Andrea Bertozzi, Ph.D. - US grants

9409484 Bertozzi The dynamics of a thin viscous film on a solid surface are modeled by a fourth order nonlinear diffusion equation for the film thickness (Greenspan, J. of Fluid Mech. 1978, vol. 84, p. 125). The project involves three aspects of this problem, singularity formation, weak solutions, and pattern formation. A singularity occurs when the thickness of the film goes to zero. Physically the singularity corresponds to a topological transition in the fluid. The study will be made via a combination of numerics and analysis the onset of such singularity formation forth order degenerate diffusion equations. This work may include collaborations with M. Brenner, L. Kadanoff, T. Dupont, and A. Bernoff. The investigator is working with Mary Pugh at the Courant Institute on the weak solution problem. This collaboration recently established sharper existence and long time behavior results for weak solutions to the thin film equation. The existence is in a regularity class that just includes a family of unique `source type' solutions analogous to the `Barenblatt' solutions of the porous media equation. Preliminary numerical computations show that in general weak solutions converge rapidly onto the `source type' solution. It is planned to complete the numerics as well as to relate these weak solutions to slip models' suggested for the spreading drop problem. Lastly, in a joint project with M. Brenner, problems in pattern formation such a gravity and temperature driven fingering instabilities in thin films will be studied. ***

HRD-9979478
Duke University


Project Advance at Duke University will recruit and develop a resilient cohort of talented first-year women students in the quantitative sciences, focusing particularly on mathematics, computer science, and statistics. As a highly selective private research university, Duke attracts women undergraduates with demonstrated excellence in science and mathematics and high self-esteem. We seek to sustain and enhance these characteristics by developing an experimental project that intentionally links pedagogical components of the first-year curriculum, introduces an innovative, interdisciplinary half-credit seminar designed to develop students' self-perceptions and identity as scientists, and establishes a structure for guided mentorships to help this talented cohort navigate successfully through science and mathematics experiences. We have targeted the first college year because of the critical role it plays in the transition from high school to the undergraduate major, in fostering women's identity as scientists and mathematicians, and in providing the skills and experiences requisite for advanced study and successful careers. This multidimensional project builds on the pedagogies of engagement and current research on processes, such as self-identity and negative stereotypes, that have been shown to facilitate or inhibit achievement. It will provide experimental data that can be broadened across the sciences and serve as a model for how research universities, in general, can foster a more welcoming environment for the retention and support of women and, thus, foster careers of distinction.

0074049
Bertozzi

A combined experimental, analytical, and computational study of
fundamental problems in the dynamics of thin viscous films and fluid
interfaces is proposed. The recent discovery of stable undercompressive
waves in driven films has created the opportunity for a unique
collaboration between experiments and mathematical theory. This research
program will include related studies of solid-liquid-vapor interfaces,
moving contact lines, and surface tension effects. Analytical and
computational studies will be integrated with a series of experiments that
includes a search for undercompressive waves in a spin coating geometry,
motion of contact lines near room- temperature critical points, and high-
speed video imaging of the dynamics of singularity formation in finite-
time rupture of fluid interfaces. Mathematical analysis will include models
for film rupture, stability of driven contact lines, and numerical analysis of
schemes for computing these problems.

Liquid films and moving contact lines arise in problems ranging from
industrial design of paints and microchip fabrication to medical
applications including contact lenses and the lining of the lung. All of
these problems involve interactions across widely different length-scales
in which the physical laws are not clearly understood. This is a
collaboration between researchers from the Mathematics (Bertozzi,
Witelski) and Physics (Behringer) Departments at Duke University and
the Mathematics Department at North Carolina State University (Shearer).
This effort combines mathematical modeling, analysis, and numerical
simulation with new laboratory experiments to study fundamental
problems in driven films and moving contact lines. Computational and
mathematical models will direct the design of experiments investigating
new phenomena in spin coating processes and dewetting films. The
program will involve undergraduates, graduate students, postdoctoral
associates, and visiting scientists from other institutions. This research will
foster curriculum developments in the Departments of Mathematics,
Physics and the Center for Nonlinear and Complex Systems at Duke
University

Abstract
Bertozzi

The Duke Mathematics Department's VIGRE program was designed with NSF's stated objectives
of "increas(ing) interdisciplinary activities involving mathematics" and "encourag(ing) activities aimed at broadening undergraduate and graduate curricula" in mind while maintaining existing disciplinary strengths. Interdisciplinary research is a perfect setting for vertical integration and
the Duke Mathematics Department has existing disciplinary strengths in algebraic geometry, analysis, differential equations, differential geometry, numerical analysis, probability, and topology.

We will use the team research model to enhance our graduate training program and our undergraduate and postdoctoral programs. Graduate and undergraduate students will confront
research problems early in their careers and will have ongoing exposure to them through their respective programs. Advanced graduate students and postdocs will take on new roles as project leaders in vertically integrated research seminar courses and in team research groups.
All students and postdocs will develop skills in computation, communication and collaboration as part of their mathematical training, thereby broadening the range of opportunities available
to them as mathematical scientists. The goal of the program is to have students and postdocs completing their programs with a broad scientific knowledge and clear understanding of the inner workings of a scientific team, as well as an academic research and teaching portfolio that will enhance their career advancement potential.

In the undergraduate program our goal is to increase students' enthusiasm for mathematics and its role in research at all levels of their training at Duke. We will increase the number of first-year and major seminars aimed at exposing students to current topics of research. Current seminar course topics include cryptography, physiology and medicine, geometry, and optimization. New topics include heart dynamics, gravitational lensing, and artificial intelligence. A new Perspectives on Science seminar aimed at first-year women interested in mathematics, statistics and computer science will expose young students to current research of senior scientists, postdocs, and graduate students.

