## Research Groups: Faculty

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.

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 research faculty and their students are interested in investigating and analyzing the actions of people and the contents of their minds in relationship to the task of learning and understanding mathematics or to crafting new or alternative approaches to the teaching and learning of mathematics. The value of the successful mathematics educator to the mathematical community is that new understanding and insights about how people learn and teach mathematics and about how to increase the effectiveness of learning mathematics among mathematical learners is added to the body of mathematics educational knowledge.

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.

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 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.