Mathematics

Recent Papers:

• William A. Bogley, Martin Edjvet, and Gerald Williams, Aspherical relative presentations all over again, to appear in Groups St. Andrews 2017 (31 pp). arXiv:1809.03460.
• Willam A. Bogley and Forrest W. Parker, Cyclically presented groups with length four positive relators, J. Group Theory 21 (2018), 911-948, published online 6/15/2018. arXiv:1611.05496
• W. A. Bogley and Gerald Williams, Coherence, subgroup separability, and metacyclic structures for a class of cyclically presented groups, J. Algebra 480 (2017) 266–297. arXiv:1606.00216
• W. A. Bogley and Gerald Williams, Efficient finite groups arising in the study of relative asphericity, Math. Z. (2016) 284:507–535. PDF arXiv:1607.01951
• W. A. Bogley, On shift dynamics for cyclically presented groups, J. Algebra 418 (2014), 154-173. PDF arXiv:1312.5382

Areas of interest and publication:

1. Structure and shift dynamics for cyclically (and symmetrically) presented groups
2. Asphericity of cell complexes and cohomology of groups
3. Relative homotopy groups and equations over groups
4. Actions on cell complexes and finiteness properties of groups
5. Combinatorial geometry and algorithmic/decidability properties of groups
6. Wild metric cell complexes and omega-group structures, in which certain infinite products can be formed subject to suitable generalizations of the associativity axiom.

Professor Bokil's general research interests are in applied mathematics, scientific computing, numerical analysis and mathematical biology. Her primary research interests are in the numerical solution of wave propagation problems. Specifically, she has conducted research on the numerical solution of Maxwell's equations using a variety of finite difference and finite element methods. Bokil is also working on several problems in mathematical ecology which involve the construction and analyses of  deterministic and stochastic models for applications in population dynamics, epidemiology and spatial ecology.

Dr. Bokil is currently a co-PI on an NSF funded project OP: Collaborative Research: Compatible Discretizations for Maxwell Models in Nonlinear Optics in collaboration with Drs Yingda Cheng at Michigan State and Fengyan Li at RPI. The objective of the collaborative research program is to make significant advances in the understanding and simulations of Maxwell models in nonlinear optics.

Dr. Bokil has also received funding via the Thomas Jefferson Fund launched by the French-American Cultural Exchange (FACE) Foundation for her project Mathematical epidemiology of viruses coinfecting plants: Modeling, Analysis and Optimal Control Strategies, with French collaborator, Professor Frederic Hamelin of Agrocampus Ouest in Rennes, France. 

Professor Cozzi investigates problems on existence, uniqueness, and regularity of solutions to partial differential equations arising in fluid mechanics.

Dascaliuc's research focuses on qualitative properties and long-time behavior of nonlinear PDE. In particular, he is interested in dynamics of the incompressible Navier-Stokes and Euler equations, especially in relation to the turbulence theory. The main goals are to derive the tenets of the heuristic theory of turbulence directly from the physical equations of motion and to understand connections between turbulence in fluids and regularity theory of the 3D Navier-Stokes and Euler equations. 

Dick's research interests include the study of factors related to mathematics achievement and participation, cognitive science as applied to the learning of advanced mathematics, uses of technology in the learning of mathematics, and mathematical discourse. He has worked extensively in the calculus curriculum reform movement. He has served on the joint AMS/MAA Committee on Research in Undergraduate Mathematics Education, the National Council of Teachers of Mathematics Research Advisory Committee, and the Advanced Placement Calculus Development Committee. He is a past co-editor of Connecting Research to Teaching for NCTM's Mathematics Teacher journal, associate editor for School Science and Mathematics, an editorial panel member for the Journal for Research in Mathematics Education, and an editorial panel member for the Mathematics Teacher Educator.

Past work has emphasized General Relativity, studying model spacetimes and their properties, as well as the interface between relativity and quantum physics. Current work investigates applications of the Octonions, the unique non-associative division algebra, to the physics of fundamental particles. Other past and present interests include Algebraic Computing, Asymptotic Structure, Non-Euclidean Geometry, Quantum Field Theory in Curved Space, and Signature Change.

Work in Mathematics Education includes directing the Vector Calculus Bridge Project, co-directing the Paradigms in Physics Project, and designing course content for the Oregon Mathematics Leadership Institute, as well as close collaboration with the Physics Education Group at OSU and membership in the ESTEME group of science educators at OSU.

Escher's current work lies in the interaction between algebraic topology and differential geometry, in particular the use of surgery theory to classify spaces of positive sectional curvature. Her previous work is connected to the field of minimal submanifolds; in particular existence and uniqueness questions of minimal isometric immersions of spherical space forms into spheres.

