# Eric M. Rains

Professor of Mathematics, Emeritus

B.A., Case Western Reserve University, 1991; B.S., 1991; M.S., 1991; Ph.D., Harvard University, 1995. Visiting Associate and Lecturer, Caltech, 2002; Professor, 2007-23, Professor, Emeritus, 2023-; Executive Officer, 2019-22.

Research Areas:
Mathematics

## Research Interests

Main current interest: Special functions, especially applications from and to noncommutative algebraic geometry. Past interests include random matrices, codes, lattices, quantum information, as well as a number of miscellaneous side projects.Eric's work covers a wide variety of research ﬁelds, spanning from quantum information theory and coding theory, to the theory of random matrices, the study of special functions, non-commutative geometry and number theory (among others). Eric's most notable contribution to quantum information theory is the so-called "additive" construction with Calderbank, Shor, Sloane. In their work, they construct binary quantum codes from combinatorial data, producing much stronger codes than previously known. In recent work joint with Poonen, he developed a model for the distribution of *p*-Selmer groups of elliptic curves, which has since been generalized by others to *p ^{l}*-Selmer groups and to the

*p*-adic part of Tate-Shafarevich groups.