James D. Meiss - US grants

This project will study transport phenomena in symplectic mappings of four or more dimensions. The goal is to partition phase space into regions bounded by partial barriers through which flux occurs. A candidate for such a region is a resonance, defined as a volume in the neighborhood of an elliptic point; its boundary will be obtained from limits of librational periodic orbits. Resonance volumes will be computed and we will determine whether resonances partition phase space. The correlation function for orbits in the neighborhood of a resonance boundary well be studied to determine whether the series for the diffusion tensor converges. Transport will be defined in terms of the flux across resonance boundaries, including branching ratios for transitions from one resonance to a neighboring one, as well as drift along a commensurability channel. To compute the latter a restricted symplectic map will be introduced and studied. Numerical techniques will include frequency filtering and finding periodic orbits.

This is a grant under the Scientific Computing Research Equipment for the Mathematical Sciences program of the Division of Mathematical Sciences of the National Sciences Foundation. This program supports the purchase of special purpose computing equipment dedicated to the conduct of research in the mathematical sciences. This equipment is required for several research projects and would be difficult to justify for one project alone. Support from the National Science Foundation is coupled with discounts and contributions from manufacturers and with substantial cost-sharing from the institutions submitting the proposal. This program is an example of academic, corporate, and government cooperation in the support of the basic research in the mathematical sciences. This equipment will be used to support five research projects in the Program in Applied Mathematics of the University of Colorado, Boulder: aspects of solutions, integrable systems, and computation, directed by Mark Ablowitz; iterative methods for the solution of nonlinear equations, directed by James Curry; discrete dynamical systems, directed by Robert Easton; dynamics of Hamitonian mappings, directed by James Meiss; the Kadomtsev- Petviashvili equation and water waves, directed by Harvey Segur.

The principal objective of the National Science Foundation Graduate Research Traineeship Program is to increase the numbers of talented U.S. undergraduates enrolling in doctoral programs in critical and emerging areas of science and engineering. Proposals were solicited from institutions whose existing facilities and staff could accommodate additional graduate students in Ph.D. programs of high quality. The program is also intended to contribute to strengthening the Nation's human resource base across all geographic sectors and among all under-represented groups. Graduate Research Traineeship awards are packages of student support. The colleges and universities that receive the awards are responsible for the selection of trainees, retention of trainees, and administration of traineeships. The University of Colorado at Boulder will initiate a focused traineeship program in the areas that couple applied, nonlinear, and computational mathematics. The initiative will integrate the tools of high-performance computing into the educational program with supercomputer facilities at Los Alamos National Laboratories (LANL) and the National Center for Atmospheric Research (NCAR), provide for summer internships at national research laboratories (LANL and NCAR), and fund an industrial laboratory seminar series. The department will continue to make strong efforts to recruit and retain students from under-represented groups.

9305847 Meiss The dynamics of four and higher dimensional symplectic mappings is of fundamental importance to understanding stability and chaos in conservative physical systems. In this proposal a combination of numerical and analytical techniques will be used. We propose to determine the domain of existence of invariant tori both by using recursive generation of the Fourier series for the tori, and by continuation of the Cantor sets from the anti-integrable limit. The goal is to develop methods for estimating practical stability boundaries and for investigating the transition to chaotic behavior. Computations will determine the robustness of the tori of various frequency vectors, leading to a generalization of the noble numbers that provide the most robust frequencies in two dimensions. A study of one dimensional, resonant tori will also be undertaken-these may be more persistent than two-tori, and form an important component of the barriers to transport. Transport in four dimensions. will be studied by numerical computation of exit time decompositions for cylinders of various homotopy types. Our goal is the development of a geometrical description of trapping regions and resonance zones and a characterization of the practical stability domain around an elliptic point. New techniques for control of transport will be developed for symplectic systems. All of the fundamental equations of physics are formulated as Hamiltonian dynamical systems. We propose to study the structure of the orbits of these systems with the motivation being to understand the problem of "transport. " This is of primary importance in such areas as particle accelerator confinement, chemical reaction rates, fluid mixing, plasma confinement in magnetic fusion devices, asteroid and planetary ring stability, etc. The basic question is: how does a system evolve from one state (e.g. a confined beam in an accelerator), to another (e.g. beam hits the tunnel wall), and how long does this take. Typically trajectories must wend their way through exotic structures such as Cantor sets and self-similar fractals, some of which exhibit a remarkable "stickiness", in order to move through the phase space. The construction and visualization of these structures requires careful computer study guided by mathematical insight. A major problem is that the systems of interest correspond to four and higher dimensional spaces--our ordinary three-dimensional intuition fails. In various applications transport is either to be encouraged (speeding up reaction rates) or discouraged (confining particles); we will investigate techniques for accomplishing both tasks. ***

