Applied Mathematics Seminar

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Bacterial motility and growth play central roles in agriculture, the environment, and medicine. While bacterial behavior is typically studied in bulk liquid or on flat surfaces, many bacterial habitats—e.g., soils, sediments, and biological gels/tissues—are complex and crowded spaces. In this talk, I will describe my group's work using tools from soft matter physics to address this gap in knowledge. In particular, using studies of _E. coli_ in transparent 3D porous media, we demonstrate how confinement in a crowded medium fundamentally alters bacterial behavior. In particular, we show how the paradigm of run-and-tumble motility is dramatically altered by pore-scale confinement, both for cells performing undirected motion and those performing chemotaxis, directed motion in response to a chemical stimulus. Our porous media also enable precisely structured multi-cellular communities to be 3D printed. Using this capability, we show how spatial variations in the ability of cells to perform chemotaxis enable populations to autonomously stabilize large-scale perturbations in their overall morphology. Finally, we show how when the pores are small enough to prevent cells from swimming through the pore space, expansion of a community via cellular growth and division gives rise to distinct, highly-complex, large-scale community morphologies. Together, our work thus reveals new principles to predict and control the organization of bacteria, and active matter in general, in complex and crowded environments.

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In two phase materials, each phase having a non-local response in time, we were surprised to discover that for appropriate driving fields the response somehow untangles at specific times, allowing one to directly infer useful information about the geometry of the material, such as the volume fractions of the phases. This rests on the existence of approximate, measure independent, linear relations between the values that Markov functions take at a given set of possibly complex points, not belonging to the interval [-1,1] where the measure is supported. The problem is reduced to simply one of polynomial approximation of a given function on the interval [-1,1]. In the context of the motivating problem, the analysis also yields bounds on the response at any particular time for any driving field, and allows one to estimate the response at a given frequency using an appropriately designed driving field that effectively is turned on only for a fixed interval of time. The approximation extends directly to Markov-type functions with a positive semi-definite operator valued measure, and this has applications to determining the shape of an inclusion in a body from boundary flux measurements at a specific time, when the time-dependent boundary potentials are suitably tailored. This is joint work with Ornella Mattei and Mihai Putinar.

 

 

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Surface gravity waves transport mass through a process referred to as Stokes drift. Finite-sized particles of arbitrary shape exhibit additional Stokes drift-like phenomena. In this talk, I explore the effects of particle inertia and shape on particle behavior in waves. I demonstrate both an angular analogue of Stokes drift for non-spherical particles and a vertical Stokes drift for settling particles. Experimental observations are compared with these analytical results, and these behaviors are discussed in the context of microplastic transport in the ocean.

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How weak is the weak turbulence? We analyze turbulence of weakly
interacting waves using the tools of information theory. It offers a
unique perspective for comparing thermal equilibrium and turbulence. The
mutual information between resonant modes in a finite box is shown to be
stationary and small in thermal equilibrium, yet to grow with time in
weak turbulence. We trace this growth to the concentration of
probability on the resonance surfaces, which can go all the way to a
singular measure. The surprising conclusion is that no matter how small
is the nonlinearity and how close to Gaussian is the statistics of any
single amplitude, a stationary phase-space measure is far from Gaussian,
as manifested by a large relative entropy. At the end I shall describe a
new class of the so-called Fibonacci models that describe resonantly
interacting waves and allow turbulence close to thermal equilibrium.
(0.1103/PhysRevLett.125.104501)

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Motivated by Enceladus cryovolcanism and possible shear heating along
its south pole fractures, we perform a theoretical and numerical study
of boundary-layer convection along a vertical heated wall in a bounded
ideal mushy region. By focusing on this simple model, we demonstrate the
existence of four regions with different regimes and scalings that are
studied asymptotically, showing a good agreement between the theory and
the numerical simulations. Close to the heated wall, the convection in
the mushy layer is similar to a rising buoyant plume abruptly stopped at
the top, leading to increased pressure and temperature in the upper
region, that could be the ignition of Enceladus' geysers.