This volume presents contributions of modern developments in analysis, PDEs and geometric analysis by leading experts, as well as distinguished junior and senior researchers. The book includes contributions from speakers at the Methusalem Colloquium, Methusalem Junior Seminar, and Geometric Analysis Seminar, all of which were held at the Department of Mathematics, Ghent University from 2023 to 2024. Additionally, it features colloquia and mini-courses delivered by visiting and invited guests. The book has two main directions:Analysis and PDEs: Covers topics in noncommutative analysis and geometry, functional and harmonic analysis, geometry and spectral methods, and applications to mathematical physics. Geometric Analysis: Covers topics in modern analytic techniques applied to elliptic, hypoelliptic, and subelliptic PDEs. A central theme of this book is the role of pseudo-differential operators in spectral analysis of geometric structures, with direct implications for index theory, noncommutative geometry, and quantization.