Applied Mathematics Colloquium Fall 2018

 
The interaction of motile microorganisms and surrounding fluids is of importance in a variety of biological and environmental phenomena including the development of biofilms, colonization of microbes in human and animal bodies, and formation of marine algal blooms. Ambient fluid flow is pervasive in microbial environments and can have profound effects on the motility of microbes, affecting fundamental microbial processes such as their ability to take up nutrients and colonize surfaces. In this study, we scrutinize the role of properties of the surrounding fluid on the spatial distribution of motile microorganisms, their collective behavior in complex fluids, and the corresponding flow structures. The outcomes of this work provide us a framework to examine the effects of surrounding fluid environment and cell motility on the accumulation of microbes.

A number of fundamental questions about the structure of matter can be formulated as problems in energy-driven pattern formation. For example, Gamow's liquid drop model addresses the shape and stability properties of atomic nuclei as arising from competition between long-range Coulomb repulsion and short-range repulsion modeled by surface tension. We discuss a related model for perpendicularly oriented dipoles in the plane, and in which perimeter (representing line tension) and regularized 3D dipolar repulsion compete under a volume constraint. Examples of such situations are Langmuir monolayers and the patterns formed in ultrathin ferromagnetic films with perpendicular anisotropy. In contrast to previously studied similar problems, the nonlocal term contributes to the perimeter term to leading order for small regularization cutoffs. For subcritical dipolar strengths we prove that the limiting functional is a renormalized perimeter and that for small cutoff lengths all minimizers are disks. For critical dipolar strength, we identify the next-order Γ-limit when sending the cutoff length to zero and prove that with a slight modification of the dipolar kernel there exist masses for which classical minimizers are not disks.

With traditional science and mathematics teaching, students struggle with fundamental concepts. For example, they cannot reason with graphs and have no feel for physical magnitudes. Their instincts remain Aristotelian: In their gut, they believe that force is proportional to velocity. With such handicaps in intuition and reasoning, students can learn only by rote. I'll describe these difficulties using mathematical and physical examples, and illustrate how street-fighting mathematics and science---the art of insight and approximation---can improve our thinking and teaching, the better to handle the complexity of the world.

http://web.mit.edu/sanjoy/www/

Liquid crystals (LCs) are anisotropic, viscoelastic fluids that can be used to direct colloids into organized assemblies with unusual optical, mechanical, and electrical properties. In past studies, the colloids have been sufficiently rigid that their individual shapes and properties have not been strongly coupled to elastic stresses imposed by the LCs. We will discuss how soft colloids (micrometer-sized shells) behave in LCs. We reveal a sharing of strain between the LC and shells, resulting in formation of spindle-like shells and other complex shapes. These results hint at previously unidentified designs of reconfigurable soft materials with applications in sensing and biology. Numerical approaches to solving this complex fluid-structure interaction problem and related efforts relevant to biolocomotion will also be discussed.

While many people say they have no rhythm, most humans when listening to music can easily discern and move to a beat. On the other hand, many of us are not so adept at actually generating and maintaining a constant beat over a period of time. Demonstrating a beat is a very complicated task. Among other things, it involves the ability of our brains to estimate time intervals and to make physical movements, for example hitting a drum, in coordination with the time estimates that we make. In this talk, I will introduce a neuromechanistic model of a beat generator, which is defined here as a group of neurons that can learn to keep a constant beat across a range of frequencies relevant to music. The model is a biophysical manifestation of two different types of models: error/correction and neural entrainment models. The goal of the talk is not just to introduce a new way of thinking of beat generation, but also to raise a series of questions about the nature of time and the role of perception in our ability to make decisions.

Dynamic wetting is crucial to processes where liquid displaces another fluid (such as air) along a solid surface, an important example being the deposition of a coating liquid onto a moving substrate. Dynamic wetting failure occurs when the displacement happens too quickly, and this leads to entrainment of the receding fluid into the advancing liquid. In coating processes this entrainment compromises the quality of the final product, so it is desirable to develop a fundamental understanding of the factors that control the onset of dynamic wetting failure. In this talk, I will discuss how the interplay between experiments and modeling has enabled progress in this area. The experiments involve measurements of the critical speed at which wetting failure occurs and flow visualizations of air entrainment. The modeling involves a combination of asymptotic analysis and two-dimensional finite element calculations that link the onset of wetting failure to limit points in families of steady-state solutions. The results reveal the mechanisms responsible for wetting failure and suggest strategies for delaying the onset of air entrainment in coating flows.

The cohesive movement of a biological population is a commonly observed natural phenomenon. With the advent of platforms of unmanned vehicles, such phenomena have attracted a renewed interest from the engineering community. This talk will cover a survey of models ranging from aggregation models in nonlinear partial differential equations to control algorithms and robotic testbed experiments. We will show how pairwise potential models are used to study biological movement and how to develop a systematic theory of such models. We also discuss how to use designer potentials to orchestrate cooperative movement in a specific patterns, many of which may not be observed in nature but could be desireable for artificial swarms. Finally we conclude with some recent related work on emotional contagion in crowds and on design algorithms for crop pollination.

