Ezra Miller, Professor
Professor Miller's research centers around problems in geometry, algebra, topology, probability, statistics, and computation originating in mathematics and the sciences, including biology, chemistry, computer science, and 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 collapse 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; Lie 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 and lungs, 3D folded protein structures, and photographs of fruit fly wings for developmental morphological studies.  Contact Info:
Teaching (Fall 2024):
 MATH 221.01, LINEAR ALGEBRA
Synopsis
 Physics 259, TuTh 11:45 AM01:00 PM
 MATH 721.01, LINEAR ALGEBRA & APPLICA
Synopsis
 Physics 259, TuTh 11:45 AM01:00 PM
Teaching (Spring 2025):
 MATH 403.01, ADVANCED LINEAR ALGEBRA
Synopsis
 Physics 205, TuTh 11:45 AM01:00 PM
 MATH 703.01, ADVANCED LINEAR ALGEBRA
Synopsis
 Physics 205, TuTh 11:45 AM01:00 PM
 Office Hours:
Office hours: Tuesday, 13:00 – 14:15 in Physics 209 or outside Thursday, 13:00 – 14:15 in Physics 209 or outside
 Education:
Ph.D.  University of California, Berkeley  2000 
ScB  Brown University (Math)  1995 
AB  Brown University (Music)  1995 
B.S.  Brown University  1995 
 Specialties:

Geometry
Algebra Applied Math Topology Mathematical Biology Probability
 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.
 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
 Keywords:
algebra • Algebra • algorithms • applications • biology • Biology • combinatorics • evolution • geometry • Geometry • neuroscience • Probability • statistics • Statistics • topology • Topology
 Curriculum Vitae
 Current Ph.D. Students
(Former Students)
 Postdocs Mentored
 Undergraduate Research Supervised
 Victoria Cheng (2013)
topic: mathematical biology  Rowena (Jingxing) Gan (2012  2015)
topic: Geometry of harmony in Impressionist music  Ezgi Kantarcı (May, 2010  September, 2010)
math grad student, USC; studied Combinatorial Game Theory: infinite misère quotients  Alan Guo (2009  2011)
PhD, MIT Computer Science; was Duke Faculty Scholar, Goldwater honorable mention; thesis topic: Combinatorial Game Theory  Nathaniel Born (2008  2009)
University of Minnesota UROP; topic: Implementing a nonoverlapping unfolding algorithm  Matthew Coudron (2008  2009)
Goldwater Scholar, University of Minnesota Mathematics Honors Program  David Molitor (2003  2007)
Goldwater Scholar, University of Minnesota Math and Economics major, Physics minor
 Representative Publications
(More 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, American Mathematical Society (AMS), 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
(2016),
pp. 198218 [arXiv:1411.6652], [1411.6652v1], [doi] [abs]
 Huckemann, S; Mattingly, JC; Miller, E; Nolen, J, Sticky central limit theorems at isolated hyperbolic planar singularities,
Electronic Journal of Probability, vol. 20
(2015),
pp. 134, Institute of Mathematical Statistics [repository], [doi] [abs]
 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
(October, 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 and Computational Geometry, vol. 39 no. 13
(January, 2008),
pp. 339388, Springer Nature, 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
(November, 2006),
pp. 229325, Springer Nature, 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
