Steven Henry Strogatz - US grants

DMS_9500948 Strogatz The proposal is to study the nonlinear dynamics of oscillator arrays, using mathematical methods of dynamical systems theory, bifurcation theory, and perturbation theory, along with numerical simulation. One project concerns arrays of superconducting Josephson junctions, and will be carried out in collaboration with an experimental group at MIT. This work is aimed at understanding the phase-locking properties of series arrays; the propagation and instabilities of kinks in one-dimensional parallel arrays; and the behavior of vortices and row-switched states in two-dimensional arrays. A second project (in collaboration with an experimental group at Georgia Tech) will address the dynamics of laser arrays. Areas of investigation will include phase-locking in slightly detuned laser arrays; controlling chaos in laser arrays; and mutual synchronization of chaotic lasers, with possible applications to private communication. The Principal Investigator and his experimental collaborators will study the dynamics of superconducting Josephson arrays and laser arrays. Josephson arrays can be used as very sensitive detectors of magnetic fields (with biomedical and astronomical applications), voltage standards, power generators at submillimeter wavelengths (a part of the spectrum that is difficult to access by other technologies), and as ultrafast switching devices and transmission lines. All of these technologies would benefit from a deeper mathematical understanding of synchronization and vortex propagation in Josephson arrays. The goals of the laser array project are to find methods for improving the phase-locking characteristics of laser arrays (to increase their power output), and to develop novel encryption schemes for private communications, based on mutual synchronization of chaotic lasers.

This Integrative Graduate Education and Research Training (IGERT) award will support the establishment of a multidisciplinary graduate training program in Nonlinear Systems that emphasizes common mathematical and theoretical ideas that find expression in the analysis of natural and engineering systems. This activity is a joint effort of thirty-six scientists and engineers from the Departments of Aerospace Engineering, Agricultural and Biological Engineering, Applied and Engineering Physics, Biometry, Chemical Engineering, Economics, Electrical Engineering, Geological Sciences, Management, Mathematics, Mechanical Engineering, Neurobiology and Behavior, Operations Research and Industrial Engineering, Physics, Physiology, and Theoretical and Applied Mechanics at Cornell University. Joining them are scientists, engineers, and business scholars from Morgan Stanley, SUNY Health Science Center, Los Alamos National Laboratory, Georgia Institute of Technology, Hewlett-Packard, and the Boyce-Thompson Institute. Members of this group of scholars have established research collaborations; their collective expertise will provide the intellectual underpinning for the training of a diverse cadre of some sixty graduate students over the five-year tenure of the award. Doctoral students in this IGERT activity will be admitted to one of Cornell's existing Graduate Fields and will take an integrated two-semester course in nonlinear science that includes both deterministic and stochastic mathematical methods, and explore applications in various disciplines. These students must also complete the PhD requirements in their respective fields, take a minor in a different field, attend weekly colloquia and seminars, and complete a summer internship in a laboratory, hospital, Wall Street firm, or industrial setting as appropriate. The goal of the program is to educate students to be fluent in several disciplines and theoretical methodologies, all of which bear on the theme of nonlinear phenomena. IGERT is a new, NSF-wide program intended to facilitate the establishment of innovative, research-based graduate programs that will train a diverse group of scientists and engineers to be well-prepared to take advantage of a broad spectrum of career options. IGERT provides doctoral institutions with an opportunity to develop new, well-focussed multidisciplinary graduate programs that transcend organizational boundaries and unite faculty from several departments or institutions to establish a highly interactive, collaborative environment for both training and research. In this first year of the program, support will be provided to seventeen institutions for new or nascent programs that collectively span all areas of science and engineering supported by NSF.

Sources of on-line information are becoming increasingly decentralized, heterogeneous, and complex even as they become correspondingly richer and more valuable. Determining the structure of these information sources is becoming key to extracting and managing the knowledge they contain. Some of these sources exhibit an explicit network structure --- the hyperlinks of the World Wide Web form an excellent example. In other domains, ranging from electronic communication to informal human social networks, subtle hidden linkage relations play a large role in determining the information flow within and between communities. The link structures of both types of environments can yield a surprising wealth of latent information about their content, making their complexity manageable.

The proposed research seeks effective mechanisms for eliciting a global understanding of link structures in information networks. A key component of this effort is the design of techniques and tools enabling a richer level of interaction with on-line information. The research focuses on the development and integration of new techniques in three areas: natural language understanding methods to uncover implicit relationships in on-line content, efficient algorithms to analyze complex networks of inter-connections, and mathematical models of the dynamics and social processes by which networked information evolves.

