The course is intended for people with some mathematical maturity, like graduate students and postdocs, that are interested in an introduction to partial differential equations and an overview of some research directions. Specific expertise in PDEs is not assumed or required.

Topics may include:

Introduction:

• Origins in physics
• First and second order equations and systems
• The main types– elliptic, parabolic and hyperbolic – and their specific features
• Linear and nonlinear PDEs
• What does it mean to solve a PDE?

Elliptic PDEs

• Laplace equation as paradigm: to what extent is it useful?
• The questions of existence and regularity of solutions; weak solution concepts
• Maximum principle and Harnack inequality
• Variational methods
• Eigenvalue problems, Green functions
• Reaction-diffusion equations, Turing mechanism
• Approximations, finite elements
• Singularities

Parabolic PDEs

• Diffusion processes
• Relations with elliptic PDEs
• Semigroups
• Stochastic aspects: Fokker-Planck equations, random processes, heat kernels and Brownian motion

Hyperbolic PDEs

• First and second order
• Wave phenomena, like traveling waves
• Singularities

Geometry and PDEs – PDEs and Geometry

• Spaces for finding solutions: Banach spaces and their geometry, convergence notions
• Convexity and generalizations
• Spaces of solutions: from uniqueness and saddle points to moduli spaces
• Applications of PDEs in geometry

Additional topics, as emerging during the course

Some of the topics are presented in [1]. Additional references may be provided during the course.

References

[1] Jürgen Jost, Partial Differential Equations, 3rd ed., Springer, 2013

This course runs from September 7 to September 11, with daily sessions from 02:30pm to 05:00pm.

Registration is open till August 31, 2026.