This project involves three seemingly unrelated parts of mathematics: the representation theory of real reductive groups, the Fourier coefficients of automorphic forms and the mathematics of entanglement in quantum computing. The threads that hold these subjects together involve the invariant theory, finite dimensional representation theory, combinatorics and the algebraic geometry of group actions. The first two subjects have played an important role in the great triumphs of mathematics in the twentieth century. The latter subject is in preparation for computing in the second half of this century. The representation theory to be studied involves finding new ways of constructing the most elusive unitary representations which we call small in this proposal. The analysis of Fourier coefficients involves the search for the "most general" multiplicity one theorem for generalized Whittaker modules. The work on entanglement involves finding useful measures of entanglement that can be used by experimental physicists in their attempt to build quantum computers.
Representation theory has its roots in nineteenth century invariant theory, early twentieth century quantum mechanics and mid-twentieth century number theory. In this first decade of the twenty first century the theory has returned to its roots. The nineteenth century invariant theory emphasized concrete questions on binary forms with algorithmic solutions. These problems have reemerged and are now being generalized to apply to quantum computation. Early quantum mechanics studied puzzling and weird measurements involving photons, electrons etc. These phenomena led to the Hilbert space approach to quantum mechanics. The philosophical debates of the early quantum mechanics have reemerged as quantum information technology. The Hilbert space approach also gave birth to representation theory, which has as one of its main applications in number theory. The Langlands program has established a goal for the twenty first century to establish a non-commutative class field theory (Wile's proof of Fermat's Last Theorem is actually proof of a special case of the Tanayama-Shimura conjecture which is a special case of the Langlands program). This project is in the interface of all of these exciting directions.