Strogatz
0078074
In several branches of science and technology, one like
would like to imitate nature's success at designing networks that
can synchronize themselves. For instance, a semiconductor laser
array generates greater collective power when it synchronizes,
but such phase-locked operation is notoriously difficult to
achieve in practice. The investigator studies the nonlinear
dynamics of oscillator networks, using mathematical methods of
dynamical systems theory, bifurcation theory, and statistical
mechanics, along with numerical simulation. Three projects
explore how synchrony emerges in a group of dissimilar
oscillators, motivated by both biological and laser applications.
Areas of investigation include the stability of partial locking
in the Kuramoto model of coupled biological oscillators; the
dynamics of biological oscillators coupled by phase-response
curves; and phase-locking in slightly detuned laser arrays. Two
other projects address the relation between the connectivity of a
network and its ability to synchronize. Small-world networks,
which combine small diameter with large clustering, are
investigated to test whether they synchronize more readily than
lattices.
The goal of this project is to develop a deeper
understanding of complex systems that are made of many
oscillating parts and that manage to synchronize themselves. For
instance, the thousands of pacemaker cells in the heart always
fire in unison, even though they are all slightly different from
one another. Unfortunately, a similar kind of coordination
sometimes happens in the brain, where it leads to epilepsy. In
both cases, nature has provided us with examples in which
millions of cells begin to act in unison. By understanding
better how this synchrony is achieved, it should be possible to
design arrays of technologically important devices that can
synchronize themselves. Such self-synchronizing systems would
have important technological applications in many areas of
national interest, including environment (atmospheric pollution
monitoring uses sensitive detectors based on arrays of
oscillators), civil infrastructure (the proper functioning of the
national power grid depends on self-synchronization of the
generator network), and nanotechnology (where arrays of millions
of microscopic mechanical oscillators are being used in new
devices).

The Cornell University IGERT Program in Nonlinear Systems supports graduate education and research in the area of complex nonlinear systems. The research component of the program will be organized around interdisciplinary groups (IRTG) comprising faculty with expertise in theoretical, computational and empirical science, who will jointly mentor graduate student fellow projects. The research areas of the initial IRTG, including areas of applications, are (i) networks (social networks, gene networks, internet, electric power grid); (ii) gene regulation (cell signaling and gene expression networks); (iii) moving machines and organisms (manual dexterity and control of locomotion); and (iv) biological pattern formation (cardiac electrophysiology).

Nonlinear science has been a role model for interdisciplinary research. Principles arising from dynamical systems theory have revealed common features in seemingly unrelated phenomena across the breadth of science and engineering. The intellectual merit of this project lies in the extension of successful strategies employed in nonlinear dynamics to confront increasingly complex systems. A primary goal of the research is to understand how systems, especially those arising in the life sciences, can be more than the sum of their parts. For example, legged locomotion and manual dexterity will be studied through a combination of mechanical devices, observation of human and animal behavior and computer models. The broader impacts of this research will be in improving the performance of robots and the treatment of physical injuries. Another theme that will be explored is how network architecture influences dynamics of a system. The concept of small world networks, developed by the founder of this IGERT Program, Steve Strogatz and his students, has already influenced research on biological, social and communication networks. Applied to the internet, the results of this research facilitate efficient web searches. In general, the program will have broad impact in developing methods to predict the dynamics of complex systems, taking full account of underlying network structures and making extensive use of experimental data.

The primary mechanism of the IGERT program is the engagement of Ph.D. students in nonlinear systems research early in their studies. The program involves students in the conceptual phases of research, and it encourages faculty to develop long term collaborations, stimulated by their joint mentorship of students in the IRTG. The most direct impact of the program is in training a new generation of scientists with broad interests and expertise. In the words of a former IGERT fellow, "graduate students who go through the IGERT program learn to speak the language of two or more fields with considerable fluency, and all students are introduced to a common mathematical foundation so that even those who do not share the language of a specific field can interact meaningfully."

IGERT is an NSF-wide program intended to meet the challenges of educating U.S. Ph.D. scientists and engineers with the interdisciplinary background, deep knowledge in a chosen discipline, and the technical, professional, and personal skills needed for the career demands of the future. The program is intended to catalyze a cultural change in graduate education by establishing innovative new models for graduate education and training in a fertile environment for collaborative research that transcends traditional disciplinary boundaries. In this sixth year of the program, awards are being made to institutions for programs that collectively span the areas of science and engineering supported by NSF.

