**Office Location:** Physics 189**Office Phone:** 660-2818**Email Address:** bray@math.duke.edu

**Specialties:**

Geometry

Analysis

Mathematical Physics

**Education:**

PhD, Stanford University (adviser: Richard Schoen), 1997

BA, BS,, Rice University, Math and Physics, summa cum laude, 1992

**Research Categories:** *Geometric Analysis, General Relativity, Theoretical Astrophysics*

**Teaching (Spring 2015):**

- Math 421.01,
*Differential geometry*Synopsis- Physics 047, TuTh 11:45 AM-01:00 PM

**Representative Publications**

- H.L. Bray and A.S. Goetz,
*Wave Dark Matter and the Tully-Fisher Relation*(Preprint, September, 2014) [arXiv:1409.7347], [7347] . - H.L. Bray, J. L. Jauregui, M. Mars,
*Time Flat Surfaces and the Monotonicity of the Spacetime Hawking Mass II*(Preprint, February, 2014) [arXiv:1402.3287 [math.DG]], [3287] [abs]. - H.L. Bray and J. L. Jauregui,
*Time Flat Surfaces and the Monotonicity of the Spacetime Hawking Mass*, Communications in Mathematical Physics (Accepted, April, 2014) [arXiv:1310.8638 [math.DG]], [8638] . - H.L. Bray and J. L. Jauregui,
*On Curves with Nonnegative Torsion*(Preprint, December, 2013) [arXiv:1312.5171 [math.DG]], [5171] . - H.L. Bray and A. R. Parry,
*Modeling Wave Dark Matter in Dwarf Spheroidal Galaxies*(Preprint, January, 2013) [html] . - H.L. Bray,
*On Wave Dark Matter, Shells in Elliptical Galaxies, and the Axioms of General Relativity*(Preprint, December 22, 2012) [html] . - H.L. Bray,
*On Dark Matter, Spiral Galaxies, and the Axioms of General Relativity*, AMS Contemporary Mathematics Volume, "Geometric Analysis, Mathematical Relativity, and Nonlinear Partial Differential Equations", vol. 599 (2013), American Mathematical Society [html] . - H.L. Bray,
*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 Hubert L. Bray and William P. Minicozzi (2011), Higher Education Press and International Press [arXiv:1101.2230v1], [pdf] [author's comments]. - H.L. Bray and M.A. Khuri,
*P.D.E.'s Which Imply the Penrose Conjecture*, Asian Journal of Mathematics, vol. 15 no. 4 (December, 2011), pp. 54, International Press [pdf] [author's comments]. - H.L. Bray and M.A. Khuri,
*A Jang Equation Approach to the Penrose Inequality*, Discrete and Continuous Dynamical Systems A, vol. 27 no. 2 (June, 2010) [arXiv:0910.4785v1] . - H.L. Bray, S. Brendle, M. Eichmair, A. Neves,
*Area-Minimizing Projective Planes in 3-Manifolds*, Communications in Pure and Applied Mathematics (2010) [arXiv:0909.1665v1] . - H.L. Bray, S. Brendle, A. Neves,
*Rigidity of Area-Minimizing Two-Spheres in Three-Manifolds*, Communications in Analysis and Geometry (2010) [arXiv:1002.2814] . - H.L. Bray and J.L. Jauregui,
*A Geometric Theory of Zero Area Singularities in General Relativity*, Asian Journal of Mathematics, vol. 17 no. 3 (September, 2013), pp. 525-560 [arXiv:0909.0522v1] . - H.L. Bray, D.A. Lee,
