# Arlie Petters

# Arlie Petters

Professor of Mathematics, Physics, and Business Administration, Duke University

MLK Visiting Scholar 2002-2004

Hosted by the Department of Physics

## Bio

Arlie Petters is the Benjamin Powell Professor and a Professor of Mathematics, Physics, and Business Administration at Duke University. Research interests: mathematical physics and astronomy, cosmology, singularity theory and differential geometry.

Dr. Petters earned his BA and MA in Mathematics and Physics (1986) from Hunter College- CUNY. He earned a PhD in Mathematics (1991) from MIT, during which he also spent two years as an Exchange Scholar at Princeton. His dissertation was titled "Singularities in Gravitational Microlensing," with Bertram Kostant (MIT) and David Spergel (Princeton) as advisors.

After completing his PhD, Dr. Petters remained at MIT for another two years as an Instructor of Pure Mathematics in the Department of Mathematics. Before joining the faculty at Duke in 1998, he was Assistant Professor of Mathematics and Co-Director of Graduate Studies in the Department of Mathematics at Princeton.

In 2009, Dr. Petters became the Benjamin Powell Professor at Duke. Dr. Petters has an impressive record of service to Duke, to the discipline, to students, and to his native Belize, where he served as Chairman of the Council of Science Advisers to the Prime Minister of Belize from 2010 to 2013. He has been Visiting Professor at the Max Planck Institute, Oxford University, and Princeton University.

The work of Dr. Petters in gravitational lensing (explaining how light bends over massive distances) and its application to the possibility of a fourth spatial dimension is world renowned--as is his personal story. His extraordinary journey from a childhood in Belize and immigration to the US in 1979 to a career spanning physics, mathematics, and business has been featured in The New York Times, Belize News, and PBS NOVA.

Dr. Petters returned to MIT as an MLK Visiting Professor hosted by the Department of Physics from 2002 to 2004. A year after his visit to MIT, Dr. Petters founded the Petters Research Institute in Belize.

## Video

## Publications

BOOKS

Scientific Reasoning: Teacher Edition

A. O. Petters (BRC, Benque-Belize, 2007).

Scientific Reasoning: Student Edition

A. O. Petters (BRC, Benque-Belize, 2007).

Algebra, Geometry, and Trigonometry: Teacher Edition

A. O. Petters (BRC, Benque-Belize, 2007).

Algebra, Geometry, and Trigonometry: Student Edition

A. O. Petters (BRC, Benque-Belize, 2007).

PSE Mathematics: Teacher-Tutor-Parent Edition

A. O. Petters (BRC, Benque-Belize, 2007).

PSE Mathematics: Student Edition

A. O. Petters (BRC, Benque-Belize, 2007).

Singularity Theory and Gravitational Lensing

A. O. Petters, H. Levine, and J. Wambsganss (Birkhäuser-Springer, Boston, 2001).

BOOKS IN PREPARATION

Gravitational Lensing and Black Holes

A. O. Petters and M. C. Werner (Springer, New York, December 2013). In preparation.

Mathematical Finance with Applications: Understanding and Building Financial Intuition

A. O. Petters and X. Dong (Springer, SUMAT Series, New York, June 2013). In preparation.

PAPERS

Lensing by Kerr Black Holes. II. Analytical Study of Quasi-Equatorial Lensing Observables

A. B. Aazami, C. R. Keeton, and A. O. Petters, J. Math. Phys. 52, 102501 (2011); astro-ph arXiv:1002.4304v1.

Lensing by Kerr Black Holes. I. General Lens Equation and Magnification Formula

A. B. Aazami, C. R. Keeton, and A. O. Petters, J. Math. Phys. 52, 092502 (2011); astro-ph arXiv:1102.4300v1.

Orbifolds, the A, D, E Caustic Singularities, and Gravitational Lensing

A. B. Aazami, A. O. Petters, and J. M. Rabin, J. Math. Phys. 51, 022501 (2011); math-ph arXiv:1004.0516v1.

Gravity's Action on Light

A. O. Petters, Notices of the AMS 57, 1392 (2010).

Mathematics of Gravitational Lensing: Multiple Imaging and Magnification

A. O. Petters and M. C. Werner, Gen. Rel. and Grav., Special Issue on Gravitational Lensing (2010); astro-ph arXiv:0912.0490v1.

