# Hubert Bray

### **Professor of Mathematics**

Director of Undergraduate Studies in the Department of Mathematics

Director of Undergraduate Studies in the Department 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*, edited by H. Bray and W. Minicozzi, Higher Education Press and International Press, 2011.

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

Bray, H. “Geometric Flows and the Penrose Inequality.” *Encyclopedia of Mathematical Physics: Five-Volume Set*, 2004, pp. 510–20. *Scopus*, doi:10.1016/B0-12-512666-2/00058-4.
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Bray, H., and P. T. Chrusciel. “The Penrose Inequality.” *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.

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

Bray, H. L., and W. P. Minicozzi. “Preface.” *Notices of the American Mathematical Society*, vol. 65, no. 11, Dec. 2018, pp. 1412–13. *Scopus*, doi:10.1090/noti1748.
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Sormani, Christina, et al. “The Mathematics of Richard Schoen.” *Notices of the American Mathematical Society*, vol. 65, no. 11, American Mathematical Society (AMS), Dec. 2018, pp. 1–1. *Crossref*, doi:10.1090/noti1749.
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Bray, H., and H. Roesch. “Proof of a Null Geometry Penrose Conjecture.” *Notices of the American Mathematical Society.*, vol. 65, American Mathematical Society, Feb. 2018.

Bray, H. L., et al. “Time Flat Surfaces and the Monotonicity of the Spacetime Hawking Mass II.” *Annales Henri Poincare*, vol. 17, no. 6, June 2016, pp. 1457–75. *Scopus*, doi:10.1007/s00023-015-0420-2.
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Martinez-Medina, L. A., et al. “On wave dark matter in spiral and barred galaxies.” *Journal of Cosmology and Astroparticle Physics*, vol. 2015, no. 12, Dec. 2015. *Scopus*, doi:10.1088/1475-7516/2015/12/025.
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Bray, H. L., and J. L. Jauregui. “On curves with nonnegative torsion.” *Archiv Der Mathematik*, vol. 104, no. 6, June 2015, pp. 561–75. *Scopus*, doi:10.1007/s00013-015-0767-0.
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Bray, H. L., and J. L. Jauregui. “Time Flat Surfaces and the Monotonicity of the Spacetime Hawking Mass.” *Communications in Mathematical Physics*, vol. 335, no. 1, Jan. 2015, pp. 285–307. *Scopus*, doi:10.1007/s00220-014-2162-2.
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Bray, H., and A. S. Goetz. *Wave Dark Matter and the Tully-Fisher Relation*. Sept. 2014.

Bray, H. L., and J. L. Jauregui. “A geometric theory of zero area singularities in general relativity.” *Asian Journal of Mathematics*, vol. 17, no. 3, 2013, pp. 525–60. *Scival*, doi:10.4310/AJM.2013.v17.n3.a6.
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Bray, H. L. “On Dark Matter, Spiral Galaxies, and the Axioms of General Relativity.” *Ams Contemporary Mathematics Volume*, vol. 599, no. Geometric Analysis, Mathematical Relativity, and Nonlinear Partial Differential Equations, American Mathematical Society, 2013.

## Pages

Bray, H. L., and A. R. Parry. “Modeling wave dark matter in dwarf spheroidal galaxies.” *9th Biennial Conference on Classical and Quantum Relativistic Dynamics of Particles and Fields (Iard 2014)*, vol. 615, 2015. *Wos-lite*, doi:10.1088/1742-6596/615/1/012001.
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Bray, H. L. *A family of quasi-local mass functionals with monotone flows*. 2006, pp. 323–29. *Scopus*, doi:10.1142/9789812704016_0030.
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Bray, H, and Morgan, F. "An isoperimetric comparison theorem for schwarzschild space and other manifolds." 2002. Full Text

Bray, H. “Black Holes and the Penrose Inequality in General Relativity.” *Proceedings of the International Congress of Mathematicians*, vol. 2, 2002, pp. 257–72.