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.

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.

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 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.