Faculty Research Groups
Analysis
Analysis encompasses a large part of mathematics and might be defined as the study
of limit processes. Thus calculus and differential equations are basic analysis courses,
and questions arising from analysis have led to the development of many other fields.
Workers in mathematical analysis at OSU conduct research in a variety of areas. There
are researchers in functional analysis, complex/harmonic analysis, several complex
variables, approximation theory, and differential equations.
Commutative Algebra
The commutative algebra group at OSU studies ideals in polynomial rings over a field.
We are especially interested in combinatorial commutative algebra, a relatively new
area in which researchers use tools from combinatorics to answer questions in algebra
and vice versa. Our work often focuses on monomial ideals, articularly understanding
their free resolutions.
Lie Theory/Representation Theory
Lie theory originated from an attempt to use continuous groups of symmetries to study
differential equations. The subject now plays an important role in many areas of mathematics,
such as mathematical physics, number theory and geometry. Members of OSU math department
study Lie theory as it relates to invariant theory, symmetries in systems of differential
equations and automorphic forms. Several OSU researchers investigate the structure
and invariants of representations of semisimple Lie groups through computational and
geometric methods.
Mathematics Education
The mathematics education research group at Oklahoma State University conducts educational
research related to teaching and learning mathematics at the undergraduate level.
Several members of our research team are involved in a statewide project to improve
mathematics instructor professional development. Our group also includes practitioners
who are serving as course coordinators and are involved with mentoring TAs and the preparation of preservice teachers.
Number Theory
Number theory is the study of all mathematics arising from the arithmetic of the ordinary
integers. It is at once one of the most ancient disciplines, with integral Pythagorean
triples appearing on ancient cuneiform tablets which are thousands of years old, and
at the same time one of the most active and modern areas of study including the most
famous problems of mathematics such as Fermat's Last Theorem, the Twin Primes Conjecture,
the Riemann Hypothesis, and the grand system of conjectures known as the Langlands
Program. Modern number theory is now also involved in vital applications throughout
society including digital communications and security systems.
Numerical Analysis
The numerical analysis group at OSU focuses mainly on the study of numerical methods
for partial differential equations. Topics we have been working on include continuous
and discontinuous Galerkin methods, the finite volume methods, a priori and a posteriori
error estimations, least squares methods, various preconditioning techniques, and
numerical implementations. We also have extended interests in other related topics
such as finite difference methods, numerical linear algebra, and large-scale computing.
Accurate and efficient numerical methods can be used to successfully simulate many
complicated physical processes in areas such as solid and fluid mechanics, surface
sciences, electromagnetism, and mathematical finance, etc.
Partial Differential Equations
The partial differential equations (PDEs) group here at OSU focuses on the analysis
and applications of several nonlinear PDEs, especially those arising in fluid mechanics,
geophysics, astrophysics, meteorology and other science and engineering practice.
The particular PDEs that the faculty members here have worked on include the Navier-Stokes
equations, the surface quasi-geostrophic equations, the Boussinesq equations, the
magnetohydrodynamics equations and other related equations. These PDEs have been at
the center of numerous analytical, experimental, and numerical investigations. One
of the most fundamental problems concerning these PDEs is whether their solutions
are globally regular or they develop singularities in a finite-time. The regularity
problem can be extremely difficult, as in the case of the 3D Navier-Stokes equations.
The global regularity problem on the 3D Navier-Stokes equations is one of the Millennium
Prize Problems. In addition, the PDEs group here is also interested in the numerical
computations and analysis of the aforementioned PDEs.
Topology
Topology research at OSU focuses on knot theory and three-dimensional manifolds, using
combinatorial, geometric and algebraic tools. Particular topics include triangulations,
embedded surfaces and both classical and quantum invariants.