Mathematics

Edward E. Allen
with Jacquelyn Fetrow and David J. John, COMPUTER SCIENCE

Algebraic and Statistical Models of Redox Signaling
Awarded $123,379 for the period 4/1/08 to 3/31/09
Source: National Institutes of Health (NIH)

An interdisciplinary research group spanning the Reynolda and Health Sciences campuses aims to develop theory, algorithms, computational tools, and research methodologies for network modeling of redox-regulated events in human cells. Recent research indicates that redox-regulated networks are central to cellular communication under a variety of normal and diseased conditions, including cancer, neurodegenerative diseases, and aging. This project will (1) identify a comprehensive set of cellular proteins modified at cysteine residues as a result of redox-dependent signaling; (2) correlate the concentration of a given cellular perturbant and its associated redox signal; 3) associate networks with particular perturbants; and 4) produce both topological and dynamic models of the cellular network associated with these pathways. These models will then be compared to other data on protein/protein interactions and kinase cascades to produce a more comprehensive model of cellular regulation and its biological outcomes.

Kenneth Berenhaut
with Errin Fulp, COMPUTER SCIENCE

Modeling Mobile Agent Populations and Movement for CEDS
Awarded $196,473 for the period 12/13/10 to 10/31/13
Source: Pacific Northwest National Laboratory (PNNL)/Battelle Memorial Institute

Mathematical models will be used to clarify the basic performance trade-offs associated with agent populations, pheromone strength, and nonregular geographies. As more advanced features are added to the system, simulation will be used to find appropriate settings. Modeling will also be used to characterize and assure the operational security of the Digital Ants system.

Jennifer Erway

  • Collaborative research: Trust-search methods for inverse problems in imaging
    Awarded $149,857 for the period 7/1/13 to 6/30/16
    Source: National Science Foundation (NSF)

This research uses linear algebra and optimization theory to develop software for processing and analyzing very large data sets. Results will be especially useful for such image-processing applications as medical imaging, low-light video surveillance, and nocturnal ecological activity monitoring where the data very large and noisy.

  • Optimization Methods for Solving the Einstein Constraint Equations
    Awarded $5,000 for the period 6/1/09 to 5/31/10
    Source: Oak Ridge Associated Universities (ORAU)

The prestigious and highly competitive Ralph E. Powe Junior Faculty Enhancement Award aims to enrich the research and professional growth of young faculty in a wide variety of fields from over 120 member universities across the USA. This research program aims to develop and implement large-scale optimization methods for solving the Einstein constraint equations. Up to now, attempts at a solution have relied on ad hoc approaches and Newton methods with restarts. This project will tailor unconstrained minimization work to solve the inequality-constrained optimization problem implicit in these equations using state-of-the-art nonlinear optimization techniques. It will also develop a class of preconditioners to help solve the linear system arising from each discretization.

  • Second-order Methods for Large-scale Optimization in Compressed Sensing
    Awarded $49,659 for the period 4/5/10 to 6/30/11, Year 3
    Source: NSF

This 3-year research program aims to develop and implement large-scale optimization methods for recovering sparse signals from a limited number of indirect observations in the context of compressed sensing. Compressed sensing theory states that sparse signals can be recovered very accurately with high probability from indirect measurements by solving an appropriate minimization problem. This emerging area allows data acquisition with fewer measurements and sampling rates in such fields as medical imaging, astrophysics, biosensing, and geophysics. Building on second-order methods for large-scale unconstrained optimization, the project will develop significantly more efficient methods for solving compressed sensing minimization problems. The algorithms are designed to be matrix-free; they do not require storage of potentially very large second-derivative matrices but use matrices only as operators for matrix-vector products. The research advances state-of-the-art methods for compressed sensing and second-order line-search and trust-region methods for bound-constrained optimization improving large-scale applications, such as image processing, as well as theoretical proofs of convergence and numerical stability of the algorithms.

