Title: Conformal symmetry breaking and self-similar spirals

Jacob Bernoulli’s Spira Mirabilis set free

Speaker: Dr. Jemal Guven (ICN, UNAM )

Abstract: Self-similar curves are a recurring motif in nature. They also arise naturally as tension-free stationary states of conformally invariant energies. Planar logarithmic spirals, for example, are associated with the simplest such energy, the conformal arc-length, and their remarkable properties follow as a consequence of this invariance and the manner of its breaking. In this talk I will construct their three-dimensional analogues explicitly. The qualitative behavior of these states is controlled by two parameters, the conserved scaling current S and the magnitude of the torque M. The corresponding conservation laws determine the curvature and the torsion. If the spiral apex is located at the origin, the conserved special conformal current vanishes. This permit the spiral trajectory to be traced in the spherical polar coordinate system adapted to the spiral apex and the torque axis. Planar logarithmic spirals occur when M and S are tuned so that 4MS = 1. More generally, the spiral exhibits internal structure, nutating between two cones aligned along the torque axis. It expands monotonically as this pattern precesses about this axis. If the spiral is supercritical (4MS > 1) the cones are identical and oppositely oriented; the torsion changes sign within the nutation cycle where the projection along the torque axis reverses. These elementary spirals provide the natural templates against which to compare spiral patterns exhibiting self-similar behavior. In particular, supercritical trajectories approximate rather well the nutating tip of the growing tendril in a climbing plant first described by Darwin. This is probably not a coincidence.

Time: Thursday, 18 July 2019, 2:30pm

Location: Lecture Room, 1st Floor, School of Theoretical Physics, DIAS, 10 Burlington Road, Dublin 4