The cornerstone of the undergraduate VIGRE program is a new two-year intensive research program, entitled `Practical Research for Undergraduates with VIGRE' (PRUV, pronounced "prove"), for advanced mathematics majors, which integrates course work, summer internships in a vertically integrated environment, and senior theses. These improvements in the undergraduate program are targeted to involve more students in research activities. We expect
that one outcome will be more senior honors theses on research problems and more students choosing to apply to graduate school in the mathematical sciences.

For the graduate program, the VIGRE grant will be combined with Duke resources to support all first year students without teaching duties. After the first year, the strongest VIGRE-eligible students will receive a one semester-per-year teaching release in order to direct more of their energy towards research. All graduate students in the graduate program will have the opportunity to work in vertically integrated research teams, including postdocs, graduate students PRUV students, and at least one mathematics professor. Those working on interdisciplinary problems
will also collaborate with at least one professor from another discipline. Current interdisciplinary activities include deforestation and aerosol dynamics, geometric computing, granular flow,
heart dynamics, liquid films, photonic band gaps, physiology and medicine,
string theory, and uncertainty in porous flow, with Duke collaborators in the School of the Environment, Computer Science, Physics, Biomedical Engineering, Cell Biology, Civil and Environmental Engineering, and the Institute for Statistics and Decision Sciences.

Graduate teacher training currently begins in the first-year with students supervising
calculus laboratories. In order to shorten time to degree and increase retention, we will postpone teaching responsibilities and teacher training by one year for all graduate students.
This will enable all first year graduate students to focus on fundamental course work required to pass the qualifying exam before the beginning of year two. The current teacher-training program will be expanded to include alternative teaching experiences for students and postdocs
including outreach to high school teachers, and mentoring of undergraduate and graduate students. A new computational requirement will be part of their first-year curriculum. The graduate students currently run a weekly seminar with talks aimed at first and second year graduate students given by graduate students and postdocs. PRUV undergraduates will be encouraged to attend this seminar.

VIGRE postdocs will play a leadership role in the overall program. Each year, for the first three years, two VIGRE postdocs will be hired for three year terms. These positions will parallel existing Duke funded Assistant Research Professors (ARP). VIGRE Postdocs will have the opportunity to play a strong leadership role in the team research groups including mentoring graduate students and undergraduates and developing a well-defined independent research program. They will also have the opportunity to take on a significant supporting role in at least one of the following activities: (1) coordinating research seminars and workshops, (2) co-supervising summer PRUV students and graduate student internships, (3) helping undergraduates and graduate students in a writing workshop.

Postdocs and students will collaborate with high school teachers in Project CHISEL `Carolina High School Educational Leadership project'. They will design modules for classroom use
that will introduce high school students to the use of mathematics in current research.

Funding for this activity was provided by the Division of Mathematical Sciences and the MPS Office for Multidisciplinary Activity.

NSF Award Abstract - DMS-0138923
Mathematical Sciences: Nonlinear Differential Equations, Mechanics and Bifurcation

Abstract

0138923 Shearer

This award supports U.S. participants in the conference on Nonlinear Differential Equations, Mechanics, and Bifurcation held at Duke University on May 20-22, 2002. The purpose of the conference is to explore connections between the fields of ordinary and partial differential equations, mechanics and bifurcation, and industrial mathematics, and to encourage the involvement of young mathematical scientists in these areas.
The speakers include established leaders in these fields, as well as promising young mathematicians and scientists. The meeting is a collaboration between the Research Triangle universities of Duke University, North Carolina State University, and the University of North Carolina at Chapel Hill.

The central theme of the conference is the role of nonlinearity in physical systems, especially elasticity, granular materials, and fluid flow. The interplay between experiments, modeling, theoretical mechanics, mathematical analysis, numerical simulation, and industrial design has proved particularly fruitful in recent years. This conference brings together scientists and mathematicians from all of these areas. The conference is designed to include lively discussion of the interplay between discoveries of new experimental phenomena, proposed mathematical models, theoretical developments and challenges, and corresponding innovations in numerical methods. Bifurcation and pattern formation are observed in experiments with solids, fluids and granular materials, and the mathematical techniques to analyze phenomena have grown quite sophisticated. This connection between experiment and theory, coupled to numerical simulation, occurs throughout the conference.

This project seeks to develop a comprehensive research and education program in the area of computational
methods and simulations of physical systems described by high order Partial Differential Equations (PDEs). The program will unify algorithmic, visualization, theoretical, and experimental efforts as well as address applications in areas of science and technology, including computer graphics, image processing, biology, and fluids.
Intellectual merit of the proposed activity
This project advances knowledge in the area of high order PDEs, with particular emphasis on curved surface data, and produces enabling technology to address fundamental problems in biology, image processing, computer graphics, and fluids in general. The novel science is in the computational
techniques, experimental research, and diverse applications addressed by a multi-disciplinary team. This
project brings together the five fields of computer science, applied mathematics, mechanical engineering, physics, and electrical and computer engineering.
Broader impacts of the proposed activity
With the increasing interest in high order PDEs, the computational tools and experience resulting from this project impact beyond the particular applications in this proposal. Students will receive unusually broad
interdisciplinary training and the workshop planned further brings experts from different fields together. New public domain software incorporating the developed algorithms enables researchers from different fields using higher order PDEs to perform state-of-the-art numerical simulations and graphics rendering of their application of interest.
Educational initiatives of this research program include: (1) new interdisciplinary training of graduate students and postdocs through co-mentoring by PIs in different fields; (2) new interdisciplinary courses in computer graphics, numerical analysis, and modeling/simulation of physical phenomena described by higher order PDEs; (3) a workshop bringing together for the first time diverse scientific researchers using high order PDEs.

