# 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

McKnight Presidential Professor

arnold@umn.edu

Numerical analysis, differential equations, mechanics, computational relativity

### Jeffrey Calder

Associate Professor

jwcalder@umn.edu

partial differential equations, numerical analysis, applied probability, machine learning, image processing and computer vision

### Maria-Carme Calderer

Professor

mcc@umn.edu

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

### Max Engelstein

Assistant Professor

mengelst@umn.edu

harmonic analysis, geometric measure theory, calculus of variations

### Hao Jia

Associate Professor

jia@umn.edu

partial differential equations, regularity, stability, large data asymptotics

### Markus Keel

Professor

keel@umn.edu

partial differential equations; real, harmonic, and functional analysis

### Yulong Lu

Assistant Professor

yulonglu@umn.edu

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

### Mitchell Luskin

Professor

luskin@umn.edu

numerical analysis, scientific computing, applied mathematics, computational physics

### Svitlana Mayboroda

McKnight Presidential Professor and Northrop Professor

svitlana@umn.edu

analysis and partial differential equations

### Peter Olver

Professor

olver@umn.edu

Lie groups, differential equations, computer vision, applied mathematics, differential geometry, mathematical physics

### Vladimir Sverak

Distinguished McKnight University Professor

sverak@umn.edu

partial differential equations

### Li Wang

Associate Professor

liwang@umn.edu

numerical analysis, scientific computing, applied analysis, kinetic theory, optimal transport, inverse problems, scientific machine learning

### Alex Watson

Assistant Professor

abwatson@umn.edu

Partial differential equations, mathematical physics, numerical analysis, computational physics, data science