Professor H.L. Bray

Professor H. L. Bray contact curriculum vitae
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More Cool Stuff

Join me at the North Carolina Math Camp this summer from July 6 - 19, or July 20 - August 2, 2025.
- The first camp does not require calculus, but is still very mathematically challenging.
- The second camp is a more advanced calculus version of the first camp, for those who are accepted.
- Join ambitious and enthusiastic high school students from around the world.
- Amazing camp counselors, most of whom are Duke math majors and graduate students.
- Learn special relativity, game theory, Fourier series, and many other fun math topics from me.
- Guest lectures from other professors on most days, on a wide variety of topics related to their research.
- Live on campus at Saint Mary's School in downtown Raleigh.
- Basketball courts, gym, fields for sports, ping pong, local trips, and more fun!

Click here for my best wishes to the Duke University graduating class of 2020! (Lower resolution version here.)
Click here to see our Pi Day celebration in 2022 (pie eating contest, digits of pi recitation contest, and pieing of 4 math professors) hosted by The Lyceum, our interdisciplinary math social club. Thanks to Fayfay Ning, the Lyceum president, for organizing this event!
Click here to see my presentation "Einstein's Greatest Ideas" for the general public of these slides that I gave at Cornell University in 2023.

Hubert Lewis Bray

Professor Bray's research 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. This motivates very interesting questions about geometric partial differential equations and the dynamics of spiral galaxies.

Professor Bray received his Ph.D. in mathematics from Stanford University in 1997 under the direction of Richard Schoen. He then spent one year as an NSF postdoc at Harvard supervised by S.-T. Yau, before going to MIT where he was an instructor, an assistant professor, and an associate professor. Professor Bray accepted an associate professorship at Columbia in 2003 and a full professorship at Duke in 2004, where he resides today as a professor of mathematics and physics. He has been married since 2004 and has six children.

Professor Bray has supervised 13 Ph.D. graduates (11 in math, 2 in physics) at Duke from 2006 to 2023. His 2017 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 the geometries of light cones and other null hypersurfaces in curved spacetimes.

ON THE JOB MARKET: Kai Xu, who will be graduating with his Ph.D. in 2025, is a truly remarkable young mathematician. With 8 very nice papers (submitted) and counting, he is already an expert on many topics in geometric analysis, including scalar and Ricci curvature, inverse mean curvature flow, and spectral geometry. With Demetre Kazaras, Kai introduced the notion of "drawstring metrics" which are relevant to understanding the stability of nonnegative scalar curvature metrics on tori. He has also proved a topological gap theorem for the π2-systole of positive scalar curvature 3-manifolds as a clever new application of inverse mean curvature flow. For a complete list of Kai's papers, visit his web page.

Another student, Yiyue Zhang (Duke Ph.D. 2021), has just begun a tenure track position at the Beijing Institute of Mathematical Sciences and Applications. Yiyue (with Sven Hirsh) proved the most general rigidity case of the case of equality of the Positive Mass Theorem. He is also an expert on scalar curvature, Ricci curvature, and 2-Ricci curvature. He and his collaborators also proved new geometric theorems which have deep implications about the formation of black holes, which was featured in Quanta Magazine. Click here to read about it.

Professor Bray has also supervised 8 undergraduates who wrote senior theses in math at Duke, from 2009 to 2021. Among them is Zhenhua Liu (Princeton Ph.D. expected in 2024) who graduated from Duke in 2019 with highest honors. His senior thesis, "Stationary one-sided area-minimizing hypersurfaces with isolated singularities" foreshadowed the amazing career he is now having, even as a graduate student.

Another former Duke math major is Daniel Stern, now a professor at Cornell, who wrote his 2014 senior thesis on classifying general relativity type actions of a certain form from a geometric perspective. Daniel's 2019 paper "Scalar curvature and harmonic maps to S^1" represents a new technique for understanding scalar curvature, a fundamental concept in geometric analysis, and hence has been very influential in the field. Among many other applications, this new approach has led to new explicit formulas for the total mass of asymptotically flat and asymptotically hyperbolic 3-manifolds, which yield the corresponding versions of the Positive Mass Theorem as corollaries.

These results were further extended by Sven Hirsch, Demetre Kazaras, and Marcus Khuri who generalized the harmonic functions used above to define a new useful tool called spacetime harmonic functions. This leads to a new explicit formula relating the total energy and total momentum of a slice of a spacetime to the local energy and momentum density of the spacetime on the spacelike slice, no matter what the second fundamental form of the slice may be. The Spacetime Positive Mass Theorem follows as a corollary. They then joined forces with Yiyue Zhang to prove purely geometric comparison theorems for Riemannian bands involving scalar curvature, Ricci curvature, and 2-Ricci curvature using these same spacetime harmonic functions.

