Christopher Peterson, PhD - US grants

Abstract

98-16091
Peterson

Collaborative Research: NO3-N Retention in Headwater Streams: Influences of Riparian Vegetation, Metabolism and Subsurface Process

Human activities have introduced massive quantities of nitrogen to both terrestrial and aquatic environments with detrimental effects on forests, rivers, and estuaries. Consequently, it is important to understand how and where natural processes retain and transform nitrogen as it moves across the landscape. Our research will focus on how headwater streams function as sites for retention of nitrogen with emphasis on microbial processes that return nitrogen to the atmosphere. The program design compares headwater streams that differ in their terrestrial settings and the degree with which streams interact with groundwater. Study sites in New Mexico, Tennessee, and North Carolina will be established to compare the influence of forest type (open or dense) and subsurface processes (streams on bedrock vs streams that exchange water with subsurface sediments) to assess how the biota of stream ecosystems effectively retain nitrogen and how these processes vary across the landscape.

abstract

The investigator and his colleagues study small rank vector bundles on projective spaces. New and simplified constructions are obtained both in finite and in zero characteristics. In the case of rank two bundles, these constructions on projective four space are valid only in positive characteristics. The investigator and his colleagues work on the question of extending these constructions to characteristic zero. This is related to the deformation theory of these bundles. They investigate whether these bundles can be deformed from positive to zero characteristic. The deformation theory of such bundles also has applications to questions regarding degenerating sums of line bundles and the existence of exotic components of the Hilbert scheme.

The investigator and his colleagues give explicit constructions of objects called vector bundles. Vector bundles are devices which encode algebraic information about huge numbers of geometric figures like curves and surfaces. With their explicit knowledge of vector bundles, the investigator and his colleagues can then construct geometric figures with desired properties. The work is done using matrices which are easily implemented on computer algebra systems. Much of the work is done over finite fields which allows the use of computers to give exact answers. Problems involving geometric modeling in the real world require approximating the answers, one method being the use of such finite fields. The project studies the interplay between geometric objects existing in the real world (over fields of characteristic zero) and their approximations over finite fields.

ABSTRACT
PI: Michael Kirby
proposal: 0434351

This proposal develops new algebro-geometric tools for data analysis, and in particular, for the investigation and understanding of massive data sets arising in computer science, where both the ambient dimension and number of samples may be large. Particular examples of interest to the team include brain-computer interface (BCI) problems and computer-vision and recognition problems. Specifically, the team is relating secant varieties (of arbitrary order) to the determination of optimal projections of data sets; they inquire what order of variety is appropriate and are investigating the relationship between varieties of different orders. Secant varieties also play a role in understanding canonical forms and decompositions of higher-order tensors, which in turn are of fundamental interest in signal-processing applications such as the BCI problem. Algebraic geometry provides a framework for developing algorithms for higher-order tensors. Finally, very recent developments in the Schubert calculus on Grassmannians have dramatically increased the potential applicability of this theory. The investigators are studying how to exploit these ideas to develop efficient algorithms for finding near-optimal projectors subject to constraints.

The research, which this team is carrying out, has distinctive impacts on the mathematical and computer science communities. By bringing together groups of researchers from seemingly disparate areas of expertise, it aims for cross-fertilization among these fields. Computer science provides the problems for which the team is developing new algorithms. Conversely, the team expects new mathematical conjectures to arise from the practical problems being addressed by this research. Furthermore, the graduate students (all working towards the Ph.D.) will receive innovative training during the course of this project that will prepare them for jobs in either academia or industry where they will have tools and preparation essentially unlike any of their peers. This should greatly enhance their ability to contribute new and innovative ideas to the team research environment. Beyond this effect on the professional communities, the applications driving the proposed research have clear and immediate impact on aspects of such disparate issues as national security and the broadening of the ability of disabled individuals to participate more fully in society.

Some of the most exciting developments in modern computational science have resulted from exploiting ideas in areas of mathematics not traditionally associated with numerical computation. Conversely, numerical techniques have been applied to solve computational problems arising in "non-traditional" fields. A good example is the fruitful interaction between computational mathematics and algebraic geometry. Singularity theory, which builds on ideas of algebraic geometry, has been embraced by computational scientists to compute paths of critical points in multi-parameter systems of differential and partial differential equations. Symmetries and group actions are used to create numerical methods for specific types of problems with significantly improved accuracy and stability properties. Techniques in algebraic geometry are also very useful for finding solutions of differential equations on manifolds, and are currently being applied to develop algorithms to compute decompositions of higher order tensors. On the other hand, numerical techniques for continuation, homotopy and symmetry provide the basis for methods in numerical algebraic geometry that are used to compute solution components of systems of polynomial equations. The ability to carry out the numerical decomposition of polynomial systems has yielded applications in mechanical engineering including the understanding and design of mechanisms that transmit, control, or constrain relative motion, robotics, control theory (pole placement), integer programming, and statistics. The development of hybrid exact/approximate methods for finding solutions of polynomial equations gives rise to issues of errors and stability that confront numerical analysts in many other contexts.

