Math Colloquium - Spring 2023 | Department of Mathematical Sciences

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January 20

Raghav Venkatraman, NYU

Host: Travis Askham

The Robustness of ENZ Device

"ENZ" devices are a class of electromagnetic devices that operate at a frequency at which one of their components has dielectric permittivity close to zero. Such devices have curious properties that have made them valuable in creating entirely new kinds of waveguides and resonators. While their analytical study in the physics literature so far has been limited to the idealized "epsilon =0" limit, the robustness of the associated effects to epsilon merely small have only been explored numerically, and an analytical understanding is lacking.

We will discuss a few different examples in which we can get complete analytical information in the setting where epsilon is merely small.

This is joint work with Bob Kohn, based on conversations with Nader Engheta.

January 27

Jonathan Jaquette, Boston University

Host: Michael Siegel

Exploring Global Dynamics and Blowup in Some Nonlinear PDEs

Conservation laws and Lyapunov functions are powerful tools for proving the global existence or stability of solutions to PDEs, but for most complex systems these tools are insufficient to completely understand non-perturbative dynamics. In this talk I will discuss a complex-scalar PDE which may be seen as a toy model for vortex stretching in fluid flow, and cannot be neatly categorized as conservative nor dissipative.

In a recent series of papers, we have shown (using computer-assisted-proofs) that this equation exhibits rich dynamical behavior existing globally in time: non-trivial equilibria, homoclinic orbits, heteroclinic orbits, and integrable subsystems foliated by periodic orbits. On the other side of the coin, we show several mechanisms by which solutions can blowup.

February 3

Ehud Yariv, Department of Mathematics, Technion - Israel Institute of Technology

Host: Michael Siegel

Flows About Superhydrophobic Surfaces

Superhydrophobic surfaces, formed by air entrapment within the cavities of hydrophobic solid substrates, offer a promising potential for hydrodynamic drag reduction. In several of the prototypical surface geometries the flows are two-dimensional, governed by Laplace’s equation in the longitudinal problem and the biharmonic equation in the transverse problem. Moreover, low-drag configurations are typically associated with singular limits. Thus, the analysis of liquid slippage past superhydrophobic surfaces naturally invites the use of both singular-perturbation methods and conformal-mapping techniques. I will discuss the combined application of these methodologies to several emerging problems in the field.

February 10

Sophie Ramananarivo, LadHyX, Ecole Polytechnique

Host: Anand Oza

Can We Tailor the Behavior of Flexible Sheets in Flows by Adding Cuts or Folds?

Lightweight compliant surfaces are commonly used as roofs (awnings), filtration systems or propulsive appendages, that operate in a fluid environment. Their flexibility allows for shape to change in fluid flows, to better endure harsh or fluctuating conditions, or enhance flight performance of insect wings for example. The way the structure deforms is however key to fulfill its function, prompting the need for control levers. In this talk, we will consider two ways to tailor the deformation of surfaces in a flow, making use of the properties of origami (folded sheet) and kirigami (sheet with a network of cuts). Previous literature showed that the substructure of folds or cuts allows for sophisticated shape morphing, and produces tunable mechanical properties. We will discuss how those original features impact the way the structure interacts with a flow, through combined experiments and theory. We will notably show that a sheet with a symmetric cutting pattern can produce an asymmetric deformation, and study the underlying fluid-structure couplings to further program shape morphing through the cuts arrangement. We will also show that extreme shape reconfiguration through origami folding can cap fluid drag.

February 17

Paul Milewski, Department of Mathematical Sciences, University of Bath

Host: Wooyoung Choi

Embedded Solitary Internal Waves

The ocean and atmosphere are density stratified fluids. This means that a wide variety of gravity waves propagate in their interior, redistributing energy and mixing the fluid, affecting global climate balances. We shall focus on a particular wave-type (the second baroclinic mode) where both observations and modelling have only recently started. Stratified fluids with narrow regions of rapid density variation with respect to depth (pycnoclines) are often modelled as layered flows. We shall adopt this model and examine horizontally propagating internal waves within a three-layer fluid, with a focus on mode-2 waves which have oscillatory vertical structure. Mode-2 waves (typically) occur within the linear spectrum of mode-1 waves (i.e. they travel at lower speeds than mode-1 waves), and thus mode-2 solitary waves are generically associated with an unphysical resonant mode-1 infinite oscillatory tail. We will present evidence that these tail oscillations can be found to have zero amplitude, thus resulting in families of localised solutions (so called embedded solitary waves) in the Euler equations. This is the first example we know of embedded solitary waves in the Euler equations, and would imply that these waves are longer lived that previously thought.

