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Research in this area offers a rare cross-over between pure mathematics and computation. In the case of Dr. Slattery, it is concerned with designing and analyzing algorithms and data structures to compute information about groups. Two important computer algebra systems (CAS) used for group theory are GAP and MAGMA.

This laboratory investigates extremal and probabilistic questions in graph theory and combinatorics, including graph coloring questions and games on graphs.  Some specific topics currently being studied include independent sets, vertex colorings, and alternating sign matrices.  

This laboratory focuses on theoretical and applied aspects of mathematics education. It generates an understanding of the nature of mathematical thinking, teaching, and learning, and uses it to create tools for improving mathematics instruction.

Current projects include:

1. Mathematical argumentation - This longitudinal research program is sponsored by the National Science Foundation and focuses on K-8 pre-service teacher education. The research addresses pre-service teachers’ knowledge development in the area of mathematical argumentation.

2. Spontaneous metacognitive thinking - This project focuses on the design of research settings that optimize opportunities to observe and document how problem solvers engage in spontaneous metacognitive activity in complex problem-solving situations. The project provides motivation for the design of research settings in which metacognitive activity and complex problem-solving processes are studied together as a co-dependent activity.

3. Equal sign and equality - This project explores the use of comparative relational strategies in the context of solving problems about the equal sign and equality in the arithmetic and algebraic problem contexts.

Randomness and nonlinearity are dominant features in many mathematical models. The interplay between the two often causes interesting behavior that we wish to understand and predict. Current projects include:

1. Hazard Mapping - Devastation caused by pyroclastic (rapid, granular, volcanic) flows can be extreme for communities situated near volcanoes. We are devising methods to draw accurate hazard maps for use in civil protection and planning. Work is in collaboration with investigators at the State University of New York at Buffalo (math, volcanology), Duke University (statistics), and National Institute of Statistical Sciences. This project is supported by the NSF.

2. Nonlinear Optics - In nonlinear, random systems interesting phenomena are often in the form of rare but important events. The performance of optical communication systems and mode-locked lasers is limited physically by noise, i.e., incoherent photons introduced during amplification. We seek to understand both how errors occur and how frequently they occur. Work is in collaboration with investigators at the State University of New York at Buffalo and Northwestern University.

3. Data Assimilation - Data assimilation is a broad term for techniques that combine noisy observations with dynamic model based predictions. Our current work is in devising methods of data assimilation that work well in systems that are both nonlinear and high-dimensional. Work is in collaboration with investigators at the University of North Carolina at Chapel Hill (math, statistics) and University of California at Los Angeles (geosciences).

Elaine Spiller’s research work intersects with the Biomathematics group’s activities, but also interfaces with the Statistics group’s work.

Electrical Impedance Tomography (EIT) is a non-invasive, portable, radiation-free, imaging modality used for monitoring hearth and lung function of patients in the ICU setting. The goal of EIT is to recover the internal structure of a body from electrical measurements taken at the surface. The conductivity and permittivity of biological tissues such as heart, lung, blood, and fat are different allowing a medical doctor to then use the EIT images for diagnostic/monitoring purposes. Additional applications include: detection and classification of breast tumors, stroke classification in brain imaging, non-destructive evaluation on concrete structures, detection of groundwater contamination, oil exploration and landmine detection. The reconstruction task is a highly ill-posed nonlinear inverse problem and requires the use of mathematical techniques from functional and complex analysis as well as computational skills. Traditional methods based on least squares minimization or linearization are not sufficient for static imaging applications (breast cancer, stroke). Our lab focuses on solving the full nonlinear problem directly (non-iteratively), in real-time, using any available information to improve the images.

GasDay forecasts natural gas consumption for about 30 energy delivery companies all over the US. We build mathematical models, perform statistical analysis of data, develop software, deliver that software to customer utilities, prepare custom research reports, and provide on-going customer support. We function like a small business within the University. Each day, we provide forecasts for nearly 20% of the natural gas in the US for residential, industrial, and commercial uses.  We are a student centered learning project, a research project, and the technology transfer project.  Students do research and publish in the areas of mathematical modeling, data mining, statistical analysis, database design, computer systems architecture, and software engineering. GasDay supports 10 graduate students and 20+ undergraduate students.  Participating faculty come from MCCS, Engineering, and Business.

