http://publications.mfo.de:80 The DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material. Thu, 16 Jul 2026 09:47:20 GMT 2026-07-16T09:47:20Z http://publications.mfo.de/themes/Mirage2/images/apple-touch-icon.png http://publications.mfo.de:80 http://publications.mfo.de/handle/mfo/4431 Spherical Subgroups in Reductive Algebraic Groups Knop, Friedrich; Röhrle, Gerhard In [KR15], we determined all spherical affine homogeneous varieties for simple algebraic groups in arbitrary characteristic. The present paper extends this classification to semisimple groups. This generalizes work done independently by Brion and Mikityuk in characteristic zero. Our primary approach relies on a lifting lemma to characteristic zero, enabling us to directly apply Brion’s and Mikityuk’s results. [MSC 2020] 20G15; 14M27; 14M17; 14L30; 20G05; Acknowledgments: Work on this paper began during a visit to the Mathematisches Forschungsinstitut Oberwolfach (MFO) under the "Oberwolfach Research Fellows" program; we thank the MFO for its support. Wed, 01 Jul 2026 00:00:00 GMT http://publications.mfo.de/handle/mfo/4431 2026-07-01T00:00:00Z Knop, Friedrich Röhrle, Gerhard In [KR15], we determined all spherical affine homogeneous varieties for simple algebraic groups in arbitrary characteristic. The present paper extends this classification to semisimple groups. This generalizes work done independently by Brion and Mikityuk in characteristic zero. Our primary approach relies on a lifting lemma to characteristic zero, enabling us to directly apply Brion’s and Mikityuk’s results. http://publications.mfo.de/handle/mfo/4426 The Kakeya conjecture Hickman, Jonathan The geometry of lines is a fundamental part of mathematics and the way we interact with the physical world. Core concepts such as distance and angle, and the accompanying theory of trigonometry, have been studied since antiquity and taught to countless generations of students. However, there are simple questions about lines which have stumped some of the greatest minds in mathematics over the last fifty years. One notable example is the Kakeya conjecture, which asks how lines which point in different directions can be packed together in a small space. Fri, 19 Jun 2026 00:00:00 GMT http://publications.mfo.de/handle/mfo/4426 2026-06-19T00:00:00Z Hickman, Jonathan The geometry of lines is a fundamental part of mathematics and the way we interact with the physical world. Core concepts such as distance and angle, and the accompanying theory of trigonometry, have been studied since antiquity and taught to countless generations of students. However, there are simple questions about lines which have stumped some of the greatest minds in mathematics over the last fifty years. One notable example is the Kakeya conjecture, which asks how lines which point in different directions can be packed together in a small space. http://publications.mfo.de/handle/mfo/4425 G-Complete Reducibility, Geometric Invariant Theory and Spherical Buildings Bate, Michael; Martin, Benjamin; Röhrle, Gerhard The aim of this textbook is to introduce readers at a graduate level to G-complete reducibility and explain some of its many applications across pure mathematics. It is based on the Oberwolfach Seminar of the same name which took place in 2022. The notion of G-complete reducibility for subgroups of a reductive algebraic group is a natural generalisation of the notion of complete reducibility in representation theory. Since its introduction in the 1990s, complete reducibility has been widely studied, both as an important concept in its own right, with applications to the classification and structure of linear algebraic groups, and also as a useful tool with applications in representation theory, geometric invariant theory, the theory of buildings, and number theory. [MSC 2020]: 20Gxx; 20E42; 14L24; 14L30 Mon, 01 Jun 2026 00:00:00 GMT http://publications.mfo.de/handle/mfo/4425 2026-06-01T00:00:00Z Bate, Michael Martin, Benjamin Röhrle, Gerhard The aim of this textbook is to introduce readers at a graduate level to G-complete reducibility and explain some of its many applications across pure mathematics. It is based on the Oberwolfach Seminar of the same name which took place in 2022. The notion of G-complete reducibility for subgroups of a reductive algebraic group is a natural generalisation of the notion of complete reducibility in representation theory. Since its introduction in the 1990s, complete reducibility has been widely studied, both as an important concept in its own right, with applications to the classification and structure of linear algebraic groups, and also as a useful tool with applications in representation theory, geometric invariant theory, the theory of buildings, and number theory. http://publications.mfo.de/handle/mfo/4422 Simulating Alzheimer's disease with a brainsphere model Kunoth, Angela; Weller, Anna; Yilmaz, Tolunay Nowadays, there are different medical imaging techniques to collect data about the progression of Alzheimer's disease in a patient's brain. These data describe different phenomena which are still