Time & Location: All talks are on Thursdays in Gibson Hall 126 at 3:30 pm unless otherwise noted. Refreshments in Gibson 426 after the talk.
Comments indicating vacations, special lectures, or change in location or time are in green.
Organizer: Gustavo Didier
The field of topology, and in particular computational topology, has produced a powerful set of tools for studying both model systems and data measured directly from physical systems. I will focus on three classes of topological tools: computational homology, topological persistence, and, very briefly, Conley index theory. To illustrate their use, I will discuss recent projects studying coupled-patch population dynamics, flickering red blood cells, and pulse-coupled neurons.
Geophysical hazards – landslides, tsunamis, volcanic avalanches, etc. – which lead to catastrophic inundation are rare yet devastating events for surrounding communities. The rarity of these events poses two significant challenges. First, there are limited data to inform aleatoric scenario models, how frequent, how big, where. Second, such hazards often follow heavy-tailed distributions resulting in a significant probability that a larger-than-recorded catastrophe might occur. To overcome this second challenge, we must rely on physical models of these hazards to “probe” the tail for catastrophic events. Typically these physical models are computationally intensive to exercise and a probabilistic hazard map relies on an expensive Monte Carlo simulation which samples a scenario model. This approach forces one to focus resources on a single scenario model that is based on one set of assumptions. We will present a surrogate-based strategy that allows great speed-up in Monte Carlo simulations and hence the flexibility to explore the impact of non-stationary scenario modeling on short term forecasts. Additionally, this approach provides a platform to perform uncertainty quantification on hazard forecasts.
When computable, a linear feedback control guarantees the stability of a steady-state flow. This can, for example, be used to stabilize an unstable periodic orbit in a flow control problem. However, as we will show, the stability region might be too small to be implemented
in practice. This motivates the use of nonlinear feedback strategies to expand the stability region. These can now be efficiently computed using the nonlinear systems toolbox on very low-dimensional, reduced-order models. Using a simple nonlinear control problem, we explore the possibility of expanding the stability region and comment on the development of low-order nonlinear state estimators.
We study quantitative and qualitative aspects of the problem of turbulent transport of a passive scalar quantity in a random 2-d velocity field. In particular, we give precise meaning to some of the multifractal structure of the dissipation times, predicted earlier in the physics and engineering literatures. The unexplained terms of the abstract will be described more precisely in the talk.
This is based on joint work with Jingyu Huang.
Metabolic systems are not simple sequential pathways but have cycles within cycles, so that everything is both upstream and downstream of everything else. In addition, there are multitudes of regulatory (allosteric) interactions in which metabolites affect the activity of distant enzymes in the network. Creating mathematical models based on the real underlying physiology and biochemistry is labor-intensive, but such models can be used to understand the systems behavior in health and disease. I will describe how my colleagues and I create and validate such models and some of our results. I will discuss how we can take account of biological variation and I will also discuss the different roles that stochasticity plays in biological systems. I will explain how the regulatory mechanisms buffer metabolism against large functional genetic mutations and why genotype may not be a good predictor of disease risk. Finally, I will discuss several new, interesting questions in pure mathematics have arisen from this work.
The Burgers equation is a basic nonlinear evolution PDE of Hamilton--Jacobi type related to fluid dynamics and growth models. I will talk about the ergodic theory of randomly forced Burgers equation in noncompact setting. The basic objects are one-sided infinite minimizers of random action (in the inviscid case) and polymer measures on one-sided infinite trajectories (in the positive viscosity case). This is joint work with Eric Cator, Kostya Khanin, and Liying Li.