# Charles Epstein (mathematician)

Charles L. Epstein is a Senior Research Scientist in the Center for Computational Mathematics at the Flatiron Institute. He is the Thomas A. Scott Professor of Mathematics Emeritus at the University of Pennsylvania, Philadelphia.

## Research areas

Charles Epstein is an analyst and applied mathematician. His research interests include partial differential equations, mathematical physics, boundary value problems, mathematical biology, population genetics, nuclear magnetic resonance and medical imaging, and numerical analysis; he has also worked in hyperbolic geometry, univalent function theory, several complex variables, microlocal analysis and index theory.

## Education and career

He was an undergraduate in mathematics at the Massachusetts Institute of Technology and graduate student at the Courant Institute, New York University, where he received his Ph.D. in 1983 under the direction of Peter Lax.
He was a postdoc with William Thurston before moving to the University of Pennsylvania, where he has been since. Epstein won a Sloan Research Fellowship in 1988.
He is currently a Senior Research Scientist in the Center for Computational Mathematics at the Flatiron Institute, New York City, and Thomas A. Scott Professor of Mathematics Emeritus at the University of Pennsylvania, Philadelphia.

## Awards and honors

In 2014, Charles Epstein became a Fellow of the American Mathematical Society "for contributions to analysis, geometry, and applied mathematics including medical imaging, as well as for service to the profession". He was a co-recipient of the Stefan Bergman Prize in 2016.

## Books

* C L Epstein, Introduction to the mathematics of medical imaging. Second edition. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. xxxiv+761 pp. ISBN 978-0-89871-642-9
* C L Epstein, The spectral theory of geometrically periodic hyperbolic 3-manifolds. Mem. Amer. Math. Soc. 58 (1985), No. 335, ix+161 pp.

## Publications

* C L Epstein, R B Melrose, G A Mendoza, Resolvent of the Laplacian on strictly pseudoconvex domains. Acta Mathematica 167 (1991), no. 1–2, 1–106.
* C L Epstein, The hyperbolic Gauss map and quasiconformal reflections. Journal für die Reine und Angewandte Mathematik 372 (1986), 96–135.
* C L Epstein, R Melrose, Contact degree and the index of Fourier integral operators. Math. Res. Lett. 5 (1998), no. 3, 363–381.
* C L Epstein, Embeddable CR-structures and deformations of pseudoconvex surfaces. I. Formal deformations. J. Algebraic Geom. 5 (1996), no. 2, 277–368.
* C L Epstein, CR-structures on three-dimensional circle bundles. Invent. Math. 109 (1992), no. 2, 351–403.
* D M Burns, C L Epstein, Embeddability for three-dimensional CR-manifolds. J. Amer. Math. Soc. 3 (1990), no. 4, 809–841.
* C L Epstein A relative index on the space of embeddable CR-structures. I. Annals of Mathematics (2) 147 (1998), no. 1, 1–59.
* C L Epstein, Asymptotics for closed geodesics in a homology class, the finite volume case. Duke Math. J. 55 (1987), no. 4, 717–757.
* C L Epstein; G M Henkin, Stability of embeddings for pseudoconcave surfaces and their boundaries. Acta Mathematica 185 (2000), no. 2, 161–237.
* C L Epstein, A relative index on the space of embeddable CR-structures. II. Annals of Mathematics (2) 147 (1998), no. 1, 61–91.
* D Burns, C L Epstein, Characteristic numbers of bounded domains. Acta Mathematica 164 (1990), no. 1–2, 29–71.
* C L Epstein, M Gage, The curve shortening flow. Wave motion: theory, modelling, and computation (Berkeley, Calif., 1986), 15–59, Math. Sci. Res. Inst. Publ., 7, Springer, New York, 1987.
* D M Burns, Jr, C L Epstein, A global invariant for three-dimensional CR-manifolds. Invent. Math. 92 (1988), no. 2, 333–348.
* C L Epstein, G M Henkin, Extension of CR-structures for 3-dimensional pseudoconcave manifolds. Multidimensional complex analysis and partial differential equations (São Carlos, 1995), 51–67, Contemp. Math., 205, Amer. Math. Soc., Providence, RI, 1997.
* C L Epstein, B Kleiner, Spherical means in annular regions. Comm. Pure Appl. Math. 46 (1993), no. 3, 441–451.
* C L Epstein, G M Henkin, Embeddings for 3-dimensional CR-manifolds. Complex analysis and geometry (Paris, 1997), 223–236, Progr. Math., 188, Birkhäuser, Basel, 2000.
* C L Epstein, Subelliptic Spin$_C$ Dirac operators. I. Annals of Mathematics (2) 166 (2007), no. 1, 183–214.