Ellen Kirkman 

  • Study of undergraduate programs in the mathematical and statistical sciences in the United States and publications of the results
    Awarded $53,846 for the period 8/1/15 to 9/30/17
    Source: National Honor Society (NSH)/ American Mathematical Society (AMS)

Dr. Kirkman will perform a comprehensive stratified random sample survey of the nation’s undergraduate mathematical and statistical sciences programs at two- and four-year institutions and write a comprehensive report available in spring 2017. Coordinated by the Conference Board for the Mathematical Sciences (CBMS), the survey tracks changes in the curriculum, pedagogy, enrollment levels, graduates, and faculty. This iteration focuses on: (1) evidence-based teaching practices; (2) distance-learning course practices; (3) instructional strategies in elementary statistics; (4) instructional strategies and delivery methods in developmental mathematics; (5) program assessment; and (6) full-time faculty in four-year departments who are not eligible for tenure.

  • Invariant Theory of Artin-Schelter Regular Algebras
    Awarded $7,000 for the period 9/1/15 to 8/31/16
    Source: Simons Foundation

This project will extend results from the classical setting of invariant theory—a finite group acting on a commutative polynomial ring—to a new setting—a finite group acting on an Artin-Schelter regular algebra—to increase our understanding of noncommutative rings with nice properties. Past work has been in collaboration with James Kuzmanovich (WFU) and James Zhang (University of Washington, Seattle), and the grant supports travel.

Frank Moore

  • Intensive Workshop in 2012 for Macaulay 2 Development
    Awarded $17,126 for the period 7/1/12 to 6/30/13
    Source: NSF

The workshop teaches participants how to implement algorithms in rapidly growing areas of algebraic geometry, such as numerical algebraic geometry, algebraic statistics, enumerative algebraic geometry, and differential graded algebras, with applications in fields as diverse as computational biology, robotics, coding theory, and string theory. Parallelization of new and existing algorithms inMacaulay2 is emphasized. The intensive workshop provides a great opportunity for graduate students and postdoctoral fellows to enhance their package-writing skills, taking advantage of the experience and expertise of more senior researchers.

  • Intensive Workshop in 2012 for Macaulay2 Package Development
    Awarded $12,050 for the period 8/11/11 to 8/10/12
    Source: National Security Agency

The workshop will implement algorithms in such rapidly growing areas as numerical algebraic geometry, algebraic statistics, enumerative algebraic geometry, and differential graded algebras with applications in fields as diverse as computational biology, robotics, coding theory, and string theory. Special attention will be paid to parallelization of new and existing algorithms in Macaulay2, a new parallel computing engine.

Jason Parsley
Conference on Applications of Geometry to Topology and Physics, November 2008, Newark, NJ 
Awarded $19,900 for the period 9/1/08 to 8/31/09
Source: NSF

The 3-day conference at Rutgers University-Newark focuses on applying geometry to problems in physics and topology. It will also celebrate the achievements of Herman Gluck, a distinguished University of Pennsylvania geometer and topologist, whose 70th birthday falls during the dates of the conference.

Robert J. Plemmons (see also Computer Science)

  • Innovations in Statistical Image Analysis and Applications to 3D Imaging for Improved SSA
    Awarded $51,856 for the period 8/15/15 to 8/14/16
    Source: AFOSR/University of New Mexico

The project works to derive statistical methods to assess the performance of Space Situational Analysis (SSA) systems that rely on optical and IR imaging and associated technologies. Highly efficient computational algorithms with near-real-time performance will be developed for two applications: ground-based polarimetric and spectral solar-reflectance (BRDF) measurements of space-object surface integrity, 3D shape, and material composition; and space-based 3D localization and tracking of space debris via rotating PSF imaging.

  • Comprehensive space-object characterization using spectrally compressive polarimetric imaging
    Awarded $70,000 for the period 7/15/13 to 7/14/14
    Source: Air Force Office of Scientific Research (AFOSR)/University of New Mexico

In collaboration with the University of New Mexico and Duke University, advanced imaging methods are used to identify and track objects in space. Space surveillance allows US space system operators to determine the capabilities of potential adversaries, to warn of an attack on a US space system, and to predict potential collisions and re-entry impact points.

