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Abstract: Verifying nonlinear stability of a laminar fluid flow against all perturbations is a classic challenge in fluid dynamics. All past results rely on monotonic decrease of a perturbation energy or a similar quadratic generalized energy. This "energy method" cannot show global stability of any flow in which perturbation energy may grow transiently. For the many flows that allow transient energy growth but seem to be globally stable (e.g. pipe flow and other parallel shear flows at certain Reynolds numbers) there has been no way to mathematically verify global stability. After explaining why the energy method was the only way to verify global stability of fluid flows for over 100 years, I will describe a different approach that is broadly applicable but more technical. This approach, proposed in 2012 by Goulart and Chernyshenko, uses sum-of-squares polynomials to computationally construct non-quadratic Lyapunov functions that decrease monotonically for all flow perturbations. I will present a computational implementation of this approach for the example of 2D plane Couette flow, where we have verified global stability at Reynolds numbers above the energy stability threshold. This energy stability result for 2D Couette flow had not been improved upon since being found by Orr in 1907. The results I will present are the first verification of global stability – for any fluid flow – that surpasses the energy method. This is joint work with Federico Fuentes (Universidad Católica de Chile) and Sergei Chernyshenko (Imperial College London).

Abstract: Schrödinger operators with delta potentials are of longstanding interest for admitting closed analytic solutions, and they receive renewed attention for connections to the Kardar–Parisi–Zhang equation. Along with a review of this background, the talk will discuss a standard model of the operators in 2D and its counterpart in Feynman–Kac formulas.

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Abstract: The Schwarzschild potential is defined as U(r)=-A/r-B/r^3, where r is the relative distance and A, B>0. In my talk I will present qualitative results on the scattering solutions of 3 body systems interacting with Schwarzschild's potential, where the 3 point masses form an isosceles triangle at all times. We first provide collision vs singularity criterion and convexities of scattering solutions. Then, we approximate the scattering and partial scattering solutions in various cases such that the error term converges to 0 as t goes to infinity. The scattering dynamics are also classified based on the value of the angular momentum. We also investigated some topological properties of the configuration space by applying the collision criterion and the scattering classification. This is a joint work with S. Ibrahim.

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Abstract: Contact tracing is an important intervention measure to control infectious diseases. We present a new approach that borrows the edge dynamics idea from network models to track contacts included in a compartmental SIR model for an epidemic spreading in a randomly mixed population. Unlike network models, our approach does not require statistical information of the contact network, data that are usually not readily available. The model resulting from this new approach allows us to study the effect of contact tracing and isolation of diagnosed patients on the control reproduction number and the number of infected individuals. We also estimate the effects of tracing coverage and capacity on the effectiveness of contact tracing. Finally, we show an extension of this model that incorporates a latent/exposed compartment.

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Abstract: Mountain pine beetle (MPB) are a bark beetle native to western North America that lay their eggs in pine trees, killing them in the process. In the last few decades, MPB have spread well beyond their historical range into Alberta and they threaten further spread North and East. Existing population models of MPB focus on capturing a single outbreak, but management goals have moved into the longer term and are also concerned with changing tree resilience. In this talk, I will present a discrete time model that couples key biological aspects of MPB population dynamics to an age structured tree model to understand how MPB dynamics will change on longer time scales. I will show that as forest resilience decreases, a fold bifurcation occurs and there is a stable fixed point with a non-zero MPB population. If the number of beetles is above the Allee threshold, the population approaches this fixed point over a long time scale with transient outbreaks driven by the age structure of the forest. I will then discuss a biologically motivated vignette and show how adding a small number of lower vigor trees can add an additional stable fixed point with a low number of beetles, and how decreasing resilience can result in outbreaks from this endemic population.

Abstract: In this talk, I will present some recent results of study on Burgers equations with time-delay and degeneracy of viscosity. The modelling equations arise from the nonlocal reaction-diffusion equations for population dynamics. When the Rankine-Hugoniot condition and the Lax's entropy condition hold, the dynamical equation possesses some sharper traveling waves, the so-called sharper viscous shock waves. The degeneracy of viscosity causes the traveling waves to be sharper, and the large time-delay makes the traveling waves to be oscillatory.