(2012), SIGMA (Symmetry, Integrability and Geometry: Methods and Application) (paper 025, 25 pages.) [math.DS/1208.0874], [DOI:10.3842/SIGMA.2013.025], [doi] [abs]
 Miller, E; Kahle, T; O'Neill, C, Irreducible decomposition of binomial ideals,
Compositio Mathematica, vol. 152 no. 6
(June, 2016),
pp. 15 pages, Oxford University Press (OUP) [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
(January, 2011),
pp. 5784, European Mathematical Publishing House, 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
(January, 2011),
pp. 363378, Elsevier BV, 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
(September, 2000),
pp. 180234, Elsevier BV [MR2001k:13028], [pdf], [doi] [abs]
 Selected Invited Lectures
 Lattice games, July, 2009, Halifax, Nova Scotia (GamesatDal)
 Applications of binomial commutative algebra, June, 2009, Norway (Abel Symposium) [speakers]
 Potential applications of commutative algebra to combinatorial game theory, April, 2009, Oberwolfach
 Equivariant transversality and Ktheoretic positivity, March, 2009, MSRI [show_workshop]
 Geometry of flags and permutations, March, 2009, Thessaloniki, Greece
 Cellular resolutions of multiplier ideals of sums, May, 2008, Mangalia, Romania
 Nonoverlapping unfolding of polyhedra, December, 2007, American Institute of Math, Palo Alto
 Combinatorics of Horn hypergeometric series, July, 2007, Tianjin, China (FPSAC plenary speaker)
 Old hypergeometric mysteries and new toric algebra, July, 2007, Medellín, Colombia
 KleimanBertini theorems for sheaf tensor products, May, 2007, CRM Montréal, Canada
 Unfolding polyhedra, October, 2006, Cincinnati, Ohio (AMS meeting plenary talk)
 Multigraded commutative algebra, September, 2006, Constanţa, Romania
 Lattice basis ideals and Horn systems, May, 2006, Luminy, France
 Positivity by degeneration, December, 2005, Taichung, Taiwan (AMSTMS)
 Gröbner geometry of quiver polynomials, September, 2005, UNAM, Mexico City
 Homological methods for hypergeometric families, July, 2005, Lisbon, Portugal
 Overview of geometric combinatorics, July, 2004, Park City, Utah (IAS/PCMI)
 Computing injective resolutions over semigroup rings, June, 2003, Kaiserslautern, Germany (MEGA)
 Positive combinatorial formulae for quiver polynomials, June, 2003, Seville, Spain (AMSRSME)
 Minors in products of matrices, April, 2003, Banff, Canada (BIRS workshop)
 Hilbert schemes of points in the plane, September, 2002, MSRI Introductory workshop
 Gröbner geometry of Schubert polynomials, July, 2002, Melbourne, Australia (FPSAC onehour talk)
 Gröbner geometry of Schubert polynomials via determinantal ideals, June, 2002, Trento, Italy
 Embedding planar graphs in staircases, May, 2001, Morelia, Mexico (AMSSMM)
 The Čech hull, December, 1999, Guanajuato, Mexico (Local Cohomology workshop)
 Alexander duality & local duality with monomial support, June, 1999, Essen, Germany
 Alexander duality for arbitrary monomial ideals, April, 1998, Kyoto, Japan (RIMS)
 Recent Grant Support
 RTG: Linked via Lfunctions: training versatile researchers across number theory, National Science Foundation, 2023/102028/09.
 R2 [Reciprocal Relationships]: Mentorships to Strengthen and Sustain STEM Teachers, National Science Foundation, 2020/102025/09.
 HDR TRIPODS: Innovations in Data Science: Integrating Stochastic Modeling, Data Representation, and Algorithms, National Science Foundation, 2019/102023/09.
