# Arlie Petters

Arlie Petters | |
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| |

Born |
Arlie Oswald Petters February 8, 1964 Stann Creek Town, British Honduras (now Dangriga, Belize) |

Alma mater |
Hunter College Massachusetts Institute of Technology Princeton University |

Known for | Mathematical Theory of Gravitational Lensing |

Awards |
Alfred P. Sloan Fellowship NSF CAREER Award Blackwell-Tapia Prize Most Excellent Order of the British Empire |

Scientific career | |

Doctoral advisors |
Bertram Kostant David Spergel |

**Arlie Oswald Petters**, MBE (born February 8, 1964) is a Belizean-American mathematical physicist, who is the Benjamin Powell Professor of Mathematics and a Professor Physics and Economics at Duke University.^{[1]} Petters is a founder of mathematical astronomy, focusing on problems connected to the interplay of gravity and light and employing tools from astrophysics, cosmology, general relativity, high energy physics, differential geometry, singularities, and probability theory.^{[2]} His monograph "Singularity Theory and Gravitational Lensing" is the first to develop a mathematical theory of gravitational lensing.
Petters is also the Dean of Academic Affairs for Trinity College of Arts and Sciences and Associate Vice Provost for Undergraduate Education at Duke
University.^{[3]}

## Biography

Petters was raised by his grandparents in the rural community of Stann Creek Town, British Honduras (now Dangriga, Belize). His mother immigrated to Brooklyn, New York, and married a U.S. citizen, with Arlie joining them when he was 14 years old.^{[4]}

Petters earned a B.A./M.A. in Mathematics and Physics from Hunter College, CUNY in 1986 with a thesis on "The Mathematical Theory of General Relativity", and began his Ph.D. at the Massachusetts Institute of Technology Department of Mathematics in the same year. After two years of doctoral studies, he became an exchange scholar in the Princeton University Department of Physics *in absentia* from MIT. Petters earned his Ph.D. in Mathematics in 1991 under advisors Bertram Kostant (MIT) and David Spergel (Princeton University). He remained at MIT for two years as an instructor of pure mathematics (1991–1993) and then joined the faculty at Princeton University in the Department of Mathematics. He was an Assistant Professor at Princeton for five years (1993–1998) before moving to Duke University.^{[3]}^{[5]}

Petters's work and life were profiled in *The New York Times*,^{[4]} on NOVA,^{[6]}
by The History Makers,
^{[7]}
and at Big Think.
^{[8]}

## Research

Petters is renowned for his pioneering work in the mathematical theory of gravitational lensing.

Over the ten-year period from 1991–2001, Petters systematically developed a mathematical theory of weak-deflection gravitational lensing, beginning with his 1991 MIT Ph.D. thesis on "Singularities in Gravitational Microlensing"
^{[9]}
and followed by the 12 papers [AP1] – [AP12] below. The papers resolved an array of theoretical problems in weak-deflection gravitational lensing covering image counting, fixed-point images, image magnification, image time delays, local geometry of caustics, global geometry of caustics, wavefronts, caustic surfaces, and caustic surfing.
^{[10]}
His work culminated with a 2001 mathematical tome [AP13] that, among other things, systematically created a framework of stability and genericity for k-plane gravitational lensing. The book drew upon powerful tools from the theory of singularities and put the subject of weak-deflection k-plane gravitational lensing on a rigorous and unified mathematical foundation.
^{[11]}

Following his 1991–2001 body of mathematical lensing work, Petters turned to more astrophysical lensing issues from 2002–2005. In collaboration with astronomers, he applied some of the mathematical theory in [AP13] to help develop a practical diagnostic test for the presence of dark substructures in galaxies lensing quasars;
^{[12]}
^{[13]}
classify the local astrometric (centroid) and photometric curves of an extended source when it crosses fold and cusp caustics due to generic lenses;
^{[14]}
^{[15]}
predict the quantitative astrometric curve's shape produced by Galactic binary lenses.
^{[14]}
^{[15]}
The classified local properties of the astrometric curves revealed a characteristic S-shape for fold crossings, parabolic and swallowtail features for cusp crossings, and a jump discontinuity for crossings over the fold arcs merging into a cusp. A formula for the size of the jump was also found.