Faridani conducts research in numerical analysis, investigating problems arising in signal processing and tomography. In signal processing he is interested in non-equidistant sampling of bandlimited functions in several variables. His research in tomography comprises questions of optimal sampling and resolution; error estimates for reconstruction algorithms in two and three dimensions; and theory and implementation of local tomography.
For publications and preprints please click here.

Finch works on inverse problems, particularly those arising in medical imaging. He has worked on x-ray computed tomography and vector tomography. He is currently working on problems arising in the medical imaging technique called thermoacoustic tomography and other forms of hybrid imaging. 

Dr. Flahive's work in number theory is principally in Diophantine approximation, with techniques from the geometry of numbers. She is also working with colleagues in computer science on projects in algebraic coding theory and also on developing and analyzing interconnection networks.

Garity's current research is focused on properties of embeddings of Cantor sets in Euclidean spaces, focusing on embeddings in R^3. Cantor sets often arise as limit sets or invariant sets of dynamical systems. The complements of such Cantor sets in R^3 are 3-manifolds with a rich end structure. The possible types of embeddings of these sets in R^n can provide information about possible dynamical systems and about the end structure of manifolds.  Recent examples include nonstandard embeddings with simply connected complement, nonstandard embeddings that are rigid, and nonstandard embeddings with both properties. 

Gibson's primary research interests are computational electromagnetics, finite element and finite difference methods, and inverse problems. Research topics coinciding with primary interests include: wave propagation modeling, homogenization, optimization and regularization techniques, high performance and parallel computing, parameter identification and sensitivity analysis, and uncertainty quantification. Current work involves finite element and finite difference methods for time-domain electromagnetics in dispersive and random media.

Guo is working on circle packing, discrete conformal geometry, hyperbolic geometry, Teichmüller space.

Higdon has worked on open boundary conditions for wave propagation problems and on issues related to the well-posedness of hyperbolic initial-boundary value problems and the stability of their numerical approximations. He is presently working on some mathematical and computational problems related to the numerical modeling of ocean circulation. This modeling involves the solution of a system of partial differential equations that describes fluid flow, as adapted to that case. In this area, Higdon first worked on some problems with stability and efficiency that can arise from the multiple time scales that are contained in the system. He is presently working on a project to develop, analyze, and test some procedures for using discontinuous Galerkin numerical methods in multi-layer models of ocean circulation.

Casualty and health actuarial mathematics;  Workers compensation insurance; Health insurance; Insurance regulatory issues; Reinsurance; Medical professional liability insurance; Loss reserve modeling. Catastrophe modeling. Financial risk modeling, Pricing models and algorithms.

Yevgeniy Kovchegov works in the field of probability and stochastic processes. His research interests include mathematics of data science, mathematical models of statistical mechanics, interacting particle systems, models of mathematical biology, stochastic self-similarity, and quantum computation. Kovchegov's work is centered around the following topics: self-similar random trees; extending the probabilistic coupling method; orthogonal polynomials in stochastic processes, probability and statistics; mixing times; quantum walks and quantum computation; chaos and fractals; coalescent processes; applications of probability theory in biological systems, geomorphology, computer and wireless networks, network coding, and environmental sciences.

Lee has made contributions to the existence, uniqueness, and continuous dependence theory for solutions to nonlinear boundary value problems. This work, which continues, is joint with R.B. Guenther of Oregon State and A. Granas of the University of Montreal. Lee has also worked on the numerical calculation of solutions to such problems. He has helped develop the extension of Sturm oscillation theory and the properties of Sturm-Liouville eigenvalue problems to higher order equations. This work is closely related to the branch of approximation theory which deals with Tchebycheff systems and with positive operator theory. It led to related work on best quadrature formulas.

Lockwood's primary research interest involves undergraduate mathematics education, particularly studying how students think about and learn combinatorial topics. She has put forth a model of student's combinatorial thinking that especially emphasizes the role of sets of outcomes in effective counting. Other research in this area includes exploring student-generated connections among counting problems through a lens of actor-oriented transfer and determining the effectiveness of systematic listing in counting. Through two NSF-funded grants, she is currently investigating students’ generalizing activity as they solve advanced counting problems, and she is studying ways in which having students engage in computational activity can help them solve counting problems more successfully. Two additional collaborations and areas of research include studying the role of examples in proof and considering the relationship between mathematical content and mathematical practices at all levels.

Solitons, spinning things, donuts, and something random on computers.

Preprints:
(2020) P. Nabelek. ``On Primitive Solutions to the KP Equation.'' (Not yet on arXiv, please contact me for a copy.)
(2020) P. Nabelek. ``Algebro-Geometric Finite Gap Solutions to the Korteweg–de Vries Equation as Primitive Solutions.'' Physica D. (in press, arXiv:1907.09667)

Publications:
(2020) P. Nabelek, V. Zakharov. ``Solutions to the Kaup–Broer system and its (2+1) dimensional integrable generalization via the dressing method.'' Physica D.
(2020) S. Dyachenko, P. Nabelek, D. Zakharov, V. Zakharov. ``Primitive Solutions to the Korteweg de--Vreis Equation.'' TMP.
(2019) K. McLaughlin, P. Nabelek. ``A Riemann–Hilbert Problem Approach to Infinite Gap Hill’s Operators and the Korteweg–de Vries Equation.'' IMRN.
(2019) P. Nabelek, D. Zakharov, V. Zakharov. ``On Symmetric Primitive Potentials.'' J. Int. Sys.