9408697 Nitsche The goal of the present project is to perform numerical investigations to develop a better understanding of vortex ring dynamics. The case is considered in which a vortex ring is formed by ejecting an amount of fluid from an opening. In the first part of the project, the focus will be on the initial formation process of axisymmetric rings. Viscous effects dominate the flow in an initial time interval and affect the vortex trajectory and total shed circulation. The effects of viscosity by adapting a successful 2-dimensional Navier Stokes solver for high Reynolds number flow to the axisymmetric case will be investigated. Inviscid effects of the flow using a vortex sheet model will also be studied in order to determine whether present similarity theory predictions can be adjusted to better predict the initial flow. The second part of the project concerns 3-dimensional dynamics of vortex rings formed at an opening. The development of a numerical method to compute 3-dimensional vortex sheet separation at an edge will enable the study of the stability of these flows, as well as the effects of nonaxisymmetric openings and nonaxisymmetric forcing. For this purpose, a 3-dimensional vortex filament method will be developed which incorporates vortex separation at a sharp edge and implements a fast summation algorithm to enable high resolution calculations. Understanding the dynamics of vortex rings is essential to understand more complicated flows such as those that occur in combustion processes, or in the airborne vortex structures presenting a hazard to aircraft. An inviscid numerical model has been developed for axisymmetric vortex rings generated at a circular opening. This model was proven by comparison with experiment to recover detailed information about the real flow. In the present work, this model will be extended to 3-dimensional flows, and will be used to study the stability of the flows, the effects of non-axisymmetric openings a nd nonaxisymmetric forcing, as well as the potential applicability of present theoretical results to predict the flow. Several of these aspects of the flow are difficult to understand experimentally or analytically, and the computations promise to give a deeper insight into the flow dynamics. In order to perform this work, current numerical tools available for 2-dimensional flows will be expanded to 3-dimensions.

The Department of Applied Mathematics of the University of Colorado Boulder will purchase SUN Microsystems hardware and associated software which will be used to support research in the mathematical sciences. The equipment will be used in 5 research projects, including computational problems related to: Wavelets and Fast Algorithms, Iterating Lie Point Symmetries, problems in geophysics, development of first order systems least squares for PDEs, and the dynamics of multidimensional mappings.

9810751 Meiss The Department of Applied Mathematics of the University of Colorado at Boulder's VIGRE program is focussed in the areas of computational and nonlinear mathematics. The program creates tetrahedral research groups consisting of a faculty mentor, a postdoctoral fellow, several graduate students, and undergraduate majors. Tetrahedra will focus on particular interdisciplinary research problems, hold seminars, write research proposals, and participate in the curriculum reform program. The curriculum will be integrated with computation using multi-layered case study modules. Affiliated faculty serve as co-advisors for Ph.D. theses of our students; the affiliated faculty program will be extended to government laboratories and technology companies. The program will create four tetrahedral groups in the areas of Dynamical Systems, Nonlinear Waves, Multilevel Computation, and Fast Algorithms and Modeling. The four facets of interaction within the groups include: teaching--as a seminar, and in the development of case study projects; learning--to develop mathematical, computational, and communication skills; discovering--to develop the techniques for formulation of useful and solvable research problems; and communicating--to collaborate in the joint production of research papers, grant proposals, and interim research reports. Computational mathematics provides the unifying theme for vertical integration of our training program. Computation will be integrated into lower division courses through Case Study Modules. Vertical integration will be implemented through the development of a multi-layered modeling course that has both lower and upper division undergraduate components as well as a graduate component. The VIGRE grant will support four Postdoctoral Fellows. They will receive mentoring from a faculty advisor and teach one course each semester for the Department. Twelve Graduate Trainees will be funded by the proposal. They will participate in the tetrahedra, research pro posal development, and the development and implementation of case study modules and receive teacher training through the Teaching and Learning Seminar and the Graduate Teacher Program. Four Undergraduate Research Experiences will be funded each year. Students will participate in one of the tetrahedra part-time during the academic year and for two months during the summer. These initiatives will be sustainable and lead to a number of permanent structural changes: successful tetrahedral research groups will endure and be emulated, case study modules will form an essential part of our curriculum, and the extended affiliated faculty will result in a continuing option for training of our students in application areas. Funding for this activity will be provided by the Division of Mathematical Sciences and the MPS Office of Multidisciplinary Activities.

