Hubert Bray
Professor of Mathematics
Professor Bray uses differential geometry to understand general relativity, and general relativity to motivate interesting problems in differential geometry. In 2001, he published his proof of the Riemannian Penrose Conjecture about the mass of black holes using geometric ideas related to minimal surfaces, scalar curvature, conformal geometry, geometric flows, and harmonic functions. He is also interested in the largescale unexplained curvature of the universe, otherwise known as dark matter, which makes up most of the mass of galaxies. Professor Bray has proposed geometric explanations for dark matter which he calls "wave dark matter," which motivate very interesting questions about geometric partial differential equations.
Professor Bray has supervised 8 math Ph.D. graduates at Duke from 2006 to 2017. He is currently supervising one math Ph.D. student and one physics Ph.D. student. His most recent Ph.D. graduate, Henri Roesch, proved a Null Penrose Conjecture, open since 1973, as his thesis. While the physical motivation about the mass of black holes is the same as for the Riemannian Penrose Conjecture, the geometry involved is almost unrecognizably different, and may be viewed as a fundamental result about null geometry.
Professor Bray has supervised 8 math Ph.D. graduates at Duke from 2006 to 2017. He is currently supervising one math Ph.D. student and one physics Ph.D. student. His most recent Ph.D. graduate, Henri Roesch, proved a Null Penrose Conjecture, open since 1973, as his thesis. While the physical motivation about the mass of black holes is the same as for the Riemannian Penrose Conjecture, the geometry involved is almost unrecognizably different, and may be viewed as a fundamental result about null geometry.
Current Appointments & Affiliations
 Professor of Mathematics, Mathematics, Trinity College of Arts & Sciences 2004
Contact Information
 189 Physics Bldg, Durham, NC 27710
 Box 90320, Durham, NC 277080320
 hubert.bray@duke.edu (919) 7578428
 Professor Bray's Main Webpage
 Background

Education, Training, & Certifications
 Ph.D., Stanford University 1997
 B.A., Rice University 1992

Previous Appointments & Affiliations
 Director of Undergraduate Studies in the Department of Mathematics, Mathematics, Trinity College of Arts & Sciences 2019  2022
 Professor of Physics, Physics, Trinity College of Arts & Sciences 2019  2022
 Professor in the Department of Physics, Physics, Trinity College of Arts & Sciences 2011  2019
 Recognition

In the News

JUL 3, 2014 Duke Today 
JUL 3, 2014 
MAR 17, 2014


Awards & Honors
 Community Service Award. Duke University’s Office of Durham and Regional Affairs. 2014
 Fellow of the AMS, Inaugural Class. The American Mathematical Society. January 2013
 Sloan Research FellowshipMathematics. Alfred P. Sloan Foundation. 2001
 1997  1999: Mathematical Sciences Postdoctoral Research Fellowship. National Science Foundation. 1997
 Expertise

Subject Headings
 Research

Selected Grants
 Time Flat Curves and Surfaces, Geometric Flows, and the Penrose Conjecture awarded by National Science Foundation 2014  2018
 Scalar Curvature, the Penrose Conjecture, and the Axioms of General Relativity awarded by National Science Foundation 2010  2014
 Geometric Analysis Applied to General Relativity awarded by National Science Foundation 2007  2010
 Scalar Curvature, Geometric Flow, and the General Penrose Conjecture awarded by National Science Foundation 2005  2008

External Relationships
 American Scholastic Convention
 Publications & Artistic Works

Selected Publications

Academic Articles

Bray, Hubert, Daniel Stern, Marcus Khuri, and Demetre Kazaras. “Harmonic Functions and The Mass of 3Dimensional Asymptotically Flat Riemannian Manifolds,” November 15, 2019.Link to Item

Bray, Hubert, and Daniel Stern. “Scalar curvature and harmonic oneforms on threemanifolds with boundary,” November 15, 2019.Link to Item

Bray, Hubert, Zhenhua Liu, Yiyue Zhang, and Feng Gui. “Proof of Bishop's volume comparison theorem using singular soap bubbles,” March 29, 2019.Link to Item

Bray, H., B. Hamm, S. Hirsch, J. Wheeler, and Y. Zhang. “Flatly foliated relativity.” Pure and Applied Mathematics Quarterly 15, no. 2 (January 1, 2019): 707–47. https://doi.org/10.4310/PAMQ.2019.v15.n2.a4.Full Text

Bray, H. L., and W. P. Minicozzi. “Preface.” Notices of the American Mathematical Society 65, no. 11 (December 1, 2018): 1412–13. https://doi.org/10.1090/noti1748.Full Text

Bray, H. L., and W. P. Minicozzi. “The mathematics of richard schoen.” Notices of the American Mathematical Society 65, no. 12 (December 1, 2018): 1349–1176. https://doi.org/10.1090/NOTI1749.Full Text

Bray, H., and H. Roesch. “Proof of a Null Geometry Penrose Conjecture.” Notices of the American Mathematical Society. 65 (February 1, 2018).

