Mary Catherine Silber - US grants

9410115 Silber The investigator will develop equivariant bifurcation theory for applications to symmetry-breaking bifurcations in hydrodynamic systems and in globally coupled oscillator arrays. The first part of the research focuses on translation symmetry breaking in hydrodynamic systems that are extended in more than one space dimension. Both the situation where translation symmetry is broken spontaneously by an attracting solution of the governing equations, and the situation where it is broken externally by perturbing the equations will be considered. In the former case, the manifestations of the underlying translation symmetry in the possible evolution of instabilities of the symmetry-broken state will be investigated. In the case of external symmetry breaking, the research focuses on the specific physical system of rotating Rayleigh-Benard convection; this is a system for which it has already been established that breaking translation symmetry can have a profound effect on certain instabilities. The second part of the research project consists of a group theoretic/dynamical systems analysis of the $S_n$-equivariant Hopf bifurcation problem. This bifurcation problem is pertinent to the dynamics of n globally-coupled identical limit cycle oscillators. This research is motivated by recent analytic and numerical studies of series arrays of Josephson junctions. Global coupling represents an important limiting case for the coupling of oscillators; this aspect of the research project should contribute to our understanding of oscillator arrays ubiquitous in the modeling of physical, chemical, and biological systems. ***

9404266 SILBER The goal of the project is to develop equivariant bifurcation theory for applications to pattern formation and for applications to globally coupled oscillator arrays. The proposed research will be carried out within the framework of finite-dimensional center manifold normal forms of bifurcation problems. The symmetries of these bifurcation problems will be exploited in their analysis and in the interpretation of the results. Special attention will also be given to developing geometric methods that exploit integrable Hamiltonian limits of the bifurcation problems. Symmetries play a central role in the evolution of pattern-forming instabilities in hydrodynamic systems. The primary goal of the proposed research is to develop mathematical methods that exploit their presence. The research on pattern formation will aid the interpretation of large-scale numerical studies of three-dimensional magnetoconvection, and laboratory studies of electroconvection in nematic liquid crystals. The research on globally coupled oscillators will contribute to our understanding of the dynamics of series arrays of Josephson junctions.

9502266 Silber This work is supported by a National Science Foundation Faculty Early Career Development Award. The research component will focus on the further development of mathematical methods of equivariant dynamical systems for application to specific physical problems. These include hydrodynamic models of rotating convection and of magnetoconvection, models of Josephson junction arrays, and models applicable to combustion or solidification problems in various two- and three- dimensional geometries. In each case, the focus is on instabilities for which there is clear evidence that symmetry plays a central role. An essential component of the proposed research is to compare theoretical results, based on model- independent group theoretic/dynamical systems methods, with those obtained by direct numerical simulation of the governing differential equations. The education component will include course development in applied dynamical systems at the undergraduate level, curriculum development at the graduate level, graduate student advising, mentoring, and organizing graduate seminars. The National Science Foundation strongly encourages the early development of academic faculty as both educators and researchers. The Faculty Early Career Development (CAREER) Program is a Foundation- wide program that provides for the support of junior faculty within the context of their overall career development. It combines in a single program the support of quality research and education in the broadest sense and the full participation of those traditionally underrepresented in science and engineering. This program enhances and emphasizes the importance the Foundation places on the development of full, balanced academic careers which include both research and education.

The Department of Engineering Sciences and Applied Mathematics
of Northwestern University propose to purchase a Hewlett Packard
multiprocessor graphics workstation and two X-terminal stations. The
equipment will be used for scientific computation for the following
research projects:

* Numerical simulation of viscous sintering
* Numerical studies of frequency conversion and light propagation
in optical systems
* Complex Structures in Pattern-Forming Systems
* Superlattice wave patterns

The principal investigator is D. Chopp. Co-principal investigators are
G. G. Luther, H. Riecke, and M. Silber. Northwestern University has
agreed to contribute 50% of the cost of the equipment.

