Math Colloquium - Fall 2021

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September 3

Michael Booty, NJIT

Location: Webex

Flows with Soluble Surfactant or Electrokinetic Effects

Surfactants are energetically favored to seek out the interface between immiscible fluids, where  they influence the dynamics of the interface by reducing its surface tension. Surfactants can also occur in dissolved form away from the interface and transfer between interfacial and bulk forms. Because surfactant molecules are large relative to solvent molecules they diffuse very slowly relative to other time scales such as the relaxation time associated with surface tension.

Electrokinetic flows, on the other hand, consist of a strong electrolyte solution, meaning that the  solute is almost completely disassociated into ions, in the presence of an electric field.  The solution is charge-neutral except near an interface between immiscible electrolytes, where the electric field induces separation of charge to occur in thin Debye layers, and the Coulomb force acting on the separated charge induces flow. 

These are both examples where a spatially narrow boundary layer or transition layer develops adjacent to a sharp fluid-fluid interface, and where the layer exerts a substantial element of control over the flow’s evolution and dynamics.  Asymptotic techniques are introduced that allow reduction of the governing equations to a more tractable form that resolves the layer dynamics but nonetheless needs numerical solution. Various physical examples will be presented. This is collaborative work with Michael Siegel and a number of past and present doctoral students and postdocs in DMS. 

Colloquium Poster - 9/3/21

September 10

Peter Baddoo, MIT

Location: CULM Lecture Hall II

New Methods for Data-Driven Modeling of Dynamical Systems

Current techniques for data-driven modeling of dynamical systems struggle to capture high-dimensional nonlinearities and rarely produce models that respect the known physical properties of the system at hand. These challenges are addressed with two novel data-driven frameworks. Firstly, the Linear and Nonlinear Disambiguation Optimization (LANDO) is a kernel learning architecture that uses data measurements to construct efficient representations of high-dimensional nonlinear systems. Furthermore, LANDO enables an accurate disambiguation of the underlying linear and nonlinear mechanisms, which produces a set of physically interpretable linear modes. Secondly, the physics-informed Dynamic Mode Decomposition (piDMD) performs a constrained optimization to learn the best-fit linear operator that respects given physical properties. piDMD incorporates a range of physical structures including symmetries, invariances, causalities, spatial locality and conservation laws. As well as enforcing known physics, piDMD can also be employed to discover unknown physics. These new methods are applied to a sequence of problems from the physical sciences, thus demonstrating their improved diagnostic, predictive and interpretative abilities.

This is a collaborative project with Profs Benjamin Herrmann (U. of Chile), Beverley McKeon (CalTech), Steven Brunton (U. of Washington) and Nathan Kutz (U. of Washington).

September 17

Guido De Philippis, NYU

Location: Webex

(Boundary) Regularity for Mass Minimising Currents

Plateau problem consists in finding a surface of minimal area among the ones spanning a given curve. It is among the oldest problem in the calculus of variations and its study lead to  wonderful development in mathematics. Federer and Fleming integral currents provide a suitably weak solution to the Plateau problem in arbitrary Riemannian manifolds, in any dimension and co-dimension. Once this weak solution has been found a natural question consists in understanding whether it is classical one. i.e. a smooth minimal surface. This is the topic of the regularity theory, which naturally splits into interior  regularity and boundary regularity. After the monumental work of Almgren, revised by De Lellis and Spadaro, interior regularity is by now well understood. Boundary regularity is instead less clear and some new phenomena appear. Aim of the talk is  to give an overview of the problem and to present some boundary regularity results we have obtained in the last years.

September 24

Stephen Shipman, LSU

Location: Webex

Embedded Eigenvalues for the Neumann-Poincaré

The PDEs governing sub-wavelength quasi-static scattering and resonance in small particles are intimately connected to the spectral theory of the Neumann-Poincaré (NP) boundary-integral operator in potential theory.  In two and three dimensions, we prove the existence of scatterers for which the NP operator has eigenvalues embedded in its continuous spectrum.  These eigenvalues arise due to a combination of a sharp point and symmetry in the scatterer.  This is joint work with Wei Li (DePaul) and Karl-Mikael Perfekt (NTNU).

October 1

Jiaoyang Huang, NYU

Location: CULM Lecture Hall II

Extreme Eigenvalues of Random $d$-regular Graphs

Extremal eigenvalues of graphs are of particular interest in theoretical computer science and combinatorics. In particular, the spectral gap, the gap between the first and second largest eigenvalues, measures the expanding property of the graph. In this talk, I will focus on random $d$-regular graphs. I'll first explain some conjectures on the extremal eigenvalue distributions of adjacency matrices of random $d$-regular graphs; some have been solved, some are still widely open. In the second part of the talk, I will give a new proof of Alon's second eigenvalue conjecture that with high probability, the second eigenvalue of a random $d$-regular graph is bounded by $2\sqrt{d-1}+o(1)$, where we can show that the error term is polynomially small in the size of the graph. This is based on a joint work with Horng-Tzer Yau.