not understood from a biological point of view. How can these data sets be combined in a mathematical model to simulate the evolution of such a neurodegenerative disease in a computer? In this snapshot, we present one possible approach to address this task with the help of graph theory and partial differential equations. Tue, 09 Jun 2026 00:00:00 GMT http://publications.mfo.de/handle/mfo/4422 2026-06-09T00:00:00Z Kunoth, Angela Weller, Anna Yilmaz, Tolunay Nowadays, there are different medical imaging techniques to collect data about the progression of Alzheimer's disease in a patient's brain. These data describe different phenomena which are still not understood from a biological point of view. How can these data sets be combined in a mathematical model to simulate the evolution of such a neurodegenerative disease in a computer? In this snapshot, we present one possible approach to address this task with the help of graph theory and partial differential equations. http://publications.mfo.de/handle/mfo/4421 How big is my slice of cheese? Meroni, Chiara In this snapshot, we introduce the study of slices of polytopes – geometric shapes with flat sides – and examine the area of these slices. This is connected to combinatorics and polynomials and is surprisingly complex, even in three dimensions. Since we are greedy humans, we conclude by finding the largest possible slice of cheese. Tue, 19 May 2026 00:00:00 GMT http://publications.mfo.de/handle/mfo/4421 2026-05-19T00:00:00Z Meroni, Chiara In this snapshot, we introduce the study of slices of polytopes – geometric shapes with flat sides – and examine the area of these slices. This is connected to combinatorics and polynomials and is surprisingly complex, even in three dimensions. Since we are greedy humans, we conclude by finding the largest possible slice of cheese. http://publications.mfo.de/handle/mfo/4415 The 4-Sample Theorem on planar graphs Améndola, Carlos; Kahle, Thomas The famous 4-Color Theorem from graph theory states that the vertices of any planar graph can be colored with four colors, so that no neighboring vertices have the same color. The 4-Sample Theorem from algebraic statistics says that the maximum likelihood estimator for a Gaussian graphical model of a planar graph exists with probability 1 if one has at least four samples. This number of necessary samples, the maximum likelihood threshold, is a new graph invariant from algebraic statistics and connected not only to parameter estimation, but also to matrix completion, the theory of filling partial matrices, and rigidity theory, which deals with stability of objects. Fri, 10 Apr 2026 00:00:00 GMT http://publications.mfo.de/handle/mfo/4415 2026-04-10T00:00:00Z Améndola, Carlos Kahle, Thomas The famous 4-Color Theorem from graph theory states that the vertices of any planar graph can be colored with four colors, so that no neighboring vertices have the same color. The 4-Sample Theorem from algebraic statistics says that the maximum likelihood estimator for a Gaussian graphical model of a planar graph exists with probability 1 if one has at least four samples. This number of necessary samples, the maximum likelihood threshold, is a new graph invariant from algebraic statistics and connected not only to parameter estimation, but also to matrix completion, the theory of filling partial matrices, and rigidity theory, which deals with stability of objects. http://publications.mfo.de/handle/mfo/4411 Transverse Foliations for Two-Degree-of-Freedom Mechanical Systems de Paulo, Naiara V.; Kim, Seongchan; Salomão, Pedro A. S.; Schneider, Alexsandro We investigate the dynamics of a two-degree-of-freedom mechanical system for energies slightly above a critical value. The critical set of the potential function is assumed to contain a finite number of saddle points. As the energy increases across the critical value, a disk-like component of the Hill region gets connected to other components precisely at the saddles. Under certain convexity assumptions on the critical set, we show the existence of a weakly convex foliation in the region of the energy surface where the interesting dynamics takes place. The binding of the foliation is formed by the index-2 Lyapunov orbits in the neck region about the rest points and a particular index-3 orbit. Among other dynamical implications, the transverse foliation forces the existence of periodic orbits, homoclinics, and heteroclinics to the Lyapunov orbits. We apply the results to the Hénon-Heiles potential for energies slightly above 1/6. We also discuss the existence of transverse foliations for decoupled mechanical systems, including the frozen Hill's lunar problem with centrifugal force, the Stark problem, the Euler problem of two centers, and the potential of a chemical reaction. NdP was partially supported by CAPES/MATH-AMSUD 88881.878892/2023-01. SK was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (No. RS-2025-16070003). A part of this work was done during SK’s visit to the Mathematisches Forschungsinstitut Oberwolfach (MFO) as an Oberwolfach Leibniz Fellow in 2020. SK cordially