Mathematical and statistical research at UVM varies from immediately practical collaboration with researchers in other areas (especially the Larner College of Medicine) to the investigation of purely theoretical mathematical or statistical problems. However, no sharp division between these two kinds of research exists in practice, and members of our department find plenty of problems of intrinsic interest while collaborating with scholars in other fields.

## Areas of Faculty Research

## 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. For more details, see Québec-Vermont Number Theory Seminar.

Christelle Vincent, Assistant Professor , works in the areas of algebraic number theory and arithmetic geometry. Her current focus is on the explicit construction of abelian varieties with complex multiplication. More precisely, one step of the construction 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 a matrix algebra over the quaternions. Her interests also include studying curves over finite fields (especially if they have many points) and algebras of Drinfield modular forms. | |

Taylor Dupuy, Assistant Professor, works in the areas of differential algebraic geometry and arithmetic geometry. More precisely, his research revolves around the following themes: Diophantine equations, deformation theory, Witt vectors and p-derivations, absolute geometry, and applied model theory. |

## 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 3 grants 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

James Bagrow, Associate Professor, is interested in understanding the underlying rules and organizing principles of complex physical and social systems. His work combines mathematical models with large-scale data analysis to better understand these systems, with a particular emphasis on network science, computational social science, and human dynamics. Other interests include data science, dynamical systems, and novel optimization and machine learning methods. He is a member of the Vermont Complex System Center and co-director of the UVM-Google OCEAN project. | |

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. | |

Chris Danforth, Professor, is the director of the Vermont Advanced Computing Core. 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 and depression using Instagram photos. | |

Taras Lakoba, 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 (Jackie) Yang, Williams Professor of Mathematics, University Distiguished Professor, and University Scholar. His research interests are nonlinear waves and their physical applications --- an important branch of applied mathematics. Nonlinear waves are prevalent in science and engineering, with examples such as water waves in the ocean and light waves in optical fibers. Mathematically, nonlinear waves are described by nonlinear partial differential equations (PDEs). He develops advanced analytical and numerical methods to solve important nonlinear PDEs, and explores applications of such theoretical results to physical problems. His current research interests are on nonlinear optics, rogue waves, parity-time symmetry and numerical methods. | |

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

Spencer Backman, Associate Professor, does research is in algebraic and geometric combinatorics, e.g. algebraic aspects of graphs, matroids, and polytopes. A common thread running through his work has been tropical geometry, a combinatorial piecewise-linear version of classical algebraic geometry, and the new perspective it lends to classical combinatorial objects. | |

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. | |

Greg Warrington, 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

Janice Bunn, Research Associate 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 UVM PI of an NIH-R01 sub-award with University of South Carolina titled "Hospital quality, Medicaid expansion and racial/ethnic disparities in maternal mortality and morbidity." | |

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. | |

Abby Crocker, Research Assistant Professor, is the Director of Research for the National Center on Restorative Justice, a federally funded partnership between the Vermont Law School, the University of Vermont, and the University of San Diego. She is an applied methodologist with a research focus on health disparities and justice-involved populations, social determinants of health, and restorative justice. Abby partners with interdisciplinary teams, across organizations, to support the use of data in addressing complex health and social concerns. | |

Erika Edwards, Research Associate Professor, is Director of Data Science for Vermont Oxford Network, a nonprofit, voluntary worldwide collaborative dedicated to improving the quality, safety, and value of care for newborns through a coordinated program of data-driven quality improvement, education, and research. She is an epidemiologist who conducts applied research using data collected on infants admitted to neonatal intensive care units. She has a secondary appointment in the Department of Pediatrics, Larner College of Medicine. | |

Mike Miller Eismeier, Assistant Professor works in the area of low-dimensional topology (such as knot theory and the study of 3-dimensional shapes up to deformation), studying low-dimensional objects using tools coming from gauge theory, a collection of partial differential equations inspired by related physical equations. Work in this area uses math from many fields, connecting topology to both algebra and analysis. | |

Richard Single, Associate 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. | |

Jean-Gabriel Young, Assistant Professor, works on Bayesian statistics and its applications to complex systems and epidemiology. His recent research focuses on inference problems in network science, including for dynamical models of networks; network reconstruction from noisy data; and the inference of high-order interactions from pairwise data. Dr. Young is also a junior faculty of the Translational Global Infectious Diseases Research Center, where he develops statistical methods for epidemiological models on networks, and of the Vermont Complex Systems Center. |