Latin American Mathematics Series – serie

This proceedings volume contains peer-reviewed, selected papers and surveys presented at the conference Spectral Theory and Mathematical Physics (STMP) 2018 which was held in Santiago, Chile, at the Pontifical Catholic University of Chile in December 2018. The original works gathered in this volume reveal the state of the art in the area and reflect the intense cooperation between young researchers in spectral theoryand mathematical physics and established specialists in this field. The list of topics covered includes: eigenvalues and resonances for quantum Hamiltonians; spectral shift function and quantum scattering; spectral properties of random operators; magnetic quantum Hamiltonians; microlocal analysis and its applications in mathematical physics. This volume can be of interest both to senior researchers and graduate students pursuing new research topics in Mathematical Physics.

This textbook provides an essential introduction to Lie groups, presenting the theory from its fundamental principles. Lie groups are a special class of groups that are studied using differential and integral calculus methods. As a mathematical structure, a Lie group combines the algebraic group structure and the differentiable variety structure. Studies of such groups began around 1870 as groups of symmetries of differential equations and the various geometries that had emerged. Since that time, there have been major advances in Lie theory, with ramifications for diverse areas of mathematics and its applications.Each chapter of the book begins with a general, straightforward introduction to the concepts covered; then the formal definitions are presented; and end-of-chapter exercises help to check and reinforce comprehension. Graduate and advanced undergraduate students alike will find in this book a solid yet approachable guide that will help them continue their studies with confidence.

This concise textbook introduces the reader to advanced mathematical aspects of general relativity, covering topics like Penrose diagrams, causality theory, singularity theorems, the Cauchy problem for the Einstein equations, the positive mass theorem, and the laws of black hole thermodynamics. It emerged from lecture notes originally conceived for a one-semester course in Mathematical Relativity which has been taught at the Instituto Superior Técnico (University of Lisbon, Portugal) since 2010 to Masters and Doctorate students in Mathematics and Physics. Mostly self-contained, and mathematically rigorous, this book can be appealing to graduate students in Mathematics or Physics seeking specialization in general relativity, geometry or partial differential equations. Prerequisites include proficiency in differential geometry and the basic principles of relativity. Readers who are familiar with special relativity and have taken a course either inRiemannian geometry (for students of Mathematics) or in general relativity (for those in Physics) can benefit from this book.

This concise textbook introduces the reader to advanced mathematical aspects of general relativity, covering topics like Penrose diagrams, causality theory, singularity theorems, the Cauchy problem for the Einstein equations, the positive mass theorem, and the laws of black hole thermodynamics. It emerged from lecture notes originally conceived for a one-semester course in Mathematical Relativity which has been taught at the Instituto Superior Técnico (University of Lisbon, Portugal) since 2010 to Masters and Doctorate students in Mathematics and Physics. Mostly self-contained, and mathematically rigorous, this book can be appealing to graduate students in Mathematics or Physics seeking specialization in general relativity, geometry or partial differential equations. Prerequisites include proficiency in differential geometry and the basic principles of relativity. Readers who are familiar with special relativity and have taken a course either inRiemannian geometry (for students of Mathematics) or in general relativity (for those in Physics) can benefit from this book.

This concise textbook gathers together key concepts and modern results on the theory of holomorphic foliations with singularities, offering a compelling vision on how the notion of foliation, usually linked to real functions and manifolds, can have an important role in the holomorphic world, as shown by modern results from mathematicians as H. Cartan, K. Oka, T. Nishino, and M. Suzuki.The text starts with a gentle presentation of the classical notion of foliations, advancing to holomorphic foliations and then holomorphic foliations with singularities. The theory behind reduction of singularities is described in detail, as well the cases for dynamics of a local diffeomorphism and foliations on complex projective spaces. A final chapter brings recent questions in the field, as holomorphic flows on Stein spaces and transversely homogeneous holomorphic foliations, along with a list of open questions for further study and research. Selected exercises at the end of each chapter help the reader to grasp the theory.Graduate students in Mathematics with a special interest in the theory of foliations will especially benefit from this book, which can be used as supplementary reading in Singularity Theory courses, and as a resource for independent study on this vibrant field of research.