9972059
Silber

The research project addresses the formation of free surface standing wave
patterns that arise when a container of fluid is vibrated vertically. This
well-studied hydrodynamic problem represents a model system for
studying pattern formation via symmetry-breaking parametric instability.
Understanding the dynamical mechanism for the experimentally observed
"superlattice patterns" is a primary focus for much of the research. These
two-dimensional wave patterns are characterized as being spatially-
periodic, with structure on two disparate length scales. The analysis will
be based on methods of equivariant bifurcation theory. The role of normal
form symmetries, and spatial/temporal resonance in the pattern selection
process will be elucidated by the analysis. Aspects of the research project
will contribute to our understanding of the effects of weak external
symmetry breaking on equivariant bifurcation problems. It will also
develop applications of some recent mathematical results on bifurcation of
periodic solutions with spatio-temporal symmetries. Finally, possible
symmetry-based methods for controlling spatio-temporal patterns will be
investigated.

The research project is aimed at a basic mathematical understanding of a
variety of experimental observations of exotic wave patterns on the
surface of a vibrated liquid. The results are expected to carry over to other
pattern-forming systems of technological interest, such as magnetic fluids
and nonlinear optical systems. The research will contribute to our basic
understanding of the effects of vibration on nonlinear systems with many
degrees of freedom; vibration that may be detrimental because it causes
instability or useful because it suppresses instability. The research will
also address control of the nonlinear pattern-formation process in certain
hydrodynamic, nonlinear optical and chemical reaction-diffusion systems.
The training of graduate students in applied mathematics is an integral part
of the research project.

9987577
Stephen Davis - Northwestern University
IGERT: Dynamics of Complex Systems in Science and Engineering

This Integrative Graduate Education and Research Training (IGERT) award supports the establishment of a multidisciplinary graduate training program of education and research on the dynamics of complex systems. Nonlinear science has increasing impact on many disciplines in the natural sciences, engineering, and medicine, and requires training that bridges traditional departmental and school boundaries. This graduate training program emphasizes the unity of fundamental concepts underlying a broad range of scientific research areas: nonlinear optics, computational neuroscience, pattern formation, chaos, ergodic theory; and applications to engineering and materials science problems: interface motion, combustion, and mixing. The goal is to prepare the students for today's rapidly evolving professional environment. Students will be equipped with the tools and intuition needed to tackle complex nonlinear problems arising in various guises and technical fields. Cross-disciplinary research and communication skills will be developed through an intensive year-long course, in which small graduate-student teams will investigate a topic guided by two faculty members with complementary perspectives. This early research experience will be followed by a thesis on a different topic; in the case of IGERT fellows the thesis project will be co-advised and cross-disciplinary. Internships will provide additional training experience. Cross-departmental research seminars and student-run seminars, regional workshops, yearly retreats, and an active visitor program will foster a highly cooperative, diverse, cross-disciplinary training environment.

IGERT is an NSF-wide program intended to meet the challenges of educating Ph.D. scientists and engineers with the multidisciplinary backgrounds 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 new, innovative models for graduate education and training in a fertile environment for collaborative research that transcends traditional disciplinary boundaries. In the third year of the program, awards are being made to nineteen institutions for programs that collectively span all areas of science and engineering supported by NSF. The intellectual foci of this specific award reside in the Directorates for Mathematical and Physical Sciences; Engineering; Computer and Information Science and Engineering; and Education and Human Resources.

Proposal: DMS-0309667
PI: David Mary Silber [m-silber@northwestern.edu]
Institution: Northwestern University
Title: Temporal and Spatio-Temporal Forcing of Oscillatory and Excitable Systems