For any quantity of interest in a system governed by nonlinear differential equations it is natural to seek the largest (or smallest) long-time average among solution trajectories. Upper bounds can be proved a priori using auxiliary functions, the best choice of which is a convex optimization. We show that the problems of finding maximal trajectories and minimal auxiliary functions are strongly dual. Thus, auxiliary functions provide arbitrarily sharp upper bounds on maximal time averages. They also provide volumes in phase space where maximal trajectories must lie. For polynomial equations, auxiliary functions can be constructed by semidefinite programming which we illustrate using the Lorenz and Kuramoto-Sivashinsky equations.

This is joint work with Ian Tobasco and David Goluskin, part of which appears in Physics Letters A 382, 382-386 (2018).

Invasive species cause global environmental and economic harm. A prime example is emerald ash borer (EAB), a wood-boring insect native to Asia and invading North America. Since its discovery near Detroit in 2002, EAB has killed untold millions of ash trees in United States and Canada and may functionally extirpate one of North America’s most widely distributed tree genera. EAB causes significant economic damage in cities, where it kills high-value ash trees that shade streets, homes, and parks. Local actions to reduce damage include surveillance to find EAB and control to slow its spread. We develop a multistage stochastic mixed-integer programming (MSS-MIP) model for optimization of surveillance, treatment, and removal of ash trees in cities. The objective is to allocate resources to surveillance and control over space and time to maximize public benefits. We develop a new cutting plane algorithm to strengthen the MSS-MIP formulation and facilitate optimal solution. We calibrate and validate our model of ash dynamics and apply the optimization model to a possible infestation in Burnsville, Minnesota. Under a belief of infestation, it is critical to apply surveillance immediately to locate EAB and then prioritize treatment of minimally infested trees followed by removal of highly infested trees.
This is a joint work with Drs. Robert Haight, Kathleen Knight and Charlie Flower from U.S. Forest Service.

Direct numerical approximation of high frequency wave propagation typically requires a very large number of unknowns and is computationally very costly. We will discuss two aspects of this type of problem formulated in frequency domain. One is the development and analysis of fast numerical algorithms of optimal computational complexity for boundary integral formulations and for variable coefficient differential equations. In the variable coefficient case the challenge is preconditioning. The algorithms are based on a separable approximation lemmas. The other aspect is analysis revealing when algorithms of this type of operator compression are possible and when they are not. We will briefly comment on analysis based on stationary phase and lower bounds for low rank approximations.

The normal alignment of circadian rhythms with the 24-hour light-dark cycle is disrupted after rapid travel between home and destination time zones, leading to sleep problems, indigestion, and other symptoms collectively known as jet lag. We have developed a new tool, called an entrainment map, to study the process of reentrainment to the light-dark cycle of the destination time zone in a model of the human circadian pacemaker. Using this 1-dimensional map, we are able to determine conditions for existence and stability of 1:1 phase-locked solutions between the intrinsic circadian oscillator and the external light-dark forcing. The map is also ideally suited to calculate the amount of time required to achieve entrainment as a function of initial conditions and the bifurcations of stable and unstable periodic solutions that lead to loss of entrainment. We calculate the reentrainment time for travel between any two points on the globe at any time of the day and year. We then use entrainment maps to explain several properties of jet lag, such as why most people experience worse jet lag after traveling east than west. We also show that the change in daylength encountered during north-south travel in the winter or summer can cause jet lag even when no time zones are crossed, contrary to the conventional wisdom that jet lag only occurs after east-west travel across multiple time zones.

The diblock copolymer equation models phase separation processes which involve long-range interactions, and therefore promote the formation of fine structure. While the model arises through a regular perturbation from the classical Cahn-Hilliard model for phase separation in binary alloys, its dynamics is considerably richer, and exhibits for example a high level of multistability. As a dissipative model, its long-term dynamics can in principle be completely described by the dynamics on its global attractor, which is comprised of equilibrium solutions and connecting orbits between them. Unfortunately, however, classical mathematical methods have so far failed at uncovering this attractor structure. In this talk, we provide an overview of how rigorous computational techniques can be used to obtain computer-assisted proofs for the existence of equilibrium solutions, curves of secondary bifurcation points, as well as heteroclinic connections. In the course of this, we uncover the formation of energy minimizers with fine structure through a homotopy from the classical Cahn-Hilliard bifurcation diagram, and it will be shown that typical solutions originating close to the homogeneous state are trapped by local minimizers of the energy, and do not in fact reach the global minimizers.

In the noisy cellular environment, expression of genes has been shown to be stochastic across organisms ranging from prokaryotic to human cells. Stochastic expression manifests as cell-to-cell variability in the levels of RNAs/proteins, in spite of the fact that cells are genetically identical and are exposed to the same environment. Development of computationally tractable frameworks for modeling stochastic fluctuations in gene product levels is essential to understand how noise at the cellular level affects biological function and phenotype. I will introduce state-of-the-art computational tools for stochastic modeling, analysis and inferences of biomolecular circuits. Mathematical methods will be combined with experiments to study infection dynamics of two viral systems in single cells. First, I will show how stochastic expression of proteins results in intercellular lysis time and viral burst size variations in the bacterial virus, lambda phage. Next, I will describe our efforts in stochastic analysis of the Human Immunodeficiency Virus (HIV) genetic circuitry. Our results show that HIV encodes a noisy promoter and stochastic expression of key viral regulatory proteins can drive HIV into latency, a drug-resistant state of the virus.