Geometric And Functional Inequalities And Recent Topics In Nonlinear Pdes

This volume contains the proceedings of the virtual conference on Geometric and Functional Inequalities and Recent Topics in Nonlinear PDEs, held from February 28–March 1, 2021, and hosted by Purdue University, West Lafayette, IN. The mathematical content of this volume is at the intersection of viscosity theory, Fourier analysis, mass transport theory, fractional elliptic theory, and geometric analysis. The reader will encounter, among others, the following topics: the principal-agent problem; Maxwell's equations; Liouville-type theorems for fully nonlinear elliptic equations; a doubly monotone flow for constant width bodies; and the edge dislocations problem for crystals that describes the equilibrium configurations by a nonlocal fractional Laplacian equation.

This volume contains the proceedings of the conference Recent Advances and New Directions in the Interplay of Noncommutative Algebra and Geometry, held from June 20–24, 2022, at the University of Washington, Seattle, in honor of S. Paul Smith's 65th birthday. The articles reflect the wide interests of Smith and provide researchers and graduate students with an indispensable overview of topics of current interest. Specific fields covered include: noncommutative algebraic geometry, representation theory, Hopf algebras and quantum groups, the elliptic algebras of Feigin and Odesskii, Calabi-Yau algebras, Artin-Schelter regular algebras, deformation theory, and Lie theory. In addition to original research contributions the volume includes an introductory essay reviewing Smith's research contributions in these fields, and several survey articles.

This volume contains the proceedings of the virtual AMS Special Session on Fractal Geometry and Dynamical Systems, held from May 14–15, 2022. The content covers a wide range of topics. It includes nonautonomous dynamics of complex polynomials, theory and applications of polymorphisms, topological and geometric problems related to dynamical systems, and also covers fractal dimensions, including the Hausdorff dimension of fractal interpolation functions. Furthermore, the book contains a discussion of self-similar measures as well as the theory of IFS measures associated with Bratteli diagrams. This book is suitable for graduate students interested in fractal theory, researchers interested in fractal geometry and dynamical systems, and anyone interested in the application of fractals in science and engineering. This book also offers a valuable resource for researchers working on applications of fractals in different fields.

This volume collects the notes of the CIME course "Nonlinear PDE’s and applications" held in Cetraro (Italy) on June 23–28, 2008. It consists of four series of lectures, delivered by Stefano Bianchini (SISSA, Trieste), Eric A. Carlen (Rutgers University), Alexander Mielke (WIAS, Berlin), and Cédric Villani (Ecole Normale Superieure de Lyon). They presented a broad overview of far-reaching findings and exciting new developments concerning, in particular, optimal transport theory, nonlinear evolution equations, functional inequalities, and differential geometry. A sampling of the main topics considered here includes optimal transport, Hamilton-Jacobi equations, Riemannian geometry, and their links with sharp geometric/functional inequalities, variational methods for studying nonlinear evolution equations and their scaling properties, and the metric/energetic theory of gradient flows and of rate-independent evolution problems. The book explores the fundamental connections between all of these topics and points to new research directions in contributions by leading experts in these fields.

This volume contains the proceedings of the AMS-EMS-SMF Special Session on Advances in Functional Analysis and Operator Theory, held July 18–22, 2022, at the Université de Grenoble-Alpes, Grenoble, France. The papers reflect the modern interplay between differential equations, functional analysis, operator algebras, and their applications from the dynamics to quantum groups to number theory. Among the topics discussed are the Sturm-Liouville and boundary value problems, axioms of quantum mechanics, $C^{*}$-algebras and symbolic dynamics, von Neumann algebras and low-dimensional topology, quantum permutation groups, the Jordan algebras, and the Kadison–Singer transforms.

This volume contains the proceedings of the AMS-EMS-SMF Special Session on Recent Advances in Diffeologies and Their Applications, held from July 18–20, 2022, at the Université de Grenoble-Alpes, Grenoble, France. The articles present some developments of the theory of diffeologies applied in a broad range of topics, ranging from algebraic topology and higher homotopy theory to integrable systems and optimization in PDE. The geometric framework proposed by diffeologies is known to be one of the most general approaches to problems arising in several areas of mathematics. It can adapt to many contexts without major technical difficulties and produce examples inaccessible by other means, in particular when studying singularities or geometry in infinite dimension. Thanks to this adaptability, diffeologies appear to have become an interesting and useful language for a growing number of mathematicians working in many different fields. Some articles in the volume also illustrate some recent developments of the theory, which makes it even more deep and useful.

This volume contains the proceedings of the Alexandre Vinogradov Memorial Conference on Diffieties, Cohomological Physics, and Other Animals, held from December 13–17, 2021, at the Independent University of Moscow and Moscow State University, Moscow, Russia. The papers are devoted to various interrelations of nonlinear PDEs with geometry and integrable systems. The topics discussed are: gravitational and electromagnetic fields in General Relativity, nonlocal geometry of PDEs, Legendre foliated cocycles on contact manifolds, presymplectic gauge PDEs and Lagrangian BV formalism, jet geometry and high-order phase transitions, bi-Hamiltonian structures of KdV type, bundles of Weyl structures, Lax representations via twisted extensions of Lie algebras, energy functionals and normal forms of knots, and differential invariants of inviscid flows. The companion volume (Contemporary Mathematics, Volume 789) is devoted to Algebraic and Cohomological Aspects of PDEs.

This volume contains the proceedings of the conference on Formal and Analytic Solutions of Diff. Equations, held from June 28–July 2, 2021, and hosted by University of Alcalá, Alcalá de Henares, Spain. The manuscripts cover recent advances in the study of formal and analytic solutions of different kinds of equations such as ordinary differential equations, difference equations, $q$-difference equations, partial differential equations, moment differential equations, etc. Also discussed are related topics such as summability of formal solutions and the asymptotic study of their solutions. The volume is intended not only for researchers in this field of knowledge but also for students who aim to acquire new techniques and learn recent results.