This volume presents modern developments in analysis, PDEs and geometric analysis by some of the leading worldwide experts, prominent junior and senior researchers who were invited to be part of the Ghent Analysis & PDE Center Methusalem Seminars from 2021 to 2022. The contributions are from the speakers of the Methusalem Colloquium, Methusalem Junior Seminar and Geometric Analysis Seminar. The volume has two main topics:

1. Analysis and PDEs. The volume presents recent results in fundamental problems for solving partial integro-differential equations in different settings such as Euclidean spaces, manifolds, Banach spaces, and many others. Discussions about the global and local solvability using micro-local and harmonic analysis methods, studies of new techniques and approaches arising from a physical perspective or the mathematical point of view have also been included. Several connected branches arising in this regard are shown.

   2. Geometric analysis. The volume presents studies of modern techniques for elliptic and subelliptic PDEs that in recent times have been used to establish new results in differential geometry and differential topology. These topics involve the intrinsic research in microlocal analysis, geometric analysis, and harmonic analysis abroad. Different problems having relevant geometric information for different applications in mathematical physics and other problems of classification have been considered.

   High quality short research announcements, communications or summaries Provide the most recent results and trends by some leading experts in analysis, PDEs and geometric analysis Serve as a platform for outstanding young researchers in various areas of analysis and PDEs to share their ideas

   Autorentext
   Duván Cardona is a Postdoctoral Fellow of the Reseacrh Foundation - Flanders (FWO) in the Department of Mathematics at Ghent University. His research interests are in Fourier Integral Operators, Pseudo-differential operators, PDEs, Control Theory and Harmonic Analysis, Spectral Theory and Index theory and Geometric Analysis. Previously, he was a PhD student at the Ghent Analysis and PDE Center. He was the receipt of the 2018 Yu-Takeuchi Prize awarded by the Colombian Academy of Sciences. He is currently serving as the president of the Scientific Board for the ICMAM Latin American (2022 - 2024).
   **Joel Restrepo is a postdoctoral researcher at the Ghent Analysis and PDE Center at Ghent University in Belgium. Previously, he was a postdoctoral researcher at different research centers in Russia and Kazakhstan. He has carried out mathematical research in a variety of topics connecting different areas of pure mathematics. His research links several techniques in complex and mathematical analysis, operator theory, harmonic analysis, PDE, etc. He has participated in many international conferences, seminars and other related events and has been involved in the organisation of several of these scientific projects as well. His most notable achievements to date include: 2019 award by the International Society for Analysis, its Applications and Computation (ISSAC) of the ISAAC life membership in view of his achievements and contributions to the potential theory and special functions.
   **Michael Ruzhansky is a Senior Full Professor of Mathematics at Ghent University in Belgium, and a Professor of Mathematics at Queen Mary University of London in the United Kingdom. His research interests mainly lie in Partial Differential Equations, Microlocal and Harmonic Analysis, and Pseudo-Differential Operators on Lie Groups and Manifolds. Previously, he had appointments at Utrecht University, Johns Hopkins University, University of Edinburgh, and Imperial College London. He is the recipient of various awards and fellowships, notably, the Ferran Sunyer i Balaguer Prize in 2014 and 2018, Daiwa Adrian Prize in 2010, and the ISAAC award in 2007. He is serving as the head of the Ghent Analysis & PDE Center of Ghent University.

   Inhalt
   - Part I Geometric Analysis. - Analysis on Noncompact Manifolds and Index Theory: Fredholm Conditions and Pseudodifferential Operators. - Singular Value Decomposition for the X-Ray Transforms on the Reduced Heisenberg Group, and a Two-Radius Theorem. - Nonlocal Functionals with Non-standard Growth. - A Variational Approach to the Hot Spots Conjecture. - Endpoint Sobolev Inequalities for Vector Fields and Cancelling Operators. - Scattering of Maxwell Potentials on Curved Spacetimes. - Part II Analysis and PDEs. - Remark on the Ill-Posedness for KdV-Burgers Equation in Fourier Amalgam Spaces. - -Convergence for the Bi-Laplace-Beltrami Equation on Hypersurfaces. - Bounded Weak Solutions with Orlicz Space Data: An Overview. - Recent Progress on the Mathematical Theory ofWave Turbulence. - Laplace-Beltrami Equation on Lipschitz Hypersurfaces in the Generic Bessel Potential Spaces. - On the Convergence Fourier Series and Greedy Algorithm by Multiplicative System. - Asymptotics of Harmonic Functions in the Absence of Monotonicity Formulas. - Semiregular Non-commutative Harmonic Oscillators: Some Spectral Asymptotic Properties. - Global Compactness, Subcritical Approximation of the Sobolev Quotient, and a Related Concentration Result in the Heisenberg Group. - A Note on Carleson-Hunt Type Theorems for Vilenkin-Fourier Series. - Self-Similar Gravitational Collapse for Polytropic Stars. - Control of Parabolic Equations with Inverse Square Infinite Potential Wells. - On Geometric Estimates for Some Problems Arising from Modeling Pull-in Voltage in MEMS. - A Note on Fractional Powers of the Hermite Operator. - Non-Standard Version of Egorov Algebra of Generalized Functions. - Density Conditions for Coherent State Subsystems of Nilpotent Lie Groups. - Space-Time Mixed Norm Estimates in Riemannian Symmetric Spaces of Non-Compact Type. - Analysis on Compact Symmetric Spaces: Eigenfunctions and Nonlinear Schrödinger Equations. - PartIII Applied Mathematics. - On Empirical Bayes Approach to Inverse Problems. - The Interferon Influence on the Infection Wave Propagation. - Machine Learning-Based Analysis of Human Motions for Parkinson's Disease Diagnostics.