9971760
Meiss

The principal investigator proposes to study the destruction of chaos and
the concurrent creation of structures (such as stable orbits and tori) in
multi-dimensional (three or more dimensions) volume-preserving and
symplectic maps. Both analytical and computational techniques will be
employed. One approach is based on the "anti-integrable" (AI) limit,
introduced by Aubry in 1992. This principle yields analytical bounds for
the existence of chaotic dynamics, as well as an efficient numerical
technique for continuation of families of periodic orbits. By extrapolation
one can also follow quasiperiodic and heteroclinic orbits as well. The
destruction of chaos is signaled by the first bifurcations in the system and
the creation of order by the bifurcations that create stable orbits and tori. A
second project is to classify the heteroclinic orbits and their bifurcations
for a multi-dimensional version of the quadratic Henon map.
Classification will be given through construction of "primary intersection
manifolds," and by determining their homology on the "fundamental
annuli." A topological classification of structures in dynamical systems is
important both in the analysis of data, and in the interpretation of
numerical experiments. It is well known that chaotic systems often have
fractal invariant sets with various topological properties. These will be
studied in this proposal through the "disconnectedness" and "discreteness"
of compact sets. The PI will develop techniques for computational
homology, yielding a definition for "lacunarity" and computational
methods for Betti numbers of resolution dependent approximations to
these sets.

Proposal #0202032
PI: J.D. Meiss
Institution: University of Colorado Boulder
Title: Geometry and Computation of Dynamics for Conservative Systems

ABSTRACT

The principal investigator proposes to study the geometry of low-dimensional dynamical systems, especially symplectic and volume-preserving maps, using both computational and analytical techniques. While much is known about the two-dimensional case, there are still many questions about the onset and development of chaos for three- and higher-dimensional systems. While most oscillators are anharmonic (have twist), twistless bifurcations occur in one-parameter families of these systems. In the proposal, the geometry of twistless bifurcations will be studied leading to an understanding of fold and cusp bifurcations in the twist. The resulting geometry of the reconnection of resonances and exotic twistless tori will be studied numerically. These should play a role in limiting the stability domains for many dynamical systems. From the other side, the destruction of chaos can be profitably studied using a limit of extreme chaos, the anti-integrable (AI) limit as a starting point. In this proposal, the PI will use the AI limit to study coupled systems of maps and chaotic boundaries. Near this limit, structures such as exotic versions of the Smale horseshoe, and other heteroclinic tangles should occur. The onset of chaos in conservative systems is signaled by the destruction of tori. These have been studied by a rescaling analysis called the renormalization transformation. The structure of this transformation for four and higher dimensional systems is only beginning to be understood. The PI proposes that recent approximate versions of this transformation will give effective numerical strategies for finding the destruction and analyzing the topology of the resulting objects.

Developing an understanding of the dynamics of conservative systems is important to applications including the design of particle accelerators, obtaining rates for simple chemical reactions, calculating confinement times for charged particles in plasma fusion devices, understanding the spectra of highly excited atomic systems, and designing efficient spacecraft trajectories in an era of lower budgets. Dynamics is such systems is often chaotic, which implies
that prediction of specific trajectories is difficult; however, chaos can be profitably utilized to improve efficiency, for example of spacecraft trajectories, by judiciously applying small course
corrections. Chaos can also dramatically affect the lifetimes of particles in confinement devices and the rates of chemical reactions. The PI proposes to develop geometrical and computational techniques that can be used to address these questions. In addition extending our understanding of chaos to higher dimensional cases will help populate the zoo of chaotic objects in multidimensional systems.