Strogatz

Many populations of biological oscillators can exhibit remarkable
collective behavior. Despite differences in their individual
natural frequencies, the oscillators spontaneously synchronize to
a common frequency. Examples include chorusing crickets,
fireflies that flash in unison, and synchronous firing of cardiac
pacemaker cells. In several other branches of science and
technology, one would like to imitate nature's success at
designing networks that can synchronize themselves. For
instance, a semiconductor laser array generates greater
collective power when it synchronizes, but such phase-locked
operation is notoriously difficult to achieve in practice. The
PI and his students study the dynamics of oscillator networks,
using mathematical methods of dynamical systems theory,
bifurcation theory, and statistical mechanics, along with
numerical simulation. One project investigates a peculiar
mathematical phenomenon that was recently discovered in a model
of cellular oscillators communicating with one another through
the exchange of a signaling molecule. Three additional projects
venture into areas where oscillator theory has rarely been
applied, namely molecular genetics, stochastic analysis, and the
interplay of human behavior and civil engineering. The goal in
each case is to answer a mathematically fascinating question that
is important in the real world. Specifically: (1) How can one
build the analog of a multicellular biological clock, using
synthetic gene circuits? (2) Can a precise biological clock be
made by synchronizing many imprecise components, and is this what
takes place in our own circadian pacemakers? (3) What caused
London's Millennium Bridge to wobble on opening day? Each of
these projects enlists the collaboration of a top experimentalist
in the relevant field.

Networks of oscillators arise throughout science and technology
and are ubiquitous in nature: lasers arrays work better when they
oscillate synchronously, structures such as buildings and bridges
can be endangered by synchronous oscillation, synchronous firing
of cardiac pacemaker cells keeps hearts beating. The
investigator mathematically explores the behavior of networks of
oscillators, studying how networks synchronize themselves.
Benefits are expected for the understanding of how rhythmically
active cells work together in tissues and organs; for the
characterization of the potentially dangerous ways that crowds of
pedestrians can inadvertently cause footbridges to shake; and for
spin-offs to technological applications involving arrays of
oscillators, such as lasers, microwave oscillators, and
superconducting Josephson junctions. By training three graduate
students through the research opportunities offered here, this
project also helps to develop human resources that are vital to
our nation's success in science and engineering.

Understanding the dynamical behavior of the Internet and other large networks is a daunting problem. Traditional approaches try to capture the Internet?s detailed behavior at a granular level, but then struggle with the system?s overwhelming complexity. At the other extreme, models that emphasize the large scale while ignoring fine details are more tractable, but frequently neglect features needed for practical engineering issues, especially when dynamical rather than static behaviors are an important concern. This research develops a happy medium through a novel fusion of analytical and computational techniques. Three important problems are explored. The first concerns the overall efficiency and stability of the Internet; the second develops methods for improving the functioning of sensor networks (large arrays of small sensors that can be used to monitor environmental changes); and the third focuses on improving the quality of large cooperative computing systems by providing incentives for users to behave in ways that benefit the whole.

The potentially transformative aspect of this research lies in its ?computational-analytic? approach to creating effective models for a wide range of complex networks. In this strategy, computational techniques are interlaced with analytical ones, using such methods as ?equation-free modeling?? from chemical engineering and numerical renormalization from statistical physics. The approach is radically interdisciplinary, with the power to unify and extend multi-scale analytical methods adapted from such diverse fields as dynamical systems theory, mathematical physics and theoretical biology. Spin-offs of the techniques created here are expected and will be investigated for other important problems involving complex networks, including routing on the Internet, synchronization in neural networks, and stability in global financial networks.

Abstract
Ramakrishna

There is a dearth of American women and underrepresented minorities (African-American, Latino-American and Native American) completing doctorates in the mathematical sciences overall, and even fewer completing their degrees at top tier programs. Our proposal, the Summer Mathematics Institute at Cornell (SMI), is aimed at underrepresented groups, but will be open to all talented students who do not have access to a full range of advanced mathematics courses at their undergraduate institutions. Each year we will run an eight-week summer program for 12-14 students. Students will take a rigorous course in algebra or real analysis, with the subjects alternating yearly to allow the possibility for some students to attend the program twice. We expect that most of the first time participants will be in between their junior and senior years, although we will also consider finishing sophomores who have already had a course in that year's topic. Students will also participate in a research project, working in groups of 3-5. While some students may do original research, we imagine that in most cases students will be engaged in the process of rediscovering results that have already been published.

To excel in a top tier research oriented doctoral program graduate students must negotiate the transition from undergraduate mathematics problem solving to engaging in the abstract reasoning required in graduate course work and learn to communicate mathematical ideas. Failure to make this transition inhibits performance in both course and seminar work and is a serious obstacle to success in graduate studies. A thorough grounding in the core subjects of real analysis and algebra is essential for pure and applied mathematicians. Inadequate preparation significantly hinders entering and completing graduate school. This situation is particularly salient in the case of minority students, many of whom begin graduate study after coming from colleges with limited resources to teach advanced math courses. Indeed, we have seen students at Cornell experience this difficulty in both our pure and applied programs. SMI will facilitate these students' transition to and success in graduate school by broadening their knowledge of mathematics, strengthening their understanding of basic concepts, enhancing their communications skills, and providing an experience of mathematical research and learning what to expect in a challenging doctoral program. SMI participation will both improve students' chances of gaining admission at a top tier graduate program and their chances of success upon enrolling in graduate school. As we anticipate that perhaps 3-5 students in each class will attend SMI for two summers, we expect that over _ve years we will train approximately 55 students. Ultimately, SMI will help widen the talent pool from which American mathematics will draw.

