Introductory presentations for DOmath 2022.

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.

It is a fair assumption that many of us in the math department enjoyed math puzzles in our youth and this helped to bring us to where we are. I know I did (and do!). I recently had to solve a classic style of problem: find the nth term of a sequence of integers. I tried everything I knew but only had a pile of scratched out notes to show for it. And then I was told about generating functions. Although not a total panacea for all things sequential, generating functions provide a staightforward blueprint for deriving nth-term formulas and more. I will present a few basic examples and some notes on the excellent book I used as a reference, but the majority of the talk will discuss my particular problem and its solution via generating functions. The main goal will be to impress upon younger grad students the power of this method where other more familiar methods fail.

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.

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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.