ABSTRACT

The investigator, together with students and colleagues, studies three problems in which temporal or spatio-temporal forcing of oscillatory or excitable systems is important: (1) parametrically excited surface wave patterns, (2) spatio-temporal local feedback in pattern forming systems, and (3) Hopf bifurcation based mechanisms for amplification of sound by inner ear hair cells. Faraday waves, excited on the free surface of a fluid, form in a wide variety of patterns depending on the fluid properties and the form of the periodic forcing function. The investigator's research program focuses on a bifurcation analysis of three- and four-wave interactions when a periodic sequence of delta-function impulses is applied to the fluid container. This idealized forcing function admits unprecedented analytic progress to be made in the linear and weakly nonlinear regimes that apply at or near onset of instability. This project probes how the periodic forcing function may be designed to favor particular patterns. In the second research project spatio-temporal feedback is used to probe the nonlinear pattern formation process, as well as to actively control it. The control of spatio-temporal patterns by local time-delayed and spatially-transformed feedback will be investigated through linear stability analysis, equivariant bifurcation theory, and numerical simulation. In the third project, models of inner ear hair cells, responsible for translating sound-induced motion into electrical signals, are analysed. The initial focus is on amphibian hair cells, for which two separate mechanisms that contribute to the cells' frequency selectivity have been identified - one due to active mechanical motions of the hair bundle and the other captured by an electrochemical model of ion channels in the hair cell body. In each model proximity to a Hopf bifurcation contributes to the amplification properties of the hair cells. The investigator's research project uses dynamical systems methods to derive a reliable reduced model, from existing detailed physiological models of the two Hopf bifurcation mechanisms, with attention to the effects of this two-stage amplification on gain and frequency selectivity. This project lays the foundation for further investigation of the effects of coupling the hair cell bundles.

Many spatially extended nonlinear systems, including hydrodynamic and laser systems, exhibit spatio-temporal chaotic behavior when subjected to external forcing. The investigator's research program will lead to a deeper understanding of how to eliminate irregular behavior in favor of spatio-temporally regular patterns. This is done through appropriate design of the temporal forcing function in the case of hydrodynamic waves, or through spatio-temporal feedback in the case of nonlinear optical and chemical systems. Careful comparison between theoretical results and results of experimental investigations will be made, providing valuable feedback to this research effort. The investigator's analysis of biophysical models of inner ear hair cells contributes to a greater understanding of how the nonlinearities in two proposed mechanisms of frequency selectivity and amplification might work together to achieve greater gain. The training of applied mathematics graduate students and postdoctoral fellows in interdisciplinary research activities is an integral part of the research effort.

Proposal: DMS - 0709232
PI: Silber, Mary
Institution: Northwestern University
Title: Bifurcation theory and delay equations: applications to controlling pattern formation and modeling protein translation

ABSTRACT

The proposed research addresses two applications of bifurcation theory
and delay differential equations: (1) autoadjusting feedback control
of oscillatory patterns, and (2) mathematical modeling of cellular
protein translation. The proposed research on controlling patterns
investigates an autoadjusting feedback control scheme aimed at
stabilizing oscillatory patterned-states. The feedback control method
exploits symmetries of the targeted pattern in such a way that it
becomes noninvasive when control is achieved. Two related case studies
will be pursued: (A) stabilization of traveling plane wave solutions
of the two-dimensional complex Ginzburg-Landau equation in the
Benjamin-Feir unstable regime, and (B) control of chemical traveling
wave patterns of the photo-sensitive Belousov-Zhabotinsky reaction in
the oscillatory regime. The mathematical relationship between these
two case studies will be elucidated by the proposed analysis. The
proposed research on mathematical modeling of protein translation is
aimed at deriving, by systematic approximation, a delay equation model of
protein translation that could then be used as a component in simple
models of synthetic gene networks involving more than one protein. The
delay model is obtained from a continuum description of the elongation
process, which ultimately shows up as a delay time in the reduced
mathematical model. The proposed research will extend the model to
incorporate the degradation of mRNA. The fidelity of the delay model
to the mechanistic one, in the case of simple gene switches and
oscillators will then be investigated using bifurcation theory, aided
by a numerical continuation package that was developed for delay
differential equations.

The proposed research will contribute to the training of graduate
students and postdoctoral fellows in interdisciplinary, applied
mathematics research. It will aid the development of feedback control
schemes for eliminating spatio-temporal chaos in chemical
reaction-diffusion systems, as well as other pattern-forming systems.
Delay differential equations frequently arise in the modeling of
biological processes, such as cellular protein translation, the
process whereby ribosomes assemble proteins, one amino acid at a time,
using the information encoded in the messenger RNA (mRNA). The
proposed research will contribute to the development of a systematic
mathematical framework for deriving reduced delay models from complex,
biologically-detailed mechanistic models. The proposed projects
represent important applications of delay differential equations and
their analysis using bifurcation theory. The analysis of these
proposed case studies are essential to the development of these
mathematical and computational tools.