The Workshop on geometry and symmetry in numerical computation will bring together experts from computational mathematics and algebraic geometry in order to explore and develop the potential in this rich interdisciplinary area. The program has been planned specifically to introduce and attract students and young investigators to this area. Each session will begin with an introductory lecture followed by four talks by leading experts. The introductory speakers will prepare a short "guide" describing some basic language and results that the audience can use during the invited talks. The one hour lectures themselves will be aimed towards an audience of advanced graduate students and researchers from different areas of mathematics. We expect the Workshop to break down disciplinary barriers and encourage cross-fertilization between researchers from algebraic geometry and numerical analysis. The lectures and discussion sections will encourage students and young researchers to become involved in the intersection of algebraic geometry and numerical analysis, and will provide stimulation and support for those already engaged in this activity. Potential outcomes range from improved methods to compute large complex physical systems governed by systems of partial differential equations, to advances in computational methods for general relativity, to new geometric methods for the analysis of large data sets, and to more efficient numerical methods for robotics and control.

The main goals of this research involve removal of nitrate (NO3) from stream water in a bacterially mediated process called denitrification where NO3 is converted to biologically inactive nitrogen gas. Specifically, the work investigates this denitrification process in films of algae and bacteria referred to as biofilms. This research will investigate when biofilms in streams and rivers are of significance in removing nitrates, how the effectiveness varies with the algal and bacterial species composition and species diversity, and under what conditions the nitrate removal by biofilm is maximized.

Input of dissolved nitrogen into aquatic systems, via agricultural runoff and atmospheric deposition of products of fossil-fuel combustion, has increased dramatically over the last century. High nitrate (NO3) content in groundwater poses significant health risks, and elevated NO3 loads discharged into the ocean from rivers are implicated in the formation of ''''dead zones'''' in coastal areas. Thus, understanding the process of this nitrate regulation and removal by bacteria is highly important.

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

The PIs will study higher secant varieties of classically studied varieties such as Segre varieties, Grassmann varieties, and Segre-Veronese varieties. These varieties correspond to parameter spaces for rank one tensors, alternating tensors, and hybrids of regular tensors and symmetric tensors, and their (closed) higher secant varieties correspond to compactifications of the parameter spaces for higher rank tensors. The main goal of the research is the classification of defective secant varieties of Segre varieties, Grassmann varieties and Segre-Veronese varieties. This is analogous to the celebrated theorem of Alexander and Hirschowitz, which asserts that higher secant varieties of Veronese varieties have the expected dimension (modulo a fully described list of exceptions). This work completed the Waring problem for polynomials which had stood for some time as an outstanding unsolved problem. There is a corresponding, conjectural complete list of defective secant varieties for Segre varieties and for Grassmann varieties. The first component of the project is on the refinement of existing methods and the development of new theoretical and algorithmic methods towards the solution of this classification problem. The second component of the project is concerned with decomposition of tensors.

In many applications, it is natural to represent a collection of data as a multi-indexed list. Alternatively, one can think of the data as a multidimensional array (sometimes called a multi-way array). For example, a digital grayscale picture can be stored as a matrix of numbers where each pixel location in the picture corresponds to a location in the matrix and the number in the matrix corresponds to the darkness of the pixel. In a similar manner, a digital color picture can be stored as a three dimensional array of numbers. A mathematical framework that includes the study of multi-way arrays, and their representations as sums of more basic objects, is through parameter spaces of tensors. This project explores problems related to tensors, tensor decomposition, tensor rank and tensor border rank from an algebro-geometric point viewpoint. These subjects have significant applications in fields as diverse as signal processing, data analysis, computational biology, combinatorics, algebraic geometry and statistics. It is the expectation, therefore, that techniques developed through this research will advance our knowledge and understanding across multiple disciplines.

