Fluid velocities and Brownian effects at nanoscales in the near-wall r egion of microchannels can be experimentally measured in an image plane parallel to the wall, using for example, an evanescent wave illumination technique combi ned with particle image velocimetry [R. Sadr et al., J. Fluid Mech. 506, 357-367 (2004)]. Tracers particles are not only carried by the flow, but they undergo r andom fluctuations, the details of which are affected by the proximity of the wa ll. We study such a system under a particle based stochastic approach (Langevin) . We present the modeling assumptions and pay attention to the details of the si mulation of a coupled system of stochastic differential equations through a Mils tein scheme of strong order of convergence 1. Then we demonstrate that a maximum likelihood algorithm can reconstruct the out-of-plane velocity profile, as spec ified velocities at multiple points, given known mobility dependence and perfect mean measurements. We compare this new method with existing cross-correlation t echniques and illustrate its application for noisy data. Physical parameters are chosen to be as close as possible to the experimental parameters.

In many applications, ranging from computer animation to biology, one wants to quantify how similar two surfaces are to each other. In the last few years, the Gromov-Haussdorff distance has been applied to this problem; this gives good results, but turns out to be very heavy computationally. This talk proposes a different approach, in which (disk-like) 2-dimensional surfaces (typically embedded in 3-dimensional Euclidean space) are first mapped conformally to the unit disk, and the corresponding conformal densities are then compared via optimal mass transportation,. This mass transportation problem differs from the standard case in that we require the solution to be invariant under global Moebius transformations. The metric we construct also defines meaningful intrinsic distances between pairs of "patches" in the two surfaces, which allows automatic alignment of the surfaces. Numerical experiments on "real-life" surfaces to demonstrate possible applications in natural sciences will be shown as well. This is joint work with Yaron Lipman.

There are two proofs of Vinogradov's Mean Value Theorem (VMVT), the harmonic analysis decoupling proof by Bourgain, Demeter, and Guth from 2015 and the number theoretic efficient congruencing proof by Wooley from 2017. While there has been some work illustrating the relation between these two methods, VMVT has been around since 1935. It is then natural to ask: What does previous partial progress on VMVT look like in harmonic analysis language? How similar or different does it look from current decoupling proofs? We talk about a classical argument due to Karatsuba that shows VMVT "asymptotically" and interpret this in decoupling language. This is joint work with Brian Cook, Kevin Hughes, Olivier Robert, Akshat Mudgal, and Po-Lam Yung.

Wavespeed selection refers to the problem of determining the long time asymptotic speed of invasion of an unstable homogeneous state by some other secondary state. This talk will review wavespeed selection mechanisms in the context of reaction-diffusion equations. Particular emphasis will be placed on the qualitative differences between wavespeed selection in systems of reaction-diffusion equations and scalar problems as well as some surprising consequences. The primary example will be a system of coupled Fisher-KPP equations that exhibit anomalous spreading wherein the coupling of two equations leads to faster spreading speeds.

Taylor-Couette flow provides one of the pre-eminent examples of bifurcation in fluid dynamics. This phrase refers to the flow between concentric rotating cylinders. If the rotation speed is sufficiently rapid, the primary rotary flow around the axis becomes unstable, leading to a steady secondary flow in approximately periodic cells. Assuming infinite cylinders and exact periodicity in his theory, Taylor obtained remarkable agreement with experiment for the onset of instability, agreement that remains unsurpassed in fluid mechanics to this day. This talk is concerned with incorporating the effect of finite-length cylinders into the theory, an issue whose importance was emphasized by Benjamin. Numerous experiments and simulations of the Navier Stokes equations all support to the following, seemingly paradoxical, observations: No matter how long the apparatus, finite-length effects greatly perturb many of the bifurcating flows but, provided the cylinders are long, hardly perturb others. We understand this paradox as a result of symmetry breaking. The relevant symmetry, which is only approximate, is a symmetry between two normal-mode flows with large, and nearly equal, numbers of cells.

Mathematical models of complex physical processes often involve large number of degrees of freedom as well as events occurring on different time scales. Therefore, direct simulations based on these models face tremendous challenge. This focus of this talk is on the Mori-Zwanzig (MZ) projection formalism for reducing the dimension of a complex dynamical system. The goal is to mathematically derive a reduced model with much fewer variables, while still able to capture the essential properties of the system. In many cases, this formalism also eliminates fast modes and makes it possible to explore events over longer time scales. The models that are directly derived from the MZ projection are typically too abstract to be practically implemented. We will first discuss cases where the model can be simplified to generalized Langevin equations (GLE). Furthermore, we introduce systematic numerical approximations to the GLE, in which the fluctuation-dissipation theorem (FDT) is automatically satisfied. More importantly, these approximations lead to a hierarchy of reduced models with increasing accuracy, which would also be useful for an adaptive model refinement (AMR). Examples, including the NLS, atomistic models of materials defects, and molecular models of proteins, will be presented to illustrate the potential applications of the methods.

Analyzing and inferring the underlying global intrinsic structures of data from its local information are critical in many fields. In practice, coherent structures of data allow us to model data as low dimensional manifolds, represented as point clouds, in a possible high dimensional space. Different from image and signal processing which handle functions on flat domains with well-developed tools for processing and learning, manifold-structured data sets are far more challenging due to their complicated geometry. For example, the same geometric object can take very different coordinate representations due to the variety of embeddings, transformations or representations (imagine the same human body shape can have different poses as its nearly isometric embedding ambiguities). These ambiguities form an infinite dimensional isometric group and make higher-level tasks in manifold-structured data analysis and understanding even more challenging. To overcome these ambiguities, I will first discuss modeling based methods. This approach uses geometric PDEs to adapt the intrinsic manifolds structure of data and extracts various invariant descriptors to characterize and understand data through solutions of differential equations on manifolds. Inspired by recent developments of deep learning, I will also discuss our recent work of a new way of defining convolution on manifolds and demonstrate its potential to conduct geometric deep learning on manifolds. This geometric way of defining convolution provides a natural combination of modeling and learning on manifolds. It enables further applications of comparing, classifying and understanding manifold-structured data by combing with recent advances in deep learning.

Many experimental systems can spend extended periods of time relative to their natural time scale in localized regions of phase space, transiting infrequently between them. This display of metastability can arise in stochastically driven systems due to the presence of large energy barriers, or in deterministic systems due to the presence of narrow passages in phase space. To investigate metastability in these different cases, we take the Langevin equation and determine the effects of small damping, small noise, and dimensionality on the dynamics and mean transition time. In finite dimensions, we show the limit of small noise and small damping do not interchange. In the limit of infinite dimensions, we argue the equivalence of the finitely-damped system and the zero-damped infinite energy Hamiltonian system.

I’ll discuss a geometric approach to graph partitioning where the optimality criterion is given by the sum of the first Laplace-Dirichlet eigenvalues of the partition components. This eigenvalue optimization problem can be solved by a rearrangement algorithm, which we show to converge in a finite number of iterations to a local minimum of a relaxed objective. This partitioning method compares well to state-of-the-art approaches on a variety of graphs constructed from manifold discretizations, synthetic data, the MNIST handwritten digit dataset, and images. I'll present a consistency result for geometric graphs, stating convergence of graph partitions to an appropriate continuum partition.