  • Supplement to Novel Imaging Tools for Improved Space Objective Identification
    Awarded $20,000 for the period 5/3/11 to 7/31/11
    Source: AFOSR/University of New Mexico

With increased deployment of ever-smaller satellites at various altitudes, present-day imaging and nonimaging capabilities are often inadequate. Compressive sampling of spectral-spatial imaging data can rapidly identify space objects by cross-constraining object information and exploiting fundamental trade-offs implicit in such data. This system performance analysis will improve compressive sensor design and information transmission and formulate computationally efficient data postprocessing algorithms for identifying space objects. An experimental program will complement and validate the project’s theory, simulation, and processing.

  • Combining Imaging and Nonimaging Observations for Improved Space Object Identification
    Awarded $29,994 for the period 12/1/09 to 11/30/10
    Source: AFOSR/University of New Mexico

Current imaging and nonimaging capabilities are often inadequate to robustly and reliably determine the properties of ever-smaller satellites deployed at varying altitudes in space. This project’s low-dimensional parametric approach jointly models the essential literal and nonliteral characteristics of space objects in terms of a relatively small set of physically motivated parameters, whose values are estimated by digital postprocessing. Polarimetric and spectral data will be used to cross-constrain the radiometric information and to reconstruct space objects by exploring fundamental trade-offs, yielding data based on a priori constraints. This approach will overcome the resolution limits of even the largest existing and foreseeable AF/DoD assets.

  • Integrated Optical-Digital Imaging Camera System: Phase III: Computation Team Research and Development
    Awarded $69,784 for the period 11/9/07 to 9/30/09
    Source: Defense Microelectronics Activity; Catholic University of America

Prior notice needed for publicity.

  • Integrated Optical-Digital Imaging Camera System
    Awarded $52,975 for the period 1/22/07 to 10/15/07
    Source: US Department of Defense / University of New Mexico

No publicity allowed.

  • Phase II: Practical Enhanced-Resolution Integrated Optical-Digital Imaging Camera
    Awarded $103,464 for the period 1/22/07 to 6/30/07
    Source: University of New Mexico
  • DTO Advanced Imaging Seedling Project, A Practical Enhanced-Resolution Integrated Optical Imaging Camera (PERIODIC) System, Supplementary Funds 
    Awarded $62,026 for the period 9/15/00 to 02/28/07
    Source: ARO

This project aims to analyze, optimize, simulate, design, and fabricate a beta prototype, integrated, optical-digital, low-profile, low-cost, array-based imaging system. Considerable progress has been made in the theoretical, computational, and design/fabrication aspects, leading to the development of very promising workable prototype systems. Successful completion of this seedling effort is expected by the end of 2006, with the help of the supplemental funds, which will support two graduate and one undergraduate student, working with Professors Pauca, Plemmons, and Torgersen. Funds will also purchase additional equipment and supplies by the design and fabrication group at Catholic University (CUA).

  • Postdetection Processing and Inverse Problems in Ground-Based Imaging
    Awarded $15,000 for the period 12/31/04 to 6/30/07
    Source: AFOSR / University of New Mexico

High-resolution images are essential to many important defense, science, engineering, law enforcement, and commercial applications. Extracting meaningful information from degraded images is especially vital for such biometric DoD applications as integrated optical imaging systems for personnel identification using the iris. This project conducts extensive, novel research in pupil phase engineering (PPE) to help develop, along with industrial partner CDM Optics Company, a reliable, easy-to-use, low-cost iris recognition system for personal verification to ensure computer network security. The primary technical goal is to make iris recognition easier to use by greatly expanding the imaging system’s iris capture volume; we estimate that our methods can increase iris capture volume more than 100 times over current systems. The design of overall optical masks is a nontrivial problem and involves the numerical solution of highly nonlinear and ill-posed optimization problems with multiple design parameters. Dr. Plemmons serves as Senior Scientific Consultant to establish a major research and development program in ground-based imaging for the Air Force Research Laboratory, including the Maui High Performance Computing Center, which houses one of the world’s largest supercomputers.