A Universal Magnification Theorem. III. Caustics Beyond Codimension Five

A. B. Aazami and A. O. Petters, J. Math. Phys. 51, 023503 (2010); math-ph arXiv:0909.5235.

A Mathematical Theory of Stochastic Microlensing. II. Random Images, Shear, and the Kac-Rice Formula

A. O. Petters, B. Rider, and A. M. Teguia, J. Math. Phys. 50,122501 (2009); astro-ph arXiv:0807.4984v2.

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 Universal Magnification Theorem II. Generic Caustics up to Codimension Five

A. B. Aazami and A. O. Petters, J. Math. Phys. 50, 082501 (2009); math-ph arXiv:0904.2236v4.

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.

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.

Light's Bending Angle due to Black Holes: From the Photon Sphere to Infinity

S. V. Iyer and A. O. Petters, Gen. Rel. and Grav. 39, 1563 (2007); gr-qc/0611086.

Testing Theories of Gravity with Black Hole Lensing C. Keeton and A. O. Petters, in Proceedings of the Eleventh Marcel Grossmann Meeting on General Relativity, ed. R. Ruffini (World Scientific, Singapore, 2006).

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.

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. I. Static, Spherically Symmetric Case

C. Keeton and A. O. Petters, Phys. Rev. D, 72, 104006 (2005); gr-qc/0511019.

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.

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.

On Relativistic Corrections to Microlensing Effects: Applications to the Galactic Black Hole

A. O. Petters, Mon. Not. R. Astron. Soc., 338, 457 (2003); astro-ph/0208500.

Wavefronts, Caustic Sheets, and Caustic Surfing in Gravitational Lensing

S. Frittelli and A. O. Petters, J. Math. Phys., 43, 5578 (2002); astro-ph/0208135.

Gravitational Microlensing Near Caustics. II: Cusps

B. S. Gaudi and A. O. Petters, Astrophys. J., 580, 468 (2002); astro-ph/0206162.

Gravitational Microlensing Near Caustics. I: Folds

B. S. Gaudi and A. O. Petters, Astrophys. J., 574, 970 (2002); astro-ph/0112531.

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).

Wavefront Singularities due to an Elliptical Potential

S. Frittelli and 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).

Center of Light Curves for Whitney Fold and Cusp

B. S. Gaudi and 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).

Fixed Points due to Gravitational Lenses

A.O. Petters and F.J. Wicklin, J. Math. Phys., 39, 1011 (1998).

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).

Properties of Point Mass Lenses on a Regular Polygon and the Problem of Maximum Number of Lensed Images

S. Mao, A. O. Petters, and H. Witt, in Proceedings of the Eighth Marcel Grossmann Meeting on General Relativity, ed. R. Ruffini (World Scientific, Singapore, 1997); astro-ph/0708111.

Some Global Results on Gravitational Lensing

A. O. Petters, in Proceedings of the Eighth Marcel Grossmann Meeting on General Relativity, ed. R. Ruffini (World Scientific, Singapore, 1997).

Curvature of Caustics and Singularities of Gravitational Lenses

A. O. Petters, in Proceedings of the Second World Congress of Nonlinear Analysts, ed. V. Lakshmikantham (Elsevier Science Ltd., Oxford, 1997).

Multiplane Gravitational Lensing III. Upper Bound on Number of Images

A. O. Petters, J. Math. Phys., 38, 1605 (1997).

Lower Bounds on Image Magnification in Gravitational Lensing

A. O. Petters, Proc. R. Soc. Lond. A, 452, 1475 (1996).

A Cusp-Counting Formula for Caustics due to Multiplane Gravitational Lensing

A. O. Petters, in Astrophysical Applications of Gravitational Lensing, eds. C. S. Kochanek and J.N. Hewitt (Klumer Academic, Dordrecht, 1996).

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).

New Caustic Phenomena for Caustics in Double-Plane Lensing

A. O. Petters and F.J. Wicklin, in Astrophysical Applications of Gravitational Lensing, eds. C. S. Kochanek and J.N. Hewitt (Klumer Academic, Dordrecht, 1996).