This concise textbook gathers together key concepts and modern results on the theory of holomorphic foliations with singularities, offering a compelling vision on how the notion of foliation, usually linked to real functions and manifolds, can have an important role in the holomorphic world, as shown by modern results from mathematicians as H. Cartan, K. Oka, T. Nishino, and M. Suzuki.The text starts with a gentle presentation of the classical notion of foliations, advancing to holomorphic foliations and then holomorphic foliations with singularities. The theory behind reduction of singularities is described in detail, as well the cases for dynamics of a local diffeomorphism and foliations on complex projective spaces. A final chapter brings recent questions in the field, as holomorphic flows on Stein spaces and transversely homogeneous holomorphic foliations, along with a list of open questions for further study and research. Selected exercises at the end of each chapter help the reader to grasp the theory.Graduate students in Mathematics with a special interest in the theory of foliations will especially benefit from this book, which can be used as supplementary reading in Singularity Theory courses, and as a resource for independent study on this vibrant field of research.

This textbook provides a comprehensive introduction to the mathematical foundations of quantum statistical physics. It presents a conceptually profound yet technically accessible path to the C*-algebraic approach to quantum statistical mechanics, demonstrating how key aspects of thermodynamic equilibrium can be derived as simple corollaries of classical results in convex analysis.Using C*-algebras as examples of ordered vector spaces, this book makes various aspects of C*-algebras and their applications to the mathematical foundations of quantum theory much clearer from both mathematical and physical perspectives. It begins with the simple case of Gibbs states on matrix algebras and gradually progresses to a more general setting that considers the thermodynamic equilibrium of infinitely extended quantum systems. The book also illustrates how first-order phase transitions and spontaneous symmetry breaking can occur, in contrast to the finite-dimensional situation. One of the unique features of this book is its thorough and clear treatment of the theory of equilibrium states of quantum mean-field models.This work is self-contained and requires only a modest background in analysis, topology, and functional analysis from the reader. It is suitable for both mathematicians and physicists with a specific interest in quantum statistical physics.

This textbook provides a comprehensive introduction to the mathematical foundations of quantum statistical physics. It presents a conceptually profound yet technically accessible path to the C*-algebraic approach to quantum statistical mechanics, demonstrating how key aspects of thermodynamic equilibrium can be derived as simple corollaries of classical results in convex analysis.Using C*-algebras as examples of ordered vector spaces, this book makes various aspects of C*-algebras and their applications to the mathematical foundations of quantum theory much clearer from both mathematical and physical perspectives. It begins with the simple case of Gibbs states on matrix algebras and gradually progresses to a more general setting that considers the thermodynamic equilibrium of infinitely extended quantum systems. The book also illustrates how first-order phase transitions and spontaneous symmetry breaking can occur, in contrast to the finite-dimensional situation. One of the unique features of this book is its thorough and clear treatment of the theory of equilibrium states of quantum mean-field models.This work is self-contained and requires only a modest background in analysis, topology, and functional analysis from the reader. It is suitable for both mathematicians and physicists with a specific interest in quantum statistical physics.

This book convenes and deepens generic results about spectral measures, many of them available so far in scattered literature. It starts with classic topics such as Wiener lemma, Strichartz inequality, and the basics of fractal dimensions of measures, progressing to more advanced material, some of them developed by the own authors.A fundamental concept to the mathematical theory of quantum mechanics, the spectral measure relates to the components of the quantum state concerning the energy levels of the Hamiltonian operator and, on the other hand, to the dynamics of such state. However, these correspondences are not immediate, with many nuances and subtleties discovered in recent years.A valuable example of such subtleties is found in the so-called “Wonderland theorem” first published by B. Simon in 1995. It shows that, for some metric space of self-adjoint operators, the set of operators whose spectral measures are singularcontinuous is a generic set (which, for some, is exotic). Recent works have revealed that, on top of singular continuity, there are other generic properties of spectral measures. These properties are usually associated with a number of different notions of generalized dimensions, upper and lower dimensions, with dynamical implications in quantum mechanics, ergodicity of dynamical systems, and evolution semigroups. All this opens ways to new and instigating avenues of research.Graduate students with a specific interest in the spectral properties of spectral measure are the primary target audience for this work, while researchers benefit from a selection of important results, many of them presented in the book format for the first time.

