Title: 

An Introduction to Computer-Assisted Proofs in Nonlinear Analysis

Abstract: 

This four-lecture series serves as an accessible introduction to computer-assisted proofs (CAPs) in nonlinear analysis. CAPs are a modern mathematical tool that blend theoretical analysis with validated numerics to establish the existence (and sometimes uniqueness or stability) of solutions to complex nonlinear problems. They have become increasingly relevant in fields such as dynamical systems, partial differential equations, and bifurcation theory, where classical analytical techniques alone may be insufficient to obtain rigorous results.

The first lecture introduces the core ideas of CAPs in the setting of finite-dimensional problems, emphasizing the Newton-Kantorovich theorem and interval arithmetic. The second and third lectures transition to infinite-dimensional systems, focusing on equilibria of the Swift-Hohenberg equation, a prototypical reaction-diffusion model used in the study of pattern formation. These lectures cover the equation posed on both bounded intervals and the Euclidean plane, incorporating tools such as Fourier series and Banach algebras of sequence spaces.

Each lecture is complemented by an interactive lab session designed to reinforce the theoretical material with computational demonstrations. Participants will explore validated numerics and rigorous solution verification for both ordinary and partial differential equations. No prior experience with CAPs is required. However, a background in differential equations and basic numerical analysis will be helpful. Participants are welcome to bring their own laptops to experiment with examples provided during the labs.

Schedule: 

Lecture I - Newton-Kantorovich Theorem, Finite dimensional problems & Interval Arithmetic

Lab I - CAPs for finite dimensional problems, Examples, Eigenvalue problems

Lecture II - Equilibria in the 1D Shift-Hohenberg equation posed on the interval, Fourier series, Banach algebra of sequences spaces

Lab II - CAPs for equilibria of a PDE on the interval with a quadratic nonlinearity

Lecture III - Homoclinic in the 1D Shift-Hohenberg equation posed on the line

Lab III - CAPs for equilibria of a PDE on the line with a quadratic nonlinearity

Lecture IV - Open problems/frontiers in computer-assisted proofs: Equilibria in the 2D Shift-Hohenberg on the plane + others