February 24

Arik Yochelis, Ben-Gurion University of the Negev

Host: Yuan-Nan Young

From Intracellular Actin Waves to Mechanism and Back: How Pattern Formation Theory Aids Biological Understanding and Applications

Self-organized patterns in the actin cytoskeleton are essential for eukaryotic cellular life. They are the building blocks of many functional structures that often operate simultaneously to facilitate, for example, nutrient uptake and movement of cells. However, identifying how qualitatively distinct actin patterns can coexist remains a challenge. In my talk, I will first overview the different aspects of modeling and show how intracellular actin waves are different from many other biological systems and why they are mathematically challenging. In the second part, I will show how using mathematical methods taken from pattern formation theory, it is possible not only to experimentally uncover the mechanism of micropinocytosis (nutrient uptake) but also demonstrate that distinct actin wave patterns can indeed coexist in the cortex of living cells. We believe that the latter is central to different cell functions as well as to synthetic biology, where minimal, yet robust systems are required to reconstitute the essential features of self-organization in the cell cortex.

Selected references:

Bernitt, Döbereiner, Gov, and Yochelis, Fronts and waves of actin polymerization in a bistable-based mechanism of circular dorsal ruffles, Nature Communications 8, 15863 (2017)

Beta, Gov, and Yochelis, Why a large scale mode can be essential for understanding intracellular actin waves, Cells 9, 1533 (2020)

Yochelis, Flemming, and Beta, Versatile patterns in the actin cortex of motile cells: Self-organized pulses can coexist with macropinocytic ring-shaped waves, Physical Review Letters 129, 088101 (2022)

March 3

Igor Belykh, Department of Mathematics and Statistics, Georgia State University

Host: Amit Bose

Modelling and Predicting Crowd-induced Bridge Instabilities

Modern pedestrian and suspension bridges are designed using industry-standard packages, yet disastrous resonant vibrations are observed, necessitating multi-million dollar repairs. In this talk, we will discuss experimentally-validated dynamical systems approaches to understanding mechanisms of the dynamical instability of bridges interacting with pedestrian loads. I will explain the fundamental mechanism behind pedestrian-induced lateral instability of bridges due to some positive feedback from uncorrelated walkers whose foot forces do not cancel each other but create a bias. I will present our theory that argues that any synchronization in the timing of pedestrian footsteps is a consequence—not a cause—of the instability; this result is consistent with observations on 30 bridges, including the Brooklyn Bridge and Golden Gate Bridge. For a wide range of systems in nature and society, our work argues more generally that this macro-scale instability may emerge from micro-scale behavior without any obvious causal synchrony.

March 10

David Shirokoff, DMS, NJIT

Overcoming Order Reduction in Runge-Kutta Methods via Weak Stage Order: Theory and Order Barriers

Due to their simplicity and stability properties, Runge-Kutta (RK) methods are some of the most widely used time-integration schemes in computational science. However, when applied to stiff problems such as PDEs with time-dependent boundary conditions, or low spatial regularity, they exhibit a drop in accuracy—known as order reduction—thereby reducing computational efficiency.

We will first revisit the origin of order reduction in PDEs which arises from the presence of a singular perturbation problem. For certain problems, order reduction can be avoided if the method satisfies weak stage order (WSO) conditions. This talk will present the first general order barrier bounds relating the WSO of a scheme to its classical order and number of stages. These bounds characterize the fundamental accuracy limit of RK methods applied to stiff problems. New necessary conditions are also established—which we use to construct new families of high WSO schemes. The key mathematical ideas are to recast WSO into a pair of orthogonal invariant subspaces and perform calculations modulo minimal polynomials. We also provide new formulas for the RK stability function through the introduction of polynomials that are orthogonal with respect to a linear functional.

March 17

Spring Recess - No Colloquium

March 24

Malena Espanol, Arizona State University

Host: Zoi-Heleni Michalopoulou

*This talk will be held virtually via WebEx*

Computational Methods for Solving Inverse Problems in Imaging

Discrete linear and nonlinear inverse problems arise from many different imaging systems. These problems are ill-posed, which means, in most cases, that the solution is very sensitive to the data. Because the data usually contain errors produced by different imaging system parts (e.g., cameras, sensors, etc.), robust and reliable regularization methods need to be developed for computing meaningful solutions. In some imaging systems, massive amounts of data are produced making the data storage and computational cost of the inversion process intractable. In this talk, we will see different imaging systems, we will formulate the corresponding mathematical models, we will introduce regularization methods, and we will show some numerical results.