October 8

Jeremy Marzuola, UNC

Location: Webex

Quantitative Bounds on Impedance-to-Impedance Operators with Applications to Fast Direct Solvers for PDEs

We will discuss A recent result with Tom Beck and Yaiza Canzani that we worked on due to a conversation with Alex Barnett.  We will discuss quantitative norm bounds for a family of operators involving impedance boundary conditions on convex, polygonal domains. Robust numerical construction of Helmholtz scattering solutions in variable media can be accomplished via a domain decomposition into a sequence of rectangles of varying scales and constructing impedance-to-impedance boundary operators on each subdomain.   This can be used to construct the Dirichlet-to-Neumann operator on the total domain. Our estimates ensure the invertibility, with quantitative bounds in the frequency, of the merge operators required to reconstruct the original Dirichlet-to-Neumann operator in terms of these impedance-to-impedance operators of the sub-domains. A key step in our proof is to obtain Neumann and Dirichlet boundary trace estimates of the impedance boundary value problem. This talk will give an overview of the setting and discuss the components of the proof of the key elliptic estimates we use. 

Given time, I will also discuss bounds for similar merge operators in the obstacle scattering problem.  

October 15

Eduardo Corona, UC Boulder

Location: Webex

A Crash Course in Boundary Integral Methods with Applications to Stokesian Suspensions

Boundary integral methods consist on re-formulating PDE boundary value problems in terms of integral operators. If this integral formulation is chosen carefully, this can reduce the dimensionality of the problem, and result in well-conditioned linear systems upon discretization. In this talk, I will explain each step involved in the application of this method to build a general simulation framework of Stokesian rigid body suspension flows, focusing on the use of fast algorithms and high performance computing. 

Particulate flows are ubiquitous in the study of self-assembly of biological structures and the design of soft materials. I will discuss some of our recent work applying this boundary integral framework to simulate Janus particle systems, that is, of particles whose surfaces exhibit two distinct physical properties. 

October 22

Edward Large, University of Connecticut

Location: Webex

A Dynamical, Radically Embodied, and Ecological Theory of Rhythm Development

Musical rhythm abilities--the perception of and coordinated action to the rhythmic structure of music--undergo remarkable change over human development. Here, I introduce a theoretical framework for modeling the development of musical rhythm. It explains rhythm development in terms of resonance and attunement, which are formalized using a general mathematical theory that includes nonlinear resonance and Hebbian plasticity.  I review the developmental literature on musical rhythm, highlighting several developmental processes related to rhythm perception and action. Next, I offer an exposition of Neural Resonance Theory and argue that elements of the theory are consistent with dynamical, radically embodied (i.e., non-representational), and ecological approaches to cognition and development. I then discuss how dynamical models, implemented as self-organizing networks of neural oscillations with Hebbian plasticity, predict  key features of music development. I conclude by illustrating how the notions of dynamic embodiment, resonance and attunement provide a conceptual language for characterizing musical rhythm development, and, when formalized in empirically-constrained dynamical models, provide a theoretical framework for generating testable empirical predictions about musical rhythm development. Predictions include the kinds of native and non-native rhythmic structures infants and children can learn, steady-state evoked potentials to native and non-native musical rhythms, and the effects of short-term (e.g., infant bouncing, infant music classes), long-term (e.g., perceptual narrowing to musical rhythm), and very-long term (e.g., music enculturation, musical training) learning on music perception-action.

October 29

Eleni Katifori, University of Pennsylvania

Location: CULM Lecture Hall II

Local Rules for Global Optimization of Distribution Networks

From the microvasculature in our own bodies to the vast river networks that span entire continents, flow networks are ubiquitous at all length scales in nature. These distribution networks are built and constantly remodeled based on rules of evolution that dictate the fate (growth or shrinkage) or the network links. In biology, flow networks play important functional roles such as delivery of nutrients, removal of waste, temperature regulation, and more.  As a result, they are under strong evolutionary pressure to optimize different cost functions, including the energetic cost to overcome viscous dissipation, the material cost to build and maintain the network, or the need to uniformly distribute nutrients across the whole tissue. In this talk we will discuss how networks that optimize these cost functions can be built by adaptive rules for the links. These adaptive rules remodel the network using only local information about the links, without an explicit knowledge about the global state of the system. Using oxygen distribution in the microvasculature as a test case, we will establish structure-function relationships for the network topology. 

November 5

Pejman Sanaei, New York Institute of Technology

Location: CULM Lecture Hall II

Fluid Structure Interaction, from Flight Stability of Wedges to Tissue Engineering and Moving Droplets on a Filter Surface

In this talk, I will present 3 problems on fluid structure interaction:

1) Flight stability of wedges: Recent experiments have shown that cones of intermediate apex angles display orientational stability with apex leading in flight. Here we show in experiments and simulations that analogous results hold in the two-dimensional context of solid wedges or triangular prisms in planar flows at Reynolds numbers 100 to 1000. Slender wedges are statically unstable with apex leading and tend to flip over or tumble, and broad wedges oscillate or flutter due to dynamical instabilities, but those of apex half angles between about 40◦ and 55◦ maintain stable posture during flight. The existence of ‘‘Goldilocks’’ shapes that possess the ‘‘just right’’ angularity for flight stability is thus robust to dimensionality.

