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Ezra Miller, Professor

Ezra Miller

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 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 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; mass-action kinetics of chemical reactions;
computational geometry, symbolic computation, and combinatorial game
theory; Lie 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 and lungs, 3D
folded protein structures, and photographs of fruit fly wings for
developmental morphological studies.

Contact Info:
Office Location:  209 Physics Bldg, 120 Science Drive, Durham, NC 27708
Email Address: send me a message
Web Pages:  http://math.duke.edu/~ezra
http://math.duke.edu/~ezra/talks.html

Teaching (Fall 2024):

  • MATH 221.01, LINEAR ALGEBRA Synopsis
    Physics 259, TuTh 11:45 AM-01:00 PM
  • MATH 721.01, LINEAR ALGEBRA & APPLICA Synopsis
    Physics 259, TuTh 11:45 AM-01:00 PM
Teaching (Spring 2025):

  • MATH 403.01, ADVANCED LINEAR ALGEBRA Synopsis
    Physics 205, TuTh 11:45 AM-01:00 PM
  • MATH 703.01, ADVANCED LINEAR ALGEBRA Synopsis
    Physics 205, TuTh 11:45 AM-01:00 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, Berkeley2000
ScBBrown University (Math)1995
ABBrown University (Music)1995
B.S.Brown University1995
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 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.

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
    • 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)

    1. 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, American Mathematical Society (AMS), ISSN 0002-9920 [doi]
    2. 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. 198-218 [arXiv:1411.6652], [1411.6652v1], [doi]  [abs]
    3. 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. 1-34, Institute of Mathematical Statistics [repository], [doi]  [abs]
    4. 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]
    5. 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. 919-941, ISSN 0894-0347 [MR2007d:13027], [math.AG/0406383], [doi]
    6. with Miller, E; Pak, I, Metric combinatorics of convex polyhedra: Cut loci and nonoverlapping unfoldings, Discrete and Computational Geometry, vol. 39 no. 1-3 (January, 2008), pp. 339-388, Springer Nature, ISSN 0179-5376 [MR2008m:52027], [math.MG/0312253], [doi]  [abs]
    7. with Knutson, A; Miller, E; Shimozono, M, Four positive formulae for type A quiver polynomials, Inventiones Mathematicae, vol. 166 no. 2 (November, 2006), pp. 229-325, Springer Nature, ISSN 0020-9910 [MR2007k:14098], [math.AG/0308142], [doi]  [abs]
    8. 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 (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]
    9. 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]
    10. 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 (January, 2011), pp. 57-84, European Mathematical Publishing House, ISSN 1435-9855 [math.AG/0808.2785], [DOI:10.4171/JEMS/244], [doi]  [abs]
    11. with Guo, A; Miller, E, Lattice point methods for combinatorial games, Advances in Applied Mathematics, vol. 46 no. 1-4 (January, 2011), pp. 363-378, Elsevier BV, ISSN 0196-8858 ([math.CO/0908.3473] [math.CO/1105.5420].) [repository], [doi]  [abs]
    12. Miller, E, The Alexander duality functors and local duality with monomial support, Journal of Algebra, vol. 231 no. 1 (September, 2000), pp. 180-234, Elsevier BV [MR2001k:13028], [pdf], [doi]  [abs]
    Selected Invited Lectures

    1. Lattice games, July, 2009, Halifax, Nova Scotia (Games-at-Dal)    
    2. Applications of binomial commutative algebra, June, 2009, Norway (Abel Symposium) [speakers]    
    3. Potential applications of commutative algebra to combinatorial game theory, April, 2009, Oberwolfach    
    4. Equivariant transversality and K-theoretic positivity, March, 2009, MSRI [show_workshop]    
    5. Geometry of flags and permutations, March, 2009, Thessaloniki, Greece    
    6. Cellular resolutions of multiplier ideals of sums, May, 2008, Mangalia, Romania    
    7. Nonoverlapping unfolding of polyhedra, December, 2007, American Institute of Math, Palo Alto    
    8. Combinatorics of Horn hypergeometric series, July, 2007, Tianjin, China (FPSAC plenary speaker)    
    9. Old hypergeometric mysteries and new toric algebra, July, 2007, Medellín, Colombia    
    10. Kleiman-Bertini theorems for sheaf tensor products, May, 2007, CRM Montréal, Canada    
    11. Unfolding polyhedra, October, 2006, Cincinnati, Ohio (AMS meeting plenary talk)    
    12. Multigraded commutative algebra, September, 2006, Constanţa, Romania    
    13. Lattice basis ideals and Horn systems, May, 2006, Luminy, France    
    14. Positivity by degeneration, December, 2005, Taichung, Taiwan (AMS-TMS)    
    15. Gröbner geometry of quiver polynomials, September, 2005, UNAM, Mexico City    
    16. Homological methods for hypergeometric families, July, 2005, Lisbon, Portugal    
    17. Overview of geometric combinatorics, July, 2004, Park City, Utah (IAS/PCMI)    
    18. Computing injective resolutions over semigroup rings, June, 2003, Kaiserslautern, Germany (MEGA)    
    19. Positive combinatorial formulae for quiver polynomials, June, 2003, Seville, Spain (AMS-RSME)    
    20. Minors in products of matrices, April, 2003, Banff, Canada (BIRS workshop)    
    21. Hilbert schemes of points in the plane, September, 2002, MSRI Introductory workshop    
    22. Gröbner geometry of Schubert polynomials, July, 2002, Melbourne, Australia (FPSAC one-hour talk)    
    23. Gröbner geometry of Schubert polynomials via determinantal ideals, June, 2002, Trento, Italy    
    24. Embedding planar graphs in staircases, May, 2001, Morelia, Mexico (AMS-SMM)    
    25. The Čech hull, December, 1999, Guanajuato, Mexico (Local Cohomology workshop)    
    26. Alexander duality & local duality with monomial support, June, 1999, Essen, Germany    
    27. Alexander duality for arbitrary monomial ideals, April, 1998, Kyoto, Japan (RIMS)    
    Recent Grant Support

    • RTG: Linked via L-functions: training versatile researchers across number theory, National Science Foundation, 2023/10-2028/09.      
    • R2 [Reciprocal Relationships]: Mentorships to Strengthen and Sustain STEM Teachers, National Science Foundation, 2020/10-2025/09.      
    • HDR TRIPODS: Innovations in Data Science: Integrating Stochastic Modeling, Data Representation, and Algorithms, National Science Foundation, 2019/10-2023/09.      
    Conferences Organized

     

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    fax: 919.660.2821

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