The Toda lattice is the prototypical discrete-space, continuous-time completely integrable Hamiltonian system.  It was introduced by Morikazu Toda in 1967 and analyzed in detail by Flaschka in 1974.  The bi-infinite Toda lattice can be solved with its associated inverse scattering transform (IST).  The IST is closely tied to the interpretation of the flow as an isospectral deformation of a bi-infinite tridiagonal matrix.  The Toda lattice has a completely integrable counterpart for finite symmetric and Hermitian (dense) matrices.  And due the the isospectral nature of the flow, it can be used as an eigenvalue algorithm. This talk has two parts. First, I will discuss the numerical computation of the IST for the Toda lattice by solving Riemann--Hilbert problems numerically.  Second, I will show that the time, called the halting time, it takes for the Toda lattice to compute the largest eigenvalue of a random matrix is universal --- the rescaled halting time converges to a universal distribution.

We study random permutations of the vertices of a hypercube  given by products of (uniform, independent) random transpositions on edges.  We establish the existence of a phase transition accompanied by emergence of cycles of diverging lengths. The problem is motivated by phase transitions in quantum spin models. (Joint work with Piotr Miłoś and Daniel Ueltschi.)

For singular operators of the form (H_{\alpha,\theta}u)_n=u_{n+1}+u_{n-1}+ \frac{g(\theta+n\alpha)}{f(\theta+n\alpha)} u_n, we prove such operators have purely singular spectrum on the set {E: \delta{(\alpha,\theta)}>L(E)\}, where f and g are both analytic functions.

The growth-optimal (Kelly) criterion almost surely leads to more capital in the long run and reaches levels of capital asymptotically faster than alternative strategies, but such outperformance may not be realized with high probability for an exceptionally long time. We will first demonstrate how the Kelly criterion arises in finance without first appealing to a logarithmic utility function, and then consider strategies based on alternative utilities that emphasize the probability of exceeding an underperforming benchmark faster than Kelly.