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January 2021
Applied math. talk: Minimization of the first nonzero eigenvalue problem for two-phase conductors with Neumann boundary conditions by Chiu-Yen Kao, CMC
Abstract: We consider the problem of minimizing the first nonzero eigenvalue of an elliptic operator with Neumann boundary conditions with respect to the distribution of two conducting materials with a prescribed area ratio in a given domain. In one dimension, we show monotone properties of the first nonzero eigenvalue with respect to various parameters and find the optimal distribution of two conducting materials on an interval under the assumption that the region that has lower conductivity is simply connected. On…
Find out more »February 2021
Applied math. talk: Searching for singularities in Navier-Stokes flows using variational optimization methods by Di Kang, McMaster University, Canada
Abstract: In the presentation we will discuss our research program concerning the search for the most singular behaviors possible in viscous incompressible flows. These events are characterized by extremal growth of various quantities, such as the enstrophy, which control the regularity of the solution. They are therefore intimately related to the question of possible singularity formation in the 3D Navier-Stokes system, known as the hydrodynamic blow-up problem. We demonstrate how new insights concerning such questions can be obtained by formulating…
Find out more »Applied Math. Talk: Complex Fluids in the Immersed Boundary Method: From Viscoelasticity to Blood Clots by Aaron Barrett, Department of Mathematics, University of Utah
The immersed boundary method was first developed in the 1970s to model the motion of heart valves and has since been utilized to study many different biological systems. While the IB method has seen countless modifications and advancements from the perspective of fluid-structure interaction, the use of a Newtonian fluid model remains a fundamental component of many implementations. However, many biological fluids exhibit non-Newtonian responses to stresses, and as such, a Newtonian fluid model falls short to fully describe the…
Find out more »Applied Math. Talk: Modeling and Simulation of Ultrasound-mediated Drug Delivery to the Brain by Peter Hinow, University of Wisconsin, Milwaukee
We use a mathematical model to describe the delivery of a drug to a specific region of the brain. The drug is carried by liposomes that can release their cargo by application of focused ultrasound. Thereupon, the drug is absorbed through the endothelial cells that line the brain capillaries and form the physiologically important blood-brain barrier. We present a compartmental model of a capillary that is able to capture the complex binding and transport processes the drug undergoes in the…
Find out more »Applied math. talk: Heatmap centrality: a new measure to identify super-spreader nodes in scale-free networks by Christina Duron, the University of Arizona
Abstract: The identification of potential super-spreader nodes within a network is a critical part of the study and analysis of real-world networks. Motivated by a new interpretation of the “shortest path” between two nodes, this talk will explore the properties of the recently proposed measure, the heatmap centrality, by comparing the farness of a node with the average sum of farness of its adjacent nodes in order to identify influential nodes within the network. As many real-world networks are often…
Find out more »March 2021
Applied math. talk: Blowup rate estimates of a singular potential in the Landau-de Gennes theory for liquid crystals by Xiang Xu, Old Dominion University.
Abstract: The Landau-de Gennes theory is a type of continuum theory that describes nematic liquid crystal configurations in the framework of the Q-tensor order parameter. In the free energy, there is a singular bulk potential which is considered as a natural enforcement of a physical constraint on the eigenvalues of symmetric, traceless Q-tensors. In this talk we shall discuss some analytic properties related to this singular potential. More specifically, we provide precise estimates of both this singular potential and its…
Find out more »Applied math. talk: Optimal control of the SIR model in the presence of transmission and treatment uncertainty by Henry Schellhorn, CGU
Abstract The COVID-19 pandemic illustrates the importance of treatment-related decision making in populations. This article considers the case where the transmission rate of the disease as well as the efficiency of treatments is subject to uncertainty. We consider two different regimes, or submodels, of the stochastic SIR model, where the population consists of three groups: susceptible, infected and recovered. In the first regime the proportion of infected is very low, and the proportion of susceptible is very close to 100%. …
Find out more »Applied math. talk: Periodic travelling waves in nonlinear wave equations: modulation instability and rogue waves by Dmitry Pelinovsky, McMaster University, Canada
Abstract: I will overview the following different wave phenomena in integrable nonlinear wave equations: (1) universal patterns in the dynamics of fluxon condensates in the semi-classical limit; (2) modulational instability of periodic travelling waves; (3) rogue waves on the background of periodic and double-periodic waves. Main examples include the sine-Gordon equation, the nonlinear Schroedinger equation, and the derivative nonlinear Schroedinger equation. For the latter equation, in collaboration with Jinbing Chen (South East University, China) and Jeremy Upsal (University…
Find out more »Applied math. talk: Hyperbolicity-Preserving Stochastic Galerkin Method for Shallow Water Equations by Dihan Dai, Department of Mathematics, University of Utah
Abstract: The system of shallow water equations and related models are widely used in oceanography to model hazardous phenomena such as tsunamis and storm surges. Unfortunately, the inherent uncertainties in the system will inevitably damage the credibility of decision-making based on the deterministic model. The stochastic Galerkin (SG) method seeks a solution by applying the Galerkin method to the stochastic domain of the equations with uncertainty. However, the resulting system may fail to preserve the hyperbolicity of the original model.…
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