Partial Differential Equations

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More about 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

Seminars

  • A weekly PDE seminar, currently held on Wednesday afternoons
  • Every year the group hosts the Rivière-Fabes Symposium, which brings leading researchers from around the world to campus for a Spring weekend to discuss particularly exciting developments in analysis and PDE

Programs

 

Faculty

Douglas Arnold

Douglas Arnold

McKnight Presidential Professor

[email protected]
Numerical analysis, differential equations, mechanics, computational relativity

Jeffrey Calder

Jeffrey Calder

Associate Professor

[email protected]
partial differential equations, numerical analysis, applied probability, machine learning, image processing and computer vision

Maria-Carme Calderer

Maria-Carme Calderer

Professor

[email protected]
applied mathematics, partial differential equations, dynamical systems, materials sciences, mathematical biology and soft-matter physics

Max Engelstein

Max Engelstein

Associate Professor

[email protected]
harmonic analysis, geometric measure theory, calculus of variations

Hao Jia Receives NSF CAREER Grant Award

Hao Jia

Associate Professor

[email protected]
partial differential equations, regularity, stability, large data asymptotics 

Markus Keel

Markus Keel

Professor

[email protected]
partial differential equations; real, harmonic, and functional analysis

Nicolai Krylov

Nicolai Krylov

Professor Emeritus

[email protected]
partial differential equations, probability

Ru-Yu Lai

Ru-Yu Lai

Associate Professor

[email protected]
inverse problems and partial differential equations

Yulong Lu headshot

Yulong Lu

Assistant Professor

[email protected]
Mathematical foundations of machine learning and data sciences, applied probability and stochastic dynamics, applied analysis and PDEs, Bayesian and computational statistics, inverse problems and uncertainty quantification

Mitch Luskin

Mitchell Luskin

Professor

[email protected]
numerical analysis, scientific computing, applied mathematics, computational physics

Svitlana Mayboroda Square

Svitlana Mayboroda

McKnight Presidential Professor and Northrop Professor

[email protected]
analysis and partial differential equations

Peter Olver

Peter Olver

Professor

[email protected]
Lie groups, differential equations, computer vision, applied mathematics, differential geometry, mathematical physics

Peter Polacik

Peter Polacik

Professor

[email protected]
partial differential equations, dynamical systems

Mikhail Safonov

Mikhail Safonov

Professor

[email protected]
partial differential equations, probability

Arnd Scheel

Arnd Scheel

Professor

[email protected]
dynamical systems, partial differential equations, applied math

Dan Spirn Headshot

Daniel Spirn

Professor

[email protected]
partial differential equations, applied mathematics

Vladimir Sverak

Vladimir Sverak

Distinguished McKnight University Professor

[email protected]
partial differential equations

Jiaping Wang

Jiaping Wang

Professor

[email protected]
differential geometry and partial differential equations

Li Wang

Li Wang

Associate Professor

liwang@umn.edu 
numerical analysis, scientific computing, applied analysis, kinetic theory, optimal transport, inverse problems, scientific machine learning

 


 

Alex Watson headshot

Alex Watson

Assistant Professor

[email protected]
Partial differential equations, mathematical physics, numerical analysis, computational physics, data science

More faculty research areas