Bray, H. L., J. L. Jauregui, and M. Mars. “Time Flat Surfaces and the Monotonicity of the Spacetime Hawking Mass II.” Annales Henri Poincare 17, no. 6 (June 1, 2016): 1457–75. https://doi.org/10.1007/s0002301504202.Full Text

Bray, H. L., and J. L. Jauregui. “On curves with nonnegative torsion.” Archiv Der Mathematik 104, no. 6 (June 29, 2015): 561–75. https://doi.org/10.1007/s0001301507670.Full Text

Bray, H. L., and J. L. Jauregui. “Time Flat Surfaces and the Monotonicity of the Spacetime Hawking Mass.” Communications in Mathematical Physics 335, no. 1 (April 1, 2015): 285–307. https://doi.org/10.1007/s0022001421622.Full Text

Bray, H., and A. S. Goetz. “Wave Dark Matter and the TullyFisher Relation,” September 2014.Link to Item

Bray, H. L. “On Dark Matter, Spiral Galaxies, and the Axioms of General Relativity (Accepted).” Ams Contemporary Mathematics Volume 599, no. Geometric Analysis, Mathematical Relativity, and Nonlinear Partial Differential Equations (2013).Link to Item

Bray, H. L., and J. L. Jauregui. “A geometric theory of zero area singularities in general relativity.” Asian Journal of Mathematics 17, no. 3 (2013): 525–60. https://doi.org/10.4310/AJM.2013.v17.n3.a6.Full Text

Bray, H. “On Wave Dark Matter, Shells in Elliptical Galaxies, and the Axioms of General Relativity,” December 22, 2012.Link to Item

Bray, H. L., and M. A. Khuri. “P.D.E.'s Which Imply the Penrose Conjecture.” Asian Journal of Mathematics 15, no. 4 (December 2011): 54.Link to Item

Bray, H. L., and M. A. Khuri. “P. D. E. 'S which imply the penrose conjecture.” Asian Journal of Mathematics 15, no. 4 (January 1, 2011): 557–610. https://doi.org/10.4310/AJM.2011.v15.n4.a5.Full Text

Bray, H., S. Brendle, M. Eichmair, and A. Neves. “AreaMinimizing Projective Planes in 3Manifolds.” Communications on Pure and Applied Mathematics 63, no. 9 (September 1, 2010): 1237–47. https://doi.org/10.1002/cpa.20319.Full Text

Bray, H. L., and M. A. Khuri. “A jang equation approach to the penrose inequality.” Discrete and Continuous Dynamical Systems 27, no. 2 (June 1, 2010): 741–66. https://doi.org/10.3934/dcds.2010.27.741.Full Text

Bray, H., S. Brendle, and A. Neves. “Rigidity of areaminimizing twospheres in threemanifolds.” Communications in Analysis and Geometry 18, no. 4 (January 1, 2010): 821–30. https://doi.org/10.4310/CAG.2010.v18.n4.a6.Full Text

Bray, H. L., and D. A. Lee. “On the Riemannian Penrose inequality in dimensions less than eight.” Duke Mathematical Journal 148, no. 1 (May 1, 2009): 81–106. https://doi.org/10.1215/001270942009020.Full Text

Bray, H., and P. Miao. “On the capacity of surfaces in manifolds with nonnegative scalar curvature.” Inventiones Mathematicae 172, no. 3 (June 1, 2008): 459–75. https://doi.org/10.1007/s002220070102x.Full Text

Bray, H., S. Hayward, M. Mars, and W. Simon. “Generalized inverse mean curvature flows in spacetime.” Communications in Mathematical Physics 272, no. 1 (May 1, 2007): 119–38. https://doi.org/10.1007/s0022000702039.Full Text

Bray, H. L., and A. Neves. “Classification of Prime 3Manifolds with Yamabe Invariant Greater than RP^3.” Annals of Mathematics 159, no. 1 (January 2004): 407–24.Link to Item

Bray, H. L. “Black Holes, Geometric Flows, and the Penrose Inequality in General Relativity.” Notices of the American Mathematical Society 49, no. 11 (2002): 1372–81.

Bray, H. L., and K. Iga. “Superharmonic Functions in R^n and the Penrose Inequality in General Relativity.” Communications in Analysis and Geometry 10, no. 5 (2002): 999–1016.

