Georgia Institute of Technology College of Sciences

Consider the following question of interest to cryptographers: A message is encoded in a binary string of length n. Consider a set of scrambling operations S (a proper subset of permutations on n bits). If a scrambling operation is applied uniformly at random from S at each step, then after how many steps will the composition of scrambling operations look like a random permutation on all the bits? This question asks for the convergence time for a random walk on the permutation group. Replace the binary string with a quantum state and scrambling operations in S with Haar random quantum channels on two bits (qudits) at a time. Broadly speaking, we study the following question: If a scrambling operation is applied uniformly at random from S at each step, then after how many steps will the composition of scrambling operations (quantum channels) look like a Haar random channel on all qudits? This question asks about the convergence time for a random walk on the unitary group. Various protocols in quantum computing require Haar random channels, which motivates us to understand the number of operations one would require to approximately implement that channel.

A fully nonlinear surface dynamics of the time dependent potential flow of ideal incompressible fluid with a free surface is considered in two dimensional geometry. Arbitrary large surface waves can be efficiently characterized through a time-dependent conformal mapping of a fluid domain into the lower complex half-plane. We reformulate the exact Eulerian dynamics through a non-canonical nonlocal Hamiltonian system for the pair of new conformal variables. We also consider a generalized hydrodynamics for two components of superfluid Helium which has the same non-canonical Hamiltonian structure. In both cases the fluid dynamics is fully characterized by the complex singularities in the upper complex half-plane of the conformal map and the complex velocity. Analytical continuation through the branch cuts generically results in the Riemann surface with infinite number of sheets including Stokes wave, An infinite family of solutions with moving poles are found on the Riemann surface. Residues of poles are the constants of motion. These constants commute with each other in the sense of underlying non-canonical Hamiltonian dynamics which provides an argument in support of the conjecture of complete Hamiltonian integrability of surface dynamics. If we consider initial conditions with short branch cuts then fluid dynamics is reduced to the complex Hopf equation for the complex velocity coupled with the complex transport equation for the conformal mapping. These equations are fully integrable by characteristics producing the infinite family of solutions, including the pairs of moving square root branch points. The solutions are compared with the simulations of the full Eulerian dynamics giving excellent agreement.

This talk is centered around a symplectic approach to eigenvalue problems for systems of ordinary differential operators (e.g., Sturm-Liouville operators, canonical systems, and quantum graphs), multidimensional elliptic operators on bounded domains, and abstract self-adjoint extensions of symmetric operators in Hilbert spaces. The symplectic view naturally relates spectral counts for self-adjoint problems to the topological invariant called the Maslov index. In this talk, the notion of the Malsov index will be introduced in analytic terms and an overview of recent results on its role in spectral theory will be given.