1067751/1067439
Gray/Kelly

Society is seeing the rapid transition of nanotechnology as it moves from discovery to commercialization.
This revolution in atomic and molecular engineering promises many environmental and human health benefits such as dramatic improvements in efficiency, reduced resource use and waste production, and astounding improvements in medical diagnostics and therapeutics. Yet, the risks posed by nanotechnology to ecological and environmental health have not been rigorously assessed, and without these data a meaningful regulatory framework to protect human and environmental health and safety and guide the development of nanomaterials cannot be formulated. The defining characteristic of nanomaterials (NMs) is their size (at least one dimension of 100 nm or less), which falls into a transitional zone between individual atoms and molecules, and bulk materials. The small sizes, novel shapes, and high surface areas promote unusual and novel physicochemical properties to NMs and make possible nearly infinite possibilities for surface functionalization, targeted reactivity and robust material development. These very features, however, open unimagined opportunity for engineering, scientific and medical applications may also pose threats to human health and ecosystem integrity. At this time it is virtually impossible to make predictions about NM environmental fate and impact. Subtle changes in size, shape and surface functionality have profound effects on chemical and physical behavior. Furthermore, the protocols to screen and then, interrogate systems at the mechanistic level and to determine dose effects rigorously are lacking. The purpose of the proposed research is to study fundamental interactions of a representative nanomaterial (various forms of nanotitania) in biological and environmental systems at increasing scale, from subcellular through ecosystem, in order to develop a testing and measurement strategy that comprehensively characterizes the ecotoxicity of nano-scaled TiO2.
The applicants propose a three-year collaborative research project that involves environmental engineers and scientists from Northwestern University (NU) and ecologists Loyola University of Chicago (LUC). This project combines the unique capabilities of nanomaterial synthesis and characterization that exist at NU with those at LUC in the field of ecology that will allow us to elucidate the effects of NM on the structure and function of benthic ecosystems. The proposed work is based on a long running collaboration among NU and LUC researchers and will be performed in novel experimental facilities well adapted to interrogating NM effects in biological systems at multiple scales.

In view of the collaborative strengths and established infrastructure surrounding this project, the intellectual merit of the proposed study promises to make critical advances in understanding the relationships between NM characteristics, environmental fate and biological consequence and to probe the mechanistic basis for ecosystem responses to NM exposure. With the ability to synthesize state-of-the-art TiO2 NMs that are currently the more widely used nanoscale materials and are likely to be used even more extensively in the future, this research will improve the characterization of the environmental, health and safety aspects of NM by detailing the physical and chemical properties of the nanotitania under study. This will be accomplished by measuring the fate, transport and environmental stability of NM under relevant conditions, and conducting ecotoxicological testing at multiple scales.

The broader impacts of the results of this study will not only inform the systematic evaluation of NM health and safety and serve as a model for developing the necessary scientific basis for meaningful policy formulation, but they will also illustrate a strategy to supply critical feedback to the design and production of environmentally-safe NMs. This research promises transformative insights that parallel the technology revolution of nanotechnology, itself and will help to guide NM development along routes of reduced ecotoxicity. They propose to move beyond the identification of biomarkers associated with single component responses (e.g., reactive oxygen species) to discover meaningful NM definitions based on properties not simply size and bioindicators that signal complex chemical and biological interactions at the system level. The results of this research will illustrate the feasibility of comprehensive ecotoxicological testing, but will also reveal the extent to which screening results are predictive of system level response. The broader impacts of this proposed collaborative research are also strongly connected to education at the pre-college, undergraduate, graduate and post-graduate levels, as well as to community level outreach. The research team will promote interdisciplinary exchange among ecologists, environmental engineers, molecular biologists and analytical chemists. Both NU and LUC have strong commitments to undergraduate education and actively involve undergrads in research. A strong mentoring environment exists at both NU and LUC and will nurture the scientific development of a diverse team of students.

This proposal concerns the development of new geometric algorithms for detecting and classifying threats from airborne biological agents and chemical agents. The investigators propose a mathematical framework centered on encoding massive data sets associated with streaming hyperspectral imagery as points on special manifolds, e.g., as representations on Grassmann, Stiefel as well as flag manifolds. In this setting, algorithms will be developed for computing descriptive statistics. A well-known example of such an algorithm was introduced by Karcher to compute the mean of a set of points on a Grassmann manifold. The investigators are primarily concerned with developing new algorithms with improved computational properties on Grassmannians, as well algorithms that can be applied to, e.g., Stiefel and flag manifolds. These algorithms will be designed to be applied to very large data sets in real time and will be evaluated using temporally-evolving hyperspectral data sets made available by the Defense Threat Reduction Agency. These include (but are not limited to) data acquired using a Fabry-Perot Interferometer and Frequency Agile Lidar.