During 2005–2007, Petters collaborated with astronomers and physicists to explore gravitational lensing
in directions beyond its traditional confines in astronomy. In a series of three mathematical physics papers (2005–2006) with the astronomer Keeton, he utilized higher-order gravitational lensing effects by compact bodies to test different theories of gravity with Einstein's general theory of relativity among them. The first two papers computed beyond the standard weak-deflection limit the first- and second-order corrections to the image positions, magnifications, and time delays due to lensing in general relativity and alternative gravitational theories describable within the PPN formalism,
^{[16]}
and even determined lensing invariants for the PPN family of models.
^{[17]}
Their findings were applied to the Galactic black hole, binary pulsars, and gravitational microlensing scenarios to make testable predictions about lensed images and their time delays.
^{[16]} The third paper took on the difficult issue of how to test hyperspace models like braneworld gravity that postulate an extra dimension to physical space. The paper developed a semi-classical wave theory of braneworld black hole lensing and
used that theory along with braneworld cosmology to predict a testable signature of microscopic
braneworld black holes on gamma-ray light.
^{[18]}
^{[19]}
Additionally, in a 2007 paper, Petters and Werner found a system of equations that can be applied to
test the Cosmic Censorship Hypothesis observationally using the realistic case of lensing by a Kerr black hole.
^{[20]}
^{[21]}

Petters's previous work (1991–2007) dealt with non-random gravitational lensing. His recent research program (2008–present) has been to develop a mathematical theory of random (stochastic) gravitational lensing. In two papers, he, Rider, and Teguia took first steps in creating a mathematical theory of stochastic gravitational microlensing. They characterized to several asymptotic orders the probability densities of random time delay functions, lensing maps, and shear maps in stochastic microlensing
and determined a Kac-Rice type formula for the global expected number of images due to a general stochastic lens system.
^{[22]}
^{[23]}
The work forms a concrete framework from which extensions to more general random maps can be made.
In two additional papers, he and Aazami found geometric universal magnification invariants of higher-order caustics occurring in lensing and caustics produced by generic general maps up to codimension five.
^{[24]}
^{[25]}
^{[26]}
The invariants hold with a probability of 1 for random lenses and thereby form important consistency checks for research on random image magnifications of sources near stable caustics.

For more information, consult Petters's official Duke University CV
^{[3]}
for a very useful road map with detailed and extensive summaries of his research papers.

Selected papers from 1991–2001:
^{[27]}

[AP1] "Morse Theory and Gravitational Microlensing," A. O. Petters, J. Math. Phys., 33, 1915 (1992).

[AP2] "Arnold's Singularity Theory and Gravitational Lensing," A. O. Petters, J. Math. Phys., 34, 3555 (1993).

[AP3] "Multiplane Gravitational Lensing I: Morse Theory and Image Counting," A. O. Petters, J. Math. Phys., 36, 4263 (1995).

[AP4] "Multiplane Gravitational Lensing II: Global Geometry of Caustics," A. O. Petters, J. Math. Phys., 36, 4276 (1995).

[AP5] "Multiplane Gravitational Lensing III. Upper Bound on Number of Images," A. O. Petters, J. Math. Phys., 38, 1605 (1997).

[AP6] "Caustics of the Double-Plane Two Point-Mass Gravitational Lens with Continuous Matter and Shear," A. O. Petters and F.J. Wicklin, Mon. Not. R. Astron. Soc., 277, 1399 (1995).

[AP7] "Lower Bounds on Image Magnification in Gravitational Lensing," A. O. Petters, Proc. R. Soc. Lond. A, 452, 1475 (1996).

[AP8] "Counting Formulas and Bounds on Number of Fixed Points due to Point-Mass Lenses," A. O. Petters and F.J. Wicklin, in Proceedings of the Eighth Marcel Grossmann Meeting on General Relativity, ed. R. Ruffini (World Scientific, Singapore, 1997).

[AP9] "Bounds on Number of Cusps due to Point Mass Gravitational Lenses with Continuous Matter and Shear," A. O. Petters and H. Witt, J. Math. Phys., 37, 2920 (1996).

[AP10] "Mathematical Aspects of Gravitational Lensing," A. O. Petters, in Proceedings of the Seventh Marcel Grossmann Meeting on General Relativity, vol. B, eds. R. T. Jantzen and G. M. Keiser (World Scientific, Singapore, 1996).

[AP11] "Fixed Points due to Gravitational Lenses," A.O. Petters and F.J. Wicklin, J. Math. Phys., 39, 1011 (1998)

[AP12] "Stable Lens Systems, Lensed Image Magnification, and Magnification Cross Sections," A. O. Petters, in Proceedings of the Ninth Marcel Grossmann Meeting on General Relativity, eds. V. Gurzadyan, R. T. Jantzen, and R. Ruffini (World Scientific, Singapore, 2001).

