## Algebra and Number Theory

Number theory researchers at UVM enjoy participation in the stimulating Québec-Vermont Number Theory Seminar, an extremely active and high-quality gathering of mathematicians from several institutions in Quebec and Vermont that has run since 1984. At least twice a month, a seminar day will present two or three speakers discussing recent work. Often these are top researchers describing cutting-edge results of major significance. Outside of the lectures seminar participants talk informally and develop collaborations. In this arena, Professors Dummit, Foote and Sands of UVM have collaborated with R. Murty and J. Labute of McGill, H. Kisilevsky, K. Murty, and X. Roblot of Concordia, and L. Simons of St. Michael's College. For more details, see Québec-Vermont Number Theory Seminar.

Richard Foote, Professor, works in the area of group theory and its applications. His early research and continuing touchstone is the Classification of Finite Simple Groups. He has explored applications of group theory to problems in algebraic number theory, topology, and digital signal processing. He is currently working with Gagan Mirchandani in UVM's School of Engineering on discrete and algebraic notions of multiresolution analysis and wavelets, and the applications of group theory and algebra to image processing and edge detection. His interests also include Complex Systems and their relations to group theory and theoretical physics. Richard Foote is coauthor (with Prof. Emeritus David Dummit) of the internationally used graduate text |

Jonathan Sands, Professor, works in the area of algebraic number theory. His current research focuses on Stark's conjectures for abelian and non-abelian Galois extensions of number fields, and generalizations of these conjectures involving towers of number fields or algebraic K-theory. These conjectures link certain algebraic invariants to L-functions which are defined analytically. Together with a number of co-authors, he has also published theoretical and computational results on the subjects of Leopoldt's conjecture and Iwasawa theory. |

Christelle Vincent, Assistant Professor , works in the areas of alegebraic number theory and arithmetic geometry. Her current focus is on the sxplicit construction of abelian varieties with complex multiplication. More precisely, one step of the constructin involves passing from a floating point approximation of certain class invariants to their exact value. To do this requires studying intersection theory and a certain embedding problem for an order of an imaginary quadratic field into matrix algebra over the quaternations. Her interests also include studying curves over finite fields (especially if they have many points) and algebras of Drinfield modular forms. |

## Analysis

J. Michael Wilson, Professor, works in the area of weighted norm inequalities involving Haar functions. This area forms part of Littlewood-Paley theory, an area of harmonic analysis that has been intensively studied over the past half-century. His work in this area has been supported by a grant from the National Science Foundation. It has applications in signal processing and to the Schrödinger operator. In addition, he is a highly skilled writer of fiction and has taught creative writing at the University. |

## Applied Mathematics

Daniel Bentil, Associate Professor, does research at the interface of Applied Mathematics and Mathematical Biology. His work, which focuses on mathematical modeling in biology and medicine, is highly interdisciplinary. In physiology, for example, he is currently working on model mechanisms for muscle contraction, aerosol deposition in the lungs, and the dynamics of hemodialysis administered to end-stage renal disease patients. Some ecological studies have involved modeling invasive species spread and host-parasite interactions. Together with his collaborators and graduate students, Daniel Bentil has been developing and analyzing mathematical models, and interpreting and comparing modeling results to real experimental data. He is very well funded and his graduate students have always had no difficulty obtaining jobs right after graduation. |

Christopher Danforth, Associate Professor, is the Flint Professor of Mathematical, Natural, and Technical Sciences at the University of Vermont. Along with Peter Dodds, he co-directs the Computational Story Lab, a group of applied mathematicians at the undergraduate, masters, phd, and postdoctoral level working on large-scale, system problems in many fields including sociology, nonlinear dynamics, networks, ecology, and physics. His research has been covered by the New York Times, Science Magazine, and the BBC among others. Recent projects include quantifying levels of happiness using Tweets & Depression using Instagram photos. |

Peter Dodds, Professor, works on problems in geomorphology, biology, ecology, and sociology, with an overriding interest in complex systems and networks. |

Taras Lakoba, Assistant Professor, applies his expertise in perturbation methods to a variety of topics in applied mathematics. Most recently, he has been interested in proving convergence of certain numerical iterative schemes for finding stationary solutions of nonlinear wave equations. In the past, he developed perturbation theories for a number of nonlinear wave equations integrable by the inverse scattering transform. Taras also worked, and still maintains interest, in fiber optics, where his expertise lies in nonlinear signal transmission, polarization effects, and noise accumulation. He was part of the team at Lucent Technologies that developed an ultra-long haul, dense wavelength-division multiplexed transmission (WDM) system in 2002. |

Jianke Yang, Professor, works in the area of nonlinear waves and their physical applications. Nonlinear waves are prevalent in science and engineering, and they are described mathematically by nonlinear partial differential equations. His recent research interest is on nonlinear wave phenomena in optics, soliton perturbation theory, as well as numerical methods for nonlinear wave equations. He is one of the top researchers in these areas in the world. |

Jun Yu, Professor, works in the area of applied mathematics with applications in biomedicine, geophysics, fluid mechanics and combustion. A major focus of his research has been on the dynamics of the intracranial system in the human brain. This problem involves a blending of fluid mechanics, elasticity, and theoretical and computational methods with both clinical and experimental aspects of human physiology. Recently, he has become involved in the study of the dynamics and thermodynamics of oceans and ice mass of the Earth, using satellite data from NASA as well as mathematical modeling techniques. At the same time he continues to do research in the area of classical fluid mechanics. There, his research focus is on nonlinearity and stability of water waves. He has examined the evolution of the weakly nonlinear solution for the case in which a parameter (Froude number) goes through its critical value and the linear solution fails. More recently, a solid combustion model was studied, and the onset of linear instability as well as the weakly nonlinear solution behavior in the presence of the linear instability was also analyzed. |

