Bücher der Reihe Tutorials, Schools, and Workshops in the Mathematical Sciences

This book is based on lectures given at the first edition of the Domoschool, the International Alpine School in Mathematics and Physics, held in Domodossola, Italy, in July 2018. These are extended versions of lectures given at the Domoschool, written by well-known experts in mathematics and physics related to General Relativity.

One of the most important questions concerning the foundations of physics, especially since the discovery of relativity and quantum theory, is the nature and role of time.

This book presents four survey articles on various aspects of open quantum systems, specifically addressing quantum Markovian processes, Feller semigroups and nonequilibrium dynamics.

This volume presents lectures given at the Summer School Wisla 18: Nonlinear PDEs, Their Geometry, and Applications, which took place from August 20 - 30th, 2018 in Wisla, Poland, and was organized by the Baltic Institute of Mathematics.

This volume collects lecture notes and research articles from the International Autumn School on Computational Number Theory, which was held at the Izmir Institute of Technology from October 30th to November 3rd, 2017 in Izmir, Turkey.

This book consists of two lecture notes on geometric flow equations (O. Schnürer) and Lorentzian geometry - holonomy, spinors and Cauchy Problems (H. Baum and T. Leistner) written by leading experts in these fields. It grew out of the summer school "Geometric flows and the geometry of space-time" held in Hamburg (2016) and provides an excellent introduction for students of mathematics and theoretical physics to important themes of current research in global analysis, differential geometry and mathematical physics

Part I: Introduction to Orthogonal Polynomials.- An Introduction to Orthogonal Polynomials.- Classical Continuous Orthogonal Polynomials.- Generating Functions and Hypergeometric Representations of Classical Continuous Orthogonal Polynomials.- Properties and Applications of the Zeros of Classical Continuous Orthogonal Polynomials.- Inversion, Multiplication and Connection Formulae of Classical Continuous Orthogonal Polynomials.- Classical Orthogonal Polynomials of a Discrete and a q-Discrete Variable.- Computer Algebra, Power Series and Summation.- On the Solutions of Holonomic Third-Order Linear Irreducible Differential Equations in Terms of Hypergeometric Functions.- The Gamma Function.- Part II: Recent Research Topics in Orthogonal Polynomials and Applications.- Hypergeometric Multivariate Orthogonal Polynomials.- Signal Processing, Orthogonal Polynomials, and Heun Equations.- Some Characterization Problems Related to Sheffer Polynomial Sets.- From Standard Orthogonal Polynomials to Sobolev Orthogonal Polynomials: The Role of Semiclassical Linear Functionals.- Two Variable Orthogonal Polynomials and Fej├⌐r-Riesz Factorization.- Exceptional Orthogonal Polynomials and Rational Solutions to Painlev├⌐ Equations.- (R, p, q)-Rogers-Szeg├╢ and Hermite Polynomials, and Induced Deformed Quantum Algebras.- Zeros of Orthogonal Polynomials.- Properties of Certain Classes of Semiclassical Orthogonal Polynomials.- Orthogonal Polynomials and Computer Algebra.- Spin Chains, Graphs and State Revival.- An Introduction to Special Functions with Some Applications to Quantum Mechanics.- Orthogonal and Multiple Orthogonal Polynomials, Random Matrices, and Painlev├⌐ Equations.

The motion of water is governed by a set of mathematical equations which are extremely complicated and intractable. This is not surprising when one considers the highly diverse and intricate physical phenomena which may be exhibited by a given body of water. Recent mathematical advances have enabled researchers to make major progress in this field, reflected in the topics featured in this volume. Cutting-edge techniques and tools from mathematical analysis have generated strong rigorous results concerning the qualitative and quantitative physical properties of solutions of the governing equations. Furthermore, accurate numerical computations of fully-nonlinear steady and unsteady water waves in two and three dimensions have contributed to the discovery of new types of waves. Model equations have been derived in the long-wave and modulational regime using Hamiltonian formulations and solved numerically.This book brings together interdisciplinary researchers working in the field of nonlinear water waves, whose contributions range from survey articles to new research results which address a variety of aspects in nonlinear water waves. It is motivated by a workshop which was organised at the Erwin Schrödinger International Institute for Mathematics and Physics in Vienna, November 27-December 7, 2017. The key aim of the workshop was to describe, and foster, new approaches to research in this field. This is reflected in the contents of this book, which is aimed to stimulate both experienced researchers and students alike.

This book presents four survey articles on various aspects of open quantum systems, specifically addressing quantum Markovian processes, Feller semigroups and nonequilibrium dynamics.

This volume presents lectures given at the Wisla 19 Summer School: Differential Geometry, Differential Equations, and Mathematical Physics, which took place from August 19 - 29th, 2019 in Wisla, Poland, and was organized by the Baltic Institute of Mathematics.

This volume presents an accessible overview of mathematical control theory and analysis of PDEs, providing young researchers a snapshot of these active and rapidly developing areas. The chapters are based on two mini-courses and additional talks given at the spring school "e;Trends in PDEs and Related Fields"e; held at the University of Sidi Bel Abbes, Algeria from 8-10 April 2019. In addition to providing an in-depth summary of these two areas, chapters also highlight breakthroughs on more specific topics such as:Sobolev spaces and elliptic boundary value problemsLocal energy solutions of the nonlinear wave equationGeometric control of eigenfunctions of Schrodinger operatorsResearch in PDEs and Related Fields will be a valuable resource to graduate students and more junior members of the research community interested in control theory and analysis of PDEs.

This volume guides early-career researchers through recent breakthroughs in mathematics and physics as related to general relativity. Chapters are based on courses and lectures given at the July 2019 Domoschool, International Alpine School in Mathematics and Physics, held in Domodossola, Italy, which was titled ¿Einstein Equations: Physical and Mathematical Aspects of General Relativity¿. Structured in two parts, the first features four courses from prominent experts on topics such as local energy in general relativity, geometry and analysis in black hole spacetimes, and antimatter gravity. The second part features a variety of papers based on talks given at the summer school, including topics like:Quantum ergosphereGeneral relativistic Poynting-Robertson effect modellingNumerical relativityLength-contraction in curved spacetimeClassicality from an inhomogeneous universeEinstein Equations: Local Energy, Self-Force, and Fields in General Relativity will be a valuable resource for students and researchers in mathematics and physicists interested in exploring how their disciplines connect to general relativity.

This volume presents lectures given at the Wis¿a 20-21 Winter School and Workshop: Groups, Invariants, Integrals, and Mathematical Physics, organized by the Baltic Institute of Mathematics. The lectures were dedicated to differential invariants ¿ with a focus on Lie groups, pseudogroups, and their orbit spaces ¿ and Poisson structures in algebra and geometry and are included here as lecture notes comprising the first two chapters. Following this, chapters combine theoretical and applied perspectives to explore topics at the intersection of differential geometry, differential equations, and category theory. Specific topics covered include:The multisymplectic and variational nature of Monge-Ampère equations in dimension fourIntegrability of fifth-order equations admitting a Lie symmetry algebraApplications of the van Kampen theorem for groupoids to computation of homotopy types of striped surfacesA geometric framework to compare classical systemsof PDEs in the category of smooth manifoldsGroups, Invariants, Integrals, and Mathematical Physics is ideal for graduate students and researchers working in these areas. A basic understanding of differential geometry and category theory is assumed.