Stephen Milne

We first recall the "sums of squares problem'', focusing on Jacobi's elliptic function approach from his epic ``Fundamenta Nova'' of 1829. We then report on our infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi's (1829) 4 and 8 squares identities to

squares, respectively, without using cusp forms such as those of Glaisher or Ramanujan for 16 and 24 squares. We derive our formulas by utilizing combinatorics to combine a variety of methods and observations from the theory of Jacobi elliptic functions, continued fractions, Hankel or Turanian determinants, Lie algebras, Schur functions, and multiple basic hypergeometric series related to the classical groups. We also note our derivation proof of the two Kac and Wakimoto (1994) conjectured identities concerning representations of a positive integer by sums of

triangular numbers, respectively. These conjectures arose in the study of Lie algebras and have also recently been proved by Zagier using modular forms. Related and subsequent work of Ken Ono and Heng Huat Chan is reviewed. We conclude with a brief discussion of our new formulas for Ramanujan's tau function, including one in terms of the Leech lattice. If time allows, we then present analogous new formulas for several other classical cusp forms that appear in quadratic forms, sphere-packings, lattices and groups.