Algebra, an immense area of mathematics, is the study of mathematical symbols and the rules for regulating these symbols. At WVU, we conduct research in commutative algebra. We focus on commutative rings and modules over them by using homological and representation theoretical techniques. The theory of commutative algebra stems from the work of eminent mathematicians such as David Hilbert and Emmy Noether, and it plays an important role in algebraic number theory and complex analysis. The field was enriched by its relation to several areas of mathematics including combinatorics, homological algebra, mathematical physics, representation theory and modern research areas such as algebraic graph theory and algebraic topology. Commutative algebra is one of the essential foundations of modern algebraic geometry. It is the main tool to study the theory of schemes.
- Ela Celikbas: Commutative Algebra, Representation Theory
- Olgur Celikbas: Commutative Algebra, Homological Algebra
- Tony Se: Commutative Algebra, Homological Algebra
Research by members of this group involves the use of mathematical analysis to solve real-world problems. Our work is motivated by applications to fluid dynamics, finance, biology, electro-magnetism, celestial mechanics and other industries. Faculty conduct research on the theory and applications of ordinary and partial differential equations, difference equations and metric spaces. The main focus lies on existence, uniqueness, regularity, stability and other properties of solutions. Functional analysis, harmonic analysis and dynamical systems provide some of the necessary tools in the investigation.
- Harumi Hattori: Partial Differential Equations, Conservation Laws and Shock Wave
- Harry Gingold: Continuous and Discrete Dynamical Systems, Mathematical Physics, Factorization of Scalar and Matrix Power Series, Foundation (Geometry)
- Dening Li: Partial Differential Equations, Shock Theory
- Charis Tsikkou: Hyperbolic and Mixed Type Partial Differential Equations, Conservation Laws
- Adrian Tudorascu: Partial Differential Equations, Optimal Transport
- Qingtian Zhang: Partial Differential Equations, Hyperbolic Conservation Law, Fluid Dynamics
Rather than defining new mathematical objects and studying their features, research in applied mathematics aims to employ or adapt existing mathematical concepts to help solve mathematical problems emerging from various fields, from physical and life sciences to engineering, social science, and, increasingly, everyday life.
Applied mathematicians at WVU are active in dynamical systems and statistics, biomathematics (modeling and simulation of cellular processes), design and optimization of cooperative robot behavior, and crowd dynamics.
- Krzysztof Ciesielski: Image Processing
- Marjorie Darrah: Algorithm Development
- Harvey Diamond: Spline Functions, Applied Probability
- Adam Halasz: Mathematical Biology, Stochastic Simulations, Swarm Robotics
- Daewa Kim: Pedestrian Dynamics, Mathematical Modeling in Social Systems
- Casian Pantea: Mathematical Biology, Dynamical Systems
Graph theory is the study of graphs (also known as networks), used to model pairwise relations between objects, while combinatorics is an area of mathematics mainly concerned with counting and properties of discrete structures. Both have applications in computer science, data science, biology, social network theory and neuroscience. They are closely related to many other areas of mathematics including algebra, probability, topology, and geometry. Infinite combinatorics is also closely related to set theory.
Behrooz Bagheri Ghavam Abadi: Graph Theory
- John Goldwasser: Combinatorics, Graph Theory
- Hong-Jian Lai: Graph Theory, Matroid Theory
- Rong Luo: Discrete Mathematics
- Kevin Milans: Combinatorics, Graph Theory, and Partially Ordered Sets
- Jerzy Wojciechowski: Graph Theory, Combinatorics
- Cun-Quan Zhang: Graph Theory, Algorithms, Bioinformatics, Data Mining
Research in mathematics education seeks to understand the nature of mathematical thinking, teaching and learning as well as how to improve mathematics instruction. WVU's mathematics education researchers focus on many different areas of mathematical thinking, such as learning of mathematical proof, learning of key topics in calculus, using gestures effectively in mathematics instruction, effective use of technology for mathematics learning and effective graduate student preparation. Others focus on K-12 outreach to teachers and students including dual credit high school courses and professional development activities for teachers.
- Marjorie Darrah: Educational Technology, K-12 Outreach
- Ben Davies: Undergraduate Mathematics Education, Proof Comprehension, Philosophy of Mathematical Practice
- Jessica Deshler: Undergraduate Mathematics Education, Gender Equity in Mathematics, Calculus Student Learning, Graduate Student Professional Development, Scholarship of Teaching and Learning in Mathematics
- Nicole Engelke: Undergraduate Mathematics Education, Calculus Student Learning, and Proof Learning
- Tim McEldowney: Undergraduate Mathematics Education, Graduate Mathematics Education, Equity in Mathematics Education
- David Miller: Undergraduate Mathematics Education, Proof Learning and Assessment
- Laura Pyzdrowski: Mathematics/STEM Education, K-12 Outreach, Distance Learning, Educational Technology
- Vicki Sealey: Undergraduate Mathematics Education, Calculus Student Learning
Many models in engineering and physical sciences can be explained and approximated using the probabilistic and statistical tools. Constructions of optimal estimates as well as powerful goodness of fit tests represent main objects of our investigation. In particular, we are concentrated on developing efficient procedures for recovering unobserved functions and probability distributions in ill-posed and statistical inverse problems, like deconvolution, demixing and error-in variable models. The famous classical moment problem and the problem of numerical approximation of the radon transform inversion have a various applications in different fields including the image analysis and Computer Tomography.
- Robert Mnatsakanov: Statistical Inverse Problems, The Hausdorff and Stieltjes Moment Problems, Applications of Atatistical Methods in Actuarial and Financial Mathematics
Set theory and general topology provide fundamental theoretical structures on which other areas of mathematics are constructed. Set theory is concerned with the systematic analysis of the most basic or fundamental concepts of mathematics and their conceptual integrity and hierarchy. General topology together with other areas of abstract mathematics provide a bridge between set theory and the applied areas. Our research activity spans the full range from set theory through general topology to abstract real analysis.