Department members engage in cutting-edge research on a wide variety of topics in mathematics and its applications. Topics continually evolve to reflect emerging interests and developments, but can roughly grouped into the following areas.

## Algebra, Combinatorics, and Geometry

__Algebra, combinatorics, and geometry__ are areas of very active research at the University of Pittsburgh.

## Analysis and Partial Differential Equations

The __research of the analysis group__ covers functional analysis, harmonic analysis, several complex variables, partial differential equations, and analysis on metric and Carnot-Caratheodory spaces.

## Applied Analysis

The department is a leader in the __analysis of systems of nonlinear differential equations and dynamical systems__ that arise in modeling a variety of physical phenomena. They include problems in biology, chemistry, phase transitions, fluid flow, flame propagation, diffusion processes, and pattern formation in nonlinear stochastic partial differential equations.

## Mathematical Biology

The __biological world__ stands as the next great frontier for mathematical modeling and analysis. This group studies complex systems and dynamics arising in various biological phenomena.

## Mathematical Finance

A rapidly growing area of __mathematical finance__ is Quantitative Behavioral Finance. The high-tech boom and bust of the late 1990s followed by the housing and financial upheavals of 2008 have made a convincing case for the necessity of adopting broader assumptions in finance.

## Numerical Analysis and Scientific Computing

The diversity of this group is reflected in its research interests: __numerical analysis of partial differential equations__, adaptive methods for scientific computing, computational methods of fluid dynamics and turbulence, numerical solution of nonlinear problems arising from porous media flow and transport, optimal control, and simulation of stochastic reaction diffusion systems.

## Topology and Differential Geometry

__Research in analytic topology__ continues in the broad area of generalized metric spaces. This group studies relativity theory and differential geometry, with emphasis on twistor methods, as well as geometric and topological aspects of quantum field theory, string theory, and M-theory.