Arlie O. Petters
Benjamin Powell Professor of Mathematics
- 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.
Petters, A. O., and H. Witt. “Bounds on Number of Cusps Due to Point Mass Gravitional Lenses with Continuous Matter and Shear.” J. Math. Phys., vol. 37, no. 2920, 1996.
Fetters, A. O. “Lower bounds on image magnification in gravitational lensing.” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 452, no. 1949, Jan. 1996, pp. 1475–90.
Petters, A. O. “A Cusp—Counting Formula For Caustics Due To Multiplane Gravitational Lensing.” Symposium International Astronomical Union, vol. 173, Cambridge University Press (CUP), 1996, pp. 281–82. Crossref, doi:10.1017/s0074180900231550. Full Text
Petters, A. O., and F. J. Wicklin. “Caustics of the double-plane two-point-mass gravitational lens with continuous matter and shear.” Monthly Notices of the Royal Astronomical Society, vol. 277, no. 4, Oxford University Press (OUP), Dec. 1995, pp. 1399–403. Crossref, doi:10.1093/mnras/277.4.1399. Full Text
Petters, A. O. “Multiplane gravitational lensing. I. Morse theory and image counting.” Journal of Mathematical Physics, vol. 36, no. 8, Jan. 1995, pp. 4263–75. Scopus, doi:10.1063/1.530961. Full Text
Petters, A. O. “Multiplane gravitational lensing. II. Global geometry of caustics.” Journal of Mathematical Physics, vol. 36, no. 8, Jan. 1995, pp. 4276–95. Scopus, doi:10.1063/1.530962. Full Text
Petters, A. O. “Arnold's singularity theory and gravitational lensing.” Journal of Mathematical Physics, vol. 34, no. 8, Jan. 1993, pp. 3555–81. Scopus, doi:10.1063/1.530045. Full Text
Levine, H. I., et al. “Applications of singularity theory to gravitational lensing. I. Multiple lens planes.” Journal of Mathematical Physics, vol. 34, no. 10, Jan. 1993, pp. 4781–808. Scopus, doi:10.1063/1.530321. Full Text
Witt, H. J., and A. O. Petters. “Singularities of the one- and two-point mass gravitational lens.” Journal of Mathematical Physics, vol. 34, no. 9, Jan. 1993, pp. 4093–111. Scopus, doi:10.1063/1.530029. Full Text
Levine, H., et al. “Applications of Singularity Theory to Gravitational Lensing.” J. Math. Phys., vol. 34, 1993.