This collaborative project among Duke University, the University of California at Los Angeles, and North Carolina State University involves research into an array of interrelated problems in the fluid dynamics of thin viscous films. It combines analytical, computational, and experimental approaches to fundamental issues concerning surfactant spreading, patterned surfaces, vibrational forcing, Marangoni flows, flows on curved surfaces, and spin-coating flows. Methods include the theory of nonlinear partial differential equations, scientific computing, mathematical modeling, asymptotics, and experiments. Connections will be made with applications such as the role of surfactants in lung physiology, and the emerging field of microfluidics. The weekly Focused Research Group meeting, a central organizing feature of the project, combines research and education by promoting discussion and interaction across disciplines, and among faculty, students, and post-docs.

The central theme of this project is the study of how surface tension forces govern the motion of micro-scale fluids in many different settings. Progress in this basic research on thin liquid films will lead to new theoretical understanding of the physics of fluid flow on a microscopic scale, which is crucial to the emerging technology of microfluidic devices. These "labs on a chip" are transforming biotechnology in much the same way that microelectronics has transformed the computing industry. Contributions from this project will help to identify fluid flows and surface properties relevant to specific devices, as well as establishing the theoretical underpinnings for future research in this area. In addition to microfluidics, research will also be undertaken in the modeling of surfactant transport in the liquid lining of the lung. In developing new theory for this important physiological application, the research group will focus on how the surfactant changes surface tension, giving rise to a force that mobilizes the fluid coating of the lung walls. Development of theory, experiments, and numerical simulations in this project will involve ongoing interactions with scientists and engineers directly connected with the applications.

AST-0442037
Bertozzi

This project supports research in mathematical algorithms and implementations to represent, manipulate, and analyze data, specifically including feature detection, and registration and characterization of natural images. By building on a class of active contour models known as image snakes, the investigators will develop a fast method for object identification, to be used on large volumes of aerial photographs. Combined with higher order nonlinear PDE-based methods for resolving piecewise linear signals, and with shape and size constraints, the final system should be particularly efficient at locking onto predefined shapes. This interdisciplinary work draws on mathematical ideas from fluid dynamics and incorporates datasets from the intelligence community.

This award is supported jointly by the NSF and the Intelligence Community. The Approaches to Combat Terrorism Program in the Directorate for Mathematical and Physical Sciences supports new concepts in basic research and workforce development with the potential to contribute to national security.

VIGRE II at UCLA (2005-2010) builds on our first five years as an NSF VIGRE department. This large-scale program enhances education at all levels of Mathematics training while increasing the interaction of education and research activities.

The program has two related goals: (1) to introduce students as early and strongly as possible to research , and (2) to broaden the scope of activities to interactions between mathematics and other scientific and industrial fields. The specifics of the UCLA VIGRE program include: (a) a new summer interdisciplinary research internship program for graduate students, (b) new student research seminars coordinated with the NSF-sponsored Institute for Pure and Applied Mathematics at UCLA, (c) continuing summer research activities for undergraduate students (REUs), (d) a continuing post-doctoral program with special faculty mentoring of both teaching and research, and (e) a developing program to enhance the opportunities for teaching experience for graduate students. The VIGRE II program will support on average 22 graduate students, four VIGRE postdocs, seven summer graduate internships, and eight undergraduate REUs per year.

UCLA has a unique environment for advancing the careers of young researchers. State of the art research in applied mathematics, including image processing, materials science, and fluid dynamics is developed with full participation of a vibrant group of graduate students and postdocs. We propose to build on this program to provide an exceptional training and mentoring experience for younger students at the undergraduate and early graduate level. This program will support, over five years, up to 60 REU students from both UCLA and Harvey Mudd College. Proposed research projects include spatio-temporal pattern analysis in crime data, experimental and analysis of slurry flows, imaging processing, and cooperative control of robotic vehicles. An additional group of 15 PhD students, in the early stages of their career, will develop a research career in applied mathematics. These students will have the opportunity to participate in summer internship projects with faculty from other departments and with industry and national laboratories. Students will receive additional training in computation as well as mathematical analysis skills necessary to succeed in a top PhD program.

The investigators propose a research training program to enhance the workforce of US citizens and permanent residents entering research study applied and computational differential equations. This program is designed for undergraduate students and graduate students in the early part of a research career. They will work both one-on-one and in small research groups with faculty mentors on cutting edge research problems in an immersive research environment. This program will support, over five years, up to 60 REU students
from both UCLA and Harvey Mudd College. These students will have the opportunity to be part of a research team in a large research university. Proposed research projects include modeling of crime in Los Angeles, laboratory and mathematical studies of mud slides, medical imaging, and coordination and tracking of autonomous vehicles. In addition, up to 15 PhD students, in the early part of their careers, will participate in this program, which includes summer graduate internship projects with faculty from other departments and with industry and national laboratories. All students in the program will have the opportunity to participate in new curricular activities aimed at developing the mathematical and computational background necessary to succeed in a top PhD program in computational and applied mathematics.