Bray, H., and F. Finster. “Curvature estimates and the Positive Mass Theorem.” Communications in Analysis and Geometry 10, no. 2 (January 1, 2002): 291–306. https://doi.org/10.4310/CAG.2002.v10.n2.a3.Full Text

Bray, H. L. “Proof of the riemannian penrose inequality using the positive mass theorem.” Journal of Differential Geometry 59, no. 2 (January 1, 2001): 177–267. https://doi.org/10.4310/jdg/1090349428.Full Text

Bray, H., K. McCormick, R. O. Wells, and X. D. Zhou. “Wavelet variations on the Shannon sampling theorem.” Bio Systems 34, no. 1–3 (January 1995): 249–57. https://doi.org/10.1016/03032647(94)01457i.Full Text

MartinezMedina, L. A., H. Bray, and T. Mattos. “On wave dark matter in spiral and barred galaxies (Submitted),” n.d.Link to Item


Book Sections

Bray, H. “On the Positive Mass, Penrose, and ZAS Inequalities in General Dimension.” In Surveys in Geometric Analysis and Relativity in Honor of Richard Schoen’s 60th Birthday, edited by H. Bray and W. Minicozzi. Beijing and Boston: Higher Education Press and International Press, 2011.Link to Item

Bray, H. “On the Positive Mass, Penrose, and ZAS Inequalities in General Dimension.” In Surveys in Geometric Analysis and Relativity in Honor of Richard Schoen’s 60th Birthday, edited by H. Bray and W. Minicozzi. Beijing and Boston: Higher Education Press and International Press, 2011.Link to Item

Bray, H. “The Positve Energy Theorem and Other Inequalities.” In The Encyclopedia of Mathematical Physics, 2005.

Bray, H. “Geometric Flows and the Penrose Inequality.” In Encyclopedia of Mathematical Physics: FiveVolume Set, 510–20, 2004. https://doi.org/10.1016/B0125126662/000584.Full Text

Bray, H., and P. T. Chrusciel. “The Penrose Inequality.” In The Einstein Equations and the Large Scale Behavior of Gravitational Fields (50 Years of the Cauchy Problem in General Relativity), edited by P. T. Chrusciel and H. F. Friedrich. Birkhauser, 2004.Link to Item

Bray, H., and R. M. Schoen. “Recent Proofs of the Riemannian Penrose Conjecture.” In Current Developments in Mathematics, 1–36. Somerville, MA: International Press, 1999.


Other Articles

Bray, Hubert, Sven Hirsch, Demetre Kazaras, Marcus Khuri, and Yiyue Zhang. “Spacetime Harmonic Functions and Applications to Mass.” Edited by Mikhail L. Gromov and H Blaine Lawson. Perspectives in Scalar Curvature. World Scientific, February 1, 2023.Link to Item

Bray, Hubert, and Amir Aazami. “Classification of Preferential Ballot Voting Methods.” Constitutional Political Economy. Springer, n.d.


Conference Papers

Bray, H. L., and A. R. Parry. “Modeling wave dark matter in dwarf spheroidal galaxies.” In 9th Biennial Conference on Classical and Quantum Relativistic Dynamics of Particles and Fields (Iard 2014), Vol. 615, 2015. https://doi.org/10.1088/17426596/615/1/012001.Full Text Link to Item

Bray, H. “Black Holes and the Penrose Inequality in General Relativity.” In Proceedings of the International Congress of Mathematicians, 2:257–72, 2002.Link to Item

Bray, H., and F. Morgan. “An isoperimetric comparison theorem for schwarzschild space and other manifolds.” In Proceedings of the American Mathematical Society, 130:1467–72, 2002. https://doi.org/10.1090/S000299390106186X.Full Text

Bray, H. “A Family of Quasilocal Mass Functionals with Monotone Flows.” In Proceedings of the 14th International Congress on Mathematical Physics, edited by J. C. Zambrini, n.d.


