# Hubert Bray

## Professor of Mathematics

### Overview

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 large-scale 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.

### Selected Grants

Time Flat Curves and Surfaces, Geometric Flows, and the Penrose Conjecture awarded by National Science Foundation (Principal Investigator). 2014 to 2018

Scalar Curvature, the Penrose Conjecture, and the Axioms of General Relativity awarded by National Science Foundation (Principal Investigator). 2010 to 2014

Geometric Analysis Applied to General Relativity awarded by National Science Foundation (Principal Investigator). 2007 to 2010

Scalar Curvature, Geometric Flow, and the General Penrose Conjecture awarded by National Science Foundation (Principal Investigator). 2005 to 2008

Bray, H. "On the Positive Mass, Penrose, and ZAS Inequalities in General Dimension." *Surveys in Geometric Analysis and Relativity in Honor of Richard Schoen’s 60th Birthday.* Ed. H Bray and W Minicozzi. Beijing and Boston: Higher Education Press and International Press, 2011.

Bray, H. "The Positve Energy Theorem and Other Inequalities." *The Encyclopedia of Mathematical Physics.* 2005.

Bray, H, and Chrusciel, PT. "The Penrose Inequality." *The Einstein Equations and the Large Scale Behavior of Gravitational Fields (50 Years of the Cauchy Problem in General Relativity).* Ed. PT Chrusciel and HF Friedrich. Birkhauser, 2004.

Bray, H, and Schoen, RM. "Recent Proofs of the Riemannian Penrose Conjecture." *Current Developments in Mathematics.* Somerville, MA: International Press, 1999. 1-36.

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

Bray, HL, Jauregui, JL, and Mars, M. "Time Flat Surfaces and the Monotonicity of the Spacetime Hawking Mass II." *Annales Henri Poincaré* 17.6 (June 2016): 1457-1475.
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Martinez-Medina, LA, Bray, HL, and Matos, T. "On wave dark matter in spiral and barred galaxies." *Journal of Cosmology and Astroparticle Physics* 2015.12 (December 1, 2015): 025-025.
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Bray, HL, and Jauregui, JL. "On curves with nonnegative torsion." *Archiv der Mathematik* 104.6 (June 2015): 561-575.
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Bray, HL, and Jauregui, JL. "Time Flat Surfaces and the Monotonicity of the Spacetime Hawking Mass." *Communications in Mathematical Physics* 335.1 (April 2015): 285-307.
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Bray, H, and Goetz, AS. "Wave Dark Matter and the Tully-Fisher Relation." (September 2014).

Bray, HL, and Jauregui, JL. "A geometric theory of zero area singularities in general relativity." *Asian Journal of Mathematics* 17.3 (2013): 525-560.
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Bray, HL. "On Dark Matter, Spiral Galaxies, and the Axioms of General Relativity (Accepted)." *AMS Contemporary Mathematics Volume* 599.Geometric Analysis, Mathematical Relativity, and Nonlinear Partial Differential Equations (2013).

Bray, H. "On Wave Dark Matter, Shells in Elliptical Galaxies, and the Axioms of General Relativity." (December 22, 2012).

Bray, HL, and Khuri, MA. "P. D. E. 'S which imply the penrose conjecture." *Asian Journal of Mathematics* 15.4 (2011): 557-610.

## Pages

Bray, HL, and Parry, AR. "Modeling wave dark matter in dwarf spheroidal galaxies." 2015. Full Text

Bray, HL. "A family of quasi-local mass functionals with monotone flows." January 1, 2006. Full Text

Bray, H. "Black Holes and the Penrose Inequality in General Relativity." International Congress of Mathematicians. Beijing, China. 2002.

Bray, H, and Morgan, F. "An isoperimetric comparison theorem for schwarzschild space and other manifolds." 2002. Full Text

Bray, H.* The Penrose Inequality in General Relativity and Volume Comparison Theorems Involving Scalar Curvature.* 1997. (PhD Thesis)