This project will concentrate on the development of analytical and computational models of crime hot spot formation, persistence and dissipation. Crime hot spots are geographical areas with clusters of criminal offenses occurring within a specified interval of time. Hot spots may consist of clusters of property crimes such as burglaries or auto thefts, or violent crimes such as homicides, which occur on time scales ranging from hours to months. Mapping of crime hot spots is important in current approaches to understanding criminal offender behavior and is a tool used increasingly by police departments and policy makers for strategic crime prevention. However, despite the availability of sophisticated digital mapping and analysis tools there is a substantial gap in the understanding of how low-level behaviors of offenders lead to aggregate crime patterns including crime hot spots. Thus, for example, it is not possible to specify exactly why directed police action at crime hot spots sometimes leads to displacement of crime in space but, surprisingly, often can also lead to hot spot dissipation and a real reduction in crime incidences. Drawing on analytical methods in statistical physics, the mathematics of swarms, and new techniques in agent-based computational modeling, formal models of offender movement and target selection will be developed and simulated in different environments. These baseline models will be extended to consider offender behavior on abstract urban street networks and then integrate both model types with Geographic Information Systems (GIS) by exploring the spatial properties of simulated crime maps. Finally, at each stage of model development, empirical tests will be conducted against spatial crime data provided by the Los Angeles, San Diego and Long Beach Police Departments. The project will help clarify the quantitative relationships between criminal behavior, criminal opportunities and policing and may provide insight into how to design better crime prevention strategies, contributing to a broader dialog on homeland security. Simultaneous development of mathematical and simulation models, as well as continuous empirical testing, will provide a guide for the experimental use of these tools in the social sciences, while the broad interdisciplinary foundation of the project will provide a model for collaboration between mathematicians and social scientists.

The investigator plans a three year research program to develop algorithms for sensor systems for the detection of chemical and biological materials.
This work builds on prior research of the investigator and her colleagues involving autonomous mobile sensors for environmental sampling and algorithms for understanding hyperspectral imagery data. The research program involves the design of multiscale, multimodal sensing and detection algorithms, using data from both standoff detection and point detection from sensors mounted on mobile autonomous platforms. This data-intensive research depends on the modes of data available and their spatio-temporal resolution, viewpoints, and spectral resolution. The work includes the design and construction of a numerical simulator for the project, that incorporates various sensing modalities and on which algorithms are tested against against field data supplied by the government. In addition, mobile sensing algorithms are validated and tested at a laboratory multi-vehicle wireless testbed involving simpler sensors as a proxy for field sensor data. The research exploits recent algorithmic advances in image analysis and reconstruction from high dimensional data. These include, but are not limited to, compressive sensing methods, total variation minimization methods, hybrid wavelet-PDE algorithms for data fusion at different scales, hybrid geometric-stochastic algorithms for real time path planning and analysis, and nonlinear filtering.

The ability to detect and analyze biological and chemical threats in real time is essential to the future security of our country. Recent advances in sensor design now allow for rapid collection of information from multiple vantage points, involving multispectral sensing modalities. Where we are lacking is the ability to rapidly process and understand evolving information from diverse platforms to accurately identify and track the threat. This challenging problem requires new ideas for mathematical algorithm design to fuse the diverse data and provide accurate detection with both a low false alarm rate and detection delay. This research program develops new methods for high performance data processing and new fast algorithms for identification, in order to optimally utilize state-of-the-art and future sensor technology.

This award supports participant travel for the Workshop on Modern Perspectives in Applied Mathematics, held at the Courant Institute of the Mathematical Sciences, 18-22 May 2009. The conference brings together leading experts in stochastic and multi-scale modeling to discuss the extent to which recent advances have enabled further progress in the field and possible lines of attack for open problems.

Specific themes discussed at the conference include:
(1) Advances in stochastic partial differential equations, fluids, mean-field games, combustion, and partial differential equations in random media.
(2) Advances in stochastic modeling techniques for the elimination of irrelevant degrees of freedom in large dynamical systems and the derivation of effective systems for suitable coarse-grained variables.
(3) Advances in approaches for quantifying information flow in complex systems, such as Bayesian hierarchical modeling and reduced-order filtering strategies of Kalman type.

The program features keynote addresses, presentations by young researchers, and an evening poster session. The three-day meeting provides graduate students and young researchers with opportunities to present their work, to learn about recent progress and open problems, and to meet other workers in the rapidly-developing field.

Conference web site:
http://www.cims.nyu.edu/events/special/

This project develops detailed mathematical theory for aggregation models. Such models involve pair-wise interaction potentials and arise in the context of biological swarming models and control theory applications for coordinated groups and consensus. These models also appear in other contexts such as materials science applications including granular flow. The research in applied differential equations involves rigorous analysis of nonlinear partial differential equations, numerical simulation of continuum and discrete models, and asymptotic analysis and modeling. Specific problems of interest include (a) a detailed understanding of the dynamics of collapse in the case of kinematic aggregations; (b) the role of fractional dissipation in such models; (c) the discrete to continuum limit; and (d) analysis of scaling properties and behaviors of discrete swarms in the limit of large numbers. The work also involves related mathematical models for crime hotspots in residential burglaries.

The design and analysis of cooperative control and algorithms for autonomous agents is an active area of research with application to surveillance of hazardous areas, perimeter patrol, and control of teams of autonomous vehicles. Some ideas for such problems can come from modeling of biological groups that provide excellent examples in nature of how many agents can interact seamlessly, sometimes over large distances with relatively short range interactions. Many well-known models for these groupings empirically exhibit a great deal of complexity. This research program is fundamental to the understanding of these problems.

This multidisciplinary project aims to develop new mathematical methods, at the interface of the theory of nonlinear partial differential equations, statistical mechanics, graph theory, and statistics, for predictability and control of urban crime. The project focuses on spatio-temporal crime patterns and includes (1) new mathematical analysis and comparisons to crime data for discrete and continuum models of crime hotspots; (2) models with spatially embedded social networks, especially with regard to gang activity; and (3) exploration of new methods of Geographic Profiling, incorporating detailed features of urban terrain and more accurate modes of criminal movement into existing models. Mathematical work on this project includes analysis of nonlinear PDE models, analysis of statistical physics models, and further development of these models to include spatial heterogeneity, different offender movement patterns, and urban street gang networks. At the same time it provides both a deeper understanding of the mechanisms behind pattern formation in urban crime and some useful algorithms and software for local law enforcement agencies.