The proposed interdisciplinary research program addresses a major challenge related to National Security, i.e., identifying and assessing chemical and biological risks in the environment from observational data in real time. New mathematical tools for exploring massive quantities of chemical and biological hyperspectral data are proposed to assist with threat detection and characterization. A primary goal of the research program is to apply these tools to exceed performance capabilities of current techniques used for classification of biological and chemical threats. It is anticipated that the results of this research program will be useful to other applications related to National Security such as detection of anomalies in data beyond hyperspectral imagery.

This proposal concerns the development of theory and algorithms for knowledge discovery in large data clouds characterized on Grassmann and Flag manifolds. This work includes applications to the detection of geometric misclassifications as well as the development of algorithms to exploit recent work on the detection of spatially-correlated Gaussian time-series. The investigators propose a mathematical framework centered on computing statistics for data clouds on Grassmann and Flag manifolds. This includes, for example, a theoretical characterization of a Schubert Variety of Best Fit. The results of the research will be applied to data sets that include, e.g., automatic identification of insects, the brain computer interface, statistical signal processing, foliage diversity in landscapes, automatic identification through hyperspectral imagery, acoustic arrays, super-resolution, and action recognition in video sequences.

The proposed interdisciplinary research program addresses a major challenge in research related to the processing and extraction of meaningful information from large collections of data. The investigators' propose new geometric and statistical tools for classifying patterns of interest. The research program provides students with a unique multidisciplinary experience and research integration in education. Goals of the research include optimizing the detection, characterization and classification of features and anomalies in signals. A Data Set Repository of a broad nature for the purpose of furthering national expertise in Geometric Data Analysis. This repository will facilitate the development of algorithms ofinterest to a range of scientific interests.

This proposal concerns the development of new mathematical algorithms for detecting and classifying threats in large data sets arising from the presence of, e.g., biological agents or chemical plumes. The investigators propose a geometric framework centered on encoding massive data sets on subspace manifolds. Exploiting the fact that a set of points of a given class can be represented as a low-dimensional subspace of a high dimensional ambient space, it is possible to capture more variability in the threats and thus characterize them with higher accuracy. There are many ways to represent data via subspaces, each leading to a rigorous notion of a manifold, e.g., the Grassmann and Stiefel manifolds. The investigators propose to use the geometry of these manifolds, either as abstract points or via constructing embeddings in Euclidean space, for representing patterns in threats. Detection and classification algorithms originally proposed for vector spaces may now be extended to algorithms over subspace manifolds.

The proposed interdisciplinary research program addresses the detection and classification of chemical and biological threats, a major challenge for National Security. Threats delivered to an urban environment or military theater, are potentially comprised of unknown substances and the goal is to detect, characterize and track the threat. Alternatively, threats may be associated with known substances and the goal is to not only detect but classify the actual material or agent by matching it to a library of signatures of known threats. The basic research to be performed will be evaluated in the context of data sets made available by the Defense Threat Reduction Agency. These include (but are not limited to) data acquired using a Fabry-Perot Interferometer, Frequency Agile Lidar, and Raman Spectroscopy. The research will be led by faculty from the Departments of Mathematics and Computer Science at Colorado State University, providing the students with unique multidisciplinary experience with research integration in education.

This research concerns the development of innovative mathematical theory and algorithms to facilitate knowledge discovery in the massive data sets generated by scientists, engineers and today's data driven society. New approaches will be developed that permit the encoding of large quantities of data in a way that enables the detection of similarities and differences buried in the volumes of information. The framework is especially useful for characterizing degrees of similarity, and discovering features or patterns that may be shared between data sets. The project focuses on the use of tools from geometry and optimization to provide effective data representations that expand the toolkit of analysts and enhances their capacity for understanding large and complex data sets. The methodology will be validated on real world data sets like extreme weather simulations or biological data sets such as those capturing the human immune response to infection by pathogens. The techniques being developed may be viewed as part of the emerging field of geometric data learning.

The mathematical approach exploits the geometric framework of the Grassmannian, the manifold that parameterizes the set of subspaces of a given dimension of a vector space. The appeal of this approach is that subspaces, as abstract points on the Grassmann manifold, are an effective tool to capture the natural variability in data observations stemming from, for example, variations in illumination, or noise. If a subspace of data intersects another subspace of data in some prescribed number of dimensions, then these abstract points should be considered to be more related than subspaces that intersect in fewer dimensions, or not at all. This type of geometric picture, when formulated in a mathematical framework, leads to the use of flag manifolds and Schubert varieties for representing and comparing data. The proposed research program addresses new problems in data driven optimization subject to geometric constraints, for example, when the feasible set is a Schubert variety. This framework allows us to extract geometric models that characterize patterns, and leads naturally to comparisons between large sets of observations based on similarity measures which are functions of angles between subspaces.