Theses and Dissertations

 Teaching & Mentoring

Recent Courses
 MATH 491: Independent Study 2023
 ETHICS 89S: Special Topic: FirstYear Seminar in Ethics 2022
 MATH 89S: FirstYear Seminar 2022
 MATH 323S: Geometry 2022
 MATH 353: Ordinary and Partial Differential Equations 2022
 MATH 421: Differential Geometry 2022
 MATH 491: Independent Study 2022
 MATH 722: Differential Geometry 2022
 MATH 723S: Geometry 2022
 ECON 394: Research Independent Study 2021
 ETHICS 89S: Special Topic: FirstYear Seminar in Ethics 2021
 MATH 89S: FirstYear Seminar 2021
 MATH 323S: Geometry 2021
 MATH 353: Ordinary and Partial Differential Equations 2021
 MATH 494: Research Independent Study 2021
 MATH 723S: Geometry 2021
 MATH 753: Ordinary and Partial Differential Equations 2021

Advising & Mentoring
 Professor Bray has supervised 10 Ph.D. graduates (9 in math, 1 in physics) at Duke from 2006 to 2021. He is currently advising 3 Ph.D. candidates in math and 1 Ph.D. candidate in physics.
 Scholarly, Clinical, & Service Activities

Presentations & Appearances
 Flatly Foliated Relativity: Gravity without Gravitational Waves. Geometry and Relativity Conference. Erwin Schroedinger Institute. August 2017 2017
 From Pythagoras to Einstein: The Geometry of Space and Time. 6th Annual Rockwell Lectures. University of Iowa. March 27, 2017 2017
 The Intersection of Geometry and Physics. 6th Annual Rockwell Lectures. University of Iowa. March 27, 2017 2017
 Geometric Aspects of Special and General Relativity. Invited Address, AMS Southeastern Sectional Meeting. American Mathematical Society. March 12, 2017 2017
 Flatly Foliated Relativity: Gravity without Gravitational Waves. Union College Math Conference, Special Session on Geometric Analysis. December 2, 2016  December 4, 2016 2016
 Part I: Flatly Foliated Relativity, Part II: The Null Penrose Conjecture. "Geometric Analysis and General Relativity" conference. Banff International Research Station. July 17, 2016  July 22, 2016 2016
 An Elliptic Theory of Gravity between Special and General Relativity. Research Seminar. Stanford University. May 13, 2016 2016
 Flatly Foliated Relativity: Gravity without Gravitational Waves. Geometry Seminar. University of California, Santa Cruz. May 11, 2016 2016
 On Wave Dark Matter and the Axioms of General Relativity. Colloquium . Mathematics Department, University of California, Santa Cruz. May 10, 2016 2016
 From Pythagoras to Einstein: The Geometry of Space and Time (General audience talk). Faculty of Mathematics Public Lectures : University of Waterloo. March 9, 2015 2015
 The Geometry of the Universe (Research Lecture) : University of Waterloo. March 9, 2015 2015
 The Science behind “Trevor the Time Traveler” (General audience talk). Faculty of Mathematics Public Lectures : University of Waterloo. March 9, 2015 2015
 Two lectures on the geometry of space and time (General audience talks). Faculty of Mathematics Public Lectures : University of Waterloo. March 9, 2015 2015
 The G. Milton Wing Lectures, Lecture 3: "Black Holes and the Monotonicity of the Hawking Mass for Time Flat Surfaces" (Research Style Talk). University of Rochester. April 25, 2014 2014
 The G. Milton Wing Lectures, Lecture 2: "On Dark Matter, Galaxies, and the Large Scale Geometry of the Universe" (Colloquium Style Talk). University of Rochester. April 24, 2014 2014
 The G. Milton Wing Lectures (Series of 3 Lectures, 1 General Public, 1 Colloquium Style, 1 Research Talk). April 23, 2014  April 25, 2014 2014
 The G. Milton Wing Lectures, Lecture 1: "From Pythagoras to Einstein: The Geometry of Space and Time" (Broad Audience Talk) : University of Rochester. April 23, 2014 2014
 The G. Milton Wing Lectures, Lecture 1: "From Pythagoras to Einstein: The Geometry of Space and Time". October 25, 2013 2013
 The G. Milton Wing Lectures, Lecture 2: "On Dark Matter, Galaxies, and the Large Scale Geometry of the Universe". October 25, 2013 2013
 The G. Milton Wing Lectures, Lecture 3: "Black Holes and the Monotonicity of the Hawking Mass for Time Flat Surfaces". October 25, 2013 2013
 The G. Milton Wing Lectures. October 25, 2013 2013
 On the Monotonicity of the Hawking Mass for Time Flat Surfaces (Research Talk). New York General Relativity Seminar. Columbia University. October 11, 2013 2013
 On Wave Dark Matter and the Geometry of Galaxies (Research Talk). JDG Conference. Lehigh University . May 24, 2013 2013
 On Dark Matter, Galaxies, and the Large Scale Geometry of the Universe (Research Talk). The 20th Southern California Geometric Analysis Seminar. UCSan Diego. February 24, 2013 2013
 On Dark Matter, Spiral Galaxies, and the Axioms of General Relativity (Research Talk). The 41st Barrett Memorial Lectures in Mathematical Relativity. University of Tennessee, Knoxville. May 12, 2011 2011
 On Dark Matter, Spiral Galaxies, and the Axioms of General Relativity (Research Talk). The 26th Annual Geometry Festival. University of Pennsylvania. April 15, 2011 2011
 From Pythagoras to Einstein: How Geometry Describes the LargeScale Structure of the Universe (Part 1) (Broad Audience Talk). Duke University Graduate Student Recruiting Weekend. Duke University. March 26, 2011 2011
 From Pythagoras to Einstein: How Geometry Describes the LargeScale Structure of the Universe (Part 2) (Broad Audience Talk). Duke University Graduate Student Recruiting Weekend. Duke University. March 26, 2011 2011
 From Black Holes and the Big Bang to Dark Energy and (maybe even) Dark Matter: Successes of Einstein's Theory of General Relativity (Broad Audience Talk) : 45 minutes. University of Tennessee. December 13, 2010 2010
 Dark Matter in Galaxies (Dark Matter Awareness Week talk) by Hubert Bray and Andriy Badin (Survey Talk). Dark Matter Awareness Week. Duke University. December 6, 2010 2010
 An Overview of General Relativity (Broad Audience Talk). Duke University Graduate/Faculty Seminar. Duke University. October 3, 2008 2008
 What Do Black Holes and Soap Bubbles Have in Common? (Broad Audience Talk). Duke University Graduate/Faculty Seminar. Duke University. September 21, 2007 2007
 Black Holes and the Curvature of Spacetime (Broad Audience Talk). Michigan State University. November 7, 2005 2005
 Negative Point Mass Singularities in General Relativity (Research Talk). Sir Isaac Newton Institute, Cambridge, England. August 2005 2005
 Black Holes, the Penrose Conjecture, and Quasilocal Mass (Research Talk). 50 Years of the Cauchy Problem in General Relativity. Cargese Summer School  Cargese, Corsica, France. August 2002 2002
 Generalization of the Hawking Mass (Research Talk). 50 Years of the Cauchy Problem in General Relativity. Cargese Summer School  Cargese, Corsica, France. August 2002 2002
 Proof of the Poincare Conjecture for 3Manifolds with Yamabe Invariant Greater than RP^3 (Research Talk). 50 Years of the Cauchy Problem in General Relativity. Cargese Summer School  Cargese, Corsica, France. August 2002 2002
 Black Holes, Minimal Surfaces, and Geometric Flows (Research Talk). Duke Math Journal Conference. Duke University. April 28, 2001 2001
 Quasilocal Mass and Black Holes in General Relativity (Research Talk) : Duke University . Geometry Seminar. April 28, 2001 2001
 Proof of the Riemannian Penrose Conjecture (Research Talk). Institute for Theoretical Physics  UC Santa Barbara. June 25, 1999 1999

