Charles L. Epstein, Ph.D.

I'm a Senior Research Scientist in the Center for Computational Mathematics at the Flatiron Institute in New York. I am the Thomas A. Scott Professor Emeritus at the University of Pennsylvania. Epstein was the founder and chair of the Graduate Group in Applied Mathematics and Computational Science at the University of Pennsylvania. His current research interests include Partial Differential Equations, Mathematical Physics, Population Genetics, Imaging Science, and Numerical Analysis; he has also worked in Hyperbolic Geometry, Univalent Function Theory, Several Complex Variables, Microlocal Analysis and Index Theory. He was a Sloan Foundation Fellow, and is a fellow of the American Association for the Advancement of Science, and the American Mathematical Society. He was a co-recipient of the Stefan Bergman Prize in 2016. He holds a Ph.D. and MS in Mathematics from New York University and an SB in Mathematics from the Massachusetts Institute of Technology. My old web-page can be found here. Download my CV.

Research

Open Wave-Guides

I have been working for several years to develop theoretical foundations and effective numerical methods to solve scattering problems defined by open wave-guides. This problem is challenging because the wave-guides are not contained in bounded sets. My work, thus far, is summarized in 3 papers posted on the arXiv:

Solving the Scattering Problem for Open Wave-Guides, I Fundamental Solutions and Integral Equations Charles L. Epstein, (2023) arXiv:2302.04353 [math-ph]

Solving the Scattering Problem for Open Wave-Guides, II Outgoing Estimates Charles L. Epstein, (2023) arXiv:2310.05816 [math-ph]

Solving the Scattering Problem for Open Wave-Guides, III Radiation Conditions, joint with Rafe Mazzeo, (2024) arXiv:2401.04674v1 [math.AP]

Coordinate complexification for the Helmholtz equation with Dirichlet boundary conditions in a perturbed half-space,Charles L. Epstein, Leslie Greengard, Jeremy Hoskins, Shidong Jiang and Manas Rachh, arXiv:2409.06988.

A numerical method for scattering problems with unbounded interfaces, Tristan Goodwill and Charles L. Epstein, arXiv:2411.11204 .


Static Type-I Superconductors

In this project, joint with Manas Rachh, we are studying superconductors as described by the London equations $$\nabla\times {B}={J},\quad \nabla\times {J}=-\frac{1}{\lambda_L^2}{B}.$$ We showed that these equations can be solved using a Debye source representation. This work is described in the paper:

Debye source representations for type-I superconductors, I The static type I case. Charles L. Epstein, Manas Rachh, Journal of Computational Physics 452 (2022) 110892.

Type-I Superconductors in the Limit as the London Penetration Depth Goes to 0. Charles L. Epstein, Manas Rachh, and Yuguan Wang (2025); arXiv:2502.18809.

Selected Papers

  1. C.L. Epstein, Envelopes of Horospheres and Weingarten Surfaces in Hyperbolic 3-Space, arXiv:2401.12115 [math.DG], 2024.
  2. Charles L. Epstein and D. M. Burns, Jr., Embeddability for three dimensional CR–manifolds, Jour. of A. M. S., 3(1990), pages 809-841.
  3. Charles L. Epstein, A relative index for embeddable CR-structures, I,II, Annals of Math., 147(1998), pages 1–59; Annals of Math., 147 (1998), pages 61–91 Erratum: A relative index for embeddable CR-structures, I, Annals of Math., 154(2001).
  4. Brian B. Avants, C. L. Epstein, M. Grossman, and J. C. Gee, Symmetric Diffeomorphic Image Registration with Cross-Correlation: Evaluating Automated Labeling of Elderly and Neurodegenerative Brain, Med Image Anal. 12, 1 (2008), 26-41.
  5. C.L. Epstein and John Schotland, The Bad Truth about Laplace’s Transform, SIAM Review, 50(2008), 504-520.
  6. Charles L. Epstein, Subelliptic Spin_C Dirac operators, I, II, III, IV. Annals of Math, 166(2007), 225-256; 166(2007), 723-777; 168(2008), 299-365; 176(2012), 1373–1380.
  7. Charles L. Epstein and Leslie Greengard, Debye Sources and the Numerical Solution of the Time Harmonic Maxwell Equations, Comm. Pure and App. Math., 63 (2010) 0413-0463
  8. Charles L. Epstein, Leslie Greengard, and Michael O’Neil, Debye Sources and the Numerical Solution of the Time Harmonic Maxwell Equations, II, Comm. Pure and App. Math., doi: 10.1002/cpa.21420, 2012, 37pp.
  9. Charles L. Epstein, and Camelia A. Pop, The Feynman-Kac formula and Harnack inequality for degenerate diffusions, Annals of Probability, 45(2017), (doi: 10.1214/16-AOP1138), 3336-3384.
  10. Charles L. Epstein, and Camelia A. Pop, Transition probabilities for degenerate diffusions arising in population genetics, Probab. Theory Relat. Fields (2018). https://doi.org/10.1007/s00440-018-0840-2
  11. Charles L. Epstein, and Camelia A. Pop, Boundary Estimates for a Degenerate Parabolic Equation with Partial Dirichlet Boundary Conditions, J. Geom. Anal., 30 (2020), DOI 10.1007/s12220-017-9874-4, 2377-2421.
  12. C.L. Epstein and Jon Wilkening, Eigenfunctions and the Dirichlet problem for the Classical Kimura Diffusion Operator, SIAM Jour. of Applied Math., (doi: 10.1137/16M1063423), 77(2017), pp. 51–81.
  13. Alex Barnett, Charles L. Epstein, Leslie Greengard, Shidong Jiang, Jun Wang On Bounds for Toeplitz Operators and the Stability of Explicit Integral Equation Methods for the Heat Equation, Pure Appl. Anal., 1(2019), 709–742. DOI: 10.2140/paa.2019.1.709
  14. C.L. Epstein and M. Rachh, Debye source representation for type-I superconductors, I, The static type I case, Journal of Computational Physics 452 (2022) 11089, pp. 1-22.

Selected Books

  1. Charles L. Epstein, The Spectral Theory of Geometrically Periodic Hyperbolic 3–manifolds, Memoirs of the American Mathematical Society, no. 335 58(1985), 161 pages.
  2. Charles L. Epstein, Introduction to the Mathematics of Medical Imaging, 2nd Ed., SIAM, Philadelphia, PA, 2008, xxxiii+761 pages.
  3. Charles L. Epstein and Rafe Mazzeo, Degenerate Diffusion Operators Arising in Population Biology, Annals of Math. Studies, No. 185, Princeton Press, Princeton, NJ, 2013, 320 pages.
  4. Charles L. Epstein, Alex Barnett, Leslie Greengard, Jeremy Magland, Geometry of the Phase Retrieval Problem: GRAVEYARD OF ALGORITHMS, https://doi.org/10.1017/9781009003919 Cambridge University Press, 2022, xii+308 pages.

Contact