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In this talk I will present two works lying at the interface between PDEs, data science, and optimal transport.

The first part concerns the dissipative Hamiltonian structure of the Vlasov-Fokker-Planck equation (VFP). VFP describes the evolution of the probability density of the position and velocity of particles under the influence of external confinement, interaction, friction, and stochastic force. It is well-known that this equation can be formally seen as a dissipative Hamiltonian system in the Wasserstein space of probability measures. Moreover, this geometric structure has possible connections to the conjectured optimal convergence rates of underdamped Langevin Monte Carlo (ULMC), a sampling algorithm known to empirically outperform the (standard) Langevin Monte Carlo. This talk will focus on a time-discrete variational scheme for VFP which we introduce to more rigorously understand the geometric structure.

The second part, based on a joint work with Dejan Slepcev, concerns the sliced Wasserstein distance. The sliced Wasserstein metric compares probability measures by taking averages of the Wasserstein distances between projections of the measures to lines. The distance has found a range of applications in statistics and machine learning, as it is easier to approximate and compute in high dimensions than the (classical) Wasserstein distance. We will focus on our characterization of the disparate behavior of the sliced Wasserstein distance near absolutely continuous and discrete measures. We will explain the connection between this characterization and the maximum mean discrepancies (MMDs) which helps understand the behavior of the distance in high dimensions. If time permits, we will also discuss a refined result on the sample complexity of the metric and the instability of sliced Wasserstein gradient flows.

Although many mathematical models in wave theory lead to hyperbolic initial-boundary value problems which are inherently local due to the finite domain-of-dependence of the solution at any point on past values, there are also examples where nonlocal operators, in particular space-time integral operators, arise. A primary example is the radiation boundary condition needed to truncate an unbounded domain to a finite one to enable numerical solutions, as well as closely-related operators for unidirectional propagation. In this work we show how to leverage results from rational function approximation theory to construct spectrally-convergent local algorithms for evaluating such operators. For an important class of problems – systems equivalent to the scalar wave equation, such as acoustics or Maxwell’s equations in homogeneous, isotropic media, we will explain the construction, analysis, and implementation of our complete radiation boundary conditions, which are in a certain sense optimal. We will also discuss other applications of these approximation methods, as well as the fundamental barriers to extending the successful methods to other systems.

Origami, the art of folding paper into intricate shapes, has growing practical applications across fields such as architectural design, therapeutics, deployable space structures, antenna design, and soft robotics. One promising yet largely unexplored area is curved tile origami, which can store elastic energy, offering opportunities to develop next-generation functional materials, structures, and actuators. In this talk I will present a general theory of curved origami and systematic design methods for constructing large-scale, complex curved origami structures. Additionally, I will present methods to accurately calculate the stored elastic energy and the folding motions of curved origami, and I will illustrate my theoretical results by presenting some complex structures I have folded. This theory has inspired applications of curved origami in fields ranging from medical devices to a vertical-axis wind turbine, to architected materials.

TBA.

Microorganisms in the ocean are exposed to a patchy resource landscape, often provided in the form of nutrient-rich sinking particles. Yet, the most abundant microorganisms in the ocean are non-motile bacteria that rely on a diffusive supply of nutrients and cannot forage for particles via swimming. How they sustain their nutrient demand remains a conundrum. We developed a mathematical framework based on a spatial Poisson point process and analytical description of particle plumes to predict the frequency with which bacteria encounter nutrient-rich plumes from sinking particles across the global oceans. We find that due to the abundance of marine particles, non-motile bacteria encounter plumes from sinking particles frequently (~15 hr) that may readily satisfy their nutrient demand. Encounter rates are sensitive to regional differences in primary productivity and particle remineralization dynamics. Our mechanistic framework challenges the common view that non-motile bacteria in the ocean rely on vanishingly low background concentrations of dissolved nutrients and instead suggests that they can thrive on intermittent nutrient bursts from passing particles and their plumes.

TBA.