. (2005). Reciprocal domains and CohenMacaulay dcomplexes in ℝ^{d}. Electronic Journal of Combinatorics , 11 (2 N), 19.
 Miller, E. (2005). Alternating formulas for Ktheoretic quiver polynomials. Duke Mathematical Journal , 128 (1), 117. Full Text
 Knutson, A, & Miller, E. (2004). Subword complexes in Coxeter groups. Advances in Mathematics , 184 (1), 161176. Full Text
 Victor Reiner, . (2004). Reciprocal domains and CohenMacaulay dcomplexes in R^d. Electronic Journal of Combinatorics , 11 (2), Note 1, 9 pp.. (Academic Article)
 Helm, D, & Miller, E. (2003). Bass numbers of semigroupgraded local cohomology. Pacific Journal of Mathematics , 209 (1), 4166.
 Miller, E. (2003). Mitosis recursion for coefficients of Schubert polynomials. Journal of Combinatorial Theory. Series A , 103 (2), 223235. Full Text
 Miller, E. (2002). CohenMacaulay quotients of normal semioroup rings via irreducible resolutions. Mathematical Research Letters , 9 (1), 117128.
 Miller, E. (2002). Planar graphs as minimal resolutions of trivariate monomial ideals. Documenta Mathematica , 7 (1), 4390.
 David Perkinson, . (2001). Eight lectures on monomial ideals. Queen's papers in pure and applied mathematics , 120 , 3105. (Academic Article)
 Miller, E, Sturmfels, B, & Yanagawa, K. (2000). Generic and cogeneric monomial ideals. Journal of Symbolic Computation , 29 (45), 691708.
 Miller, E. (2000). The Alexander duality functors and local duality with monomial support. Journal of Algebra , 231 (1), 180234.
 Miller, EN. (2000). Icosahedra constructed from congruent triangles. Discrete and Computanional Geometry , 24 (23), 437451.
 Ezra Miller, . (1998). Multiplicities of ideals in Noetherian rings. Beiträge zur Algebra und Geometrie , 39 (1), 4751. (Academic Article)

Other Articles
 Miller, E, & Sturmfels, B. (1999, January 1). Monomial ideals and planar graphs. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) , 1719 , 1928. Full Text

 Teaching & Mentoring

Recent Courses
 MATH 393: Research Independent Study 2018
 MATH 394: Research Independent Study 2018
 MATH 403: Advanced Linear Algebra 2018
 MATH 494: Research Independent Study 2018
 MATH 221: Linear Algebra and Applications 2017
 MATH 493: Research Independent Study 2017
 MATH 501: Introduction to Algebraic Structures I 2017
 MATH 602: An Introduction to Commutative Algebra and Algebraic Geometry 2016
 MATH 79090: Minicourse in Advanced Topics 2016
 Scholarly, Clinical, & Service Activities