This book convenes and deepens generic results about spectral measures, many of them available so far in scattered literature. It starts with classic topics such as Wiener lemma, Strichartz inequality, and the basics of fractal dimensions of measures, progressing to more advanced material, some of them developed by the own authors.A fundamental concept to the mathematical theory of quantum mechanics, the spectral measure relates to the components of the quantum state concerning the energy levels of the Hamiltonian operator and, on the other hand, to the dynamics of such state. However, these correspondences are not immediate, with many nuances and subtleties discovered in recent years.A valuable example of such subtleties is found in the so-called “Wonderland theorem” first published by B. Simon in 1995. It shows that, for some metric space of self-adjoint operators, the set of operators whose spectral measures are singularcontinuous is a generic set (which, for some, is exotic). Recent works have revealed that, on top of singular continuity, there are other generic properties of spectral measures. These properties are usually associated with a number of different notions of generalized dimensions, upper and lower dimensions, with dynamical implications in quantum mechanics, ergodicity of dynamical systems, and evolution semigroups. All this opens ways to new and instigating avenues of research.Graduate students with a specific interest in the spectral properties of spectral measure are the primary target audience for this work, while researchers benefit from a selection of important results, many of them presented in the book format for the first time.

This textbook offers a concise introduction to symplectic and contact geometry, with a focus on the relationships between these subjects and other topics such as Lie theory and classical mechanics.Organized into four chapters, this work serves as a stepping stone for readers to delve into the subject, providing a succinct and motivating foundation. The content covers definitions, symplectic linear algebra, symplectic and contact manifolds, Hamiltonian systems, and more. Prerequisite knowledge includes differential geometry, manifolds, algebraic topology, de Rham cohomology, and the basics of Lie groups. Quick reviews are included where necessary, and examples and constructions are provided to foster understanding.Ideal for advanced undergraduate students and graduate students, this volume can also serve as a valuable resource for independent researchers seeking a quick yet solid understanding of symplectic and contact geometry.

This textbook offers a concise introduction to symplectic and contact geometry, with a focus on the relationships between these subjects and other topics such as Lie theory and classical mechanics.Organized into four chapters, this work serves as a stepping stone for readers to delve into the subject, providing a succinct and motivating foundation. The content covers definitions, symplectic linear algebra, symplectic and contact manifolds, Hamiltonian systems, and more. Prerequisite knowledge includes differential geometry, manifolds, algebraic topology, de Rham cohomology, and the basics of Lie groups. Quick reviews are included where necessary, and examples and constructions are provided to foster understanding.Ideal for advanced undergraduate students and graduate students, this volume can also serve as a valuable resource for independent researchers seeking a quick yet solid understanding of symplectic and contact geometry.

This volume convenes selected, peer-reviewed papers presented at the international workshop dedicated to Dr. Jorge Hounie on the occasion of his 75th birthday, held in Serra Negra, Brazil, from July 31 to August 4, 2023. The papers in this volume cover areas that include several complex variables, Cauchy-Riemann geometry, and partial differential equations.An Emeritus at the Federal University of São Carlos (UFScar), Brazil, Dr. Hounie has made significant contributions to partial differential equations, complex variables, harmonic analysis, and involutive structures. He has also been a kind and great mentor to numerous graduate students and postdocs who have gone on to pursue successful academic careers. Born in Bahia Blanca, Argentina, he completed his PhD studies at Rutgers University in 1974 and joined UFSCar as a Full Professor in 1995. He is a Full Member of the Brazilian Academy of Sciences.Within this book, readers will encounter a collection of cutting-edge research papers reflecting Dr. Hounie's research interests, valuable for both experienced researchers and graduate students alike.

This volume convenes selected, peer-reviewed papers presented at the international workshop dedicated to Dr. Jorge Hounie on the occasion of his 75th birthday, held in Serra Negra, Brazil, from July 31 to August 4, 2023. The papers in this volume cover areas that include several complex variables, Cauchy-Riemann geometry, and partial differential equations.An Emeritus at the Federal University of São Carlos (UFScar), Brazil, Dr. Hounie has made significant contributions to partial differential equations, complex variables, harmonic analysis, and involutive structures. He has also been a kind and great mentor to numerous graduate students and postdocs who have gone on to pursue successful academic careers. Born in Bahia Blanca, Argentina, he completed his PhD studies at Rutgers University in 1974 and joined UFSCar as a Full Professor in 1995. He is a Full Member of the Brazilian Academy of Sciences.Within this book, readers will encounter a collection of cutting-edge research papers reflecting Dr. Hounie's research interests, valuable for both experienced researchers and graduate students alike.