Outreach & Engaged Scholarship
 Presenter to Harrison Do's 6th Grade Science Classes. Magellan Charter School. November 14, 2016 2016
 Presenter to Harrison Do's 6th Grade Science Classes. Magellan Charter School. November 3, 2015 2015
 Presenter to Harrison Do's 6th Grade Science Classes. Magellan Charter School. March 6, 2015 2015
 Math and Science Tutor, Organizer of Mathletes Program. Emily K Center. 2011  2016 2011  2016

Service to the Profession
 Organizer (with Otis Chodosh, Greg Galloway, and Pengzi Miao). Special Session in "Geometric Analysis and General Relativity" . American Mathematical Society. March 10, 2017  March 12, 2017 2017
 Organizer (with Andrew Cooper). AMS Special Session on Geometric Analysis. American Mathematical Society. November 8, 2014  November 9, 2014 2014
 Organizer (with Greg Galloway, Rafe Mazzeo, and Natasa Sesum). Park City Mathematics Institute Summer Program on Geometric Analysis. June 30, 2013  July 20, 2013 2013
 Organizer (with Mark Stern, Lenny Ng, Carla Cederbaum, Luca DiCerbo, and Chris Cornwell). 27th Annual Geometry Festival. April 27, 2012  April 29, 2012 2012
 Organizer and Presenter at Duke University (along with Andriy Badin). Dark Matter Awareness Week. December 6, 2010 2010
 Organizer (with Benoit Charbonneau, Dick Hain, and Patrick Eberlein). 23rd Annual Geometry Festival held at Duke University. April 25, 2008  April 27, 2008 2008
 Organizer (with Richard Schoen and Jim Isenberg). AIM / Stanford Relativity Workshop. 2002 2002
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