Applied Math Seminar

Synopsis:

The dynamics of numerous physical systems, such as spins and qubits, are series of rotations, that is, walks in the manifold of the rotation group. This provokes a natural question: How likely and under what conditions will such walks return to the origin, such that the physical system will return to its initial state?     The surprising result is that almost every walk in SU(2) or SO(3) – even a highly convoluted one – will preferentially return to the origin simply by traversing the walk twice in a row and uniformly scaling all rotation angles. We will see how this question arose from the problem of rolling bodies along arbitrary paths on the plane [1,2]. Then, we will explain how its “two-period” solution followed from the distribution of random rotations and Minkowski’s theorem, and why traversing the walk only once almost never suffices to return [3]. An ideal conclusion would be the audience proposing how this property of SU(2) can be used.

[1] Sobolev et al. Solid-body trajectoids shaped to roll along desired pathways. Nature, 2023.
[2] Eckmann et al. Tumbling Downhill Along a Given Curve. AMS Notices, 2024.
[3] Eckmann et al. Walks in Rotation Spaces Return Home When Doubled and Scaled. arXiv:2502.14367.

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To maintain controlled flight, animals and aerial vehicles alike must execute continuous trimming adjustments to compensate for internal and external perturbations. Engineers solve the problem of flight stabilization through the use of proportional-integral-derivative (PID) controllers. Flying animals likely make use of a similar control strategy, using visual, inertial, and proprioceptive cues to determine and correct for deviations from the desired flight path. The fruit fly Drosophila melanogaster is a convenient model organism for studying flight control in animals; in addition to elegantly handling external perturbations such as impending collisions or sudden gusts, Drosophila are robust to remarkably large asymmetries in body morphology—even after having lost half a wing, flies retain the ability to fly straight. This performance is all the more impressive considering the controller responsible (i.e., the fly’s brain) is smaller than a sesame seed. In this talk, I will present findings illuminating the neural implementation of the algorithms we believe govern flight control in Drosophila. Prior work has identified a population of neurons descending from the brain to the body, the DNg02s, that collectively regulate wing stroke amplitude over a large dynamic range and appear to be involved in visually-mediated flight stabilization. Here, we reveal the precise function of these neurons in flight control and their downstream connectivity to flight muscles.

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In this talk, we describe the classical magneto-static approach to the theory of type-I
superconductors. The magnetic field and the current in type-I superconductors are
related by the London equations and tend to decay exponentially inside the supercon-
ducting material with support of the fields contained primarily in \(\mathcal O(\lambda L)\) neighborhood
of the superconductor. We present a Debye source based integral representation for
the numerical solution of the London equations, and demonstrate the efficacy of our
approach for moderate values of \(\lambda L\) on complex three dimensional geometries. How-
ever, for typical materials \(\lambda L \sim \mathcal O(10^{-7})\), which makes the PDE and integral equation
increasingly difficult to solve in the limit \(\lambda L \to 0\) due to the presence of two different
length scales in the problem. We derive a limiting PDE and a corresponding integral
equation, and show that the solutions of this limiting PDE and integral equations are
\(\mathcal O(\lambda L)\) accurate as compared to the corresponding solutions of the London equations
and the Debye source integral equations respectively. We demonstrate the effective-
ness of this asymptotic approach both in terms of speed and accuracy through several
numerical examples.

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Waveform inversion seeks to estimate an inaccessible heterogeneous medium by using sensors to probe the medium with signals and measure the generated waves. It is an inverse problem for a hyperbolic system of equations, with the sensor excitation modeled as a forcing term and the heterogeneous medium described by unknown, variable coefficients. The traditional formulation of the inverse problem, called full waveform inversion (FWI), estimates the unknown coefficients via nonlinear least squares data fitting. For typical band limited and high frequency data, the data fitting objective function has spurious local minima near and far from the true coefficients. This is why FWI implemented with gradient based optimization can fail, even for good initial guesses. We propose a different approach to waveform inversion: First, use the data to "learn" a good algebraic model, called a reduced order model (ROM), of how the waves propagate in the unknown medium. Second, use the ROM to obtain a good approximation of the wave field inside the medium. Third, use this approximation to solve the inverse problem. I will give a derivation of such a ROM for a general first order hyperbolic system satisfied by all linear waves in lossless media (sound, electromagnetic or elastic). I will describe the properties of the ROM and will use it to solve the inverse problem for sound waves. 

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