In many fields of engineering and applied science the problem is to extract relevant information from a signal or image. Certainly this describes the problem of identifying cyber attackers in data networks, spotting objects of interest in closed-circuit TV records, and classifying anomalies in medical images. The goal of this project is to develop a signal processing theory for detecting and classifying images that have undergone geometric transformations. Practical and topical examples are medical features viewed in variable magnification and orientation, and images of people in arbitrary orientations in crowded scenes. The solution to classification problems under such imaging conditions will advance medical practice and national defense. As a broader impact, the project prepares students for careers in mathematics and electrical engineering, with an expertise in signal processing and imaging science.

This project develops a theory of matched manifold detectors, based on a universal manifold embedding that extracts a subspace basis from an image. The basis itself codes for the coordinate transformation of the image, but its span is invariant to the transformation. Consequently the extracted subspace is an invariant statistic for detection, and the basis is a covariant statistic for the parameters of the transformation. Classification is then a problem of subspace matching on a Grassmann manifold, and identification of coordinate transformation is a problem of analyzing a subspace basis in a Stiefel manifold. We aim to adapt this theory to other problems where a transformation group turns out an orbit of images, all of which are to be classified as equivalent. The objective is to develop a theory of signal processing on manifolds that is as broad in its scope and as precise in its methodologies as modern subspace signal processing. Such a theory will augment statistical reasoning with geometrical reasoning, and bring new mathematical methods into play.

This research program concerns the development of new mathematical algorithms for identifying critical information in data sets that has the potential to reveal biological or chemical threats to humans. The starting point for this work is the observation that modeling a threat over variations in its appearance serves as the foundation of robust detection algorithms. Mathematical tools from geometry and topology enable the design of algorithms for extracting information from sets of data that enhance traditional processing methods, leading to smarter sensors. The graduate students trained in this program will earn doctorates in mathematics while the undergraduates will have the opportunity to be mentored on topics of national interest early in their careers.

This research addresses the development of mathematical modeling algorithms for data on abstract manifolds. In contrast to data clouds captured by sensors generating points that exist as a configuration in Euclidean space, this project encodes data as points on matrix manifolds. The mathematical framework requires only three features: the geometric context, the notion of distance between two points, and the ability to compute the perturbation of one point in the direction of another. Additionally, one can exploit information related to the statistics of points on matrix manifolds. This research program aims to explore the geometry of these spaces for the purposes of anomaly detection and characterization.

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.

Increased global demand for plastic products in recent decades has greatly increased the amount of plastic waste entering the environment. Most plastics are persistent and will enter the environment at the end of their life cycle if not recycled or reused. Up to 94% of manufactured plastics end up in landfills or in the environment. In the environment, large plastic materials can be broken down through physical and chemical processes into small particles (0.1 to 5 mm). These microplastics can persist in the environment for long periods of time where they can fundamentally change how organisms interact with each other and with their physical environment. These changes can, in turn, change how microplastics move between ecosystems, and their fate in the environment. This has drawn the attention of scientists and environmental managers around the world. The goals of this project are to advance understanding of how ecosystems and communities of organisms respond to the presence of microplastics, and to provide information in support of better management of plastic pollution and problems arising from it. Moreover, this project will provide undergraduate and graduate students with research and career training in the fields of science, technology, engineering, and mathematics. This project will also provide opportunities for talented low-income high school students to participate in research and gain an understanding of the potential environmental impacts of plastic pollution, thus, enhancing a sense of environmental stewardship.

This research will employ innovative designs and approaches developed based on observations and results of previous research to address two main objectives and test four detailed hypotheses. Objectives include, 1) characterize the movement of microplastics across terrestrial and aquatic ecosystems and 2) characterize the influence of algal colonization and biofilm development on microplastics surfaces on particle deposition rate, zooplankton feeding behavior and selectivity, phytoplankton community structure, primary productivity, and nutrient availability in freshwater systems. Hypotheses include, 1) deposition rates of suspended microplastics ingested and egested by aquatic organisms will be faster than natural abiotic depositional processes; 2) microplastics are transferred from aquatic to terrestrial ecosystems via emerging aquatic insects ingested by birds; 3) algal colonization and biofilm development on microplastic surfaces will increase deposition rates of suspended microplastics, alter phytoplankton community structure and primary productivity, and decrease nutrient availability in freshwater ecosystems; 4) in the presence of biofilm-covered microplastics, zooplankton will change feeding locations to areas of high concentrations of biofilm-covered microplastics and selectively feeding on them.

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.