 Conferences Organized
 Steering Committee Member : Triangle Lectures in Combinatorics. 2010, Steering Committee Member : Triangle Lectures in Combinatorics, 2010  present
 Summer school on Applicable Combinatorial Commutative Algebra, Director, July, 2015  August, 2016
 SAMSI yearlong program on Challenges in Computational Neuroscience, Local Scientific Coordinator, 2015  2016
 Conference on algebraic combinatorics and group actions, Scientific Committee Member, August, 2014  July, 2016
 Organizer : Oberwolfach miniworkshop: Asymptotic statistics on stratified spaces (29 Sep  3 Oct 2014). February 29, 2012  October 03, 2014, Organizer : Oberwolfach miniworkshop: Asymptotic statistics on stratified spaces (29 Sep  3 Oct 2014), 
 20132014 SAMSI yearlong program on LowDimensional Structure in HighDimensional Systems (LDHD), Directorate Organizer, 2011  2014
 Directorate Organizer : SAMSI Summer Program on Neuroimaging Data Analysis (NDA). December 15, 2012  July 01, 2013, Directorate Organizer : SAMSI Summer Program on Neuroimaging Data Analysis (NDA), June 2013
 Scientific Committee Member : Experimental and theoretical methods in algebra, geometry, and topology, Mangalia, Romania (June 2013). December 12, 201, Scientific Committee Member : Experimental and theoretical methods in algebra, geometry, and topology, Mangalia, Romania (June 2013), 
 Organizer : 20122013 MSRI yearlong program on Commutative Algebra. 2009  2013, Organizer : 20122013 MSRI yearlong program on Commutative Algebra, 2009  2013
 Organizer (with Sonja Mapes and Christine Berkesch) : Triangle Lectures in Combinatorics, 11 February 2012. July 01, 2011  July 01, 2012, Organizer (with Sonja Mapes and Christine Berkesch) : Triangle Lectures in Combinatorics, 11 February 2012, 
 Lead Organizer : MBI: Workshop on statistics, geometry, and combinatorics on stratified spaces arising from biological problems (May 2012). December 1, Lead Organizer : MBI: Workshop on statistics, geometry, and combinatorics on stratified spaces arising from biological problems (May 2012), 
 Program Committee Member : 2011 Formal Power Series and Algebraic Combinatorics (FPSAC), Reykjavík, Iceland. December 06, 2009  July 01, 2011, Program Committee Member : 2011 Formal Power Series and Algebraic Combinatorics (FPSAC), Reykjavík, Iceland, 
 Working Group leader and administrator : SAMSI Working Group: Analysis of data sampled from stratified spaces. September, 2010  June, 2011, Working Group leader and administrator : SAMSI Working Group: Analysis of data sampled from stratified spaces, September, 2010  June, 2011
 Organizer : Triangle Lectures in Combinatorics. September 25, 2010, Organizer : Triangle Lectures in Combinatorics, September 25, 2010
 Working group session organizer (with William Graham, UGA) : Combinatorial Lie theory and applications. October, 2009, Working group session organizer (with William Graham, UGA) : Combinatorial Lie theory and applications, October, 2009
 Program Committee Member : 2009 Formal Power Series and Algebraic Combinatorics (FPSAC), Research Institute for Symbolic Computation (RISC), Linz, Aus, Program Committee Member : 2009 Formal Power Series and Algebraic Combinatorics (FPSAC), Research Institute for Symbolic Computation (RISC), Linz, Aus, July, 2009
 Organizer : Triangle Lectures in Combinatorics. February 06, 2009, Organizer : Triangle Lectures in Combinatorics, February 06, 2009
 Scientific Committee Member : Exploratory workshop on combinatorial commutative algebra and computer algebra, Mangalia, Romania. May, 2008, Scientific Committee Member : Exploratory workshop on combinatorial commutative algebra and computer algebra, Mangalia, Romania, May, 2008
 Organizer : Workshop on Combinatorial Game Theory, Banff International Research Station (BIRS). January, 2008, Organizer : Workshop on Combinatorial Game Theory, Banff International Research Station (BIRS), January, 2008
 Workshop on Rigidity and Polyhedra, American Institute of Mathematics, Organizer, December, 2007
 AMS Special session on Geometric Combinatorics, Organizer, October, 2006
 15th Romanian National School of Algebra, Constanţa, Romania, Organizer, September, 2006
 Algebraic Geometry and Combinatorics session at the Summer Institute in Algebraic Geometry (Seattle, WA), Organizer, August, 2005
 Special session on Modern Schubert Calculus, Organizer, October, 2004
 2004 IAS/Park City Math Institute Summer Session on Geometric Combinatorics, Organizer and Steering Committee member, July, 2004
 Special session on Combinatorial Commutative Algebra and Algebraic Geometry, Organizer, May, 2003
