- Analysis of Diffuse Interface Models
- Computer Assisted Proofs in PDEs
- Control Theory
- Evolution Equations and Dynamical Systems
- Free Boundary Problems
- Game Theory and Convex Optimization
- Geometric Analysis and Riemannian geometry
- Inverse Problems for Partial Differential Equations
- Mathematical Models for Suspension Bridges
- Noncommutative Analysis
- Nonlinear Diffusions
- Nonlinear Elliptic PDEs and Degenerate Equations
- Shape Optimization
Computer Assisted Proofs in PDEs (G. Arioli)
Computer assisted proofs do not differ in principle to traditional proofs, but only in execution.
A proof is called computer assisted if it consists in finitely many elementary operations, but their number is so large that, although each step may be written down explicitly, it is only practical to perform such operations with a computer.
The are two main reasons to use computer assisted proofs: they allow to solve problems otherwise too hard, and they provide detailed qualitative information.
Computer assisted proofs have been applied e.g. to prove symmetry properties of solution of elliptic PDEs, the bifurcation graph of dissipative PDEs, existence and stability of solutions of hyperbolic PDEs.