Mathematics of criminality is an emerging topic in applied mathematics with interest on a global scale and direct relevance to U.S. homeland security. This focused research group involves interactions between researchers whose primary expertise lies within very different fields -- mathematics, physics, anthropology, and criminology -- so that pattern formation of criminal activity is dissected and understood from very different viewpoints and perspectives. The project addresses algorithm development for analyzing real field data and agent-based simulation tools for urban crime. The research will also develop new models for urban crime and carry out mathematical analysis of these models. The project involves training of students and postdoctoral scholars at all levels, including a significant undergraduate component. Ph.D. students and postdoctoral scholars will also obtain valuable mentoring experience necessary for development of their research careers. The work includes direct interaction with local law enforcement agencies and the Institute for Pure and Applied Mathematics.

AWARD


Proposal Title: Real-time adaptive imaging algorithms for atomic force microscopy
Principal Investigator: Bertozzi, Andrea
Institution: University of California-Los Angeles
Proposal No: 0940417

This research project forges an interdisciplinary intellectual partnership between UCLAs Applied Mathematics program and Lawrence Berkeley Laboratorys Molecular Foundry, with the goal of designing transformative computation-based methods for real-time data acquisition and analysis in atomic force microscopy (AFM). The work combines expertise in (a) high-precision AFM instrumentation for imaging and force spectroscopy experiments, (b) the pioneering use of AFM to investigate the dynamics of oxidation, crystallization, and assembly of inorganic and macromolecular systems with (c) advanced algorithms for real-time mobile data acquisition and state-of-the-art image processing algorithms.

The research focuses on two case studies: Potassium bromide oxidation, an important process in understanding tropospheric chemistry, and S-layer protein array formation on lipid bilayers, an in vitro model of microbial membrane development. Both problems have dynamic behavior on a time-scale too fast for current AFM imaging technologies. The work involves state-of-the-art algorithm development involving compressive sensing, image inpainting, image segmentation and deblurring, combined with real-time tip steering using ideas from recent work in control theory and mobile sensors. In addition, to the new algorithm development, the project addresses new scientific results for the example problems, and a modular software package for control of the AFM sensor that could be adapted for diverse AFM imaging applications.

The research program involves the training of two graduate students, one in mathematics and one in microscopy, in cutting-edge interdisciplinary science. Additionally, undergraduate students are involved in algorithm software and hardware implementation. The impact on science is profound namely the ability to observe biological and chemical processes at higher speeds at the level of detail of AFM and to further increase the resolution and imaging power of existing AFM hardware technologies, through software and control methodologies. The Molecular Foundry is a user facility providing support to nanoscience researchers in academic, government and industrial laboratories around the world. Thus, advances made through this research program have an immediate user audience through the many researchers visiting the Foundry. Technology developed under this research program is also disseminated to commercial AFM manufacturers.

The current state of knowledge in the area of complex networked systems lacks formal theories that can accurately predict, and hence help control the behavior of such systems. The scientific objective of this project is to develop a theoretical basis to understand, analyze, and control emergent behavior in complex systems operating in uncertain environments. In order to validate the theoretical ideas, the project specifically considers complex swarm robotic systems and aims to apply the proposed analytical framework to such systems. The central approach uses the mathematical framework of stochastic differential equations wherein the swarm is treated as a dissipative Hamiltonian system coupled by nonlinear interacting potentials. In this framework, the influence of uncertainties (noise) is treated mathematically using both the discrete and continuum formulations to be obtained from a Fokker-Planck approach.

The focus of this research is the development of a theoretical basis for modeling of complex systems and their analysis based upon given control inputs and inherent system uncertainty. This framework facilitates modeling the complex interactions, higher-dimensionality, nonlinearity, and uncertainty in complex systems as well as the control of desirable emergent behavior. Potential future applications of this research include power grids and communication networks. Furthermore, this project has broader impacts that include promoting teaching, training, and learning; broad dissemination to enhance understanding; and involvement of the underrepresented groups. Undergraduate and graduate students are mentored and trained via courses and involvement in research and outreach activities. Broad dissemination is achieved through special sessions, workshops, and tutorials at conferences that target both the dynamic systems and control as well as mathematics communities. Participation of undergraduates and students from underrepresented groups is facilitated via summer projects, summer camps, and other programs such as Women in Science and Engineering and Emerging Ethnic Engineers.

The Gulf of Mexico Oil Spill is perhaps the most significant environmental disaster in US history with a significant portion of the US coastline affected by the approaching crude. Cleanup of beaches is important for both environmental reasons and for the tourist trade. This research program addresses fundamental questions of the dynamics of oil and sand of direct relevance to the cleanup problem. The work builds on recent studies of the Principal Investigator on mathematical models for oil-sand mixtures on slopes, including work on critical inclination angles for separation of oil-sand mixtures and modeling of shear-induced migration in oil/particulate films. The current research effort addresses such basic questions as whether there is a critical angle of incline for beach sand dunes that result in oil collecting in the dune vs. flowing to the bottom of the dune. The study of periodic patterns of shearing due to waves is also important for this research project.

The dynamics of particulates (e.g. sand) in oil is a complex process involving hindered settling dynamics and dynamics of the fluid such as shear. Recently the research group of the PI has found the dominant physics for particle-oil film mixtures on an incline and can explain quantitatively the bifurcation that occurs between regimes of particle settling downstream of the flow and clear fluid separating out from the flow. The equilibrium theory compares shear-induced migration due to the bulk flow properties with hindered settling due to gravity, and matches well with laboratory experiments. The current research program develops dynamic theories for particle-sand mixtures on inclines of particular relevance to the current crude oil spill. In addition to basic time-dependent flow problems the study considers oil-water mixtures and periodic time dependent shear such as what might result from wave motion and tidal forces on beaches.