Presentations & Appearances
 Topology for statistical analysis of brain artery images. August 12, 2017 2017
 What is a QR code?. August 1, 2017 2017
 What is a barcode for persistent homology with multiple parameters?. July 31, 2017 2017
 Algebraic data structures for topological summaries. July 27, 2017 2017
 Topology for statistical analysis of brain artery images. June 23, 2017 2017
 Fruit fly wing veins as embedded planar graphs. June 13, 2017 2017
 Algebra for topology in biology and statistics. May 27, 2017 2017
 Algebraic data structures for topological summaries. January 26, 2017 2017
 Data structures for real multiparameter persistence. November 12, 2016 2016
 Lessons in topology from statistics of brain arteries. November 10, 2016 2016
 Algebraic data structures for topological summaries. November 9, 2016 2016
 Data structures for real multiparameter persistence. September 12, 2016 2016
 Real applied multigraded commutative algebra. August 18, 2016  August 24, 2016 2016
 Poset encodings of modules with actions of real cones. July 14, 2016 2016
 Topology for statistical analysis of brain artery images. June 14, 2016 2016
 Homological algebra for real multigraded modules. June 2, 2016 2016
 Effective algebra and geometry for images with varied topology. May 24, 2016 2016
 Data structures for real multiparameter persistence. May 1, 2016 2016
 Interactions between topology, biology, and imaging. December 3, 2015 2015
 Irreducible decomposition of binomial ideals. December 3, 2015 2015
 Multigraded modules from fruit flies. November 23, 2015 2015
 Multigraded modules from fruit flies. November 20, 2015 2015
 Lessons in topology from statistics of brain arteries. November 19, 2015 2015
 Multigraded moduli from fruit flies. November 3, 2015 2015
 Topology for statistical analysis of brain artery images. October 20, 2015 2015
 Persistent homology for biological images. September 28, 2015 2015
 Biological applications of persistent homology: interactions with statistics, geometry, and algebra. September 11, 2015 2015
 Multigraded moduli from fruit flies via stratified space statistics. July 7, 2015 2015
 Persistent interactions between biology, topology, and statistics. June 9, 2015 2015
 Applying persistent homology to brain artery and vein imaging. March 21, 2015 2015
 Applying persistent homology to brain artery and vein imaging. January 30, 2015 2015
 Persistent homology analysis of brain artery trees. January 11, 2015 2015
 Applying persistent homology to brain artery and vein imaging. November 12, 2014 2014
 Topological definition of sticky means in arbitrary metric spaces. October 3, 2014 2014
 Topology for statistical analysis of brain artery images. June 27, 2014 2014
 Applying persistent homology to brain artery and vein imaging. May 20, 2014 2014
 Topology for statistical analysis of brain artery images. May 15, 2014 2014
 Topological analysis of geometrically stratified data. May 12, 2014 2014
 Topology for statistical analysis of brain artery images. April 11, 2014 2014
 Applying persistent homology to brain artery imaging. March 10, 2014 2014
 Asymptotics of sampling from topologically stratified spaces. March 4, 2014 2014
 Binomial irreducible decomposition. January 25, 2014 2014
 Binomial irreducible decomposition. November 9, 2013 2013
 Binomial irreducible decomposition. October 12, 2013 2013
 Binomial irreducible decomposition. July 9, 2013 2013
 How do quivers and stratifications apply to biology?. April 27, 2013 2013
 Geometric statistics on stratified spaces. March 7, 2013 2013
 Sticky central limit theorems at singularities. March 6, 2013 2013
 Biological applications of geometric statistics on stratified spaces. February 14, 2013 2013
 Statistics without addition or division. February 6, 2013 2013
 Alexander duality and total positivity: a cluster/commutative algebra connection. December 7, 2012 2012
 Geometry for samples of metric branched structures. October 11, 2012 2012
 NSF funding from the inside. September 25, 2012 2012
 Combinatorics and topology of fibers in total positivity. May 25, 2012 2012
 Sticky central limit theorems at singularities. May 23, 2012 2012
 Sticky central limit theorems at singularities. April 24, 2012 2012
 Subword complexes and fibers of totally nonnegative parametrizations. September 24, 2011 2011
 Sticky central limit theorems at singularities. August 15, 2011 2011
 Sticky central limit theorems at singularities. August 4, 2011 2011
 How primary decomposition in commutative monoids is wrong. June 9, 2011 2011
 Stratified statistics for evolutionary biology. April 30, 2011 2011
 Contractibility via subword complexes in the context of homogeneous spaces. April 23, 2011 2011
 Geometric statistics: sampling from singular spaces. March 26, 2011 2011
 How primary decomposition in commutative monoids is wrong. March 19, 2011 2011
 Sticky central limit theorems on polyhedral spaces. March 12, 2011 2011
 Unfolding polyhedra. February 18, 2011 2011
 Statistics on data sampled from stratified spaces. January 14, 2011 2011
 Statistics on spaces of phylogenetic trees. December 3, 2010 2010
 Combinatorics of phylogenetic trees. December 2, 2010 2010
 Decompositions of commutative monoid congruences and binomial ideals. November 2, 2010 2010
 Decompositions of monoids and binomial ideals (5 hours of lectures). September 1, 2010 2010
 Unfolding convex polyhedra. July 1, 2010 2010
 Face rings of simplicial complexes with singularities. June 1, 2010 2010
 Lattice point methods for combinatorial games. May 1, 2010 2010
 Unfolding convex polyhedra. May 1, 2010 2010
 Lattice point methods for combinatorial games. April 1, 2010 2010
 Lattice point methods for combinatorial games. March 1, 2010 2010
 Unfolding convex polyhedra. February 1, 2010 2010
 Primary decomposition of binomial ideals (8 hours of lectures). January 1, 2010 2010
 Ktheory of toric varieties. December 1, 2009 2009
 Biological and chemical applications of polyhedral geometry and algebra. October 1, 2009 2009
 Equivariant transversality and Ktheoretic positivity. October 1, 2009 2009
 Combinatorics in cohomology of toric varieties. September 1, 2009 2009
 Unfolding polyhedra. September 1, 2009 2009
 What is a toric variety?. September 1, 2009 2009
 Lattice games. July 1, 2009 2009
 Applications of binomial commutative algebra. June 1, 2009 2009
 Binomial ideals. June 1, 2009 2009
 Combinatorial primary decomposition of binomial ideals. April 1, 2009 2009
 Equivariant positivity via transversality in Ktheory of G/P. April 1, 2009 2009
 Potential applications of commutative algebra to combinatorial game theory. April 1, 2009 2009
 Topological CohenMacaulay criteria for monomial ideals. April 1, 2009 2009
 Equivariant transversality and Ktheoretic positivity. March 1, 2009 2009
 Geometry of flags and permutations. March 1, 2009 2009
 Unfolding polyhedra. March 1, 2009 2009
 Unfolding polyhedra. March 1, 2009 2009
 Positivity from transversality and transitivity. January 1, 2009 2009
 Transversality in homogeneous spaces. December 1, 2008 2008
 Cohomology rings of flag and toric varieties. October 1, 2008 2008
 Topological CohenMacaulay criteria for monomial ideals. September 1, 2008 2008
 Cellular resolutions of multiplier ideals of sums. May 1, 2008 2008
 Combinatorics of Horn hypergeometric series. April 1, 2008 2008
 Gröbner geometry of Schubert polynomials. February 1, 2008 2008
 Metric geometry and unfoldings of polyhedra. February 1, 2008 2008
 Unfolding polyhedra. February 1, 2008 2008
 Metric geometry and unfoldings of polyhedra. January 1, 2008 2008

