Applied Math Seminar

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Active materials such as bacteria, molecular motors and eukaryotic cells continuously transform chemical energy taken from their surroundings to mechanical work. Dense active matter shows mesoscale turbulence, the emergence of chaotic flow structures characterised by high vorticity and self-propelled topological defects. I shall describe the physics of active defects, discussing active microfluidics, active disclinations and examples of topological defects in biological systems.

 

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Placing a two-dimensional lattice on another with a small rotation gives rise to periodic “moire” patterns on a superlattice scale much larger than the original lattice. This effective large-scale fundamental domain allows phenomena such as the fractal Hofstadter butterfly in the spectrum of Harper’s equation to be observed in real crystals. Experimentalists have more recently observed new correlated phases at the “magic” twist angles predicted by theorists.

We will give mathematical and computational models to predict and gain insight into new physical phenomena at the moiré scale including our recent mathematical and experimental results for twisted trilayer graphene.

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I will discuss some applications of random walks and PDEs in graph-based learning, both for theoretical analysis and algorithm development. Graph-based learning is a field within machine learning that uses similarities between datapoints to create efficient representations of high-dimensional data for tasks like semi-supervised classification, clustering and dimension reduction. There has been considerable interest recently in semi-supervised learning problems with very few labeled examples (e.g., 1 label per class). The widely used Laplacian regularization is ill-posed at low label rates and gives very poor classification results. In the first part of the talk, we will use the random walk interpretation of the graph Laplacian to precisely characterize the lowest label rate at which Laplacian regularized semi-supervised learning is well-posed. At lower label rates, we will show how our random walk analysis leads to a new algorithm, called Poisson learning, that is probably more stable and informative than Laplace learning. We will also briefly discuss some recent Lipschitz regularity results for graph Laplacians that have applications to improving spectral convergence rates.

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In this talk, I will introduce capillary disks - hydrophobic disks at the capillary scale whose weight is supported on the fluid interface by virtue of hydrostatics and capillarity.  

I will begin by presenting direct measurements of the attractive force between two  capillary disks.  It is well known that objects at a fluid interface may interact due to the mutual deformation they induce on the free surface, however few direct measurements of such forces have been reported. In the present work, we characterize how the attraction force depends on the disk radius, mass, and relative spacing, and rationalize our findings with a scaling analysis.

When such disks are then deposited on a vibrating fluid bath, the relative vertical motion of the disk and the interface leads to the generation of outwardly propagating capillary waves. We demonstrate that when the rotational symmetry of an individual particle is broken, the particles can steadily self-propel along the interface and interact with each other via their collective wavefield, forming a myriad of cooperative dynamic states. Our discovery opens the door to further investigations of this active system with fluid-mediated interactions at intermediate Reynolds numbers. 

Ongoing work and future directions will be discussed.

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We analyze a non-convex stochastic descent algorithm for streaming PCA due to Erkki Oja (1982). Despite its simplicity, this algorithm has resisted optimal analysis outside of the rank-one case. We show that given access to a sequence of i.i.d. d x d symmetric matrices, Oja's algorithm obtains an accurate approximation to the subspace of the top k eigenvalues of their expectation using a number of samples scaling polylogarithmically in d. This matches the performance of an optimal offline algorithm for the same problem.

Our analysis is based on new concentration results for products of random matrices, which allow us to obtain strong bounds on the tails of the random matrices which arise in the course of the algorithm's execution. The argument relies on the optimal uniform smoothness properties of the Schatten trace class obtained by Ball, Carlen, and Lieb.

Based on joint work with Huang, Tropp, and Ward.

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The active Brownian particle (ABP) model describes a swimmer,
synthetic or living, whose direction of swimming is a Brownian motion.
The swimming is due to a propulsion force, and the fluctuations are
typically thermal in origin.  We present a 2D model where the
fluctuations arise from nonthermal noise in a propelling force acting
at a single point, such as that due to a flagellum.  We take the
overdamped limit and find several modifications to the traditional ABP
model.  Since the fluctuating force causes a fluctuating torque, the
diffusion tensor describing the process has a coupling between
translational and rotational degrees of freedom.  An anisotropic
particle also exhibits a noise-induced drift.  We show that these
effects have measurable consequences for the long-time diffusivity of
active particles, in particular adding a contribution that is
independent of where the force acts.  This is joint work with Jiajia
Guo.

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Solution manifolds induced by transport-dominated problems
such as hyperbolic conservation laws typically exhibit nonlinear
structures. This means that traditional model reduction methods based
on linear approximations in subspaces are inefficient when applied to
these problems. This presentation discusses model reduction methods
for constructing nonlinear reduced models that seek approximations on
manifolds, rather than in subspaces, and so lead to efficient
dimensionality reduction even for transport-dominated problems. First,
we will discuss an online adaptive approach that exploits locality in
space and time to efficiently adapt piecewise linear approximations of
the solution manifolds. Second, we present an approach that derives
reduced approximations that are nonlinear by explicitly composing
global transport dynamics with locally linear approximations of the
solution manifolds. The compositions can be interpreted as
one-hidden-layer neural networks. Numerical results demonstrate that
the proposed approaches achieve speedups even for problems where
traditional, linear reduced models are more expensive to solve than
the high-dimensional, full model.
 

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Abstract: Contact tracing mobile applications are clear candidates enabling us to slow down an epidemics and keep the society running while holding the health risks down. Most of the currently discussed and developed mobile applications aim to notify individuals who were recently in a significant contact with an individual who tested positive. The contacted individuals would then be tested or/and put in isolation. In our work, we aim to quantify the epidemiological gain one would obtain if, additionally, individuals who were recently in contact could exchange messages of information. With such a message passing the risk of contracting the infection could be estimated with much better accuracy than simple contact tracing. Our results show that in some range of epidemic spreading (typically when the manual tracing of all contacts of infected people becomes practically impossible, but before the fraction of infected people reaches the scale where a lock-down becomes unavoidable), this inference of individuals at risk could be an efficient way to mitigate the epidemic. Our approaches translate into fully distributed algorithms that only require communication between individuals who have recently been in contact. We conclude that probabilistic risk estimation is capable of enhancing the performance of digital contact tracing and should be considered in the currently developed mobile applications.

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Diverse multi-cellular animals encode a breathtaking diversity of natural behaviors. Non local interactions in traditional nervous systems make the study of underlying origins of behavior in animals difficult (and fascinating). It is a well-known fact that simple dynamical systems can also encode perplexing complexity with purely local update rules. In this talk, using a variety of toy models and systems, we will explore how complex behavior can arise in non-neuronal ensembles; or in short "how do animals with no brains (neurons), decide, compute or think?"

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