The thrust of this Computational and Applied Mathematics program is engaging students starting and finishing the critical transition point from undergraduate to PhD student in high quality university level research. Students experience both the development of independent research projects and the milestones needed to get admitted to and succeed in a top PhD program. They participate in summer research modules on topics such as crime modeling, fluid dynamics experiments and modeling, robotics and control, medical imaging, cancer stem cells, bone growth, remote sensing applications, alcohol biosensors, photovoltaic cells, and algorithm design for microscopy. The program involves faculty from Mathematics in collaboration with faculty in Medicine, Anthropology, Engineering, Chemistry, and other disciplines. The project includes a training program for postdocs and junior faculty to learn how to involve pre-PhD students in publication-level research. The training program is based at UCLA and includes undergraduate and masters student participation from nearby colleges and universities.

The program goal is to directly address diversity and access to top level PhD programs in computational and applied mathematics. The proposed program provides
(a) summer fellowships for undergraduates to participate in research in computational and applied mathematics; (b) summer research fellowships for masters students from non-PhD granting institutions; (b) a summer traineeships for faculty from non-PhD granting institutions to gain experience supervising undergraduate research and to collaborate with research faculty based at UCLA; (c) first year PhD fellowships to recruit and train a more diverse group of PhD students (d) postdoctoral traineeships for recent PhDs with an interesting in mentoring undergraduates on research problems;
(e) summer research mentorships for postdocs to work with younger students on research modules.

The investigators intend to generate new and effective mathematical algorithms and methodologies in sensor systems for the detection of chemical and biological materials. Next, they intend to transfer this technology directly to those working towards reducing the threat to the homeland of biological and chemical attack. The new techniques they will use come primarily from information science, image science and physics, involving harmonic analysis, machine learning, optimization and partial differential equations. In particular they intend to provide useful algorithms for multi-component aerosol unmixing for active sensing using LiDAR and for mixtures of vapors in passive sensing. They will use ideas and algorithms recently developed, broadly speaking, from compressive sensing and L1 related optimization which were applied to hyperspectral imaging (recently used by Navy SEALS in the Bin Laden take down), unmixing, template matching, anomaly detection, clustering, change detection and endmember computation. They will improve relevant classical learning techniques, such as support vector machine, using their optimization techniques. They will also use ideas from machine learning with nonlocal means with prior information, in order to segment and identify objects in data collected from all sorts of sensors. Finally, they will factor in physics, such as plume dissipation, as part of the prior information needed to do spatial segmentation and identification.

The US government has been developing laser-based sensors for locating and classifying aerosols in the atmosphere at safe standoff ranges for more than a decade. There is a need to distinguish aerosols of biological origin from indifferent materials such as smoke and dust. Often, mixtures of aerosols are present and it is important to decide whether a threat exists. This project is intended to resolve data containing such a mixture into their separate components. Some success has already been obtained here by the investigators. This is an example of what this work concerns. A chemical and/or biological contamination might occur on the ground or in the air. The problem is to determine the presence of and concentration of chemical and biological threats and to track the dynamics of the cloud. The research done here is relevant to all the sensor modalities used in this type of threat detection. These include state-of-the-art LiDAR sensors, infrared radiometry and hyperpectral spensors. Plume tracking through the atmosphere is particularly important in a potential threat situation. The type of work proposed here is basic to our nation's security, given the threat posed by chemical and biological WMD's.

The research project synthesizes theory, computation and experiments to create multiscale models of dense particulate flows, ranging from microstructural dynamics of nearly contacting particles to the macrorheological properties of dense suspensions. The theoretical core of the project includes both mechanistic models of nearly contacting groups of particles, and continuum modeling and analysis of the dynamics of dense suspensions. The research develops new Stokesian Dynamics algorithms to resolve the slow particle interactions in dense suspensions, and uses simulations of how particles collectively reconfigure on the micro-scale to accommodate applied external strains to construct a continuum model for suspension rheology. New theory is supported by experiments carried out in the Applied Mathematics Laboratory at UCLA to test predictive models of suspension rheology.

Particle laden flows are important in many industrial applications including oil and gas extraction, coal processing, mining of minerals, and waste-water treatment. Results from this basic research project can be used to develop quantitative models for such diverse applications as spiral separators in the mining industry, cell separation in inertial microfluidic devices and aggregation of microbes. The project supports both graduate student research and undergraduate research involving experiments and theory.

Big data is important right now; and extreme scale hardware, programming models and storage solutions are being developed. However the connection between transformational algorithms and the big data sets needs to be addressed. Researchers talk often about the upcoming "big data problem," yet skim over the relevant algorithms that can truly attack the problems in a concrete fashion, often because the traditional means of data analysis are not prevalent in the high-performance computing (HPC) world, and in concert, the HPC expertise is not prevalent in the data world. This project will solidify this connection, making the "big data" problem real by attacking the issues through identifiable concrete algorithmic research and by implementing the methods on the latest hardware platforms. Recently the principal investigator has developed scalable desktop algorithms that bridge that gap, leveraging fast spectral solvers to compute solutions of sparse classification problems for big data. This project will build on these scalable algorithms to implement them on several large-scale platforms and will address important application areas, for which desktop computing is insufficient. Examples of application areas for this project include high dimensional hyperspectral video data for chemical and biological agents, a problem of importance to homeland security, and statistical analysis of spatio-temporal multimodal crime data, and large-scale social network analysis.

This project focuses on a new class of data-clustering algorithms that are designed to solve variants of the minimum cut problem on graphs for big data applications such as hyperspectral video data analysis, statistical analysis of spatio-temporal multimodal crime data, and large-scale social network analysis. Semi-supervised and unsupervised machine learning problems are included in the class of problems considered. The graph mincut problem is equivalent to total variation minimization on a graph and is a popular model for machine learning applications, except for its computational complexity. Building on ideas such as diffuse interfaces and dynamic thresholding, originally developed for physical sciences models and subsequently transferred to low dimensional image processing applications, this project will develop methods to solve the true graph cut problem by leveraging recent advances in scalable spectral graph algorithms. New codes for these methods will be developed for large parallel architectures. The research will advance both theoretical algorithmic issues and application areas.