[AP13] *Singularity Theory and Gravitational Lensing*, A. O. Petters, H. Levine, and J. Wambsganns
(Birkhauser, Boston, 2001)

## Social Outreach

Petters has given back significantly to the African-American community, which has included him serving as Director of the Reginaldo Howard Memorial Scholarship program at Duke University.
^{[28]}
He has also been active in the African-American community particularly through his mentoring, recruiting, and lecturing.
^{[29]}
^{[30]}
He has received several community service awards for his social outreach.
^{[31]}
^{[32]}
Petters is also the first tenured African-American professor in Mathematics at Duke University.
^{[31]}
He is very involved in the Belizean community and founded the Petters Research
Institute
^{[33]}
in 2005 to help train Belizean young people in STEM fields and foster STEM entrepreneurship.^{[34]}
^{[35]}
He has also written five books, three of which were science and mathematics problem-solving books for Belizean students.
^{[36]}
^{[37]}
Some of his entrepreneurial work was conducted while he was a Professor of Business Administration at Duke’s Fuqua School of Business (2008–2017).^{[3]}
Petters also served the Government of Belize as Chairman of the Council of Science Advisers to the Prime Minister of Belize (2010–2013).^{[38]}^{[3]}

## Awards and honors

Petters is the recipient of numerous awards and honors,
^{[39]}
which includes
an Alfred P. Sloan Research Fellowship in Mathematics (1998),
^{[40]}
a CAREER award
from the National Science Foundation (1998),
^{[41]}
and being the first winner of a Blackwell-Tapia Prize (2002).
^{[42]}
He was selected in 2006 by the National Academy of Sciences to be
part of a permanent Portrait Collection of Outstanding African-Americans
in Science, Engineering, and Medicine.
^{[43]}
In 2008 Petters was also included among the Human Relations Associates' list of
"The Twenty-Five Greatest Scientists of African Ancestry," which went back as early as
the 18th century.
^{[44]}
He received an honorary Doctor of Science from his alma mater Hunter College in 2008.
^{[45]}
Petters was named by the Queen of England in 2008 to membership in the Most Excellent Order of the British Empire.
^{[46]}
In recognition of his scientific accomplishments and service to society, Petters's birthplace—Dangriga,
Belize—honored him in 2009 with the *Dr. Arlie Petters Street*.
^{[47]}
He became in 2011 the first Belizean to receive the Caribbean American Heritage Award for Excellence in Science and Technology.
^{[48]} In 2012 he became a fellow of the American Mathematical Society^{[49]} and the first Belizean American to be Grand Marshal of the
Central American Day Parade in L.A., where he received honors from the mayor of L.A. and COFECA.^{[50]}^{[3]}

