Oberwolfach Seminars
http://publications.mfo.de/handle/mfo/483
Vol. 33(2005)-Sat, 15 Jun 2024 06:38:15 GMT2024-06-15T06:38:15ZMetric Algebraic Geometry
http://publications.mfo.de/handle/mfo/4118
Metric Algebraic Geometry
Breiding, Paul; Kohn, Kathlén; Sturmfels, Bernd
Metric algebraic geometry combines concepts from algebraic geometry and differential geometry. Building on classical foundations, it offers practical tools for the 21st century. Many applied problems center around metric questions, such as optimization with respect to distances.
After a short dive into 19th-century geometry of plane curves, we turn to problems expressed by polynomial equations over the real numbers. The solution sets are real algebraic varieties. Many of our metric problems arise in data science, optimization and statistics. These include minimizing Wasserstein distances in machine learning, maximum likelihood estimation, computing curvature, or minimizing the Euclidean distance to a variety.
This book addresses a wide audience of researchers and students and can be used for a one-semester course at the graduate level. The key prerequisite is a solid foundation in undergraduate mathematics, especially in algebra and geometry.
This is an openaccess book.
Oberwolfach Seminar: Metric Algebraic Geometry 2322b; Oberwolfach Seminar: Metric Algebraic Geometry 2322b
Tue, 27 Feb 2024 00:00:00 GMThttp://publications.mfo.de/handle/mfo/41182024-02-27T00:00:00ZBreiding, PaulKohn, KathlénSturmfels, BerndMetric algebraic geometry combines concepts from algebraic geometry and differential geometry. Building on classical foundations, it offers practical tools for the 21st century. Many applied problems center around metric questions, such as optimization with respect to distances.
After a short dive into 19th-century geometry of plane curves, we turn to problems expressed by polynomial equations over the real numbers. The solution sets are real algebraic varieties. Many of our metric problems arise in data science, optimization and statistics. These include minimizing Wasserstein distances in machine learning, maximum likelihood estimation, computing curvature, or minimizing the Euclidean distance to a variety.
This book addresses a wide audience of researchers and students and can be used for a one-semester course at the graduate level. The key prerequisite is a solid foundation in undergraduate mathematics, especially in algebra and geometry.
This is an openaccess book.Interfaces: Modeling, Analysis, Numerics
http://publications.mfo.de/handle/mfo/4076
Interfaces: Modeling, Analysis, Numerics
Bänsch, Eberhard; Deckelnick, Klaus; Garcke, Harald; Pozzi, Paola
These lecture notes are dedicated to the mathematical modelling, analysis and computation of interfaces and free boundary problems appearing in geometry and in various applications, ranging from crystal growth, tumour growth, biological membranes to porous media, two-phase flows, fluid-structure interactions, and shape optimization.
We first give an introduction to classical methods from differential geometry and systematically derive the governing equations from physical principles. Then we will analyse parametric approaches to interface evolution problems and derive numerical methods which will be thoroughly analysed. In addition, implicit descriptions of interfaces such as phase field and level set methods will be analysed. Finally, we will discuss numerical methods for complex interface evolutions and will focus on two phase flow problems as an important example of such evolutions.
Sun, 01 Jan 2023 00:00:00 GMThttp://publications.mfo.de/handle/mfo/40762023-01-01T00:00:00ZBänsch, EberhardDeckelnick, KlausGarcke, HaraldPozzi, PaolaThese lecture notes are dedicated to the mathematical modelling, analysis and computation of interfaces and free boundary problems appearing in geometry and in various applications, ranging from crystal growth, tumour growth, biological membranes to porous media, two-phase flows, fluid-structure interactions, and shape optimization.
We first give an introduction to classical methods from differential geometry and systematically derive the governing equations from physical principles. Then we will analyse parametric approaches to interface evolution problems and derive numerical methods which will be thoroughly analysed. In addition, implicit descriptions of interfaces such as phase field and level set methods will be analysed. Finally, we will discuss numerical methods for complex interface evolutions and will focus on two phase flow problems as an important example of such evolutions.Tropical and Logarithmic Methods in Enumerative Geometry
http://publications.mfo.de/handle/mfo/4072
Tropical and Logarithmic Methods in Enumerative Geometry
Cavalieri, Renzo; Markwig, Hannah; Dhruv, Ranganathan
This book is based on the lectures given at the Oberwolfach Seminar held in Fall 2021. Logarithmic Gromov-Witten theory lies at the heart of modern approaches to mirror symmetry, but also opens up a number of new directions in enumerative geometry of a more classical flavour. Tropical geometry forms the calculus through which calculations in this subject are carried out. These notes cover the foundational aspects of this tropical calculus, geometric aspects of the degeneration formula for Gromov-Witten invariants, and the practical nuances of working with and enumerating tropical curves. Readers will get an assisted entry route to the subject, focusing on examples and explicit calculations.