The MCTP: Colorado Advantage proposal will build on existing programs and efforts within the department as well as on campus (e.g. SMART program). MCTP: Colorado Advantage will continue to attract numbers of underrepresented students to the mathematical sciences by actively collaborating with the Colorado Diversity Initiative. Support from NSF-funded MCTP: Colorado Advantage will allow the department to develop and implement strategies for sustainability, such as time for re-tasking existing department resources, working with the Deans (Engineering and Arts & Sciences Deans) and Provost to gain additional resources as well as working with successful donors capable of endowing undergraduate scholarships. The 17 faculty members in the department strongly endorse this MCTP proposal. The intellectual merit of this proposal is to introduce a large number of undergraduates to the excitement of research and to stimulate their interest in furthering their mathematical education. The ability of the Department to meet this objective can be inferred from its record with its previous VIGRE grant, which began in 1999, and trained 54 undergraduates.

Among the broader impacts of the MCTP: Colorado Advantage program are that it will significantly increase the number of students who take more advanced mathematics courses; the number of majors who have the transformative opportunity of working on serious longer term projects and research projects; and the number of students who graduate from the Department of Applied Mathematics and are well prepared for graduate school and the scientific workforce. Further its multiplier effect on other undergraduates at the University through the creation of a culture of undergraduate research activity will have longer-term implications for the University as a whole. An additional impact will be on graduate students and faculty who become involved in these critical mentorship activities as they pursue their own careers. Finally, the MCTP: Colorado Advantage proposal is a model that can be used at any research university that has a graduate program, faculty committed to undergraduate education and a will to transform itself.

Abstract

Regular, quasiperiodic motion is ubiquitous in dynamical systems with sufficient symmetry. A prominent example occurs in the Hamiltonian or symplectic case, where these "invariant tori" persist---even for nearly-integrable motion, as is explained by "KAM theory." The destruction of tori in the two-dimensional case is explained by Aubry-Mather theory and renormalization results. However, a concomitant understanding of the destruction of tori upon perturbation in higher dimensions has proved elusive. In this proposal, the implications of integrability, due to symmetries and invariants, of volume-preserving dynamics will be investigated. The loss of integrability under perturbation will be studied by a combination of analytical (Aubry's anti-integrable limit, Fourier series) and numerical (invariant manifold and continuation) techniques. Tori are both created and destroyed by bifurcations, and a study of the normal forms for codimension-one and two bifurcations of fixed points will lead to classification possible phenomena. Transport will be investigated numerically with the goal of developing analytical measures of flux and transport distributions. In a second project, the PI will investigate bifurcations in nonsmooth systems appropriate to the modeling of chemical reactions, the systematic simplification of these systems by center manifold reduction, as well as the study of transport caused by weak coupling of chaotic motion to regular motion.

Conservative dynamical models are used in designing particle accelerators, obtaining rates for simple chemical reactions, calculating confinement times in plasma fusion devices, understanding the spectra of highly excited atomic systems, and designing efficient spacecraft trajectories. Dynamics in such systems is often chaotic and prediction of individual trajectories is difficult; nevertheless, chaos can be profitably utilized, for example, to improve efficiency of spacecraft trajectories, by judiciously applying small course corrections, or to enhance the lifetimes of particles in confinement devices and the rates of chemical reactions. Volume-preserving dynamics models the flow of incompressible fluids and magnetic fields and a quantitative understanding of chaos in these systems is crucial for the development of efficient mixing in microscale bioreactors as well as of predictive planetary scale weather models. Most of our current theoretical understanding is limited to the two-dimensional case that is appropriate for flows in rapidly rotating or thin layers of fluid. While this has been useful in the understanding of such phenomena as the trapping of nutrients in gulf stream rings, the formation of the ozone hole and the creation of vortex-induced mixing in sinuous tubes, even in these systems, three-dimensional, chaos-induced transport needs to be understood. The PI seeks to develop analytical and computational methods for the study of regular and chaotic volume-preserving motion both to contribute broadly to our fundamental understanding of the richness of the behavior of low-dimensional deterministic evolution, and, to relate it to mixing and transport.

This grant provides funding for the travel expenses of graduate students, postdocs, and individuals without other sources of support to attend the conference ?Dynamics Days US 2013? on nonlinear dynamics and chaos which will be held in the Marriott City Center in downtown Denver, Colorado, in January 3-6 of 2013. The conference webpage is http://amath.colorado.edu/conferences/DDays2013.
Dynamics Days is an annual international conference that has been running in the US for more than 30 years. During this time, Dynamics Days has served as one of the main venues where recent developments in and applications of nonlinear dynamics are disseminated. This year, the conference is aimed at promoting cross-fertilization across different fields of nonlinear dynamics, and will address topics such as geophysical and astrophysical fluid dynamics, neuroscience, synchronization and self-organization, nonlinear dynamics of biological systems, dynamics on and of complex networks, dynamic fracture, granular media, and pattern formation. The conference will have 21 invited speakers, a similar number of contributed talks, and poster sessions.

