The Lieb-Robinson theorem states that locality is approximately preserved in the dynamics of quantum lattice systems. Whenever one has finite-dimensional constituents, observables evolving in time under a local Hamiltonian will essentially grow linearly in their support, up to exponentially suppressed corrections. In this work, we formulate Lieb-Robinson bounds for general harmonic systems on g...

We investigate the asymptotic decay of the total correlation function h (1,2) in molecular fluids. To this end, we expand the angular dependence of h (1,2) and the direct correlation function c (1,2) in the Ornstein-Zernike equation in a complete set of rotational invariants. We show that all the harmonic expansion coefficients h l1l2l) (r) are governed by a common exponential decay length and ...

Let f : M → N be a homotopy equivalence between closed negatively curved manifolds. The fundamental existence results of Eells and Sampson [5] and uniqueness of Hartmann [15] and Al’ber [1] grant the existence of a unique harmonic map h homotopic to f . Based on the enormous success of the harmonic map technique Lawson and Yau conjectured that the harmonic map h should be a diffeomorphism. This...

In this thesis, first, joint with Longzhi Lin, we establish estimates for the harmonic map heat flow from the unit circle into a closed manifold, and use it to construct sweepouts with the following good property: each curve in the tightened sweepout, whose energy is close to the maximal energy of curves in the sweepout, is itself close to a closed geodesic. Second, we prove the uniqueness for ...

Planar, underactuated, biped walkers form an important domain of applications for hybrid dynamical systems. This paper presents the design of exponentially stable walking controllers for general planar bipedal systems that have one degree of freedom greater than the number of available actuators. The within-step control action creates an attracting invariant set—a two dimensional zero dynamics ...

We prove that a quasiconformal map of the sphere S admits a harmonic quasi-isometric extension to the hyperbolic space H, thus confirming the well known Schoen Conjecture in dimension 3.

Effects of geometric constraints on a steady flow potential are described by an elliptic-hyperbolic generalization of the harmonic map equations. Sufficient conditions are given for global triviality. MSC2000 : 58E20, 58E99, 75N10.

We describe the duality between different geometries which arises by considering the classical and quantum harmonic map problem. To appear in “Essays on Mirror Manifolds II”. e-mail address: [email protected] e-mail address: [email protected] 1