Sun, 01 Jan 2023 00:00:00 GMThttp://publications.mfo.de/handle/mfo/40722023-01-01T00:00:00ZCavalieri, RenzoMarkwig, HannahDhruv, RanganathanThis book is based on the lectures given at the Oberwolfach Seminar held in Fall 2021. Logarithmic Gromov-Witten theory lies at the heart of modern approaches to mirror symmetry, but also opens up a number of new directions in enumerative geometry of a more classical flavour. Tropical geometry forms the calculus through which calculations in this subject are carried out. These notes cover the foundational aspects of this tropical calculus, geometric aspects of the degeneration formula for Gromov-Witten invariants, and the practical nuances of working with and enumerating tropical curves. Readers will get an assisted entry route to the subject, focusing on examples and explicit calculations.New Techniques in Resolution of Singularities
http://publications.mfo.de/handle/mfo/4071
New Techniques in Resolution of Singularities
Abramovich, Dan; Frühbis-Krüger, Anne; Temkin, Michael; Włodarczyk, Jarosław
Resolution of singularities is notorious as a difficult topic within algebraic geometry. Recent work, aiming at resolution of families and semistable reduction, infused the subject with logarithmic geometry and algebraic stacks, two techniques essential for the current theory of moduli spaces. As a byproduct a short, a simple and efficient functorial resolution procedure in characteristic 0 using just algebraic stacks was produced.
The goals of the book, the result of an Oberwolfach Seminar, are to introduce readers to explicit techniques of resolution of singularities with access to computer implementations, introduce readers to the theories of algebraic stacks and logarithmic structures, and to resolution in families and semistable reduction methods.
Sun, 01 Jan 2023 00:00:00 GMThttp://publications.mfo.de/handle/mfo/40712023-01-01T00:00:00ZAbramovich, DanFrühbis-Krüger, AnneTemkin, MichaelWłodarczyk, JarosławResolution of singularities is notorious as a difficult topic within algebraic geometry. Recent work, aiming at resolution of families and semistable reduction, infused the subject with logarithmic geometry and algebraic stacks, two techniques essential for the current theory of moduli spaces. As a byproduct a short, a simple and efficient functorial resolution procedure in characteristic 0 using just algebraic stacks was produced.
The goals of the book, the result of an Oberwolfach Seminar, are to introduce readers to explicit techniques of resolution of singularities with access to computer implementations, introduce readers to the theories of algebraic stacks and logarithmic structures, and to resolution in families and semistable reduction methods.Wave Phenomena
http://publications.mfo.de/handle/mfo/4025
Wave Phenomena; Mathematical Analysis and Numerical Approximation
Dörfler, Willy; Hochbruck, Marlis; Köhler, Jonas; Rieder, Andreas; Schnaubelt, Roland; Wieners, Christian
This book presents the notes from the seminar on wave phenomena given in 2019 at the Mathematical Research Center in Oberwolfach.
The research on wave-type problems is a fascinating and emerging field in mathematical research with many challenging applications in sciences and engineering. Profound investigations on waves require a strong interaction of several mathematical disciplines including functional analysis, partial differential equations, mathematical modeling, mathematical physics, numerical analysis, and scientific computing.
The goal of this book is to present a comprehensive introduction to the research on wave phenomena. Starting with basic models for acoustic, elastic, and electro-magnetic waves, topics such as the existence of solutions for linear and some nonlinear material laws, efficient discretizations and solution methods in space and time, and the application to inverse parameter identification problems are covered. The aim of this book is to intertwine analysis and numerical mathematics for wave-type problems promoting thus cooperative research projects in this field.
Sun, 01 Jan 2023 00:00:00 GMThttp://publications.mfo.de/handle/mfo/40252023-01-01T00:00:00ZDörfler, WillyHochbruck, MarlisKöhler, JonasRieder, AndreasSchnaubelt, RolandWieners, ChristianThis book presents the notes from the seminar on wave phenomena given in 2019 at the Mathematical Research Center in Oberwolfach.
The research on wave-type problems is a fascinating and emerging field in mathematical research with many challenging applications in sciences and engineering. Profound investigations on waves require a strong interaction of several mathematical disciplines including functional analysis, partial differential equations, mathematical modeling, mathematical physics, numerical analysis, and scientific computing.