In the last decades, Dynamics Days has become established as an excellent venue for interactions between specialists in different fields and between young and established researchers. The topics of the talks cover a variety of areas with overlapping interests in nonlinear dynamics. This variety attracts participants from various fields in mathematics and physics, and also a significant number of participants with backgrounds in other fields such as engineering, biology, and geology. Experienced researchers will benefit from the exposure to new techniques and potential collaborators in diverse fields. Financial support will be available to students and young researchers, who will be able to give contributed talks or present in a poster session. These young researchers will especially benefit from the format of the conference, gaining a broad perspective of current research in interdisciplinary areas. The participation of students and young researchers from underrepresented groups will be particularly encouraged.

The persistence of quasiperiodic motion on codimension-one tori in nearly-integrable volume-preserving maps is explained by KAM theory. However, the robustness of these tori and the existence of remnants upon destruction are understood only in two-dimensions. The PI proposes to study tori of three-dimensional maps, and to generalize the residue criterion discovered by Greene and the anti-integrable limit discovered by Aubry. Studies will include symmetry reduction, invariance, and the loss of integrability for general, structure-preserving maps and flows. An important application is the optimization of mixing in open duct flows used in the continuous blending of materials. Our current understanding of the mixing process is, for the most part, limited to flows that are in essence two-dimensional and either closed or recycling. Transport in three-dimensional systems can be quantified by the flux through the destroyed structures, computed using a generalized action based on Lagrangian forms, thereby obtaining accurate and computationally efficient volume fluxes. The PI and students will use the concept of transitory dynamics to quantify and optimize transport in open flows. The extension to episodic and more general time-dependence will clarify the definition of Lagrangian coherent structures in aperiodic dynamics.

The complexity of patterns obtained by mixing a passive scalar in a fluid can be observed by anyone pouring cream into hot coffee. That this process is not fully understood is perhaps less obvious. If the flow is sufficiently turbulent then mixing is rapid and uniformity is not hard to achieve. If, however, the flow is slow, on a small scale, or viscous, then mixing is much more difficult. Yet, such processes are important to many applications including the development of micrometer scale bioreactors and effective mixing of polymer and granular materials. A predictive theory for laminar mixing would also contribute to the understanding of climate modeling and pollution dispersal in the atmosphere as well as nutrient dispersal and spawning efficiencies for sea life. Mixing in laminar flows proceeds by stretching and folding due to chaotic motion that gives rise to fine-scale structure where diffusion is effective. Any measure of mixing requires quantification of chaos and its concomitant transport. Chaotic motion in incompressible fluids has some similarities to that in conservative dynamics. The later models are used to predict the lifetime of particles in accelerators, obtain rates for simple chemical reactions, calculate confinement times in plasma fusion devices, understand the spectra of highly excited atomic systems, and design efficient spacecraft trajectories. For chaotic dynamics, prediction of specific trajectories is difficult; nevertheless, chaos can be profitably utilized, for example, to improve efficiency of spacecraft trajectories, by judiciously applying small course corrections, or to enhance the lifetimes of particles in confinement devices and the rates of chemical reactions. In this study, chaos will be used to optimize mixing with the goal of obtaining practical designs for open, three-dimensional, mixing devices.

Time-series data arise in a wide array of engineered systems, including network traffic, vibration sensors on machine tools, acoustic sensors on reactor containment vessels, and many other examples. The development of efficient and effective methods to characterize the patterns in such data has widespread utility in engineering, commerce and other fields. Methods for characterizing patterns in these streams could be used to detect malware attacks on a network, a lathe bearing that is degrading, or an impending containment failure in a reactor. Common challenges include observability - situations when sensors are expensive or difficult to deploy, or when they perturb the behavior under examination - as well as high information content, noise, and rapid regime shifts. The ultimate goal of this EArly-Grant for Exploratory Research (EAGER) project is to use computational topology, the fundamental mathematics of shape, to deal with these challenges. Shape is perhaps the roughest notion of structure and can be particularly robust to contamination of the signal. The specific goal of this study is to develop new methods for identifying and categorizing the temporal patterns associated with the regime shifts a stream of data.

