Iron Rings, Doctor Honoris Causa Raoul Bott, Carl Herz, and a Hidden Hand

The degree of Doctor of Sciences, honoris causa, was conferred on Raoul Bott by McGill University in 1987. Much of the work to make this happen was done by Carl Herz. Some of the author's personal recollections of both professors are included, along with some context for the awarding of this degree and ample historical tangents. Some cultural aspects occurring in the addresses are elaborated on, primarily, the Canadian engineer's iron ring. This paper also reprints both the convocation address of Raoul Bott and the presentation of Carl Herz on that occasion.Comment: An edited and reformatted version of this paper, with an additional photo, will appear in a volume dedicated to Raoul Bott. The author hopes to expand on some aspects of this preprint in future version

Chiral magnet models and boundary condition geometry in Skyrmion electronics

Field theoretic techniques are used to relate (i) the Landau-Lifschitz approach to Skyrmion devices based on Dzyaloshinskii-Moriya (D-M) chiral magnets, and (ii) the mathematical approaches to quantum magnetism. This results in a geometric understanding of micromagnetic singularities and boundary conditions without the usual thin-film assumptions.First author draf

On the Topological Characterization of Near Force-Free Magnetic Fields, and the work of late-onset visually-impaired Topologists

The Giroux correspondence and the notion of a near force-free magnetic field are used to topologically characterize near force-free magnetic fields which describe a variety of physical processes, including plasma equilibrium. As a byproduct, the topological characterization of force-free magnetic fields associated with current-carrying links, as conjectured by Crager and Kotiuga, is shown to be necessary and conditions for sufficiency are given. Along the way a paradox is exposed: The seemingly unintuitive mathematical tools, often associated to higher dimensional topology, have their origins in three dimensional contexts but in the hands of late-onset visually impaired topologists. This paradox was previously exposed in the context of algorithms for the visualization of three-dimensional magnetic fields. For this reason, the paper concludes by developing connections between mathematics and cognitive science in this specific context.Comment: 20 pages, no figures, a paper which was presented at a conference in honor of the 60th birthdays of Alberto Valli and Paolo Secci. The current preprint is from December 2014; it has been submitted to an AIMS journa

Dzyaloshinskii-Moriya chiral magnets and boundary conditions in Skyrmion electronics

Skyrmion-based electronic devices are a subset of spintronic nanodevices based on chiral materials (1, 2). The Dzyaloshinskii-Moriya (D-M) interaction is a chiral magnetic interaction which models chiral magnetic materials showing particular promise for extending CMOS compatible Skyrmionel ectronics at scales where silicon devices can no longer compete. There are several approaches to realizing such materials in practice. One is to focus on realizing D-M interactions as a fundamental problem in materials science supported by first principles quantum field theoretic models incorporating Majorana spinnors. Another very successful approach is to extend phenomenological micro-scale models of magnetism based on the Landau-Lifschitz-Gilbert (LLG) equation to the nanoscale by incorporating spin-torque coupling. However, this phenomenological approach obscures ties to more fundamental physics and the resulting boundary conditions can be a mystery. The present work uses well established mathematical techniques to show how Majorana spinnors and Skyrmions can appear in phenomenological models. There are three key aspects in this geometric/topological approach: • The first are Weitzenboeck identities and the Gaffney inequality (3). In electromagnetic theory, they enable us to study the distinction between Maxwell and Lame eigenmodes of cavity resonators; in micromagnetics they enable us to rewrite exchange energy in terms of fewer squares. • The second set of tools is familiar from the investigation of instantons; namely the identification of suitable divergence terms which enable one to rewrite a Hamiltonian in terms of the fewest number of squares. It is in this later step that the Majorana spinnors emerge without considerations of quantum mechanics and the Skyrmion solutions become apparent in a broader geometric context than the customary thin film scenarios. • Third, is the geometric observation that the LLG equation projects the magnetization vector so as to leave its length invariant. This enables us to consider the Hamiltonian of the system modulo the rescaling of the magnetization vector. As a result of this geometric reformulation, a clearer understanding of the use of the LLG equation at the nanoscale emerges as well as a more geometric connection to the underlying quantum phenomena. Finally, the role of chirality emerges more cleanly and it points to the role of topology in the possibility of near reversible computing generating a minimum of entropy and heat (4, 5, 6).First author draf

Inter-winding Distributed Capacitance and Guitar Pickup Transient Response

Simple RLC circuit models of guitar pickups do not account for audible features that characterize the pickup. Psycho-acoustic experiments reveal that any acoustically accurate model has to reproduce the first 30 milli-seconds of the transient response with extreme precision. The proposed model is impractical for simple-minded model reduction or brute force numerical simulations yet, by focusing on modeling electromagnetic details and exposing a connection to spectral graph theory, a framework for finding the transient response to sufficient accuracy is exposed.Comment: Four pages, no figures. This paper is associated to a conference presentation given at CEFC 2014 in Annecy France; the posted preprint is from October 2014, and the data for the final publication can be found belo

Lower and upper bounds for the Rayleigh conductivity of a perforated plate

International audienceLower and upper bounds for the Rayleigh conductivity of a perforation in a thick plate are usually derived from intuitive approximations and by physical reasoning. This paper addresses a mathematical justification of these approaches. As a byproduct of the rigorous handling of these issues, some improvements to previous bounds for axisymmetric holes are given as well as new estimates for inclined perforations. The main techniques are a proper use of the variational principles of Dirichlet and Kelvin in the context of Beppo-Levi spaces. The derivations are validated by numerical experiments in the two-dimensional axisymmetric case and the full three-dimensional one