*On the Riemannian Penrose Inequality in Dimension Less Than Eight*, Duke Mathematical Journal, vol. 148 no. 1 (2009), pp. 81-106 [arXiv:0705.1128v1], [pdf] . - H.L. Bray, P. Miao,
*On the Capacity of Surfaces in Manifolds with Nonnegative Scalar Curvature*, Inventiones Mathematicae, vol. 172 no. 3 (June, 2008) [arXiv:0707.3337v1] . - H.L. Bray, S. Hayward, M. Mars, W. Simon,
*Generalized Inverse Mean Curvature Flows in Spacetime*, Communications in Mathematical Physics, vol. 272 no. 1 (May, 2007), pp. 119-138 [arXiv:gr-qc/0603014v1] . - H.L. Bray and A. Neves,
*Classification of Prime 3-Manifolds with Yamabe Invariant Greater than RP^3*, Annals of Mathematics, vol. 159 no. 1 (2004), pp. 407--424 [p09] . - H.L. Bray,
*A Family of Quasi-local Mass Functionals with Monotone Flows*, in Proceedings of the 14th International Congress on Mathematical Physics, Lisbon, Portugal, 2003, edited by Jean-Claude Zambrini (2003) [Family%20of%20Quasi-local%20Mass%20Functionals%20with%20Monotone%20Flows&f=false] . - H. L. Bray and F. Finster,
*Curvature Estimates and the Positive Mass Theorem*, Communications in Analysis and Geometry, vol. 10 no. 2 (2002), pp. 291--306 [arXiv:math/9906047v3] . - H.L. Bray and K. Iga,
*Superharmonic Functions in R^n and the Penrose Inequality in General Relativity*, Communications in Analysis and Geometry, vol. 10 no. 5 (2002), pp. 999--1016 . - H. L. Bray and F. Morgan,
*An Isoperimetric Comparison Theorem for Schwarzschild Space and Other Manifolds*, Proceedings of the American Mathematical Society, vol. 130 no. 5 (2002), pp. 1467--1472 [pdf] . - H. L. Bray,
*Proof of the Riemannian Penrose Inequality Using the Positive Mass Theorem*, Journal of Differential Geometry, vol. 59 no. 2 (2001), pp. 177--267 [arXiv:math/9911173v1], [pdf] . - H. L. Bray and R.M. Schoen,
*Recent Proofs of the Riemannian Penrose Conjecture*, in Current Developments in Mathematics, 1999 (Cambridge, MA) (1999), pp. 1--36, Int. Press, Somerville, MA . - H.L. Bray,
*The Positve Energy Theorem and Other Inequalities in GR*, in The Encyclopedia of Mathematical Physics (2005) . - H.L. Bray 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 (2004), Birkhauser [arXiv:gr-qc/0312047v2] . - H.L. Bray,
*Black Holes, Geometric Flows, and the Penrose Inequality in General Relativity*, Notices of the American Mathematical Society, vol. 49 no. 11 (2002), pp. 1372--1381 . - H.L. Bray,
*Black Holes and the Penrose Inequality in General Relativity*, in Proceedings of the International Congress of Mathematicians, Beijing, China, 2002, vol. 2 (2002), pp. 257-272 [arXiv:math/0304261v1] . - H.L. Bray, K. McCormick, R.O. Wells, Jr., Xiao-dong Zhou,
*Wavelet Variations on the Shannon Sampling Theorem*, Biosystems, vol. 34 (1995), pp. 249-257, Elsevier Science Ireland [science] [author's comments]. - H.L. Bray,
*The Penrose Inequality in General Relativity and Volume Comparison Theorems Involving Scalar Curvature*(1997) (thesis, Stanford University.) [arXiv:0902.3241v1] .