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).

The PI will spend the academic year 2009-2010 in the Climate Group in the Geophysical Sciences Department at the University of Chicago. Primary goals of the immersion year include (1) gaining a scientific overview of key physical components of the climate system, including how they interact, their characteristic temporal and spatial scales, and how they are modeled; (2) gaining experience with paleoclimate proxy data, including how they are obtained, how they are interpreted quantitively, their uncertainties and their quality; (3) developing research collaborations with climate scientists at the University of Chicago aimed at understanding the role of coupling and feedback in climate change. To ensure a broad overview of key components of the climate system, the PI has identified three research project areas that interface with the research activities of faculty and their students in the Geophysical Sciences Department at the University of Chicago. These projects concern (1) mathematical modeling of the ice component of the climate system, which is important to understand for predicting sea level rise due to global warming, (2) mechanisms of abrupt climate change in the last glacial cycle that are related to thermohaline circulation in the deep Atlantic, which would be constrained by new paleoclimate proxy records of North Atlantic sea-surface temperature and salinity changes, and (3) investigations of system level climate models of extreme glaciation events in earth's distant past including models of `Snowball Earth'.


Additional activities include (1) expanding the PI's participation in the Climate Group journal club, which she joined in January 2009, by leading some of the journal club discussions and proposing topics of investigation that relate to her mathematical expertise in dynamical systems and bifurcation theory, (2) attending weekly departmental seminars to gain a broad perspective on geophysical sciences research, (3) auditing graduate courses on `Climate Dynamics' and `Global Climate Models', and (4) attending a number of professional meetings and workshops related to climate science. Expected outcomes of the immersion year include (1) developing sustainable interdisciplinary research collaborations with leading climate scientists, (2) identifying relevant projects for applied mathematics Ph.D. research, and (3) developing a seminar course on `Challenges of Modeling the Climate and Climate Change' aimed at Northwestern University advanced undergraduate and graduate students.

Jones, DMS-0940363
Camp, DMS-0940267
Danforth, DMS-0940271
Fung, DMS-0940272
Golden, DMS-0940249
Holland, DMS-0940241
Kostelich, DMS-0940314
McGehee, DMS-0940366
Pierrehumbert, DMS-0940261
Silber, DMS-0940262
Tung, DMS-0940342
Zeeman, DMS-0940243

The investigators form a "Mathematics and Climate Research
Network." This is a framework for an intensive effort aimed at
bringing to bear the full power of modern applied mathematics and
statistics on the prediction and understanding of the Earth's
climate. The investigators focus on three key themes: (1)
Dynamics of Climate, (2) Climate Process Modeling, and (3) Data
Analysis and Data Assimilation. Dynamics of Climate addresses
critical climate processes and their interactions. Climate
Process Modeling undertakes the modeling of climate components
that have been underrepresented in extant climate models, such as
the multi-scale material structure of sea ice. Data Analysis and
Data Assimilation develops mathematical tools for analyzing
climate data and assimilating them in current climate models.
The Research Network aims to be a national resource, with
participants at thirteen U.S. universities. The investigators
work together as a virtual community that holds regular weekly
"webinars" and working meetings over the Internet. This
multi-year effort is expected to help in defining a research area
of "climate mathematics" and in educating a new generation of
mathematical researchers to meet the scientific challenges
associated with a changing climate.