March 31

Dejan Slepcev, Department of Mathematics, Carnegie Mellon University

Host: Brittany Hamfeldt

Variational Problems on Random Structures: Analysis and Applications to Data Science

Modern data-acquisition techniques produce a wealth of data about the world we live in. Extracting the information from the data leads to machine learning/statistics tasks such as clustering, classification, regression, dimensionality reduction, and others. Many of these tasks seek to minimize a functional, defined on the available random sample, which specifies the desired properties of the object sought.

I will present a mathematical framework suitable for studies of asymptotic properties of such, variational, problems posed on random samples and related random geometries (e.g. proximity graphs). In particular we will discuss the passage from discrete variational problems on random samples to their continuum limits. Furthermore we will discuss how tools of applies analysis help shed light on algorithms

April 7

Good Friday - No Colloquium

April 14

Jeffrey Sachs, Merck

Host: Casey Diekman

Applying Math to Help Improve and Extend Life: Examples from Industry

Mathematical and Data Sciences are being applied across a broad range of industries, informing decisions by distilling into information data from across disparate domains, scales of time and space, and sources. The introduction will give a brief overview of applications (potential careers) from a broad array of healthcare sectors (pharmaceutical, biomedical equipment, digital health, insurance, etc.). The remainder of the presentation will show applications of math in the pharmaceutical industry, where a very broad range of mathematical and computational sciences show up: classical applied math such as differential and partial differential equations, discrete mathematics, visualization, network inference, (stochastic) optimization, control theory, probability, dynamical systems, inverse problems, image processing, pattern recognition/AI, natural language processing, and, of course, numerical analysis and simulation.

Applications highlighted will make use of ODE- and generalized-linear-model-based non-linear mixed effect models, stochastic simulation, machine-learning network inference, and other pharmacometric methods in oncology, Alzheimer’s Disease, and vaccines. Each example will start with the decision that needs to be made and the question that modeling could help answer. The modeling and results will be explained together with their impact on drug/vaccine discovery and development.

The seminar will include but will not focus on mathematical details, and will not have proofs, code, or detailed exposition of numerical methods.

April 21

Gwynn Elfring, Department of Mechanical Engineering, University of British Columbia

Host: Enkeleida Lushi

Active Matter in Inhomogeneous Environments

Active matter is a term used to describe matter that is composed of a large number of self-propelled active ‘particles’ that individually convert stored or ambient energy into systematic motion. Examples include a flock of birds, a school of fish, or at smaller scales a suspension of bacteria or even the collective motion within a human cell. When viewed collectively, active matter is an out-of-equilibrium material. This talk focuses on active matter systems where the active particles are very small, for example bacteria or chemically active colloidal particles. The motion of small active particles in homogeneous Newtonian fluids has received considerable attention, with interest ranging from phoretic propulsion to biological locomotion, whereas studies on active bodies immersed in inhomogeneous fluids are comparatively scarce. In this talk I will show how the dynamics of active particles can be dramatically altered by the introduction of fluid inhomogeneity and discuss the effects of spatial variations of fluid density, viscosity, and other fluid complexity.

April 28

Kasso Okoudjou, Tufts University

Host: Christina Frederick

*Please note that the seminar will be at 2 PM, instead of our usual time*

Optimal Point Distributions on the $d-$dimensional Unit Sphere

Finding optimal configurations of point masses under the action of energy functionals defined on $d-$dimensional Euclidian space appears in several fields such as numerical integration, coding theory, and chemistry. The problem often involves minimizing a functional of the form $\iint_{S^{d-1}\times S^{d-1}}f(x, y)d\mu(x)d\mu(y)$ over all probability measures $\mu$ defined on the unit sphere $S^{d-1}$, or its discrete version, minimizing $\sum_{k\neq \ell}f(\varphi_k, \varphi_\ell)$ over all sets of $N$ vectors $\Phi=\{\varphi_k\}_{k=1}^N\subset S^{d-1}$ for some function $f$.

In this talk, we will consider the case where $f(x, y)=|\langle x, y\rangle|^p$ for $p\in [0, \infty]$ which is referred to as the $p^{th}$ frame potentials. We will motivate this special case with applications from frame theory, spherical designs, and the Zauner conjecture in quantum information theory. After a survey of recent results concerning the minimizers of the $p^{th}$ frame potentials, we will focus on the interplay between the continuous and the discrete problems, especially when the dimension $d$ is small.

Updated: March 10, 2023