# Eric M. Rains

Eric Michael Rains (born 23 August 1973) is an American mathematician specializing in coding theory and special functions, especially applications from and to noncommutative algebraic geometry.

## Biography

Eric Rains was 14 when he began classes at Case Western Reserve University in 1987. He graduated with bachelor's degrees in computer science and physics and a master's degree in mathematics at age 17.
By means of a Churchill Scholarship he studied mathematics and physics at the University of Cambridge for the academic year 1991–1992, receiving a Certificate of Advanced Study in Mathematics. He received his PhD in 1995 from Harvard University with thesis Topics in Probability on Compact Lie Groups under the supervision of Persi Diaconis. From 1995 to 1996, Rains worked at the IDA's Center for Communications Research (CCR) in Princeton. From 1996 to 2002 he was a researcher for AT&T Labs. From 2002 to 2003 he returned to the CCR in Princeton. In 2003, Rains became a full professor at the University of California, Davis. From 2007-2023, Rains was a full professor at Caltech and served as the Executive Officer of the Caltech Mathematics Department from 2019 to 2022. As of Fall 2023, Rains is a professor emeritus at Caltech, per their website.
In the fall of 2006 he was a visiting professor at the University of Melbourne. He is the co-author with Gabriele Nebe and Neil J. A. Sloane of the 2006 book Self-Dual Codes and Invariant Theory.
In 2007, Rains was a plenary speaker at the Western Sectional meeting of the American Mathematical Society (AMS). In 2010 he was an invited speaker at the International Congress of Mathematicians in Hyderabad. He was elected a Fellow of the AMS in the class of 2018 for "contributions to coding theory, the theory of random matrices, the study of special functions, non-commutative geometry and number theory".

## Selected publications

* Calderbank, A. R.; Rains, E. M.; Shor, P. W.; Sloane, N. J. A. (1997). "Quantum Error Correction and Orthogonal Geometry". Physical Review Letters. 78 (3): 405–408. arXiv:quant-ph/9605005. Bibcode:1997PhRvL..78..405C. doi:10.1103/PhysRevLett.78.405. S2CID 15326700.
* Rains, E.M. (1998). "Shadow bounds for self-dual codes". IEEE Transactions on Information Theory. 44: 134–139. doi:10.1109/18.651000. S2CID 32909885.
* Calderbank, A.R.; Rains, E.M.; Shor, P.M.; Sloane, N.J.A. (1998). "Quantum error correction via codes over GF(4)" (PDF). IEEE Transactions on Information Theory. 44 (4): 1369–1387. doi:10.1109/18.681315. S2CID 1215697. (This article has over 1200 citations.)
* Rains, E.M. (1999). "Nonbinary quantum codes". IEEE Transactions on Information Theory. 45 (6): 1827–1832. arXiv:quant-ph/9703048. doi:10.1109/18.782103. S2CID 1738302.
* Rains, E. M. (1999). "Rigorous treatment of distillable entanglement". Physical Review A. 60 (1): 173–178. arXiv:quant-ph/9809078. Bibcode:1999PhRvA..60..173R. doi:10.1103/PhysRevA.60.173. S2CID 16329605.
* Rains, E. M. (1999). "Bound on distillable entanglement". Physical Review A. 60 (1): 179–184. arXiv:quant-ph/9809082. Bibcode:1999PhRvA..60..179R. doi:10.1103/PhysRevA.60.179. S2CID 14090725.
* Bennett, Charles H.; DiVincenzo, David P.; Fuchs, Christopher A.; Mor, Tal; Rains, Eric; Shor, Peter W.; Smolin, John A.; Wootters, William K. (1999). "Quantum nonlocality without entanglement". Physical Review A. 59 (2): 1070–1091. arXiv:quant-ph/9804053. doi:10.1103/PhysRevA.59.1070. ISSN 1050-2947. S2CID 15282650.
* Baik, Jinho; Rains, Eric M. (2000). "Limiting distributions for a polynuclear growth model with external sources". Journal of Statistical Physics. 100 (3/4): 523–541. doi:10.1023/A:1018615306992. S2CID 17786797.
* Odlyzko, A. M.; Rains, E. M. (2000). "On longest increasing subsequences in random permutations". Contemporary Mathematics. 251: 439–452. doi:10.1090/conm/251/03886. ISBN 9780821811481.
* Rains, E.M. (2001). "A semidefinite program for distillable entanglement" (PDF). IEEE Transactions on Information Theory. 47 (7): 2921–2933. doi:10.1109/18.959270. S2CID 6115879.
* Rains, E. M.; Sloane, N. J. A. (1998). "Self-Dual Codes". Handbook of Coding Theory. Vol. I. North Holland. pp. 177–294. arXiv:math/0208001. Bibcode:2002math......8001R. ISBN 0444500871.
* Rains, E. M.; Sloane, N. J. A.; Rains, Eric M.; Svore, Krysta M. (2006). "A logarithmic-depth quantum carry-lookahead adder". Quantum Information & Computation. 6 (4): 351–369. arXiv:quant-ph/0406142. Bibcode:2004quant.ph..6142D.
* Borodin, Alexei; Rains, Eric M. (2005). "Eynard–Mehta Theorem, Schur Process, and their Pfaffian Analogs". Journal of Statistical Physics. 121 (3–4): 291–317. arXiv:math-ph/0409059. Bibcode:2005JSP...121..291B. doi:10.1007/s10955-005-7583-z. S2CID 119599225.
* Rains, Eric M. (2010). "Transformations of elliptic hypergeometric integrals" (PDF). Annals of Mathematics. 171 (1): 169–243. doi:10.4007/annals.2010.171.169. JSTOR 27799200.
* Poonen, Bjorn; Rains, Eric (2012). "Random maximal isotropic subspaces and Selmer groups". Journal of the American Mathematical Society. 25 (1): 245–269. arXiv:1009.0287. doi:10.1090/S0894-0347-2011-00710-8. ISSN 0894-0347.