**Current Ph.D. Students**(Former Students)- Henri Roesch
- Andrew Goetz

**Postdocs Mentored**- Carla Cederbaum (July, 2011 - January, 2013)
- Ye Li (2009 - 2011)
- Dan Lee (2005 - 2008)
- Fernando Schwartz (2005 - 2008)

**Selected Invited Lectures***The G. Milton Wing Lectures*, April 23 - 25, 2014, University of Rochester (Series of 3 Lectures, 1 General Public, 1 Colloquium Style, 1 Research Talk) [html]*The G. Milton Wing Lectures, Lecture 1: "From Pythagoras to Einstein: The Geometry of Space and Time"*, April 23, 2014, University of Rochester (Broad Audience Talk) [pdf]*The G. Milton Wing Lectures, Lecture 2: "On Dark Matter, Galaxies, and the Large Scale Geometry of the Universe"*, April 24, 2014, University of Rochester (Colloquium Style Talk) [pdf]*The G. Milton Wing Lectures, Lecture 3: "Black Holes and the Monotonicity of the Hawking Mass for Time Flat Surfaces"*, April 25, 2014, University of Rochester (Research Style Talk - link to paper) [pdf]*On the Monotonicity of the Hawking Mass for Time Flat Surfaces*, October 11, 2013, New York General Relativity Seminar, Columbia University (Research Talk)*On Wave Dark Matter and the Geometry of Galaxies*, May 24, 2013, JDG Conference, Lehigh University (Research Talk) [html]*On Dark Matter, Galaxies, and the Large Scale Geometry of the Universe*, February 24, 2013, The 20th Southern California Geometric Analysis Seminar, UC-San Diego (Research Talk, slides only, but video of the similar talk below is available) [pdf]*On Dark Matter, Spiral Galaxies, and the Axioms of General Relativity*, May 12, 2011, The 41st Barrett Memorial Lectures in Mathematical Relativity, University of Tennessee, Knoxville (Research Talk, with video) [available here]*On Dark Matter, Spiral Galaxies, and the Axioms of General Relativity*, April 15, 2011, The 26th Annual Geometry Festival at the University of Pennsylvania (Research Talk - slides only - 100MB; the talk above was very similar and has video) [pdf]*Dark Matter in Galaxies (Dark Matter Awareness Week talk) by Hubert Bray and Andriy Badin*, December 6, 2010, Duke University (Survey Talk) 112 minutes total - Part I (66 minutes): Bray describes the astronomical evidence for dark matter in galaxies. Part II (21 minutes, 66 minutes in): Badin describes searches for dark matter particles. Part III (25 minutes, 87 minutes in): Bray describes his work on a general relativity approach to dark matter described by a scalar field satisfying the Klein-Gordon equation as a possible explanation for spiral patterns in galaxies [video.html]*From Black Holes and the Big Bang to Dark Energy and (maybe even) Dark Matter: Successes of Einstein's Theory of General Relativity*, December 13, 2010, University of Tennessee (Broad Audience Talk) 45 minutes [video.html]*From Pythagoras to Einstein: How Geometry Describes the Large-Scale Structure of the Universe (Part 1)*, March 26, 2011, Duke University Graduate Student Recruiting Weekend (Broad Audience Talk) [video.html]*From Pythagoras to Einstein: How Geometry Describes the Large-Scale Structure of the Universe (Part 2)*, March 26, 2011, Duke University Graduate Student Recruiting Weekend (Broad Audience Talk) [video.html]*An Overview of General Relativity*, October 3, 2008, Duke University Graduate/Faculty Seminar (Broad Audience Talk) [video.html]*What Do Black Holes and Soap Bubbles Have in Common?*, September 21, 2007, Duke University Graduate/Faculty Seminar (Broad Audience Talk) [video.html]*Black Holes and the Curvature of Spacetime*, November 7, 2005, Michigan State University (Broad Audience Talk) [html]*Negative Point Mass Singularities in General Relativity*, August, 2005, Sir Isaac Newton Institute, Cambridge, England (Research Talk) [available here]*Generalization of the Hawking Mass*, August, 2002, 50 Years of the Cauchy Problem in General Relativity, Cargese, Corsica (Research Talk) [available here]*Proof of the Poincare Conjecture for 3-Manifolds with Yamabe Invariant Greater than RP^3*, August, 2002, 50 Years of the Cauchy Problem in General Relativity, Cargese, Corsica (Research Talk) [available here]*Black Holes, the Penrose Conjecture, and Quasi-local Mass*, August, 2002, 50 Years of the Cauchy Problem in General Relativity, Cargese, Corsica (Research Talk) [available here]*Black Holes, Minimal Surfaces, and Geometric Flows*, April 28, 2001, Duke Math Journal Conference, Duke University (Research Talk) [video.html]*Quasi-local Mass and Black Holes in General Relativity*, April 28, 2001, Geometry Seminar, Duke University (Research Talk) [video.html]*Proof of the Riemannian Penrose Conjecture*, June 25, 1999, Institute for Theoretical Physics, UC Santa Barbara (Research Talk) [available here]