http://publications.mfo.de/handle/mfo/483 Vol. 33(2005)- Tue, 07 Apr 2026 09:53:18 GMT 2026-04-07T09:53:18Z http://publications.mfo.de/handle/mfo/4291 Variational and Information Flows in Machine Learning and Optimal Transport Li, Wuchen; Schmitzer, Bernhard; Steidl, Gabriele; Vialard, François-Xavier This book is based on lectures given at the Mathematisches Forschungsinstitut Oberwolfach on “Computational Variational Flows in Machine Learning and Optimal Transport”. Variational and stochastic flows on measure spaces are ubiquitous in machine learning and generative modeling. Optimal transport and diffeomorphic flows provide powerful frameworks to analyze such trajectories of distributions with elegant notions from differential geometry, such as geodesics, gradient and Hamiltonian flows. Recently, mean field control and mean field games offered a general optimal control variational view on learning problems. The four independent chapters in this book address the question of how the presented tools lead us to better understanding and further development of machine learning and generative models. Tue, 01 Jul 2025 00:00:00 GMT http://publications.mfo.de/handle/mfo/4291 2025-07-01T00:00:00Z Li, Wuchen Schmitzer, Bernhard Steidl, Gabriele Vialard, François-Xavier This book is based on lectures given at the Mathematisches Forschungsinstitut Oberwolfach on “Computational Variational Flows in Machine Learning and Optimal Transport”. Variational and stochastic flows on measure spaces are ubiquitous in machine learning and generative modeling. Optimal transport and diffeomorphic flows provide powerful frameworks to analyze such trajectories of distributions with elegant notions from differential geometry, such as geodesics, gradient and Hamiltonian flows. Recently, mean field control and mean field games offered a general optimal control variational view on learning problems. The four independent chapters in this book address the question of how the presented tools lead us to better understanding and further development of machine learning and generative models. http://publications.mfo.de/handle/mfo/4290 Stochastic Geophysical Fluid Dynamics Flandoli, Franco; Holm, Darryl; Hussein, Amru; Saal, Martin This book is based on an Oberwolfach Seminar, and it intends to give an introduction and an overview to several directions of the recent research on stochastic geophysical fluid dynamics and to discuss different geophysical stochastic partial differential equations. Tue, 01 Jul 2025 00:00:00 GMT http://publications.mfo.de/handle/mfo/4290 2025-07-01T00:00:00Z Flandoli, Franco Holm, Darryl Hussein, Amru Saal, Martin This book is based on an Oberwolfach Seminar, and it intends to give an introduction and an overview to several directions of the recent research on stochastic geophysical fluid dynamics and to discuss different geophysical stochastic partial differential equations. http://publications.mfo.de/handle/mfo/4155 Free Boundary Problems in Fluid Dynamics Ai, Albert; Alazard, Thomas; Ifrim, Mihaela; Tataru, Daniel This book, originating from a seminar held at Oberwolfach in 2022, introduces to state-of-the-art methods and results in the study of free boundary problems which are arising from compressible as well as from incompressible Euler’s equations in general. A particular set of such problems is given by gaseous stars considered in a vacuum (modeled via the compressible Euler equations) as well as water waves in their full generality (seen as recasts of incompressible irrotational Euler equations). This is a broad research area which is highly relevant to many real life problems, and in which substantial progress has been made in the last decade. Oberwolfach Seminar 2243a: Free Boundary Problems in Fluid Dynamics Thu, 20 Jun 2024 00:00:00 GMT http://publications.mfo.de/handle/mfo/4155 2024-06-20T00:00:00Z Ai, Albert Alazard, Thomas Ifrim, Mihaela Tataru, Daniel This book, originating from a seminar held at Oberwolfach in 2022, introduces to state-of-the-art methods and results in the study of free boundary problems which are arising from compressible as well as from incompressible Euler’s equations in general. A particular set of such problems is given by gaseous stars considered in a vacuum (modeled via the compressible Euler equations) as well as water waves in their full generality (seen as recasts of incompressible irrotational Euler equations). This is a broad research area which is highly relevant to many real life problems, and in which substantial progress has been made in the last decade. 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.; [Open Access] Oberwolfach Seminar: Metric Algebraic Geometry 2322b; Oberwolfach Seminar: Metric Algebraic Geometry 2322b Tue, 27 Feb 2024 00:00:00 GMT http://publications.mfo.de/handle/mfo/4118 2024-02-27T00:00:00Z 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. [Open Access] 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 GMT http://publications.mfo.de/handle/mfo/4076 2023-01-01T00:00:00Z 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. 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 GMT http://publications.mfo.de/handle/mfo/4072 2023-01-01T00:00:00Z 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. 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 GMT http://publications.mfo.de/handle/mfo/4071 2023-01-01T00:00:00Z 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. 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 GMT http://publications.mfo.de/handle/mfo/4025 2023-01-01T00:00:00Z 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. 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 GMT http://publications.mfo.de/handle/mfo/1371 2018-01-01T00:00:00Z 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. 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 GMT http://publications.mfo.de/handle/mfo/1324 2017-01-01T00:00:00Z 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.