Partial Differential Equations

Partial differential equations (PDE) arise in a wonderful variety of circumstances. A scientist weighs area-specific properties and laws to find that a PDE encodes some interesting aspects of their problem.  

Research in PDE can be motivated by an important range of questions which arise:

• Are there important phenomena that currently lie hidden from the scientific theory which can be elucidated and investigated by the analysis of the PDE?  
• Does the scientific modeling which produced this PDE make sense, or might the modeling be incomplete or inconsistent?  
• Can numerical simulation of the problem be established or improved using novel analysis of the PDE?  

Research in partial differential equations at Minnesota takes up these opportunities and challenges. The PDE group here spans and blends a tremendous variety of tools within analysis, geometry, probability, and applied mathematics.

PDE research in the School of Mathematics develops new analytical technology to advance the world’s understanding of the many different types of PDE and the phenomena encoded by these PDE in fields ranging from dynamical systems to differential geometry, from geometric measure theory to general relativity. For example, the group has developed fundamental analytical tools that illuminate parabolic, elliptic, hyperbolic, and dispersive PDE.

Further examples of applied areas where Minnesota PDE hosts cutting edge work include:

• Fluid dynamics
• Image processing
• Inverse problems
• Waves in disordered media
• Pattern formation in complex physical systems
• Numerical analysis
• Materials science
• Math biology
• Machine learning