This research training group focuses on the interface of dynamical systems, probability theory, and partial differential equations. The project provides training opportunities for undergraduate, graduate, and postdoctoral participants and aims to further the creativity, leadership, and professional development of postdoctoral fellows and faculty. The project includes outreach to local K-12 schools; research experiences for undergraduates, both during the academic year and summer, including partnerships with faculty at institutions that serve many students from groups underrepresented in the mathematical sciences; summer schools for graduate students and postdoctoral fellows; and a communication seminar. The program especially emphasizes communication, both because of its importance in reaching the wider public, and for its vital role within mathematics itself. Because probability, dynamics, and partial differential equations straddle the divide that often exists between pure and applied mathematics, these subjects provide an excellent training ground for effective communication.

Dynamical systems, probability theory, and partial differential equations have rich interconnections and span the entire spectrum of mathematical activities from theory to applications. However, such interconnections are rarely explored in undergraduate or graduate education. This project aims to highlight these connections, which have proven so fruitful at the research frontier, and thereby to broaden the perspective of the next generation of mathematicians. The research activities will introduce students to problems relevant to societal and environmental issues. The emphasis on communication will enhance the professional development of participants, whether they find themselves working in education, industry, or research. The summer schools and research experiences for undergraduates will attract eligible participants nationwide. The educational modules created as part of the outreach and modeling activities will be available online to schools, universities, and the general public. The outreach activities will invite high school students to join in the excitement of mathematical discovery, while giving graduate students an opportunity to act as mentors.

The researchers propose to create a center of data science for improved decision-making that combines expertise from computer science, information science, mathematics, operations research, and statistics. Their goal is to pursue basic research that will contribute to the theoretical foundations of data science. The research topics chosen have applications that can benefit society as a whole and integrate the perspectives of the disciplines that the project brings together. The five concrete research directions proposed are: Privacy and Fairness, Learning on Social Graphs, Learning to Intervene, Uncertainty Quantification, and Deep Learning. The aim of the Center is to advance knowledge in these areas and to broaden the range of disciplines and perspectives that can provide contributions to these challenging issues. The researchers plan to incorporate the community beyond Cornell through online seminars, workshops, and student conferences.

The research findings will provide an urgently needed foundation for data science in several topic areas of importance to society. As the center is placed at the intersection of multiple disciplines, the intellectual merit spans all disciplines involved and findings may translate to new algorithms and approaches in each one of them.

The research focus spans five core areas.

1. Privacy and Fairness. As data science becomes pervasive across many areas of society, and as it is increasingly used to aid decision-making in sensitive domains, it becomes crucial to protect individuals by guaranteeing privacy and fairness. The investigators propose to research the theoretical foundations to providing such guarantees and to surface inherent limitations.

2. Learning on Social Graphs. Many of the fundamental questions in applying data science to the interactions between individuals and larger social systems involve the social networks that underpin the connections between individuals. The researchers will develop new techniques for understanding both the structure of these networks and the processes that take place within them.

3. Learning to Intervene. Data-driven approaches to learning good interventions (including policies, recommendations, and treatments) inspire challenging questions about the foundations of sequential experimental design, counterfactual reasoning, and causal inference.

4. Uncertainty Quantification. Quantifying uncertainty about specific predictions or conclusions represents a key need in data science, especially when applied to decision-making with potential consequences to human subjects. The researchers will develop statistical tools and theoretical guarantees to assess the uncertainties of predictions made by popular algorithms in data science.

5. Deep Learning. Deep Learning algorithms have made impressive advances in practical settings. Although their basic building blocks are well understood, there is still ambiguity about what they learn and why they generalize so well. There are indications that they may learn data manifolds and that the type of optimization algorithm influences generalization.

Advances in our theoretical understanding of these phenomena requires combined efforts from optimization, statistics, and mathematics but could lead to insights for all aspects of data science.

Funds for the project come from CISE Computing and Communications Foundations, MPS Division of Mathematical Sciences, MPS Office of Multidisciplinary Activities, and Growing Convergent Research. (Convergence can be characterized as the deep integration of knowledge, techniques, and expertise from multiple fields to form new and expanded frameworks for addressing scientific and societal challenges and opportunities. This project promotes Convergence by bringing together communities representing many disciplines including mathematics, statistics, and theoretical computer science as well as engaging communities that apply data science to practical research problems.)