Department of Mathematics

The Ohio State University

100 Math Tower

231 W 18th Avenue

Columbus, OH 43210

Telephone: (614) 292-7049

Email:
milne@math.ohio-state.edu

- Ph.D. University of California, San Diego, 1976.

Thesis advisor: Professor Adriano M. Garsia

Thesis: Peano curves and smoothness of functions - A.B. San Diego State University, 1972.

Current work involves algebraic combinatorics, classical analysis
(elliptic functions, continued fractions, and multiple basic
hypergeometric series), Lie algebras, analytic number theory,
and their relationships with one another. This research studies higher
dimensional analogs of classical special functions which are inspired
by certain aspects of mathematical physics and the unitary groups
*U(n)*. Applications include topics in combinatorics
(summation theorems, symmetric functions identities, combinatorics of
the symmetric group and weighted inversion statistics), *q*-series,
additive number theory (constructing one-to-one correspondences between
different sets of partitions of *n*), analytic number
theory (sums of squares identities via elliptic functions, continued
fractions, and multiple basic hypergeometric series), Lie algebras
(generalizations of the Macdonald identities with extra free
parameters),
applications to raising operators of row type for Macdonald
polynomials, and multivariable orthogonal polynomials.

- Currently supported by a two year National Security Agency Grant for January 2008 through January 2010. ``Analytic Number Theory and Multiple Basic Hypergeometric Series.
- Continuous support by N.S.F. or N.S.A. from 1976 through Fall, 2004. Then, 2006 - 2010. Previously awarded National Security Agency grants for: June 1, 1988 - May 31, 1989; September 1, 1991 through February 28, 1994; June 1, 1993 through November 30, 1997; January 17, 1997 through July 31, 1999; and, November 17, 1998 through November 16, 2000. These grants provided Milne one month Summer salary and also partially supported (during both Summer and/or academic year) most of Milne's Ph.D. students during their thesis work. Continuous support by N.S.F. from 1976 through Spring, 1991.

- A very recent article by Ivars Peterson in
*Science News*just featured Milne's sums of squares work in (PNAS online reprint) and (Kluwer online abstract) and consequences it has had in the subsequent investigations of Don Zagier and Ken Ono. (The reference is: [Ivars Peterson,*Surprisingly Square:~Mathematicians take a fresh look at expressing numbers as the sums of squares*, Science News, June 16, 2001; Vol. 159, No. 24, p. 382--383].) This long-established (since 1927) weekly publication has reported on important developments in all areas of science (including mathematics) for many years. For references and sources used in this article see the following Science News URL:~ (online references and sources). For a text only version of the article, see the URL:~ (text of SN article).

- (March 14-16, 2003), AMS meeting in Baton Rouge, LA at LSU. Invited hour address: ``A new look at sums of squares, Jacobi elliptic functions, and Ramanujan's tau function.'' Note that this is Milne's second AMS Invited Hour Address. The first was in 1984.

- Infinite families of exact sums
of squares formulas, Jacobi elliptic functions, continued fractions,
and Schur functions,
The Ramanujan Journal
**6**(2002), 7--149.

- Glenn Lilly (1991)
- John Newcomb (1993)
- Gaurav Bhatnagar (1995)
- Sheldon Degenhardt (1996)
- Verne Leininger (1997)
- Jon Stadler (1997)
- Jon Breitenbucher (2001)
- Eric Conrad (2002)