## Combinatorics and Graph Theory

Puck Rombach, Assistant Professor, works in the area of graph theory. Her work bridges gaps between the pure and applied sides of graph/network theory. She has recently worked on problems related to graph coloring, random graphs, algorithms and complexity, graph representations of matroids, crime network modeling, and core-periphery/centrality detection in networks. |

Jeffrey Dinitz, Professor, works in the area of combinatorial designs. He is the managing editor-in-chief of the Journal of Combinatorial Designs as well as the co-editor of the CRC Handbook of Combinatorial Designs. Together with a number of different coauthors he has published numerous articles in many areas of combinatorial designs, including one-factorizations, Room Squares, triple systems, Latin squares, MOLS and transversal designs, graph decompositions, Howell designs, starters, balanced tournament designs, graph labeling, list coloring, and hill-climbing and orderly algorithms. He has also written papers on applications of combinatorial designs to networking and storage disk arrays. He has been on the faculty since 1980 and holds a secondary appointment in the Department of Computer Science. |

Greg Warrington, Associate Professor is an algebraic combinatorialist who has also been conducting research relating to gerrymandering. My combinatorics research focuses on polynomials that encode information about algebraic, geometric and representation-theoretic objects. An individual coefficient of such a polynomial (say from the family of q,t-Catalan polynomials or Kazhdan-Lusztig polynomials) would typically record the dimension of a vector space. I've also been looking into ways to identify gerrymanders that do not depend on the geometric shape of individual districts but rather on how votes are distributed among those districs. |

## Statistics and Biostatistics

Taka Ashikaga, Professor, is Director of the Biometry Facility of the College of Medicine. Dr. Ashikaga provides collaborative support to investigators in several areas of interest including breast cancer, hormone replacement, resistance training as it pertains to coronary patients, knee joint displacement, forest sciences, and smoking prevention. He is also involved with both the Vermont Cancer Center and General Clinical Research Center, which support the areas of disease detection, prevention and treatment. |

Janice Bunn, Research Assistant Professor, provides collaborative support to a wide variety of investigators throughout the University of Vermont. Her research efforts include the application of statistical techniques in group randomized clinical trials based on both nested cohort and nested cross-sectional experimental designs. She also provides statistical support for a wide range of individually randomized clinical trials in her capacity as the biostatistician for the General Clinical Research Center and works with the Department of Physical Therapy and the Office of Clinical Trials Research. In addition, she is working on the application of structural equation modeling techniques for clustered samples. |

Jeffrey Buzas, Professor, works in both theoretical and applied statistics with applications to biostatistics, epidemiology, economics and geosciences. His research efforts have focused on the study of the effects of covariate measurement error in regression models, which are used extensively in a wide variety of disciplines such as epidemiology, engineering, environmental research, biology, and physics. He is also currently Co-Investigator of a funded intervention study with the neonatology group in the UVM College of Medicine. The study, titled "Evidence-based Surfactant Therapy for Preterm Infants" is funded through the Agency for HealthCare Research and Quality of the U.S. Department of Health and Human Services. |

Peter Callas, Research Associate Professor, conducts epidemiologic research and statistical analysis at the University of Vermont College of Medicine and at Fletcher Allen Health Care (the academic medical center at UVM). He is currently involved in projects on a variety of topics, including evaluation of a telemedicine system; prevention and treatment of nicotine, alcohol, and other addictions; and investigation of genetic aspects of thrombosis and hemostasis. He has previously worked with the Department of Orthopedics and Rehabilitation and with the Department of Psychiatry on a study of factors predicting return to work among patients with low back pain. |

Chip Cole, Professor, works in the area of cancer biostatistics. His methodological research is focused on the joint evaluation of quality of life and survival in cancer clinical trials. He also has a general interest in statistical methodology related to cancer research, including survival analysis, longitudinal data analysis, and the handling of missing data. In addition to his methodological activities, Dr. Cole is an active co-investigator. With the International Breast Cancer Study Group, he works with breast cancer researchers to conduct world-wide clinical trials of treatments for breast cancer. With the Polyp Prevention Study Group, a North American consortium, he works on large-scale clinical trials of chemo-preventive agents for colorectal cancer. |

Erika Edwards, Research Assistant Professor, is Manager of Statistical Reporting for the Vermont Oxford Network, an international non-profit voluntary collaboration of health care professionals improving the quality and safety of medical care for newborn infants and their families. She conducts research on very low birth weight and other preterm infants in the Vermont Oxford Network and Neonatal Encephalopathy registries. She is also involved in epidemiologic research on adolescent health, substance use and abuse, physical activity, and metabolic syndrome using large existing datasets. |

Richard Single, Assistant Professor, conducts research in the area of statistical genetics which involves population genetic analyses, comparative genomics, methods for disease association studies, determining properties of measures of overall linkage disequilibrium (LD), and tests for LD and Hardy-Weinberg equilibrium at the individual haplotype/genotype level. An emphasis on the human Major Histocompatibility Complex (MHC) of chromosome 6 has led to the study of microsatellite and SNP markers that are predictive of classical HLA genes due to their influence on the immune response for bone marrow transplantation. He also works in the areas of program evaluation and medical biostatistics with faculty members in the College of Education and Social Services and College of Medicine. |