## References

- ↑ Duke University
- ↑ Mathematicians of the African Diaspora
- 1 2 3 4 5 6 Curriculum Vitae of Arlie O. Petters, Department of Mathematics, Duke University
- 1 2 Claudia Dreifus. A CONVERSATION WITH: ARLIE PETTERS: A Journey to Bridge Math and the Cosmos,
*The New York Times*, May 27, 2003. - ↑ Arlie Petters at the Mathematics Genealogy Project
- ↑
*NOVA Science Now* - ↑ The History Makers
- ↑ Big Think
- ↑ A. O. Petters, Ph.D. Thesis, MIT, Department of Mathematics (1991): "Singularities in Gravitational Microlensing."
- ↑ Section A, "Research Period 1991–2001," Curriculum Vitae of Arlie O. Petters, Department of Mathematics, Duke University
- ↑
*Singularity Theory and Gravitational Lensing*, A. O. Petters, H. Levine, and J. Wambsganss (Birkhauser, Boston, 2001). Amazon.com - ↑ "Identifying Lenses with Small-Scale Structure I. Cusp Lenses," C. Keeton, S. Gaudi, and A. O. Petters, Astrophys. J., 598, 138 (2003); astro-ph/0210318.
- ↑ "Identifying Lenses with Small-Scale Structure II. Fold Lenses," C. Keeton, S. Gaudi, and A. O. Petters , Astrophys. J., 635, 35 (2005); astro-ph/0503452.
- 1 2 "Gravitational Microlensing Near Caustics I: Folds," B. S. Gaudi and A. O. Petters, Astrophys. J., 574, 970 (2002); astro-ph/0112531.
- 1 2 "Gravitational Microlensing Near Caustics II: Cusps," B. S. Gaudi and A. O. Petters, Astrophys. J., 580, 468 (2002); astro-ph/0206162.
- 1 2 "Formalism for Testing Theories of Gravity Using Lensing by Compact Objects. I. Static, Spherically Symmetric Case," C. Keeton and A. O. Petters, Phys. Rev. D, 72, 104006 (2005); gr-qc/0511019.
- ↑ "Formalism for Testing Theories of Gravity Using Lensing by Compact Objects. II. Probing Post-Post-Newtonian Metrics," C. Keeton and A. O. Petters, Phys. Rev. D 73, 044024 (2006); gr-qc/0601053
- ↑ "Formalism for Testing Theories of Gravity Using Lensing by Compact Objects. III. Braneworld Gravity," C. Keeton and A. O. Petters, Phys. Rev. D 73, 104032 (2006) gr-qc/0603061
- ↑ "Scientist Predict How to Detect a Fourth Dimension," Duke News Archived September 29, 2009, at the Wayback Machine.
- ↑ "Magnification Relations for Kerr Lensing and Testing Cosmic Censorship," M. C. Werner and A. O. Petters, Phys. Rev. D 76, 064024 (2007); gr-qc/0706.0132
- ↑ "Seeking Objects Weirder than Black Holes," Duke News
- ↑ "A Mathematical Theory of Stochastic Microlensing I. Random Time Delay Functions and Lensing Maps," A. O. Petters, B. Rider, and A. M. Teguia, J. Math Phys., 50, 072503 (2009); astro-ph arXiv:0807.0232v2
- ↑ "A Mathematical Theory of Stochastic Microlensing II. Random Images, Shear, and the Kac-Rice Formula," A. O. Petters, B. Rider, and A. M. Teguia (2008); astro-ph arXiv:0807.4984
- ↑ "A Universal Magnification Theorem for Higher-Order Caustic Singularities," A. B. Aazami and A. O. Petters, J. Math. Phys. 50, 032501 (2009); astro-ph arXiv:0811.3447v2
- ↑ "A Universal Magnification Theorem II. Caustics up to Codimension Five," A. B. Aazami and A. O. Petters, 50, 082501 (2009); math-ph arXiv:0904.2236v4
- ↑ National Science Foundation, Science360 News Server, April 16, 2009
- ↑ Publications of Arlie O. Petters, Department of Mathematics, Duke University:
- ↑ The Reginaldo Howard Memorial Scholarship program, Duke University
- ↑ The Chronicle, Math Professor Reaches out to Minorities Archived September 7, 2008, at the Wayback Machine.
- ↑
Arlie O. Petters, Curriculum Vitae--
*Service to Underrepresented Minority Community,*Department of Mathematics, Duke University - 1 2 Duke Magazine, "Star Professor" Archived August 29, 2008, at the Wayback Machine.
- ↑
Arlie O. Petters, Curriculum Vitae--
*Awards and Honors* - ↑ Petters Research Institute, Dangriga, Belize
- ↑ 7 News Belize
- ↑ Computer assembly course for Belizean students by Petters Research Institute and the Belize Defence Force
- ↑ Amandala News
- ↑ Publications of Arlie O. Petters, Department of Mathematics, Duke University
- ↑ Government of Belize Official Press Release, April 28, 2010,
- ↑ Arlie Petters, official Curriculum Vitae, Department of Mathematics, Duke University
- ↑ Alfred P. Sloan Foundation, list of past fellows Archived 2011-05-19 at the Wayback Machine.
- ↑ National Science Foundation, FY 98 CAREER Awardes (New Jersey)
- ↑ SIAM News
- ↑ National Academy of Sciences, African-American History Program Archived June 29, 2010, at the Wayback Machine.
- ↑ Human Relations Associates, The Twenty-Five Greatest Scientists of African Ancestry "Archived copy". Archived from the original on 2012-02-22. Retrieved 2009-09-22.
- ↑ Hunter College News 2008
- ↑ London Gazette, Duke Today Archived September 19, 2008, at the Wayback Machine.
- ↑ Duke Research
- ↑ Channel 5 News, Belize
- ↑ List of Fellows of the American Mathematical Society, retrieved 2013-05-05.
- ↑ COFECA

## External links

*NOVA scienceNOW*https://www.pbs.org/wgbh/nova/sciencenow/3411/04.html- New York Times https://www.nytimes.com/2003/05/27/science/a-conversation-with-arlie-petters-a-journey-to-bridge-math-and-the-cosmos.html
- National Academy of Sciences https://web.archive.org/web/20100629121852/http://www.nasonline.org/site/PageServer?pagename=AAHP_Database_Bio_Petters
- Mathematicians of the African Diaspora http://www.math.buffalo.edu/mad/PEEPS/petters_arlie.html