Unpublished Manuscripts:
(2014) P. Nabelek, D. Pickrell. ``Harmonic Maps and the Symplectic Category.'' (arXiv:1404.2899)

One focus of Ossiander's research is the development of central limit theorems for sums of random functions. Results in this area are intimately connected with the properties of continuous Gaussian and product-Gaussian processes. The exploration of central limit theory in this general setting has involved the development of exponential bounds for the tail probabilities of sums of random variables. Interesting applications include the calculation of rates of convergence of classes of statistical estimators and the construction of stochastic models for physical systems.

Malgo Peszynska's research interests are in applied mathematics and computational modeling of real life phenomena. Originally trained in pure mathematics, she came to applied mathematics projects through her interest in parallel and high performance computing. Since her PhD she has worked on models of flow and transport using mathematical and numerical analysis as well as computer simulations to understand these phenomena better across the various time and spatial scales. She is involved in a variety of interdisciplinary projects with academic, national lab, and industry collaborators from hydrology, oceanography, statistics, environmental, petroleum, civil and coastal engineering, physics, and materials science. Her research projects were funded by NSF, DOE-NETL and by DOE; her current projects include the 2015-2020 NSF-DMS 1522734 "Phase transitions in porous media across multiple scales", and 2019-2022 NSF-DMS 1912938 on "Modeling with Constraints and Phase Transitions in Porous Media". She believes in "paying it forward" and is engaged in external and university service via editorial work and conference organization: she is the President of Pacific Northwest SIAM Section and Chair of the Organizing Committee for SIAM PNW 2017, the first Biennial conference of the Section, collocated with PNWNAS. In the past she served as Chair (2011-12) and Program Officer (2009-10) of SIAM Activity Group on Geosciences and in other functions.

Clayton Petsche's research includes the study of algebraic and arithmetic dynamical systems on varieties and analytic spaces, as well as the theory of height functions over global fields.

Pohjanpelto works on the theory and applications of generalized symmetries of differential equations. He has studied the structure of symmetries of the electromagnetic field and applied symmetries in the construction of conservation laws and classification of group invariant solutions. He has also used variational bicomplexes to study the correspondence of generalized symmetries of equations in physical and potential formulation.

Applied and computational mathematics, in two distinct research tracts: (1) the application of statistical physics ideas, data sciences, and dynamics on probability distributions, to the understanding complex non-equilibrium physical and human systems and to the development of forecasting tools for these systems; (2) the role of ocean and ocean transport in climate dynamics and in nearshore processes. Example research problems in tract 1: resilience and uncertainty quantification in natural and human complex systems, using data and physical models. Adaptive time series methods for climate and financial data. Extreme/rare events. Dimension and uncertainty reduction in complex systems. Example research problems in tract 2: oil spill dynamics, wave breaking dynamics, wave-generated transport, the role of oceans in global climate. Sediment dynamics. Outside of these tracts he has also done research on wavelets, bone dynamics, blood cell dynamics, public choice and elections, solitary waves, high performance computing.

Schmidt is currently most interested in: natural extensions for continued fractions for various Fuchsian groups (with C. Kraaikamp, and various third co-authors:  I. Smeets,  H. Nakada, and W. Steiner;  with K. Calta; and, most recently with P. Arnoux) and connections between the ergodic theory of billards and 1-forms on algebraic curves and, with the ergodic theory and arithmetic of generalized continued fractions. Recent results include diophantine approximation results for flow directions on translation surfaces (with Y. Bugeaud and P. Hubert, and more recently again with P. Hubert);  a new proof with A. Fisher of a beautiful result of Moeckel.  

Research interests include singular or degenerate nonlinear evolution equations and partial differential equations, variational inequalities, free-boundary problems, and related applications to initial-boundary-value problems of mechanics and diffusion. Current work is focused on the development of multiphysics mathematical models of coupled fluid-solid dynamics, deformable porous media, and upscaled models of transport and flow in heterogeneous media. - Panta Rhei  

Professor Swisher is a researcher in the areas of number theory and combinatorics.   Her current work is focused on mock and quantum modular forms, partition theory, and hypergeometric series, including the interplay between them.

Thomann's research is primarily in problems in partial differential equations arising from fluid mechanics. He also collaborates with colleagues in other departments, as well as in the Mathematics Department, in the development of mathematical models to problems in Ecology, Oceanography, Hydrology and management of Natural Resources.