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Abstract: Malignant gliomas are highly invasive brain tumors. Recent attention has focused on their capacity for network-driven invasion, whereby mitotic events can be followed by the migration of nuclei along long thin cellular protrusions, termed tumour microtubes (TM). Here I develop a mathematical model that describes this microtube-driven invasion of gliomas. I show that scaling limits lead to well known glioma models as special cases such as go-or-grow models, the PI model of Swanson, and the anisotropic model of Swan.

Numerical simulations are used to compare between the models. (Joint work with N. Loy, K.J. Painter, R. Thiessen).

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Abstract: In many biological systems, it is essential for individuals to gain information from their local environment before making decisions. In particular, through sight, hearing or smell, animals detect the presence of other individuals and adjust their behavior accordingly. Interestingly, this feature is not only restricted to higher level species, such as animals, but is also found in cells. For example, some human immune cells are able to interact non- locally by extending long thin protrusions to detect the presence of chemicals or signaling molecules. Indeed, the process of gaining information about the surrounding environment is intrinsically non-local and mathematically this leads to non-local advection terms in continuum models.

In this talk, I will focus on a class of nonlocal advection-diffusion equations modeling population movements generated by inter and intra-species interactions. I will show that the model supports a great variety of complex spatio-temporal patterns, including stationary aggregations, segregations, oscillatory patterns, and irregular spatio-temporal solutions. However, if populations respond to each other in a symmetric fashion, the system admits an energy functional that is decreasing and bounded below, suggesting that patterns will be asymptotically stable. I will describe novel techniques for using this functional to gain insight into the analytic structure of the stable steady state solutions. This process reveals a range of possible stationary patterns, including regions of multi-stability. These will be validated via comparison with numerical simulations.

Abstract: This is joint work with Sonae Hadama (Kyoto). We consider a system of Schrodinger equations with the Hartree interaction, which is a simplified mean-field model for fermions. The equation may be rewritten for operators, where the trace class corresponds to the case of finite total mass (L^2). Lewin and Sabin proved stability of some translation-invariant stationary solutions from physics, using the Strichartz estimate for the free equation in the Schatten class, where the mass is merely p-th power summable with respect to the number of particles for some p>1. In this case, the perturbation argument for the Duhamel integral is not so easy as in the scalar case, because the Schatten class is not simply embedded into or interpolated with the space-time Lebesgue norms for the Strichartz estimate. We propose some framework to solve the equation in the Schatten class. The main novelties are norms for propagators corresponding to the best constants of the Strichartz estimate in the Schatten class, and a Schatten version of the Christ-Kiselev lemma for the Duhamel integral on operators.

Abstract: A Glass network is a system of first order ODEs with discontinuous right hand side coming from step function terms. The "ON/OFF" switching dynamics from the step functions makes Glass networks effective at modelling switching behaviour typical of gene and neural networks. They also have potential application as models of true random number generators (TRNGs) in electronic circuits. As random number generators, it is desirable for networks to behave as irregularly as possible to thwart potential hacking attempts. Thus, a measure of irregularity is necessary for analysis of proposed circuit designs. The cybersecurity industry wants bit sequences generated by the circuit to have positive entropy. The nature of the discontinuities allows for Glass networks to be transformed into discrete time dynamical systems, where discrete maps represent transitions through boxes in phase space, where all possible box transitions are represented using a directed graph called the transition Graph (TG). Dynamics on the TG naturally allows for the network dynamics to be represented by shift spaces with an alphabet of symbols representing boxes. For shift spaces, entropy is used to gauge dynamical irregularity. As a result it is a perfect measure for the application to TRNGs. Previously it was shown that the entropy of the TG acts as an upper bound for the entropy of the actual dynamics realized by the network. By considering more dynamical information from the continuous system we have shown that the TG can be reduced to achieve more accurate entropy upper bounds. We demonstrate this by considering examples and use numerical simulations to gauge the accuracy of our improved upper bounds.