The goal of this book is to present a comprehensive introduction to the research on wave phenomena. Starting with basic models for acoustic, elastic, and electro-magnetic waves, topics such as the existence of solutions for linear and some nonlinear material laws, efficient discretizations and solution methods in space and time, and the application to inverse parameter identification problems are covered. The aim of this book is to intertwine analysis and numerical mathematics for wave-type problems promoting thus cooperative research projects in this field.Mathematical Theory of Evolutionary Fluid-Flow Structure Interactions
http://publications.mfo.de/handle/mfo/1371
Mathematical Theory of Evolutionary Fluid-Flow Structure Interactions
Kaltenbacher, Barbara; Kukavica, Igor; Lasiecka, Irena; Triggiani, Roberto; Tuffaha, Amjad; Webster, Justin T.
This book is devoted to the study of coupled partial differential equation models, which describe complex dynamical systems occurring in modern scientific applications such as fluid/flow-structure interactions. The first chapter provides a general description of a fluid-structure interaction, which is formulated within a realistic framework, where the structure subject to a frictional damping moves within the fluid. The second chapter then offers a multifaceted description, with often surprising results, of the case of the static interface; a case that is argued in the literature to be a good model for small, rapid oscillations of the structure. The third chapter describes flow-structure interaction where the compressible Navier-Stokes equations are replaced by the linearized Euler equation, while the solid is taken as a nonlinear plate, which oscillates in the surrounding gas flow. The final chapter focuses on a the equations of nonlinear acoustics coupled with linear acoustics or elasticity, as they arise in the context of high intensity ultrasound applications.
Mon, 01 Jan 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13712018-01-01T00:00:00ZKaltenbacher, BarbaraKukavica, IgorLasiecka, IrenaTriggiani, RobertoTuffaha, AmjadWebster, Justin T.This book is devoted to the study of coupled partial differential equation models, which describe complex dynamical systems occurring in modern scientific applications such as fluid/flow-structure interactions. The first chapter provides a general description of a fluid-structure interaction, which is formulated within a realistic framework, where the structure subject to a frictional damping moves within the fluid. The second chapter then offers a multifaceted description, with often surprising results, of the case of the static interface; a case that is argued in the literature to be a good model for small, rapid oscillations of the structure. The third chapter describes flow-structure interaction where the compressible Navier-Stokes equations are replaced by the linearized Euler equation, while the solid is taken as a nonlinear plate, which oscillates in the surrounding gas flow. The final chapter focuses on a the equations of nonlinear acoustics coupled with linear acoustics or elasticity, as they arise in the context of high intensity ultrasound applications.K-Theory for Group C*-Algebras and Semigroup C*-Algebras
http://publications.mfo.de/handle/mfo/1324
K-Theory for Group C*-Algebras and Semigroup C*-Algebras
Cuntz, Joachim; Echterhoff, Siegfried; Li, Xin; Yu, Guoliang
This book gives an account of the necessary background for group algebras and crossed products for actions of a group or a semigroup on a space and reports on some very recently developed techniques with applications to particular examples. Much of the material is available here for the first time in book form. The topics discussed are among the most classical and intensely studied C*-algebras. They are important for applications in fields as diverse as the theory of unitary group representations, index theory, the topology of manifolds or ergodic theory of group actions.
Sun, 01 Jan 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13242017-01-01T00:00:00ZCuntz, JoachimEchterhoff, SiegfriedLi, XinYu, GuoliangThis book gives an account of the necessary background for group algebras and crossed products for actions of a group or a semigroup on a space and reports on some very recently developed techniques with applications to particular examples. Much of the material is available here for the first time in book form. The topics discussed are among the most classical and intensely studied C*-algebras. They are important for applications in fields as diverse as the theory of unitary group representations, index theory, the topology of manifolds or ergodic theory of group actions.Moduli Spaces of Riemannian Metrics
http://publications.mfo.de/handle/mfo/529
Moduli Spaces of Riemannian Metrics
Tuschmann, Wilderich; Wraith, David J.
This book studies certain spaces of Riemannian metrics on both compact and non-compact manifolds. These spaces are defined by various sign-based curvature conditions, with special attention paid to positive scalar curvature and non-negative sectional curvature, though we also consider positive Ricci and non-positive sectional curvature. If we form the quotient of such a space of metrics under the action of the diffeomorphism group (or possibly a subgroup) we obtain a moduli space. Understanding the topology of both the original space of metrics and the corresponding moduli space form the central theme of this book. For example, what can be said about the connectedness or the various homotopy groups of such spaces? We explore the major results in the area, but provide sufficient background so that a non-expert with a grounding in Riemannian geometry can access this growing area of research.