This project is driven by the need to better understand the
Earth's climate system. Climate is the result of many
geophysical and chemical processes in the Earth's atmosphere,
oceans and biosphere. These processes evolve in time over many
scales, ranging from minutes to centuries, and interact in
multiple ways, most often nonlinearly. Feedback mechanisms, many
of which are poorly understood, further complicate the picture.
Because there is only one Earth, climate cannot be studied by
systematic experimentation; the only approach available to
climate researchers is through computational experiments. These
experiments are based on mathematical models, which must be
simple enough not to exceed the capabilities of today's advanced
computer architectures, while still incorporating the physical
and chemical processes that are essential for realistic climate
outcomes. The expertise of mathematical scientists in designing,
assessing and interpreting these models is critical. This
"Mathematics and Climate Research Network" helps engage the
mathematical sciences community to address the mathematical and
statistical issues of our changing climate. The Research Network
takes full advantage of current information technology;
communication and collaboration among the participants takes
place mostly remotely over the Internet.

The investigator and her students will contribute to mathematical modeling efforts for desert ecosystems, which may be vulnerable to desertification under climate change. These models address the underlying mechanisms behind the spatial patterns of vegetation biomass that occur when certain semi-arid ecosystems are stressed by decreased precipitation. Interfacing self-organizing spatial effects for vegetation with the enormous variability of precipitation that define drylands suggests new directions for fundamental pattern formation research. The research will also contribute a mathematical framework for testing the robustness of the ecological proposals that the spatial patterns may serve as early warning signs of tipping points associated with climate change.

The research will be developed within a framework of deterministic and stochastic mathematical ecological models, with its objectives enhanced by the availability of rich satellite data that can be used to test model predictions. The first objective is to compare, qualitatively and quantitatively, a class of pattern-forming reaction-diffusion vegetation models to determine the robust transitions between distinct vegetation pattern states that may occur as the model system approaches its trivial desert state. This analysis will be based in bifurcation theory. The second objective is to develop and analyze vegetation models with temporally variable precipitation inputs, and spatially variable drainage networks. This will lead to the development of stochastic models with non-Gaussian noise. Model reduction methods will be used to further determine the consistency between model frameworks. Satellite image data of dry-land ecosystems will inform and verify the modeling effort. The project also has an education component with the goal of training undergraduate and graduate students in interdisciplinary applied mathematics research.

The investigator and her students will contribute to mathematical modeling efforts for desert ecosystems, which may be vulnerable to desertification under climate change. These models address the underlying mechanisms behind the spatial patterns of vegetation biomass that occur when certain semi-arid ecosystems are stressed by decreased precipitation. Interfacing self-organizing spatial effects for vegetation with the enormous variability of precipitation that define drylands suggests new directions for fundamental pattern formation research. The research will also contribute a mathematical framework for testing the robustness of the ecological proposals that the spatial patterns may serve as early warning signs of tipping points associated with climate change.

The research will be developed within a framework of deterministic and stochastic mathematical ecological models, with its objectives enhanced by the availability of rich satellite data that can be used to test model predictions. The first objective is to compare, qualitatively and quantitatively, a class of pattern-forming reaction-diffusion vegetation models to determine the robust transitions between distinct vegetation pattern states that may occur as the model system approaches its trivial desert state. This analysis will be based in bifurcation theory. The second objective is to develop and analyze vegetation models with temporally variable precipitation inputs, and spatially variable drainage networks. This will lead to the development of stochastic models with non-Gaussian noise. Model reduction methods will be used to further determine the consistency between model frameworks. Satellite image data of dry-land ecosystems will inform and verify the modeling effort. The project also has an education component with the goal of training undergraduate and graduate students in interdisciplinary applied mathematics research.

This project is an interdisciplinary Research Training Grant (RTG) on Quantitative Biological Modeling. Postdoctoral fellows, graduate students, and undergraduates will gain experience with mathematical modeling, numerical methods, and modern data analysis/statistical tools relevant to biological problems. Trainees will perform individual research in an interdisciplinary environment focusing on the mathematical modeling of current problems in experimental and computational biology. Each trainee will benefit from having two mentors, one from applied mathematics and one from a biological discipline. Trainees will gain knowledge and experience beyond that of their own research areas via regular group meetings organized around specific biological themes, and research internships outside Northwestern University will broaden trainee perspectives. Graduate students and postdoctoral fellows will gain pedagogical experience through creation of new course materials in collaboration with participating faculty. Trainees and undergraduates will also receive coaching in written and oral communication as part of the program. A major goal of this project is to expand the preparation of applied mathematics students and postdoctoral fellows for interdisciplinary research in the life sciences and for their subsequent careers.

