Many major hypotheses about the geology of pre-Mesozoic eras rely critically on paleomagnetic data. Magnetic remanence in rocks records the ancient magnetic field and provides information on plate tectonics and the history of the geodynamo. For the remanence to be useful, it must remain stable as the rocks are subjected to temperature increases, pressure, and changes in the chemical environment. Suitable rocks become increasingly rare as their age increases. In pre-Mesozoic rocks, many of the most promising magnetic recorders do not fit the standard model for good paleomagnetic recorders because the magnetic minerals interact with each other through magnetic fields. If these interactions are not properly accounted for, paleomagnetic measurements can be misleading. The effect of interactions is complicated by thermal relaxation of the magnetic moments in which the magnetization jumps over energy barriers between stable states. The greatest challenge facing models of interacting magnets is to find these energy barriers.
The object of this proposal is to harness some powerful methods from numerical algebraic geometry to calculate the magnetic effects of particle interactions and thermal fluctuations. The equilibrium equations for interacting magnets can be expressed as systems of polynomial equations. Methods will be developed to find all the stable states and transition states for such systems using homotopy continuation. The time dependence of the magnetization can then be expressed as a set of kinetic equations. There will be three main applications of this theory: 1. A theory of remanence acquisition and demagnetization for titanomagnetite inclusions in sub-aerial basalts and Precambrian shields. These titanomagnetites are often networks of strongly interacting magnetite grains separated by ilmenite lamellae. The effect of particle interactions and grain growth on the accuracy of paleointensity measurements will be determined. 2. A model of isothermal hysteresis measurements of sedimentary systems involving single-domain and superparamagnetic particles that may have significant interactions. These include remagnetized carbonates, sediments with fossil magnetotactic bacteria, and sediments with greigite or pyrrhotite. Such systems can carry important information on ancient environments and human impacts. 3. A model of measurements involving oscillating fields, including anhysteretic remanent magnetization (ARM) and alternating field demagnetization of remanence. These measurements are common tools in paleomagnetic analysis.
The principal goal of this award was to develop mathematical methods and software aimed at various applications, specifically some coming from geomagnetism. Through this award, two software packages were developed and three scientific publications were written (two about the software packages, one about a new technique called decoupling). The first software package, Paramotopy, was developed by D. Brake, D. Bates (the PI), and M. Niemerg. This sophisticated package enables the user to use many processors (computer clusters) to solve many variations of the same problem very efficiently. More specifically, given a polynomial system that depends on some parameters, Paramotopy allows users to solve the polynomial system at many points in the parameter space in a manner more efficient than any other software package currently available. This sort of problem comes up quite frequently, in areas including geophysics, string theory, biochemistry, robotics, and others. Paramotopy will soon become a module of the Bertini software package (described below). The second package, BertiniLab, was developed by D. Bates, A. Newell, and M. Niemerg. This package is an inteface between the popular Mathworks software package MATLAB and the highly specific software package Bertini (developed by D. Bates, J. Hauenstein, A. Sommese, and C. Wampler), which is capable of solving a particular kind of problem - polynomial systems - in a novel way. MATLAB is used by many mathematicians, scientists, and engineers, inside academia and outside. BertiniLab fills a significant hole in the large set of tools already available in MATLAB. Once Paramotopy is incorporated into Bertini, users of BertiniLab will have access to this powerful new tool. These two software packages open the door for geophysicts (and others) to solve mathematics problems that were previously intractable. However, even with very highly optimized software, some problems are still beyond the reach of current methods. In particular, some problems coming from geomagnetism require the solution of huge, highly-structured polynomial systems. The number of variables and equations in such problems is well beyond what could be handled before the work performed this award. The decoupling technique, described next, opens the door for solving much larger problems. Decoupling, in a nutshell, breaks a large, highly-structured mathematics problem into a large (but manageable) number of small, nearly identical mathematics problems. More specifically, it breaks a large polynomial system into a set of small polynomial systems that differ only in coefficients and variable labels. These smaller problems can be handled effectively by Paramotopy (described above), and the solutions can then be combined to provide the solutions of the original, large problem with a little more computational effort. This new technique significantly simplifies problems of a particular type, such as those coming from geomagnetism. The intellectual merit of the work performed under this award lies in the development of these three mathematical tools. The software packages are freely available (though BertiniLab requires MATLAB), and their source code is publicly available. See the MATLAB Central File Exchange for BertiniLab and paramotopy.com for Paramotopy. The two software packages and decoupling are the subjects of three scientific articles currently under review at leading computational mathematics journals. The broader impacts associated with this award include the training of multiple graduate students at Colorado State University, the potential use of the methods described above in a wide variety of application areas, and the creation of a monthly middle school and high school student math circle program at Colorado State University.