Service to the Profession
 Editorial Board Member. Journal of Combinatorial Theory, Series A. 2014  2022 2014  2022
 AMS Fellows Selection Committee. 2013  2015 2013  2015
 Directorate Organizer : SAMSI Summer Program on Neuroimaging Data Analysis (NDA). December 15, 2012  July 1, 2013 2012  2013
 Organizer : Oberwolfach miniworkshop: Asymptotic statistics on stratified spaces (29 Sep  3 Oct 2014). February 29, 2012  October 3, 2014 2012  2014
 AMS Short Course Subcommittee. 2012  2015 2012  2015
 AMS Southeast Section Program Committee. 2012  2014 2012  2014
 Lead Organizer : MBI: Workshop on statistics, geometry, and combinatorics on stratified spaces arising from biological problems (May 2012). December 12, 2011  June 1, 2012 2011  2012
 Scientific Committee Member : Experimental and theoretical methods in algebra, geometry, and topology, Mangalia, Romania (June 2013). December 12, 2011  July 1, 2013 2011  2013
 Organizer (with Sonja Mapes and Christine Berkesch) : Triangle Lectures in Combinatorics, 11 February 2012. July 1, 2011  July 1, 2012 2011  2012
 Directorate Organizer : 20132014 SAMSI yearlong program on LowDimensional Structure in HighDimensional Systems (LDHD). 2011  2014 2011  2014
 Editorial Board Member. Discrete Mathematics. 2011 2011
 Organizer : Triangle Lectures in Combinatorics. September 25, 2010 2010
 Working Group leader and administrator : SAMSI Working Group: Analysis of data sampled from stratified spaces. September 2010  June 2011 2010  2011
 Steering Committee Member : Triangle Lectures in Combinatorics. 2010 2010
 Program Committee Member : 2011 Formal Power Series and Algebraic Combinatorics (FPSAC), Reykjavík, Iceland. December 6, 2009  July 1, 2011 2009  2011
 Working group session organizer (with William Graham, UGA) : Combinatorial Lie theory and applications. October 2009 2009
 Program Committee Member : 2009 Formal Power Series and Algebraic Combinatorics (FPSAC), Research Institute for Symbolic Computation (RISC), Linz, Austria. July 2009 2009
 Organizer : Triangle Lectures in Combinatorics. February 6, 2009 2009
 Editorial Board Member. Advances in Mathematics. 2009 2009
 Editorial Board Member. SIAM Journal on Discrete Mathematics. 2009  2017 2009  2017
 Organizer : 20122013 MSRI yearlong program on Commutative Algebra. 2009  2013 2009  2013
 Scientific Committee Member : Exploratory workshop on combinatorial commutative algebra and computer algebra, Mangalia, Romania. May 2008 2008
 Editorial Board Member. Beiträge zur Algebra und Geometrie. 2008 2008
 Organizer : Workshop on Combinatorial Game Theory, Banff International Research Station (BIRS). January 2008 2008
 Associate Editor. Discrete Mathematics. 2007  2010 2007  2010

Service to Duke
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