Constrained Energy Minimization in Liquid Crystal Models
DateMonday, October 22, 2018 - 12:30pm
AbstractLiquid crystals are a distinct phase of matter existing between the chaos of isotropic liquids and the order of crystalline solids. In addition to being partially ordered, they manifest sensitivity to changes in temperature, concentration, or electric and magnetic fields. These properties combined render liquid crystals useful in various optical and biological applications. We present two models describing each of these types of applications. One is formed by bent-core molecules, a shape that endows the material with electric responsiveness. The second is composed of disc-like molecules that form rings when added to a solution, and these in turn aggregate into interesting geometrical shapes. An important question in both setups is how the dominant mechanism - switching in the first model and shape formation in the second - is affected by specific system parameters. We formulate both models as energy minimization problems allowing us to use several variational tools. We emphasize how we can deal with challenges that arise from constraints and nonlinearities. Our results address existence, uniqueness, and computation of solutions to the ensuing partial differential equations, which in tun shed light on the physical mechanisms observed.
Cell migration from birth to death: Modeling and analyzing the motion of cells in tissues and tumors
DateThursday, October 25, 2018 - 12:30pm
AbstractBeginning momentarily after we are conceived through to our final days, cells migrate within our bodies. From embryonic development to the progression of many diseases including cancer, cell migration plays an essential role in maintaining our health. To understand the mechanisms and forces involved in migration related to early embryonic development, eye and retina development, wound healing, and cancer growth, I have developed continuum mechanical models with free boundaries and reaction-diffusion equation models of the spread of tissues and tumors. Mathematical analysis and numerical simulations of the models indicate conditions for traveling wave and similarity under scaling solutions, and data and image analysis of experimental data has facilitated the estimation of model parameter values that are physically relevant. In this talk, I will give examples of biological cell migration problems that I work on as well as an in-depth look at some of the mathematical analysis that has arisen from the model equations.
Quantum decoherence illustrated via spontaneous emission of a two level atom
DateFriday, October 26, 2018 - 12:00pm
Mathematical Models, Parameter Identification, and Uncertainty
DateFriday, October 26, 2018 - 3:00pm
Abstract: Many mathematical models, such as those commonly used to quantitatively describe various biological processes, contain a large number of rate constants. The components of the state vector usually are not directly observable, and first-principles estimates of the rate constants rarely are available. Instead, one relies on time series that are functions of the state vector to validate the model. This talk will discuss the following question: if values of model parameters can be found that fit the observed data, then what confidence can we place in predictions from the model? The predictions depend on the model parameters, for which there may or may not be unique estimates that correspond to a given set of observations; this is the identifiability problem. I will give examples from simple SIR models to more complicated models of prostate cancer.