Math @ Duke

Ezra Miller, Professor of Mathematics and Statistical Science
 Contact Info:
 Education:
PhD  University of California, Berkeley (Math)  2000 
ScB  Brown University (Math)  1995 
AB  Brown University (Music)  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 • 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)
 with Megan Owen and Scott Provan, Polyhedral computational geometry for averaging metric phylogenetic trees
(Submitted, 2012) (43 pages.) [math.MG/1211.7046] [abs]
 with Igor Pak, Metric combinatorics of convex polyhedra: cut loci and nonoverlapping unfoldings,
Discrete and Computational Geometry, vol. 39 no. 13
(2008),
pp. 339388 [MR2008m:52027], [math.MG/0312253]
 with Allen Knutson and Mark Shimozono, Four positive formulae for type A quiver polynomials,
Inventiones Mathematicae, vol. 166 no. 2
(2006),
pp. 229325 [MR2007k:14098], [math.AG/0308142]
 with Allen Knutson, Gröbner geometry of Schubert polynomials,
Annals of Mathematics (2), vol. 161 no. 3
(2005),
pp. 12451318 [MR2006i:05177], [math.AG/0110058]
 with Bernd Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics, vol. 227
(2005),
pp. xiv+417, SpringerVerlag, New York, ISBN 0387223568 [MR2006d:13001]
 with Laura Felicia Matusevich and Uli Walther, Homological methods for hypergeometric families,
Journal of the American Mathematical Society, vol. 18 no. 4
(2005),
pp. 919941 (electronic) [MR2007d:13027], [math.AG/0406383]
 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
 Combinatorics in geometry and algebra with applications to the natural sciences, National Science Foundation, DMS1001437, 2010/072015/06.
 Conferences Organized
 Triangle Lectures in Combinatorics, Steering Committee Member, 2010  present
 Oberwolfach miniworkshop: Asymptotic statistics on stratified spaces (29 Sep  3 Oct 2014), Organizer, 
 20132014 SAMSI yearlong program on LowDimensional Structure in HighDimensional Systems (LDHD), Directorate Organizer, 2011  2014
 SAMSI Summer Program on Neuroimaging Data Analysis (NDA), Directorate Organizer, June 2013
 Experimental and theoretical methods in algebra, geometry, and topology, Mangalia, Romania (June 2013), Scientific Committee Member, 
 20122013 MSRI yearlong program on Commutative Algebra, Organizer, 2009  2013
 Triangle Lectures in Combinatorics, 11 February 2012, Organizer (with Sonja Mapes and Christine Berkesch), 
 MBI: Workshop on statistics, geometry, and combinatorics on stratified spaces arising from biological problems (May 2012), Lead Organizer, 
 2011 Formal Power Series and Algebraic Combinatorics (FPSAC), Reykjavík, Iceland, Program Committee Member, 
 SAMSI Working Group: Analysis of data sampled from stratified spaces, Working Group leader and administrator, September, 2010  June, 2011
 Triangle Lectures in Combinatorics, Organizer, September 25, 2010
 Combinatorial Lie theory and applications, Working group session organizer (with William Graham, UGA), October, 2009
 2009 Formal Power Series and Algebraic Combinatorics (FPSAC), Research Institute for Symbolic Computation (RISC), Linz, Austria, Program Committee Member, July, 2009
 Triangle Lectures in Combinatorics, Organizer, February 06, 2009
 Exploratory workshop on combinatorial commutative algebra and computer algebra, Mangalia, Romania, Scientific Committee Member, May, 2008
 Workshop on Combinatorial Game Theory, Banff International Research Station (BIRS), Organizer, 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


dept@math.duke.edu
ph: 919.660.2800
fax: 919.660.2821
 
Mathematics Department
Duke University, Box 90320
Durham, NC 277080320