A topological approach to time series analysis is distinct from standard methods of the machine learning and stream-mining communities, which typically use probabilistic approaches and often implicitly assume linearity. This project seeks to extract nonlinear structure not necessarily visible in a regresssion or spectral approach. Indeed, a regime shift need not correspond to a change in the frequency content of a signal, but could nevertheless be represented as a shift in the homology (e.g., Betti numbers) of the embedded signal. A goal is to develop techniques useful to engineers and scientists for the detection of incipient system failure or rapid evaluation of state changes from hidden causes. Existing algorithms of computational topology often require lengthy computations, especially for large data sets in many dimensions. However, since not all of those variables may be observable, one may have to reconstruct the full dynamics from partial measurements--e.g., using the process called delay-coordinate embedding. This project seeks rapid evaluation of Betti numbers based on incomplete, partial embeddings. A novel aspect is that the dynamics gives rise to a multivalued map on a simplicial complex, a "witness map." Selection of multiple parameters in the algorithms will be based on persistent homology, previously developed only for the analysis of static data sets and for a single parameter. The ultimate goal is robust and rapid regime detection for a limited data stream from a "black-box" source.

This project is about a new way to detect and classify shifts in the patterns of measurements taken by sensors. Subtle shifts in the output from a vibration sensor on a bridge abutment, for instance, can indicate that a crack is developing in that structure. Identifying these shifts can be a real challenge because modern sensors can generate so much information, and so quickly. Existing approaches to this use the mathematics of statistics: averages, variability, and the like. This project uses techniques from topology, the branch of mathematics that is concerned with shape, to detect these kinds of shifts.

This project will develop techniques to assess and characterize temporal patterns in nonstationary time-series data. The project will compute aspects of the homology---e.g., the first few Betti numbers---of the stream "on the fly" to obtain a signature of its regime. To mitigate the computational burden of traditional techniques, this project uses a simplicial complex based on a small, representative set of "landmarks," with the remaining data, the "witnesses," defining the connections. New techniques include a generalization of the oft-used false near-neighbor method to evaluate the fidelity of the topology of the reconstruction, efficient selection of landmarks and choice of witness relation, classification of structure through multi-parameter persistent homology of embedded data, and development of a map on the witness complex to obtain a dynamical signature of each regime. The goals include efficient detection of regime shifts, the development of a catalog of signatures within regimes, and the detection and mitigation of noise. The classification techniques will be applicable, for example, to the detection of failure modes in manufacturing systems, malware attacks on a network, incipient heart problems, or an impending containment failure.

The conference "Dynamics Days 2018" will take place in Denver, Colorado in January 4-6, 2018. Dynamics Days is an international conference focused on chaos, nonlinear dynamics, and their applications. This award provides funds to defray travel costs for graduate and undergraduate students and postdocs without other sources of support. This year the conference will showcase research in areas including networks, fluid dynamics, mixing, topological time series analysis, nonlinear waves, dynamic mode decomposition, and biological systems. The conference will have 15 invited speakers, a similar number of contributed speakers, and poster sessions. The conference webpage is http://www.colorado.edu/amath/ddays-2018/

Dynamics Days is an international conference on applications of nonlinear dynamics and chaos which has been running in the US for 35 years. During this time, it has established itself as one of the main venues for the discussion of new ideas in nonlinear dynamics. The conference is characterized by covering a wide area of applications, which promotes the exchange of ideas and dissemination of results across different fields. The variety of topics attracts participants with diverse backgrounds, including physics, mathematics, engineering, and biology. Travel awards will be offered to graduate and undergraduate students and postdocs without other sources of support. These participants will be able to present their results in contributed talks or poster sessions, and will gain a broad perspective of interdisciplinary research in the field. The participation of underrepresented groups will be encouraged.

Anyone who has poured cream into hot coffee has observed the complex patterns that occur during the mixing of two fluids. It is perhaps surprising that the underlying process is not fully understood, especially when the fluid motion is laminar, i.e., either it is slow or the viscosity is high; then uniform, efficient mixing is hard to achieve. Such laminar processes are important to many applications including the development of micrometer scale bioreactors, effective mixing of polymer and granular materials, spreading of pollutants in the atmosphere, and even nutrient dispersal and spawning efficacy for life in the sea. Laminar flows can cause chaotic motion of advected particles, yielding rapid loss of accuracy for prediction of individual trajectories that turn out to be extremely sensitive to the miniscule changes in environment. The investigator and his colleagues study the design of efficient mixers by developing an understanding of the causes of this sensitivity, and by developing methods to globally optimize stirring protocols. The mathematics of these models is closely related to that, ubiquitous for conservative motion in physics. Techniques the investigator develops are used to predict the lifetime of particles in accelerators, obtain rates for elemental chemical reactions, calculate confinement times in plasma devices, understand the spectra of highly excited atomic systems, and predict asteroid and spacecraft trajectories. Graduate students will be trained through participation in this research project.


