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Ezra N Miller, Professor of Mathematics and Statistical Science

Ezra N Miller
Contact Info:
Office Location:  209 Physics
Office Phone:  (919) 660-2846
Email Address: send me a message
Web Page:


PhDUniversity of California, Berkeley (Math)2000
ScBBrown University (Math)1995
ABBrown University (Music)1995

Applied Math
Mathematical Biology
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 • geometry • combinatorics • algorithms • statistics • neuroscience • evolution • applications

    Curriculum Vitae
    Current Ph.D. Students   (Former Students)

    Postdocs Mentored

    Undergraduate Research Supervised

    • Victoria Cheng (2013)
      topic: mathematical biology 
    • Rowena (Jingxing) Gan (2012 - present)
      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)
      MIT Computer Science grad student; 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. with Megan Owen and Scott Provan, Polyhedral computational geometry for averaging metric phylogenetic trees (Submitted, 2012) (43 pages.) [math.MG/1211.7046]  [abs]
    2. with Igor Pak, Metric combinatorics of convex polyhedra: cut loci and nonoverlapping unfoldings, Discrete and Computational Geometry, vol. 39 no. 1-3 (2008), pp. 339-388 [MR2008m:52027], [math.MG/0312253]
    3. with Allen Knutson and Mark Shimozono, Four positive formulae for type A quiver polynomials, Inventiones Mathematicae, vol. 166 no. 2 (2006), pp. 229-325 [MR2007k:14098], [math.AG/0308142]
    4. with Allen Knutson, Gröbner geometry of Schubert polynomials, Annals of Mathematics (2), vol. 161 no. 3 (2005), pp. 1245-1318 [MR2006i:05177], [math.AG/0110058]
    5. 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]
    6. with Laura Felicia Matusevich and Uli Walther, Homological methods for hypergeometric families, Journal of the American Mathematical Society, vol. 18 no. 4 (2005), pp. 919-941 (electronic) [MR2007d:13027], [math.AG/0406383]
    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

    • Combinatorics in geometry and algebra with applications to the natural sciences, National Science Foundation, DMS-1001437, 2010/07-2015/06.      
    Conferences Organized

    ph: 919.660.2800
    fax: 919.660.2821

    Mathematics Department
    Duke University, Box 90320
    Durham, NC 27708-0320