This volume brings together ideas from several areas of mathematics that have traditionally been rather disparate. The conference at The Fields Institute which gave rise to these proceedings was intended to encourage such connections. One of the key interactions occurs between dynamical systems and algorithms, one example being the by now classic observation that the QR algorithm for diagonalizing matrices may be viewed as the time-1 map of the Toda lattice flow.Another link occurs with interior point methods for linear programming, where certain smooth flows associated with such programming problems have proved valuable in the analysis of the corresponding discrete problems. More recently, other smooth flows have been introduced which carry out discrete computations (such as sorting sets of numbers) and which solve certain least squares problems. Another interesting facet of the flows described here is that they often have a dual Hamiltonian and gradient structure, both of which turn out to be useful in analyzing and designing algorithms for solving optimization problems. This volume explores many of these interactions, as well as related work in optimal control and partial differential equations.
Resonant geometric phases for soliton equations by M. S. Alber and J. E. Marsden Schur flows for orthogonal Hessenberg matrices by G. S. Ammar and W. B. Gragg Sub-Riemannian optimal control problems by A. M. Bloch, P. E. Crouch, and T. S. Ratiu Systems of hydrodynamic type, connected with the toda lattice and the Volterra model by O. I. Bogoyavlenskii The double bracket equation as the solution of a variational problem by R. W. Brockett Integration and visualization of matrix orbits on the connection machine by J.-P. Brunet A list of matrix flows with applications by M. T.-C. Chu The Gibbs variational principle, gradient flows, and interior-point methods by L. E. Faybusovich Optimization techniques on Riemannian manifolds by S. T. Smith On the number of real roots of a sparse polynomial system by B. Sturmfels Gradient flows for local minima of combinatorial optimization problems by W. S. Wong.