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).

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).

Multiplane Gravitational Lensing II: Global Geometry of Caustics

A. O. Petters, J. Math. Phys., 36, 4276 (1995).

Multiplane Gravitational Lensing I: Morse Theory and Image Counting

A. O. Petters, J. Math. Phys., 36, 4263 (1995).

Singularities and Gravitational Lensing

H. Levine and A. O. Petters, in Passion des Formes: Hommage `a Rene' Thom , vol. 1, ed. M. Porte (E.N.S. Edition, Fontenay-St Cloud, 1994).

Applications of Singularity Theory to Gravitational Lensing

H. Levine, A. O. Petters, and J. Wambsganss, J. Math. Phys., 34, 4781 (1993).

Singularities of the One and Two Point Mass Gravitational Lens

H. Witt and A. O. Petters, J. Math. Phys., 34, 4093 (1993).

New Caustic Singularities in Multiple Lens Plane Gravitational Lensing

H. Levine and A. O. Petters, Astron. Astrophys., 272, L17 (1993).

Arnold's Singularity Theory and Gravitational Lensing

A. O. Petters, J. Math. Phys., 34, 3555 (1993).

Morse Theory and Gravitational Microlensing

A. O. Petters, J. Math. Phys., 33, 1915 (1992).

An Analytical Approach to Quasar Variability due to Microlensing

A. O. Petters and D. Spergel, in Gravitational Lenses, eds. R. Kayser, T. Schramm, and L. Nieser (Lecture Notes in Physics 406, Springer, Berlin, 1992).

Morse Theory and Gravitational Microlensing

A. O. Petters, in Gravitational Lenses, eds. R. Kayser, T. Schramm, and L. Nieser (Lecture Notes in Physics 406, Springer, Berlin, 1992).

## At MIT

Duke Mathematician Seeks Fourth Dimension

By Eileen McCluskey

First published in Technology Review (Jan/Feb 2009)

Arlie Petters talks with Belize student Leah Waight.

Arlie Petters, a professor of pure mathematics, physics, and business administration at Duke University, is a man of many dimensions. World renowned for his work in gravitational lensing, which explains how light bends over massive distances, Petters is now applying that theory at a new astrophysical frontier: he's seeking to prove that the universe has a fourth spatial dimension.

Using newly available data, Petters and astronomer Charles Keeton of Rutgers University are searching for evidence of tiny black holes in the solar system. Finding these black holes would support a recent cosmological theory that envisions the visible universe as a membrane adrift in a larger, higher-dimensional universe. Petters and Keeton theorize that "brane-world" black holes would cause passing gamma rays to bend in a specific way.

Proving brane-world theory is a tough challenge, but Petters is used to them. In 1979, he immigrated from Belize to the U.S., where he moved in with relatives. Shortly after he arrived, while he was studying mathematics and physics at Hunter College in New York, household tensions forced him to find a new place to live. "I was 18 years old and penniless," he recalls. "I had to find a way to stay in the U.S. or give up my education and return to Belize." With the help of a professor, Petters landed a full scholarship and completed an accelerated bachelor's and master's program in 1986.

"I left Hunter College as a star," says Petters. He laughs. "Then I came to MIT. MIT took me to new levels. I saw intellectual intensity there of the first order." After earning his PhD, he taught at MIT and Princeton before Duke recruited him in 1998.

In 2005 he founded the Petters Research Institute in Belize to "teach young people to use technical know-how to start businesses and raise their standard of living." At the institute, children as young as 10 learn how to assemble working computers. Older students delve into higher-level mathematics to understand finance, hurricanes, tumor growth rates, structural engineering, and more.

Petters has received numerous academic and humanitarian honors, plus an order of chivalry from Queen Elizabeth. In 2006, his portrait was added to the U.S. National Academy of Sciences' Portrait Gallery of Distinguished African-American Scientists. He is also the first African-American to be tenured in Duke's mathematics department.

In his spare time, Petters enjoys watching movies-"everything from independent films to Disney cartoons." He and his wife, Marcia, a compliance and risk manager with Blue Cross Blue Shield, live in North Carolina and Belize.