Applies mathematical modeling to address basic science questions regarding nonrespiratory functions of the [mammalian] pulmonary circulation. Current projects include:

1. Pulmonary disposition of quinones. The working hypothesis is that metabolic changes precede the remodeling that occurs in pulmonary hypertension. Quinones are used as probes to investigate changes in the pulmonary endothelium redox status/signaling. Work is in collaboration with ÏòÈÕ¿ûÊÓƵBiomedical Engineering and Zablocki/Medical College of Wisconsin researchers.

2. Structure/function studies of the pulmonary circulation. Investigation of how structural changes that result in several biological models of pulmonary hypertension (chronic hypoxia exposure, monocrotaline exposure, thoracic irradiation) impact upon the hemodynamic functioning of the pulmonary circulation. 1 Ph.D. student working on the impact of supernumerary vessel recruitment may have upon normal pulmonary function.

This laboratory designs, develops, and optimizes imaging systems and requisite experimental protocols and image analysis tools for investigating cardio-pulmonary physiology pathophysiology of small laboratory animals. Close collaboration with Prof. Krenz’s group as well as scientists and clinicians at Zablocki VA Medical Center and Medical College of Wisconsin. Ongoing funded projects include:

1. Micro-Angiography Investigation of Pulmonary Hemodynamics - Following completion of this instrument, its hardware, experimental protocols, and image analysis tools are being modified to determine capillary flow, its regional distribution, and changes therein following lung injury or treatment in rats. Supported by NHLBI, Co-Investigator.

2. Angiogenesis in the Bronchial Circulation using SPECT (single-photon emission computed-tomography) – Micro-CT and SPECT to monitor angiogenesis in the bronchial circulation of rats. Supported by NHLBI, Co-principal investigator.

3. Micro-SPECT Hardware and Software Design – Construction of a flexible system for rapid, high-resolution imaging of small animals using radiopharmaceuticals targeted at specific molecular functions. Supported by the Keck Foundation.

4. Redox Status of Lungs in Rats Exposed to Chronic Oxidative Stress with Micro-SPECT – Use molecular imaging radiopharmaceuticals and pharmacokinetic modeling to assess early lung injury. Supported by NHLBI, Co-Investigator.

5. Reducing Patient Dose in Spiral and Conebeam CT – Simulations designed to determine optimal scanning parameters. Supported by NIBIB, Co-Investigator with U. of Iowa.

The long-term goal of the Rowe Functional Magnetic Resonance Image Analysis Lab research efforts is to develop a unified mathematical model for functional magnetic resonance imaging (fMRI). This mathematical model is to include the fundamental physics of the nuclear magnetic resonance signal, the compensation for biological signals not of interest, the preprocessing of the MR images, the statistical modeling of complex-valued time series, and the statistically significant determination of focal brain activation. This mathematical model will extract the most information from the acquired data in the most efficient way possible. This unified model will contain important physiological information that may not be available by other means or is only available by more time-consuming elaborate means. This model will allow us to address important fundamental neuroscience questions with fMRI. The Rowe fMRI Analysis Lab's primary research efforts are focused on working deeper in the data acquisition and processing stream while expanding the unified model at each step.

My primary research interests lie in the developments of the statistical models in high-dimensional data structures with application to biological sciences, including, but not limited to genomics and proteomics.

In particular my doctoral research focuses on the dimension reduction and the functional data analysis in non-Gaussian frameworks, which has been applied in the context of analysis of high-dimensional gene expression data; and the main focus of my research during the postdoctoral program, was on the modeling of the large spherical data structures with an application in protein structure prediction and classification. My current research interests include:

• Bioinformatics
• Machine Learning
• Proteomics
• Functional Data Analysis and Skewed Distributions
• Nonparametric and Semiparametric Methods

This group develops and applies new statistical methods in a variety of applications. Projects include Bayesian hypothesis testing using a decision theoretic approach. A doctoral student is developing a decision theoretic methodology for testing multiple and multi-sided hypotheses to produce more powerful test procedures. The idea is to incorporate prior information in order to produce more powerful tests which will have application in gene expression data analysis.