Thursday 27 February 2025, at 11 am in Room 3014 (3rd Floor) - U5 Building, Prof. Marco Bertola (Concordia University, Montreal, Canada) will hold the following seminar
Title: Optimal lightning rods in the upper half plane and the focusing Nonlinear Schrödinger equation
Abstract: I will describe a problem in potential theory of the upper half plane (i.e. “electrostatics” in two dimensions) which, in its simplest form can be formulated as follows; Given a finite set of points E (“anchors’’) in the upper half plane and in the presence of a uniform electric field in the imaginary direction, find a connected set $K$ that “grounds’’ all the anchors to zero potential and connects with the real axis such that the energy stored by the induced charge is minimal. The problem is a variation of the celebrated problem (formulated by Pólya under the suggestion of Chebotarev) of finding connected sets of minimal (logarithmic) capacity containing a finite set of anchors. We will discuss how the optimal set (“lightning rod’’) is described as a set of so-called critical trajectories of quadratic differentials and how this can lead to their effective computation and plotting. The problem, beside its intrinsic allure, is relevant in the study of “solitons condensates” for the focusing Nonlinear Schrödinger Equation (fNLS), where the set describes optimal spectral support of the solitons so that the average intensity is minimal.
All interested are invited to participate