# Arlie O. Petters

## Benjamin Powell Professor of Mathematics

### Overview

**Mathematical Physics***Mathematics*- tools form differential geometry, singularities, and probability theory*Physics*- problems connected to the interplay of gravity and light (gravitational lensing, general relativity, astrophysics, cosmology)My current research in mathematical physics is on gravitational lensing, which is the study of how gravity acts on light. In particular, I utilizing weak and strong deflection gravitational lensing to characterize the geometry of spacetime around black holes, test theories of gravity, and probe the nature of dark matter on galactic scales. I employ tools from astrophysics, cosmology, general relativity, high energy physics, and a variety of mathematical fields (e.g., differential geometry, singularities, and probability theory).

A mathematical theory of gravitational lensing is presented in the monograph:

*Singularity Theory and Gravitational Lensing*(A. O. Petters, H. Levine, and J. Wamsbganss).Two layman articles about my research are at:

*Ripple Effect*(Scott Huler).

*Prescription lens brings spinning black holes into focus*(Ashley Yeager).**Mathematical and Scientific Methods in Business Administration***Mathematical finance with applications**Entrepreneurship and business innovation in STEM fields (developing world)*By current business administration activities are three-fold. First, I am co-authoring a text on Mathematical Finance with Xiaoying Dong, who is an Adjunct Assistant Professor in our department and a trader for over 20 years. This book is aimed at first year graduate students from mathematics, economics, physics, computer science, and engineering. Second, at Duke's Fuqua School of Business I supervise the finance concentration research projects of Executive M.B.A. students. These projects cover a variety of topics: company valuations, derivatives, portfolio theory, mergers and acquisitions, etc. Third, I am involved with sustainable business and environmentally friendly applications of Science, Technology, Engineering, and Mathematics (STEM) in a developing-world setting that integrates education and entrepreneurship. These efforts are being piloted in Belize in collaboration with the Petters Research Institute and through my appointment with Fuqua. The overall goal is to research innovative ways to help drive national development through applications of STEM tools.

Aazami, A. B., and A. O. Petters. “A universal magnification theorem for higher-order caustic singularities.” *Journal of Mathematical Physics*, vol. 50, no. 3, Apr. 2009. *Scopus*, doi:10.1063/1.3081055.
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Keeton, C. R., and A. O. Petters. “Testing theories of gravity with black hole lensing.” *11th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Gravitation and Relativistic Field Theories Proc. of the Mg11 Meeting on General Relativity*, Dec. 2008, pp. 1719–21.

Iyer, S. V., and A. O. Petters. “Light's bending angle due to black holes: From the photon sphere to infinity.” *General Relativity and Gravitation*, vol. 39, no. 10, Oct. 2007, pp. 1563–82. *Scopus*, doi:10.1007/s10714-007-0481-8.
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Werner, M. C., and A. O. Petters. “Magnification relations for Kerr lensing and testing cosmic censorship.” *Physical Review D Particles, Fields, Gravitation and Cosmology*, vol. 76, no. 6, Sept. 2007. *Scopus*, doi:10.1103/PhysRevD.76.064024.
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Keeton, C. R., and A. O. Petters. “Formalism for testing theories of gravity using lensing by compact objects. III. Braneworld gravity.” *Physical Review D Particles, Fields, Gravitation and Cosmology*, vol. 73, no. 10, May 2006. *Scopus*, doi:10.1103/PhysRevD.73.104032.
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Keeton, C. R., and A. O. Petters. “Formalism for testing theories of gravity using lensing by compact objects. II. Probing post-post-Newtonian metrics.” *Physical Review D Particles, Fields, Gravitation and Cosmology*, vol. 73, no. 4, Mar. 2006. *Scopus*, doi:10.1103/PhysRevD.73.044024.
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Keeton, C. R., et al. “Identifying lenses with small-scale structure. II. Fold lenses.” *Astrophysical Journal*, vol. 635, no. 1 I, Dec. 2005, pp. 35–59. *Scopus*, doi:10.1086/497324.
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Keeton, C. R., and A. O. Petters. “Formalism for testing theories of gravity using lensing by compact objects: Static, spherically symmetric case.” *Physical Review D Particles, Fields, Gravitation and Cosmology*, vol. 72, no. 10, Nov. 2005. *Scopus*, doi:10.1103/PhysRevD.72.104006.
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Keeton, C., et al. “Identifying Lensing by Substructure II. Fold Lenses.” *Astrophysical Journal*, vol. 635, Nov. 2005.

Keeton, C. R., et al. “Identifying lenses with small-scale structure. I. Cusp lenses.” *Astrophysical Journal*, vol. 598, no. 1 I, Nov. 2003, pp. 138–61. *Scopus*, doi:10.1086/378934.
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