Direct and Inverse Problems with Applications

This volume presents the extended abstracts from the 2024 Summer School organized by the Ghent Analysis and PDE Center. The school focused on equipping participants with a broad spectrum of mathematical tools for addressing both direct and inverse problems across various fields. Through a combination of lectures, problem-solving sessions, and collaborative discussions, the program fostered the development of innovative methods and techniques. The lectures also include broader related topics in mathematical analysis and partial differential equations, offering a comprehensive perspective on current research directions in the field.

Fourier algebras and homomorphisms. - Dispersion phenomena and applications to evolution equations. - Subriemannian geometry and analysis of hypoelliptic PDE. - Stability results for Sobolev, logarithmic Sobolev, and related inequalities. - Semiconcavity, viscosity solutions and the square distance in Carnot groups. - Direct and inverse nonstationary scattering problems for Dirac-type system. - Quantitative homogenisation for differential equations with highly anisotropic partially degenerating coefficients. - Rigidity results for evolution PDEs on homogeneuos Lie groups. - An overview of dualities in non-commutative harmonic analysis. - Colombeau type extensions, assymptotic scales. - Analytical solutions to the Laplace equation on a hemispherical domain. - Uniform spectral asymptotics for high-contrast periodic media. - Asymptotic mean value formulas for the -Laplacian in the Euclidean space and in the Heisenberg Group. - Global solutions for a class of nonlinear evolution equations in supercritical spaces. - An index transform method for solutions of the boundary value problems in a wedge.