Massive populations, or swarms, of low-cost autonomous robots have the potential to collectively perform tasksover very large domains and time scales, succeeding even in the presence of failures, errors, and disturbances. It is becoming feasible to create robotic swarms in practice due to ongoing advances in computing, sensing, actuation, power, control, and 3D printing technologies. In recent years, the miniaturization of these technologies has led to many novel robot platforms for swarm applications, including micro aerial vehicles. However, it remains a challenge to reliably control arbitrary numbers of such resource-constrained robots in unknown environments where global information and communication are limited or undependable. This research project aims to overcome this challenge by developing a rigorous framework for the scalable control of robotic swarms in realistic environments. The framework combines techniques from the fields of fluid dynamics, signal reconstruction, control theory, and optimization. This work provides a theoretically grounded approach for automatically programming robotic swarms to perform a diverse set of tasks of wide benefit to society, including environmental monitoring and exploration, disaster recovery, security operations, and even biomedical imaging and targeted cancer therapies at the nanoscale.

This project develops a formal methodology for analyzing and controlling the spatiotemporal dynamics of robotic swarms that are to be deployed in complex unknown environments. The designed robot control policies incorporate stochastic behaviors such as random encounters with environmental features and produce target collective behaviors within a specified degree of confidence. The confidence estimates are computed using a novel application of vortex methods, originally derived for fluid dynamic models and recently adapted to obtain continuum limits of discrete swarm models that incorporate pairwise interaction rules for maintenance of group structure. The control approach uses new computational algorithms for compressive sensing to reconstruct scalar environmental fields from sparse robot sensor data and to design efficient strategies for robot data collection. The methodology is demonstrated with a case study on designing control policies for micro aerial vehicles that are tasked to pollinate a crop field. Both computer models and testbed field experiments are used to validate theoretical predictions for the confidence estimates on system performance. Beyond robotics, the project provides analytical tools for a deeper understanding of the complex macroscopic behaviors of systems that can be represented with similar models, including non-well-mixed chemical reaction networks and natural swarms such as social insect colonies.

The US continued achievement at the forefront of science and technology requires a significant investment in new research in information technology to tackle the most challenging problems created by the vast data footprint created by digital recording of human activity. This project develops novel models and methods for forecasting human activity in time and space using sparse, heterogeneous data. The goals are very general and are focused on predicting and filling in missing data. An example of the type of data this project addresses would be a year's worth of geotagged Twitter data from a major city along with other informative geospatial information from that region. This project combines expertise of senior scientists in both Mathematics and Anthropology. The project develops analytical tools for understanding a diverse array of cyber-geospatial-temporal datasets. While focused on basic research, the project has tremendous potential to impact national security. This three-year project trains postdocs, graduate students, and undergraduate researchers. The mentees will be trained in research, in presentation of their work in written and spoken formats, with an emphasis on refereed journal publications and conference presentations. They will also be connected to future employers and will be given career advice throughout the length of their training.

The project focuses on information technology at the interface between large-scale cultural, social and behavioral processes and the situational conditions that lead to the expression of specific behaviors. This work extends a general conceptualization of text-based topic modeling to handle diverse collections of data types. The project develops methods to detect situational probabilistic effects through spatially-explicit topic modeling. One goal is to organize situational effects into different categories: (a) relatively stationary (e.g., the spatially discrete, but temporally stable role that the physical airport plays in driving airport related topics), (b) intermittent (e.g., discrete holidays) and (c) ephemeral (e.g., Foursquare). Another goal is temporal forecasting while a third goal is filling in missing information from a latent space. The research approach focuses on algorithms that are flexible enough to extend to a variety of datasets. The work interweaves several very useful models and algorithms for large data including self-exciting point process models for temporal information, soft topic modeling such as nonnegative matrix factorization and latent Dirichlet allocation for linear mixture models of data, hard clustering methods built around total variation minimization on graphs and graph Laplacians, and data fusion methods to combine these ideas in which latent space information is studied for forecasting and filling in missing information.

This award provides funding for a three-year REU site to support twelve undergraduate students for eight summer weeks each year to perform frontier level research. Student participants will learn to build mathematical models for problems from across the sciences. Research projects will focus on three areas: (1) Biology: how fungal spores cooperate to create networks; (2) Social sciences: mathematical patterns underlying urban crime; and (3) Physics: predicting the catastrophes that occur when industrial or geophysical slurries start to flow. Students will be recruited from outside the spheres normally reached by mathematics REU activities, including community college students and mathematics education students, and an existing university framework will be utilized to attract under-represented groups. Research activities are carefully designed to ensure that students with widely differing levels of experience are empowered to contribute vitally to each project. Additional outreach efforts will be made to enrich teaching of mathematical modeling in high school and middle school classrooms across California. Mathematics education students recruited to the REU site will create new lesson plans and teaching materials based on their research experiences and test them in real classrooms. These lesson plans will then be shared with teachers for wider use.

