Ezra Miller, Professor

Contact Info:

Office Location:	209 Physics
Office Phone:	(919) 660-2846
Email Address:
Web Page:	http://math.duke.edu/~ezra

Education:

PhD	University of California, Berkeley (Math)	2000
ScB	Brown University (Math)	1995
AB	Brown University (Music)	1995

Specialties:

Geometry
Algebra
Applied Math

Research Interests: Combinatorial aspects of geometry and algebra, with applications

Professor Miller's research centers around problems in geometry, algebra, topology, combinatorics, and computation originating in mathematics and the sciences, including biology, physics, chemistry, computer science, medical imaging, and statistics. A unifying idea in this research has been to isolate or exploit combinatorial structures that govern or arise from continuous contexts. For example, if a continuous process carries underlying discrete data, then those data might be harnessed to produce algorithms for the continuous process. On theother hand, the goal could be to understand the combinatorics rather than the geometry; the geometry then serves as a vehicle for interpolating between different interpretations of the combinatorics.

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 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, branching structures in evolutionary biology and medical imaging; mass-action kinetics of chemical reactions; computational geometry, symbolic computation, and combinatorial game theory; and statistics of object data, sampled from markedly non-Euclidean spaces.

Areas of Interest:

algebraic varieties, polyhedra, manifolds

commutative algebra and representation theory

combinatorics of games, simpicial complexes, trees

algorithms in algebra and geometry

applications to biology, computer science, chemistry, physics, statistics

Keywords:

algebra • geometry • combinatorics • algorithms • statistics • applications

Curriculum Vitae

Current Ph.D. Students   (Former Students)

• Chris O'Neill  
• Kangkang Wang  

Postdocs Mentored

• Christine Berkesch (August, 2011 - present)  
• Anne Shiu (September, 2010 - August, 2011)  
• Sonja Mapes (2009 - present)  
• Megan Owen (January, 2009 - 2010)  
• Stephen Griffeth (2006 - 2009)  
• Drew Armstrong (2006 - 2009)  
• Milena Hering (2006 - 2008)  
• Alexander Yong (2005 - 2008)  
• Calin Chindris (2005 - 2008)  
• Jeremy Martin (2003 - 2005)  

Undergraduate Research Supervised

• Ezgi Kantarcı (May, 2010 - September, 2010)
  applying to math grad school 
• 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 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]
2. 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]
3. 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]
4. 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]
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. The Alexander duality functors and local duality with monomial support, Journal of Algebra, vol. 231 no. 1 (2000), pp. 180-234 [MR2001k:13028], [pdf]

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.      
• Triangle Lectures in Combinatorics, National Science Foundation, DMS-1000130, 2010/03-2011/03.      

Conferences Organized

• Triangle Lectures in Combinatorics, Steering Committee Member, 2010 - present  
• 2012-2013 MSRI year-long program on Commutative Algebra, Organizer, 2009 - present  
• Experimental and theoretical methods in algebra, geometry, and topology, Mangalia, Romania (June 2013), Scientific Committee Member, -  
• 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), Principal 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