2) Tissue engineering: In a tissue-engineering scaffold pore lined with cells, nutrient-rich culture medium flows through the scaffold and cells proliferate. In this process, both environmental factors such as flow rate, shear stress, as well as cell properties have significant effects on tissue growth. Recent studies focused on effects of scaffold pore geometry on tissue growth, while in this work, we focus on the nutrient depletion and consumption rate by the cells, which cause a change in nutrient concentration of the feed and influence the growth of cells lined downstream.

3) Moving droplets on a filter surface: Catalysts are an integral part of many chemical processes. They are usually made of a dense but porous material such as activated carbon or zeolites, which provides a large surface area. Liquids that are produced as a byproduct of a gas reaction at the catalyst site are transported to the surface of the porous material, slowing down transport of the gaseous reactants to the catalyst active site. One example of this is in a sulphur dioxide filter, which converts gaseous sulphur dioxide to liquid sulphuric acid. Such filters are used in power plants to remove the harmful sulphur dioxide that would otherwise contribute to acid rain. Understanding the dynamics of the liquid droplets in the gas channel in a device is critical in order to maintain performance and durability of the catalyst assembly. Our goal is to develop a mathematical model using the Immersed Boundary Method to quantify the droplet movement on the filter surface.

November 12

Alex Barnett, Flatiron Institute

Location: CULM Lecture Hall II

Fresnel Diffraction for Starshade Modeling and the Nonuniform FFT

There are plans to launch a 30-meter diameter "starshade" (occulter) into space to block almost all direct light a space telescope receives from a star, while allowing a faint exoplanet orbiting that star to be imaged. Designing a good shape for this hard-edged occulter that can create the needed 10^{-10} shadow region involves Fresnel scalar wave diffraction. I present a new method for such a diffraction task, exploiting the nonuniform FFT, that is around 2 to 5 orders of magnitude faster than state-of-the-art methods for the starshade application. In the 2nd half of the talk I will explain more generally how the nonuniform FFT works, some of its other applications in applied mathematics, and some tricks we used to accelerate our own FINUFFT software library.

The 2nd half of the talk is joint work with J Magland and L af Klinteberg.

November 19

James Yorke, University of Maryland

Location: CULM Lecture Hall II

The Equations of Nature and the Nature of Equations

Most of this work is in collaboration with Sana Jahedi and Tim Sauer. The final part on Lyapunov functions is joint with Naghmen Akhaven. It applies my work with Jahedi and Sauer.

Systems of M equations in N unknowns are ubiquitous in mathematical modeling. These systems, often nonlinear, are used to identify equilibria of dynamical systems in ecology, genomics, control, and many other areas. Structured systems, where the variables that are allowed to appear in each equation are pre-specified, are especially common. For modeling purposes, there is a great interest in determining circumstances under which physical solutions exist, even if the coefficients in the model equations are only approximately known.

More specifically, a structured system of equations F(x) = c where F : R^N → R^M is a system of M equations in which it is specified which of the N variables are allowed to appear in each coordinate function Fj (. . .). Our goal in this article is to describe the properties that will hold for almost every F that satisfies the structure, and in particular the global properties of solutions for structured systems of C^∞ functions. 

As an application of these ideas I will show examples of Lotka-Volterra systems of differential equations. One example has 14 species. We show that 3 must die out exponentially fast. The technique is rather unique. We produces a fleet of 24 Lyapunov functions, each of which gives information about which species will die out.

December 3

Eric Keaveny, Imperial

Location: Webex

Coordinated Motion of Active Filaments on Spherical Surfaces

Filaments (slender, microscopic elastic bodies) are prevalent in biological and industrial settings.  In the biological case, the filaments are often active, in that they are driven by motor proteins, with the prime examples being cilia and flagella.  For cilia in particular, which can appear in dense arrays, their resulting motions are coupled through the surrounding fluid, as well as through the surfaces to which they are attached.  In this talk, I present numerical simulations exploring the coordinated motion of active filaments and how it depends on the driving force, density of filaments, and the attached surface.  These simulations take advantage of a computational framework we developed for fully 3D filament motion that combines unit quaternions, implicit geometric time integration, and quasi-Newton methods, and interfaces with matrix-free methods for hydrodynamic interactions.  On planar surfaces, the filaments synchronise and align their beating.  On spheres held fixed, defects in the vector field of cilia displacements must be present and either polar or azimuthal beating with two defects is observed.  The defects, however, induce changes in filament motion, and when the sphere is released, we observe non-trivial dynamics of the spherical surface.  This motion feeds back to all filaments resulting in the emergence of a new collective whirling state with metachronal behaviour along the equator.

Updated: November 30, 2021