Research performed as part of this project will focus on mathematical models in fields including neuroscience, developmental biology, evolutionary biology, and ecology. Models will encompass state-of-the art mathematical, computational, and statistical data-analysis methods appropriate for these application areas. New high-throughput experimental techniques have produced a wealth of data from biological systems in recent years, and major efforts are now underway to integrate these datasets to understand fundamental biological mechanisms and functionality. Within an interdisciplinary environment, trainees participating in this project will analyze imaging, sequencing and other data to address pressing biological questions. Co-mentorships and programmatic oversight of students by interdisciplinary faculty, creation of new courses, and development of a new vertical and peer teaching structure will greatly expand undergraduate, graduate, and postdoctoral training in applied mathematics in the high impact area of biological modeling. This will lead to a new generation of mathematicians, cross-trained and attuned to the particular needs of the field. The project will further connect faculty across Northwestern University and other research institutions, strengthening research progress in the focus areas.

Data science is making an enormous impact on science and society, but its success is uncovering pressing new challenges that stand in the way of further progress. Outcomes and decisions arising from many machine learning processes are not robust to errors and corruption in the data; data science algorithms are yielding biased and unfair outcomes, as concerns about data privacy continue to mount; and machine learning systems suited to dynamic, interactive environments are less well developed than corresponding tools for static problems. Only by an appeal to the foundations of data science can we understand and address challenges such as these. Building on the work of three TRIPODS Phase I institutes, the new Institute for Foundations of Data Science (IFDS) brings together researchers from the Universities of Washington, Wisconsin-Madison, California-Santa Cruz, and Chicago, organized around the goal of tackling these critical issues. Members of IFDS have complementary strengths in the TRIPODS disciplines of mathematics, statistics, and theoretical computer science, and a proven record of collaborating to push theoretical boundaries by synthesizing knowledge and experience from diverse areas. Students and postdoctoral members of IFDS will be trained to be fluent in the languages of several disciplines, and able to bridge these communities and perform transdisciplinary research in the foundations of data science. In concert with its research agenda, IFDS will engage the data science community through workshops, summer schools, and hackathons. Its diverse leadership, committed to equity and inclusion, proposes extensive plans for outreach to traditionally underrepresented groups. Governance, management, and evaluation of the institute will build on the successful and efficient models developed during Phase I.

To address critical issues at the cutting edge of data science research, IFDS will organize its research around four core themes. The complexity theme will synthesize various notions of complexity from multiple disciplines to make breakthroughs in the analysis of optimization and sampling methods, develop tools for assessing the complexity of data models, and seek new methods with better complexity properties, to make complexity a more powerful tool for understanding and inventing algorithms in data science. The robustness theme considers data that contains errors or outliers, possibly due to an adversary, and will design methods for data analysis and prediction that are robust in the face of these errors. The theme on closed-loop data science tackles the issues of acquiring data in ways that reveal the information content of the data efficiently, using strategic and sequential policies that leverage information gathered already from past data. The theme on ethics and algorithms addresses issues of fairness and bias in machine learning, data privacy, and causality and interpretability. The four themes intersect in many ways, and most IFDS researchers will work in two or more of them. By making concerted progress on these fundamental fronts, IFDS will lower several of the barriers to better understanding of data science methodology and to its improved effectiveness and wider relevance to application areas. Additionally, IFDS will organize and host activities that engage the data science community at all levels of seniority. Annual workshops will focus on the critical issues identified above and others that are sure to arise over the next five years. Comprehensive plans for outreach and education will draw on previous experience of the Phase I institutes and leverage institutional resources at the four sites. Collaborations with domain science researchers in academia, national laboratories, and industry, so important in illuminating issues in the fundamentals of data science, will continue through the many channels available to IFDS members, including those established in the TRIPODS+X program. Relationships with other institutes at each IFDS site will further extend the impact of IFDS on domain sciences and applications.

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.