Thu, 01 Jan 2015 00:00:00 GMThttp://publications.mfo.de/handle/mfo/5292015-01-01T00:00:00ZTuschmann, WilderichWraith, David J.This book studies certain spaces of Riemannian metrics on both compact and non-compact manifolds. These spaces are defined by various sign-based curvature conditions, with special attention paid to positive scalar curvature and non-negative sectional curvature, though we also consider positive Ricci and non-positive sectional curvature. If we form the quotient of such a space of metrics under the action of the diffeomorphism group (or possibly a subgroup) we obtain a moduli space. Understanding the topology of both the original space of metrics and the corresponding moduli space form the central theme of this book. For example, what can be said about the connectedness or the various homotopy groups of such spaces? We explore the major results in the area, but provide sufficient background so that a non-expert with a grounding in Riemannian geometry can access this growing area of research.Dispersive Equations and Nonlinear Waves
http://publications.mfo.de/handle/mfo/528
Dispersive Equations and Nonlinear Waves
Koch, Herbert; Tataru, Daniel; Vişan, Monica
The first part of the book provides an introduction to key tools and techniques in dispersive equations: Strichartz estimates, bilinear estimates, modulation and adapted function spaces, with an application to the generalized Korteweg-de Vries equation and the Kadomtsev-Petviashvili equation. The energy-critical nonlinear Schrödinger equation, global solutions to the defocusing problem, and scattering are the focus of the second part. Using this concrete example, it walks the reader through the induction on energy technique, which has become the essential methodology for tackling large data critical problems. This includes refined/inverse Strichartz estimates, the existence and almost periodicity of minimal blow up solutions, and the development of long-time Strichartz inequalities. The third part describes wave and Schrödinger maps. Starting by building heuristics about multilinear estimates, it provides a detailed outline of this very active area of geometric/dispersive PDE. It focuses on concepts and ideas and should provide graduate students with a stepping stone to this exciting direction of research.
Wed, 01 Jan 2014 00:00:00 GMThttp://publications.mfo.de/handle/mfo/5282014-01-01T00:00:00ZKoch, HerbertTataru, DanielVişan, MonicaThe first part of the book provides an introduction to key tools and techniques in dispersive equations: Strichartz estimates, bilinear estimates, modulation and adapted function spaces, with an application to the generalized Korteweg-de Vries equation and the Kadomtsev-Petviashvili equation. The energy-critical nonlinear Schrödinger equation, global solutions to the defocusing problem, and scattering are the focus of the second part. Using this concrete example, it walks the reader through the induction on energy technique, which has become the essential methodology for tackling large data critical problems. This includes refined/inverse Strichartz estimates, the existence and almost periodicity of minimal blow up solutions, and the development of long-time Strichartz inequalities. The third part describes wave and Schrödinger maps. Starting by building heuristics about multilinear estimates, it provides a detailed outline of this very active area of geometric/dispersive PDE. It focuses on concepts and ideas and should provide graduate students with a stepping stone to this exciting direction of research.Positional Games
http://publications.mfo.de/handle/mfo/527
Positional Games
Hefetz, Dan; Krivelevich, Michael; Stojaković, Miloš; Szabó, Tibor
This text is based on a lecture course given by the authors in the framework of Oberwolfach Seminars at the Mathematisches Forschungsinstitut Oberwolfach in May, 2013. It is intended to serve as a thorough introduction to the rapidly developing field of positional games. This area constitutes an important branch of combinatorics, whose aim it is to systematically develop an extensive mathematical basis for a variety of two player perfect information games. These ranges from such popular games as Tic-Tac-Toe and Hex to purely abstract games played on graphs and hypergraphs. The subject of positional games is strongly related to several other branches of combinatorics such as Ramsey theory, extremal graph and set theory, and the probabilistic method. These notes cover a variety of topics in positional games, including both classical results and recent important developments. They are presented in an accessible way and are accompanied by exercises of varying difficulty, helping the reader to better understand the theory. The text will benefit both researchers and graduate students in combinatorics and adjacent fields.
Wed, 01 Jan 2014 00:00:00 GMThttp://publications.mfo.de/handle/mfo/5272014-01-01T00:00:00ZHefetz, DanKrivelevich, MichaelStojaković, MilošSzabó, TiborThis text is based on a lecture course given by the authors in the framework of Oberwolfach Seminars at the Mathematisches Forschungsinstitut Oberwolfach in May, 2013. It is intended to serve as a thorough introduction to the rapidly developing field of positional games. This area constitutes an important branch of combinatorics, whose aim it is to systematically develop an extensive mathematical basis for a variety of two player perfect information games. These ranges from such popular games as Tic-Tac-Toe and Hex to purely abstract games played on graphs and hypergraphs. The subject of positional games is strongly related to several other branches of combinatorics such as Ramsey theory, extremal graph and set theory, and the probabilistic method. These notes cover a variety of topics in positional games, including both classical results and recent important developments. They are presented in an accessible way and are accompanied by exercises of varying difficulty, helping the reader to better understand the theory. The text will benefit both researchers and graduate students in combinatorics and adjacent fields.