Three recommended books that students can read to get into some of these subjects are:

"Differential Geometry" by John Oprea (for undergraduates), "Semi-Riemannian Geometry" by Barrett O'Neill (for graduate students), and "Geometric Relativity" by Dan Lee (for advanced graduate students). Professor Lee collaborated with Professor Bray on the proof of the Riemannian Penrose Conjecture in dimensions less than eight as a postdoc at Duke from 2005 to 2008.

Demetre Kazaras, now a professor at Michigan State, was as a postdoc at Duke from 2020 to 2023. During that time, he helped Professor Bray lead his research group which included Yiyue Zhang (math Ph.D. 2021), Ben Hamm (physics Ph.D. 2021), Sven Hirsch (math Ph.D. 2023), James Wheeler (physics Ph.D. 2023), Michael Lin (math Ph.D. exp. 2024), and Kai Xu (math Ph.D. exp. 2025). Professor Marcus Khuri (SUNY - Stony Brook) has also co-supervised the group with Professor Bray since 2020 via weekly zoom meetings.

Professor Bray's other graduate students who have done and continue to do fascinating work in geometric analysis and mathematical relativity include Hangjun Xu (math Ph.D. 2014), Mau-Kwong "George" Lam (math Ph.D. 2011), Graham Cox (math Ph.D. 2011), Jeff Jauregui (math Ph.D. 2010), and Nicholas Robbins (math Ph.D. 2007). A full list of Professor Bray's graduate students is here.

Professor Bray is also very interested in dark matter and its role in the universe. James Wheeler (physics Ph.D. 2023), now a postdoc at the University of Michigan working with Lydia Bieri, has done fantastic work modeling the universe since the Big Bang and the resulting power spectrum for the cosmic microwave background radiation which would be predicted given different assumptions about the role of dark matter in the early universe. He is exploring whether or not different, more geometric models of dark matter can explain the Hubble tension, refering to the discrepancy between measurements of the Hubble constant via supernovas versus the cosmic microwave background radiation, the latter of which makes assumptions about the nature of dark matter. James has also found a new, mathematically natural definition of black holes in terms of the abstract boundary of a spacetime and has improved our understanding of how naked singularities can form in spherically symmetric spacetimes. James work follows in the footsteps of three other former Ph.D. students of Professor Bray's who studied dark matter as well: Ben Hamm (physics Ph.D. 2021), Andrew Goetz (math Ph.D. 2015), and Alan Parry (math Ph.D. 2013).

Hubert Lewis Bray (1970 - ) is named after his dad's dad, Hubert Evelyn Bray (1889 - 1978), who was also a mathematician. The original Hubert Bray received the first Ph.D. awarded by Rice University in 1918 and joined the faculty of the mathematics department thereafter. He was chairman from 1935 to 1957, secretary of the faculty from 1935 to 1959, and chairman of the Committee on Outdoor Sports from 1920 to 1959, a job analogous to being the athletic director today. He was initially asked to serve in this job in 1920 because, as a mathematician, he could be trusted to reliably average the multiple stop watches used to determine the track times at Rice track meets. Upon his first retirement in 1959 he was named "Trustees' Distinguished Professor of Mathematics" for his long service to Rice. He continued to teach classes until 1970 at age 81. His grandson, Hubert Lewis Bray, attended Rice as an undergraduate from 1988 to 1992 and won the Hubert E. Bray Prize in Mathematics, awarded annually to the outstanding junior mathematics major at Rice. This 1927 photo captures some of the history of the Rice Mathematics Department, showing the entire department - faculty, staff, and graduate students - all 11 of them, including Mandelbrojt who was visiting as a guest lecturer. From left to right: E.R.C. Miles, David Widder, Miss Alice Dean , S. Mandelbrojt (visiting from France), Nat Edmondson, Arthur Copeland, H.E. Bray, May Hickey (Maria), G.C. Evans (namesake for Evans Hall, the math building at UC Berkeley), R. N. Haskell, and J. Gergen, then a graduate student, who later became department chairman at Duke from 1937 to 1966. In this photo, Hubert Evelyn Bray (seated) and Jess Nealy, the head football coach, appear together with the Cotton Bowl Trophy. This video reflects on the first one hundred years of Rice University. For more information about Hubert Lewis Bray as a person, click here.

The above image shows a small section of the Veil Nebula, as it was observed by the NASA/ESA Hubble Space Telescope. This section of the outer shell of the famous supernova remnant is in a region known as NGC 6960 or, more colloquially, the Witch's Broom Nebula. Many of the atoms that make up our world were created inside stars which later exploded, the aftermaths of which would have looked something like this.