  • A Practical, Enhanced-Resolution, Integrated Optical-Digital Imaging Camera (PERIODIC) System
    Awarded $63,486 for the period 05/23/05 to 8/30/06
    Source: United States Department of Energy (DOE)

This project aims to design an end-to-end optimized, compact, integrated, digital camera system with a modular architecture. Novel interferometric enhancements of optical resolution and use of information theory as an optimization tool will lead to imaging-system designs that maximize information throughput. Surveillance imaging systems will be developed for intelligence agency applications.

  • Innovative Computational Methods for Inverse Problems in Optical Imaging
    Awarded $51,255 for the period 6/29/05 to 2/28/06
    Source: ARO

Dr. Plemmons and colleagues will investigate and implement innovative approaches to pupil-mask design to control the depth-of-focus of imaging systems. The traditional imaging system suffers from limited depth-of-focus that can be extended most simply by reducing the pupil size. This approach is undesirable, because the light flux and resolution are both decreased. Instead, the investigators plan to use multiple parameters that can control pupil phase variation rather finely. They foresee vigorous technology transfer to the military and industrial arenas and generalization of the theoretical concepts to other domains of imaging.

Sarah Raynor
Asymptotic behavior of solutions to nonlinear dispersive equations
Awarded $35,000 for the period 9/1/12 to 8/31/17
Source: Simons Foundation

Most partial differential equations are far too complex to solve explicitly, except perhaps to find a few special solutions. Given a physical problem expressed as a partial differential equation, Dr. Raynor uses real analysis energy estimates, degree theory, regularity theory, and harmonic analysis to study the qualitative behavior of solutions: Do they exist? Are they unique, or can more than one be guaranteed to exist with the same input? How are they affected by a slight change in the input? Can they collapse? Can they blow up?

Stephen Robinson  
SEARCDE 2012 Conference
Awarded $24,832 for the period 9/15/12 to 8/31/13
Source: NSF

The Southeastern-Atlantic Regional Conference on Differential Equations (SEARCDE) brings together established and new researchers to exchange ideas. It has met annually since 1981 and in 2012, was held at Wake Forest. Funding helped to defer the travel expenses of advanced graduate and undergraduate students and recent PhD recipients and four invited plenary speakers as well as staffing and supplies.

Jeremy Rouse

  • 2016 Automorphic Forms Workshop
    Awarded $14,400 for the period 7/24/15 to 7/23/16
    Source: National Security Agency (NSA)

The Automorphic Forms Workshop is an internationally recognized and respected conference that Wake Forest will host from 7-10 March 2016. Featuring panel discussions among number theorists, it encourages the participation of undergraduate, graduate, postdoctoral, and junior faculty researchers. NSA funding will support their travel and lodging expenses.

  • REU Site: Wake/Davidson Experience in Number Theory Research
    Awarded $258,354 for the period 5/1/15 to 4/30/18
    Source: NSF

Jeremy Rouse and Katherine Thompson will run a 9-week summer research experience for undergraduate students at Wake Forest in 2015, 2016, and 2017. Each summer, eight students, chosen by nationwide search, will experience the joys and challenges of conducting original research in number theory and writing an original, publishable research paper. They will explore various topics in elementary number theory, quadratic forms, elliptic curves, and modular forms. A rich professional development program will prepare them to present their research at the University of Georgia’s Mock AMS conference and the AMS/MAA Joint Math Meetings.

  • Distribution of the Fourier coefficients of modular forms and arithmetic applications
    Awarded $51,530 for the period 7/16/10 to 7/31/12
    Source:NSF/University of Illinois

This 3-year research program studies the distribution of the Fourier coefficients of modular forms (both Archimedean and p-adic), arithmetic dynamics, and quadratic forms. It will focus on understanding the number of representations of an integer as a sum of squares and generalizing Bhargava and Hanke’s famous 290 theorem.