Professor Miller's research centers around problems in geometry, algebra, topology, combinatorics, statistics, probability, 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 contraction 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.

Office Location: | 209 Physics Bldg, 120 Science Drive, Durham, NC 27708 |

Office Phone: | (919) 660-2846 |

Email Address: | |

Web Pages: | http://math.duke.edu/~ezrahttp://math.duke.edu/~ezra/cv.html |

**Teaching (Spring 2021):**

- MATH 403.01,
*ADVANCED LINEAR ALGEBRA*Synopsis- Physics 154, TuTh 12:00 PM-01:15 PM

- MATH 403.02,
*ADVANCED LINEAR ALGEBRA*Synopsis- Online ON, TuTh 12:00 PM-01:15 PM

- MATH 703.01,
*ADVANCED LINEAR ALGEBRA*Synopsis- Physics 154, TuTh 12:00 PM-01:15 PM

- MATH 703.02,
*ADVANCED LINEAR ALGEBRA*Synopsis- Online ON, TuTh 12:00 PM-01:15 PM

**Office Hours:****Office hours:**Monday, 16:15 – 17:15 on Zoom

Thursday, 13:15 – 14:30 on Zoom from 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 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
- algebraic varieties, polyhedra, manifolds

**Keywords:**algebra • Algebra • algorithms • applications • biology • Biology • combinatorics • evolution • geometry • Geometry • neuroscience • Probability • statistics • Statistics • topology • Topology

**Current Ph.D. Students**

**Postdocs Mentored**- Justin V. Allman (2015 - present)
- Adam Jaeger (2015 - present)
- Christine Berkesch (August, 2011 - June, 2013)
- Anne Shiu (September, 2010 - August, 2011)
- Sonja Mapes (2009 - 2011)
- 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**- 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

- Victoria Cheng (2013)

**Representative Publications**- Miller, E,
*Fruit flies and moduli: Interactions between biology and mathematics*, Notices of the American Mathematical Society, vol. 62 no. 10 (January, 2015), pp. 1178-1184, American Mathematical Society (AMS), ISSN 0002-9920 [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. 198-218 [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 (January, 2015), Institute of Mathematical Statistics [repository], [doi] [abs] - 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] - 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] - with Miller, E; Pak, I,
*Metric combinatorics of convex polyhedra: Cut loci and nonoverlapping unfoldings*, Discrete & Computational Geometry, vol. 39 no. 1-3 (January, 2008), pp. 339-388, Springer Nature, ISSN 0179-5376 [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. 229-325, Springer Nature, ISSN 0020-9910 [MR2007k:14098], [math.AG/0308142], [doi] [abs] - with Gopalkrishnan, M; Miller, E; Shiu, A,
*A projection argument for differential inclusions, with applications to persistence of mass-action kinetics*, Symmetry, Integrability and Geometry: Methods and Applications, vol. 9 (August, 2013), SIGMA (Symmetry, Integrability and Geometry: Methods and Application) (paper 025, 25 pages.) [math.DS/1208.0874], [DOI:10.3842/SIGMA.2013.025], [doi] [abs] - Kahle, T; Miller, E; O'Neill, C,
*Irreducible decomposition of binomial ideals*, Compositio Mathematica, vol. 152 no. 6 (June, 2016), pp. 1319-1332, Oxford University Press (OUP) [arXiv:1503.02607], [1503.02607], [doi] [abs] - 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] - 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] - 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]

- Miller, E,

**Selected Invited Lectures***Lattice games*, July, 2009, Halifax, Nova Scotia (Games-at-Dal)*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 K-theoretic 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*Kleiman-Bertini 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 (AMS-TMS)*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 (AMS-RSME)*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 one-hour talk)*Gröbner geometry of Schubert polynomials via determinantal ideals*, June, 2002, Trento, Italy*Embedding planar graphs in staircases*, May, 2001, Morelia, Mexico (AMS-SMM)*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***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, 2019/10-2022/09.*Algebraic and Geometric Methods In Data Analysis*, National Science Foundation, 1702395, 2017/09-2020/08.

**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 year-long 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 mini-workshop: Asymptotic statistics on stratified spaces (29 Sep - 3 Oct 2014). February 29, 2012 - October 3, 2014, Organizer : Oberwolfach mini-workshop: Asymptotic statistics on stratified spaces (29 Sep - 3 Oct 2014), -
- 2013-2014 SAMSI year-long program on Low-Dimensional Structure in High-Dimensional Systems (LDHD), Directorate Organizer, 2011 - 2014
- Directorate Organizer : SAMSI Summer Program on Neuroimaging Data Analysis (NDA). December 15, 2012 - July 1, 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 : 2012-2013 MSRI year-long program on Commutative Algebra. 2009 - 2013, Organizer : 2012-2013 MSRI year-long program on Commutative Algebra, 2009 - 2013
- Organizer (with Sonja Mapes and Christine Berkesch) : Triangle Lectures in Combinatorics, 11 February 2012. July 1, 2011 - July 1, 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 6, 2009 - July 1, 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 6, 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