The modeling topics involve different application areas, including newly emerging interfaces between mathematics and fungal biology and between mathematics and criminology. Specific projects will include: (1) Analyzing spore cooperation and competition within microfluidic chambers, and modeling these interactions using evolutionary game theory; (2) Looking for scaling laws in the spatial and temporal patterns of gang violence in East Los Angeles; (3) Analyzing the flow of particle-laden fluid under gravity using the theory of shocks. REU participants will study a broad spectrum of modeling methods including differential equations, stochastic and agent based models, and scaling theory. All projects require that modeling be fused with real data, which in two of the projects will be collected by students themselves through experiments. Participants will receive training in model fitting, image analysis, scaling, data analysis, and error estimation. The program is designed to create immersive research experiences for students ranging from students early in their college careers to advanced undergraduates contemplating graduate school, as well as future expert teachers. A final goal of the program is the creation of field-tested learning materials based on the REU projects that will be taught to hundreds of middle school and high school teachers annually.

A confluence of technologies is transforming the biological, environmental, and social sciences into data-intensive sciences. Indeed, with the data now produced every day, there exists an unprecedented opportunity to revolutionize the journey of scientific discovery. By harnessing these data, one can advance the understanding of human conditions, behaviors, and their underlying mechanisms and social outcomes, enabling a spectrum of new and transformative research and practice. Fundamental new approaches across computing, mathematics, engineering, and sciences are critically needed, and future scientists must be accordingly trained in these emergent cutting-edge methods. This National Science Foundation Research Traineeship (NRT) award to the University of California, Los Angeles will address this demand by training graduate students at the intersections of data science, mathematics, cryptography, artificial intelligence, genomics, behavior science, and social science. The traineeship program anticipates training one hundred twenty (120) PhD students, including fifty (50) funded trainees, from the social, biological, mathematical and computational sciences and engineering, through a unique and comprehensive training opportunity.

This cross-disciplinary traineeship program has four research areas: genomics and genetics; brain imaging and image analysis; mobile sensing and individual behaviors; and social networks. These areas are interconnected through three core themes: mathematical modeling and network analysis, scalable machine learning and big data analytics, and biomedical applications and social outcomes. At the nexus of these research areas and core themes, this traineeship program provides novel interdisciplinary graduate education to advance both graduate student training and scientific research. Key features of the traineeship include novel curricula; cross-disciplinary laboratory rotations between engineering, life science, and social science; new foundational classes at the intersections of data science, mathematics, artificial intelligence, behavior science, and social science; summer internships at research institutes, big data firms, and hospitals and translational clinical settings; career, ethics, and technical communication skills development; and outreach to minority, women, and high school students with a distinct focus on groups traditionally underrepresented in STEM PhD programs.

The NSF Research Traineeship (NRT) Program is designed to encourage the development and implementation of bold, new potentially transformative models for STEM graduate education training. The program is dedicated to effective training of STEM graduate students in high priority interdisciplinary research areas through comprehensive traineeship models that are innovative, evidence-based, and aligned with changing workforce and research needs.

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.

The current pandemic of coronavirus disease 2019 (COVID-19) has upended the daily lives of more than a billion people worldwide, and governments are struggling with the task of responding to the spread of the disease. Uncertainty in transmission rates and the outcomes of social distancing, "shelter-at-home" executive orders, and other interventions have created unprecedented challenges to the United States health care system. This project will address these issues directly using advanced mathematical modeling from dynamical systems, stochastic processes, and networks. The mathematical models, which are formulated with the specific features of COVID-19 in mind, will provide insights that are critical to people on the front lines who need to make recommendations for intervention strategies and human-behavior patterns to best mitigate the spread of this disease in a timely manner. The project will train a postdoctoral scholar, a PhD student, and two undergraduate students in the research needed to solve these complex problems.

The standard approach for epidemic modeling, at the community scale and larger, is compartmental models in which individuals are in one of a small number of states (for example, susceptible, infected, recovered, exposed, latent), with individuals moving between states. The COVID-19 epidemic can be modeled in this way, with resistance as part of the dynamics. The simplest examples of such models for large populations are coupled ordinary differential equations that describe the fraction of a population in each of the states. To model the stochasticity of infection and latency, models with self-exciting point processes can be fit to real-world data. This project compares the dynamical systems and stochastic models of relevance to COVID-19 transmission. The models also incorporate network structure for the transmission pathways. The project extends prior research on contagions on multilayer networks by incorporating multiple transmission methods and coupling between the spread of the contagion itself and human behavior patterns. The project leverages high-resolution societal mixing patterns in epidemics, as they influence both (1) observations and demographics of who has been diagnosed with COVID-19 and (2) who transits the disease, sometimes without being diagnosed.


This award is co-funded with the Applied Mathematics program and the Computational Mathematics program (Division of Mathematical Sciences), and the Office of Multidisciplinary Activities (OMA) program.

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.

This project develops new mathematical algorithms and models involving knowledge graphs. A knowledge graph represents what is known about a subject in the form of labeled nodes and edges. More than simply labeled data, knowledge graphs organize data according to high-level meanings and assign globally unique identification to each node in the graph to match real-world entities. Much work on knowledge graphs treats databases and queries. In contrast, in the context of threat detection, this project focuses on algorithms that identify latent information in the graph and predictive models associated with data on the graph. The project will involve a combination of mathematical methods for subgraph isomorphism detection, time series analysis, agent-based and multiscale modeling, and pattern recognition. The project will train a postdoctoral scholar, PhD student, and six undergraduate researchers through involvement in the research.

This project brings together several different focused problems with large, multimodal, complex datasets. The data is organized into a knowledge graph in which additional information is added and absorbed as it becomes available. This project considers three types of knowledge graphs each for different applications: (1) knowledge graphs constructed from complex multi-part narratives; (2) knowledge graphs constructed from heterogeneous online content; and (3) knowledge graphs associated with large-scale human interaction dynamics such as a global pandemic. For (1), algorithms will be designed to identify important causal subgraphs. For (2), the project aims to identify threats in space and time based on templated patterns. For (3), desired goals are both a predictive ability for actions from a micro to macro scale along with tools to assess potential impact versus cost of preventative measures, from local to regional to country-wide scale.

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.