This book focuses on the computational aspects of graph convexity, with a particular emphasis on path convexity within graphs. It provides a thoughtful introduction to this emerging research field, which originated by adapting concepts from convex geometry to combinatorics and has experienced substantial growth.The book starts with an introduction of fundamental convexity concepts and then proceeds to discuss convexity parameters. These parameters fall into two categories: one derived from abstract convexity studies and another motivated by computational complexity. Subsequent chapters explore geometric convexity within graphs, examining various graph classes such as interval graphs, proper interval graphs, cographs, chordal graphs, and strongly chordal graphs. The text concludes with a study of the computation of convexity parameters across different convexity types, including practical applications in areas like game theory.Compact and straightforward, this work serves as an ideal entry point for students and researchers interested in pursuing further research in the field of convexity.The English translation of this book, originally in Portuguese, was facilitated by artificial intelligence. The content was later revised by the authors for accuracy.

This book focuses on the computational aspects of graph convexity, with a particular emphasis on path convexity within graphs. It provides a thoughtful introduction to this emerging research field, which originated by adapting concepts from convex geometry to combinatorics and has experienced substantial growth.The book starts with an introduction of fundamental convexity concepts and then proceeds to discuss convexity parameters. These parameters fall into two categories: one derived from abstract convexity studies and another motivated by computational complexity. Subsequent chapters explore geometric convexity within graphs, examining various graph classes such as interval graphs, proper interval graphs, cographs, chordal graphs, and strongly chordal graphs. The text concludes with a study of the computation of convexity parameters across different convexity types, including practical applications in areas like game theory.Compact and straightforward, this work serves as an ideal entry point for students and researchers interested in pursuing further research in the field of convexity.The English translation of this book, originally in Portuguese, was facilitated by artificial intelligence. The content was later revised by the authors for accuracy.

This book presents recent research results on Lyapunov stability and attraction for semigroup actions in a pedagogical format, providing the reader with numerous modern ideas and mathematical formulations for dynamical concepts in the transformation group theory.In recent decades, many fundamental concepts of dynamical systems have been extended to the general framework of transformation semigroups. Limit sets, attractors, isolated invariant sets, prolongational limit sets, and stable sets now have semigroup theoretical analogues. This monograph consolidates recent advancements in this field in a way that makes it accessible to graduate students. An effort was made to relate the presented results to important recurrence notions, for contextual clarity.A rudimentary understanding of group theory and topology, including the concepts of semigroup action, orbit, fiber bundle, compactness, and connectedness, is a prerequisite for reading this text. As a valuable resource for research projects and academic dissertations on topological dynamics, geometry, and mathematical analysis, this work can potentially open new avenues for further research.

This book presents recent research results on Lyapunov stability and attraction for semigroup actions in a pedagogical format, providing the reader with numerous modern ideas and mathematical formulations for dynamical concepts in the transformation group theory.In recent decades, many fundamental concepts of dynamical systems have been extended to the general framework of transformation semigroups. Limit sets, attractors, isolated invariant sets, prolongational limit sets, and stable sets now have semigroup theoretical analogues. This monograph consolidates recent advancements in this field in a way that makes it accessible to graduate students. An effort was made to relate the presented results to important recurrence notions, for contextual clarity.A rudimentary understanding of group theory and topology, including the concepts of semigroup action, orbit, fiber bundle, compactness, and connectedness, is a prerequisite for reading this text. As a valuable resource for research projects and academic dissertations on topological dynamics, geometry, and mathematical analysis, this work can potentially open new avenues for further research.

This textbook explores differential geometrical aspects of the theory of completely integrable Hamiltonian systems. It provides a comprehensive introduction to the mathematical foundations and illustrates it with a thorough analysis of classical examples.This book is organized into two parts. Part I contains a detailed, elementary exposition of the topics needed to start a serious geometrical analysis of complete integrability. This includes a background in symplectic and Poisson geometry, the study of Hamiltonian systems with symmetry, a primer on the theory of completely integrable systems, and a presentation of bi-Hamiltonian geometry.Part II is devoted to the analysis of three classical examples of integrable systems. This includes the description of the (free) n-dimensional rigid-body, the rational Calogero-Moser system, and the open Toda system. In each case, ths system is described, its integrability is discussed, and at least one of its (known) bi-Hamiltonian descriptions is presented.This work can benefit advanced undergraduate and beginning graduate students with a strong interest in geometrical methods of mathematical physics. Prerequisites include an introductory course in differential geometry and some familiarity with Hamiltonian and Lagrangian mechanics.