Jim Nolen : Bumps in the road: stability and fluctuations for traveling waves in an inhomogeneous medium- Undergraduate Seminars ( 211 Views )
If a partial differential equation has coefficients that vary with respect to the independent (spatial) variables, how do the fluctuations in the coefficients effect the solution? In particular, if these fluctuations have a statistical structure, can anything thing be said about the statistical behavior of the solutions? I'll consider these questions in the context of a scalar reaction diffusion equation. Without the variable coefficients, the equation admits stable traveling wave solutions. It turns out that the stability of wave-like solutions persists in a heterogeneous environment, and this fact can be used to derive a central limit theorem for the wave when the environment has a certain statistical structure.
I'll try to explain two interesting mathematical issues: First, how can one prove stability of the wave-like solution in this general setting, since spectral techniques don't seem applicable? Second, how can one use the structure of the problem to say something about how randomness in the environment effects the solution?
At the turn of this century it was realized that social and communication networks were best modeled by graphs that were "small worlds" and/or had power law degree distributions. I will discuss two examples. The first is a situation where physicist's mean field arguments give the wrong answer about the spread of epidemic. The second, inspired by a gypsy moth outbreak in the late 1980s in NY leads to chaotic behavior. I will concentrate on what is true rather than why, so the talk should be accessible to a wide audience.
Stephen Schecter : Stability of traveling waves for a class of reaction-diffusion systems that arise in chemical reaction models- Undergraduate Seminars ( 237 Views )
I'll discuss rigorous nonlinear stability results for traveling waves in a class of reaction-diffusion systems that arise in chemical reaction models. The class includes systems in which there is no diffusion in some equations. The results are detailed enough to show, for example, that the results of adding some heat or adding some reactant to a combustion front are different.
The addition of renewable energy sources, whose power production cannot be scheduled, has created increasing gaps between instantaneous electricity supply and electricity demand. Sometimes the grid is oversupplied with energy, requiring zero-marginal-cost sources of power to be shut or energy to be bled off of the grid. Other times there is insufficient electricity, requiring high-marginal-cost sources of electricity to be switched on or consumers to curtail their demand. The current state of the grid has led various utilities and power consumers deploy capital-intensive energy storage, such as lithium-ion batteries, to better-match grid supply with grid demand. We present a method to add large-scale energy storage to the power grid using only sensors, software modifications to the control systems of large industrial refrigeration systems, and mathematical optimization. Our talk will address the required instrumentation, the physics necessary to understand applicable thermal constraints, and numerical methods used to determine a mathematically optimal-discharge schedule. We further discuss the economics of the US power grid, "war stories"of doing complex mathematics in a large industrial setting and the effects of various Federal Energy Regulatory Commission and California Public Utility Commission on our efforts.
I will discuss a continuous relaxation of the Cheeger cut problem on a weighted graph, and show how the relaxation is actually equivalent to the original problem. Then I will introduce an algorithm which experimentally is very efficient at approximating the solution to this problem on some clustering benchmarks. I will also give a heuristic variant of the algorithm which is faster but often gives just as accurate clustering results. This is joint work with Xavier Bresson, inspired by recent papers of Buhler and Hein, and Goldstein and Osher, and by an older paper of Strang.
Mathematics is being used in many ways to improve the analysis and interpretation of DNA and other molecules that can affect our health. I will describe how math was used to identify genes associated with tumor progression, and to develop methods to identify and quantify genetic variations without expensive and time-consuming sequencing. resulting in a rapid, economical test for transplant compatibility, a cancer therapy, and numerous clinical diagnostic assays. I will also discuss some surprising mathematical connections discovered in the course of this work.