This award supports theoretical research and education that will use insights into the fundamental structures of theoretical models to develop sophisticated new methods for extracting information from experiments and simulations. The PI will focus on three different topics: fitting nonlinear models to data, extracting universal scaling laws from critical systems, and identifying order parameter interactions in high temperature superconductors.
(1) Fitting models to data. Systems biologists, climate modelers, economists, and most experimentalists fit their data to models. The PI has discovered that these multiparameter models all have a common, fascinating underlying structure. They are sloppy, with only a few parameter combinations that determine the fit to the data; the model predictions form a multidimensional hyper-ribbon in data space; methods for finding best fits move along geodesics on this hyper-ribbon. The PI will use these insights to develop new algorithms for finding optimal fits to data, which promise to be both faster and much more reliable than existing methods.
(2) Extracting universal scaling laws. The PI is developing SloppyScaling, a flexible, expressive software environment for exploring universality and scaling underlying continuous transitions, avalanches, and other fractal, self-similar behavior. They use it to dramatically extend the scope of these theories, systematically extracting universal scaling forms for systems with multiple control variables, corrections to scaling, and crossovers between different universality classes.
(3) Order parameters in high temperature superconductors. The PI is extracting the multiple competing order parameter fields in high temperature superconductors directly from experimental scanning-probe data. By studying how they respond to one another and to dirt and disorder, they will learn how they couple together and help piece together the puzzle of the underlying mechanism. This research project may have broad impact on other disciplines. It may improve the way we extract predictions from models and model information from data. The PI has a track record of excellent, successful women students, and the projects will provide interdisciplinary training for the graduate students involved.
NONTECHNICAL SUMMARY This award supports theoretical research and education that is aimed at improving the methods scientists use to compare theory and experiment. The PI will do so in three contexts.
(1) Magnets. A piece of iron in a magnetic field of increasing strength, will magnetize in a series of "avalanches." This is why magnets hold on to the refrigerator: they magnetize the metal wall of the refrigerator in the right direction so as to attract it. The PI will study the magnetic crackling noise as magnetic regions with the magnetic order oriented in different directions rearrange into another magnet. In principle theory can explain all properties about these avalanches and crackling noise - the kinds of shapes the avalanches make in space and time, for example. The PI is developing a software package to aid experimentalists and simulators in making full use of these theories.
(2) High temperature superconductors. The high-temperature superconductors are amazingly complicated: lots of different kinds of order seem to be competing, and it is a theoretical challenge to disentangle which features are most important for determining the superconducting properties. At sufficiently low temperatures, superconductors have an unusal kind of order that results in an electronic state of matter that can conduct electricity without losses. Sophisticated experiments on a high temperature superconductor reveal high-resolution images of the surface of one superconductor, and has found elaborate, complex patterns closely related to the superconductivity. The PI has been developing tools for extracting the competing fields out of his data, and will use them to gain quantitative understanding of how they work together.
(3) Fitting models to data. A theoretical model does not usually directly predict the behavior of an experiment - one needs to give it some information about the experimental system. Thus the theory of fluids demands that we measure the viscosity and density of the air, the air speed, and the wing geometry, before it will make predictions about the drag on an airplane. Sometimes these parameters can not be determined separately, but are used to fit the data - climate models used to study global warming, econometric models used to predict how our economy works, and models of how cells work include lots of constants that are hard or impossible to directly measure. The PI has discovered that most multiparameter models share many common features; for example, they are sloppy, with many parameter combinations being very poorly determined by the data they are fit to. By using sophisticated mathematics normally used to study general relativity, the PI is using these common features to improve the way theoretical models are fit to experimental data.
This research project may have broad impact on other disciplines. It may improve the way we extract predictions from models and model information from data. The PI has a track record of excellent, successful women students, and the projects will provide interdisciplinary training for the graduate students involved.
Our group has been pursuing how mathematics and theoretical physics can be used in understanding how the DNA and membranes of our cells work, how magnets crackle, how well we can build superconducting colliders, and how simple mathematical models can be used to describe the complicated world around us. Our biological work has focused on two systems. First, we collaborated withour colleague Michelle Wang to theoretically explain her experiments with twisted DNA. We found a way to explain her experiments that drew on ideas of how water droplets condense in clouds. Second, we worked with Sarah Veatch to understand her amazing experiments on membranes from living cells -- which showed that they were near the cusp of separating into two fluids (like oil and water separate after shaking salad dressing). Our theories suggest that the cells may have evolved to sit near this transition in order to make it easy for proteins to group together -- and we have used ideas from string theoryto calculate the force between proteins caused by Sarah's transition. Our work on magnets develops new tools for studying the shapes and sizes of the avalanches that happen when the magnet is magnetized. Similar `avalanche' physics has been used to explain earthquakes. Our work on superconductors is estimating how well new materials can theoretically work in helping accelerate electrons in superconducting particle accelerators and X-ray sources. Finally, we have been learning how models fit to data. The models we've studied range from systems biology, to insect flight, to quantum wave functions, to the magnet positions in accelerators. We use beautiful mathematics in high dimensional spaces to understand weird features of how these models act when fit to data. Using these ideas, we've found powerful new methods to do these fits more quickly, and also we have new insight into why these relatively simple models can capture the behavior of a much more complex real world.
