Graduate Student Colloquium: Chuck Akemann (UCSB), 'Computer scientists solve long standing mathematics problem'

Event Date: 

Thursday, January 9, 2014 -
3:30pm to 4:30pm

Event Location: 

  • 6635 South Hall

Event Contact: 

Daryl Cooper

Email: cooper@math.ucsb.edu

Dick Kadison and Iz Singer wrote a paper in 1959 that posed several interesting questions about the continuous linear functionals on the normed vector space B(H) = the space of continuous linear maps from a Hilbert space H onto itself. The last of these problems was solved in a June 2013 paper by three computer scientists. The original form of the problem involved linear functionals that only could exist by appeal to the Axiom of Choice. Computer science folks have more sense than to get involved in such things, so how did it come about? Over a period of years it was shown that there was were several equivalent problems that involved only finite dimensional complex matrices. The purpose of this talk is to use this example to illustrate the usefulness of alternate ways to view problems. (The Riemann Hypothesis is another, much more famous, problem that has generated this same phenomenon, except that the solution still eludes us.) The Kadison-Singer problem has the advantage of being solved by methods that would have been completely foreign to the originators of the problem. Further, the version of the problem that was solved can be explained in terms of undergraduate linear algebra. Review before lecture please: C^n denotes n-dimensional complex Euclidean space, with standard basis e_1, Ée_n. Square n by n matrices define linear mappings from C^n to itself via this basis by matrix multiplication in the usual way. We are only interested in a matrix that is self adjoint, i.e. equal to its conjugate transpose. We further restrict attention to those self adjoint matrices with non-negative eigenvalues (called the positive matrices). Solved version of the Kadison-Singer Problem. Given some positive, rank one matrices that sum to the identity matrix, and each has its single non-zero eigenvalue bounded by t