Subsonic fluids are incompressible and the resulting flows are volume-preserving. Though chaotic motion in incompressible fluids is similar to that in Hamiltonian or symplectic systems, there are profound geometrical differences due to the lack of a canonical pairing between momenta and coordinates. The investigator studies how the geometry of such dynamics changes when it is "nearly" symplectic, leading to novel elliptic structures and to a discovery of the violation of the exponential quasi-stability of nearly-integrable systems implied by Nekhoroshev's theory. The studies include the development of techniques for understanding transport through destroyed invariant tori and the long-time correlations engendered by regular islands and accelerator modes. Mixing due to chaotic advection is caused by stretching and folding, which promotes homoclinic tangles and structure so fine that diffusivity can be effective even when small. The investigator and his students will model this by a finite sequence of stirring events that determine a mixing protocol. Optimization techniques under geometric and energy constraints are used to extremize Sobolev norms that give measures of mixing.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

Eruptions generated by sunspots --- large concentrations of magnetic field on the visible surface of the Sun --- can have a number of dire impacts on Earth-based technological systems, crippling satellites and power grids, among many other things. With enough advance notice, the effects of these events can be mitigated, but predicting them is a real challenge. In current operational practice, this is accomplished by human forecasters examining images of the Sun, classifying each sunspot according to a taxonomy developed in the 1960s, and then using look-up tables of historical probabilities to say whether or not it will erupt in the next 24 hours. Recently, there has been a burst of work on machine-learning methods to automate this task. To date, the "features" used in these approaches have been predominately physics-based: the gradient of the magnetic field, for instance, or the sum of its strength over high-flux regions. The main objective of this 3-year research project is to leverage algorithms based on the fundamental mathematics of shape --- topology and geometry --- to improve the performance of these methods. The specific plan is to use these powerful techniques to extend the relevant feature set to include characteristics of the magnetic field that are based purely on the geometry and topology of 2D magnetogram images. Although this approach ignores the 3D structure of the full electromagnetic fields, it can enhance the predictive skill of machine learning systems. Preliminary results show clear topological changes emerging in magnetograms of a 2017 sunspot more than 24 hours before it flared, as well as clear improvements in the accuracy scores of a neural-net based flare prediction method that employs these shape-based features. Better predictions of solar flares could allow operators of power grids, airlines, communications satellites, and other critical infrastructure systems to mitigate the effects of these potentially destructive events. The broader impacts of this project also include the development of the STEM workforce through the training of graduate students at the University of Colorado at Boulder, as well as education and outreach, including community lectures, development of large-scale, online courses and public lecture series. The interdisciplinary nature of the project will deepen the contact between the fields of space weather, applied mathematics, and computer science, bringing researchers, students, and post-docs from both fields into productive new collaborations. The collaboration with the Space Weather Technology, Research, and Education Center at the University of Colorado offers unique opportunities to factor in real-world forecasting constraints and set the stage for transitioning the results to operational status.

For the first time, this 3-year research project would provide systematic quantitative measures of the shape of 2D magnetic structures in the Sun?s photosphere for the purposes of solar flare prediction. In a sense, this amounts to a mathematical systemization of the venerable McIntosh and Hale classification systems. This approach differs from current studies in the solar physics community that model the magnetic field-line structure: it uses topology to address the structure of two-dimensional sets. The analysis is restricted to photospheric magnetic field structures; the goal is to extract a formal characterization of shape that can be leveraged by machine learning to improve flare prediction. The considered addition of geometry into these methods by the project team is essential if they are to capture the full richness and physical relevance of the structures important in the evolution of a sunspot. This research project will point the way forward to a more robust set of features for machine-learning-based eruption prediction architectures. The research and EPO agenda of this project supports the Strategic Goals of the AGS Division in discovery, learning, diversity, and interdisciplinary research.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.