thanks the MFO for its excellent support and stimulating working atmosphere. PS acknowledges the support of the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai and the 2022 National Foreign Experts Program. PS was partially supported by FAPESP (2016/25053-8) and CNPq (306106/2016-7). PS was partially supported by the National Natural Science Foundation of China (grant number W2431007). PS thanks the support of the Shenzhen International Center for Mathematics - SUSTech. AS thanks the Instituto de Matemática Pura e Aplicada (IMPA) for the post-doc position. Part of this work was conducted during visits to the Southern University of Science and Technology (SUSTech) and the Kongju National University (KNU). AS thanks both institutes for their hospitality.; [MSC 2020] Primary 37J55; Secondary 53D35. Sun, 01 Mar 2026 00:00:00 GMT http://publications.mfo.de/handle/mfo/4411 2026-03-01T00:00:00Z de Paulo, Naiara V. Kim, Seongchan Salomão, Pedro A. S. Schneider, Alexsandro We investigate the dynamics of a two-degree-of-freedom mechanical system for energies slightly above a critical value. The critical set of the potential function is assumed to contain a finite number of saddle points. As the energy increases across the critical value, a disk-like component of the Hill region gets connected to other components precisely at the saddles. Under certain convexity assumptions on the critical set, we show the existence of a weakly convex foliation in the region of the energy surface where the interesting dynamics takes place. The binding of the foliation is formed by the index-2 Lyapunov orbits in the neck region about the rest points and a particular index-3 orbit. Among other dynamical implications, the transverse foliation forces the existence of periodic orbits, homoclinics, and heteroclinics to the Lyapunov orbits. We apply the results to the Hénon-Heiles potential for energies slightly above 1/6. We also discuss the existence of transverse foliations for decoupled mechanical systems, including the frozen Hill's lunar problem with centrifugal force, the Stark problem, the Euler problem of two centers, and the potential of a chemical reaction. http://publications.mfo.de/handle/mfo/4407 Homogeneous Structures: Model Theory meets Universal Algebra Many fundamental mathematical structures, such as the rationals or the random graph, are homogeneous, meaning that local isomorphisms extend to global automorphisms. Such structures arise as limits of classes of finite structures and encode these classes in a single object. This viewpoint has proved fruitful in model theory, universal algebra, and computer science, with applications to constraint satisfaction, automata theory, and verification. Homogeneous structures have rich automorphism groups, which makes them interesting for topological dynamics. For many applications, however, automorphism groups do not store enough information about the homogeneous structure, and one must instead consider polymorphism clones. Universal algebra has recently achieved major results for polymorphism clones on finite structures, culminating in the 2017 resolution of the Feder--Vardi dichotomy conjecture. An analogous conjecture for homogeneous structures remains open despite growing structural insights. Wed, 01 Jan 2025 00:00:00 GMT http://publications.mfo.de/handle/mfo/4407 2025-01-01T00:00:00Z Many fundamental mathematical structures, such as the rationals or the random graph, are homogeneous, meaning that local isomorphisms extend to global automorphisms. Such structures arise as limits of classes of finite structures and encode these classes in a single object. This viewpoint has proved fruitful in model theory, universal algebra, and computer science, with applications to constraint satisfaction, automata theory, and verification. Homogeneous structures have rich automorphism groups, which makes them interesting for topological dynamics. For many applications, however, automorphism groups do not store enough information about the homogeneous structure, and one must instead consider polymorphism clones. Universal algebra has recently achieved major results for polymorphism clones on finite structures, culminating in the 2017 resolution of the Feder--Vardi dichotomy conjecture. An analogous conjecture for homogeneous structures remains open despite growing structural insights. http://publications.mfo.de/handle/mfo/4406 Mini-Workshop: Hyperbolic meets Stochastic Geometry The mini-workshop brought together researchers from hyperbolic geometry and stochastic geometry with the aim of advancing the emerging field of hyperbolic stochastic geometry. It focused on understanding how negative curvature fundamentally influences the behaviour of random geometric models. Particular emphasis was placed on limit theorems, phase transitions, and scaling phenomena that differ substantially from those observed in Euclidean settings. The program combined survey lectures, research presentations, and discussion sessions to link geometric methods with probabilistic techniques tailored to hyperbolic spaces. As a result, the workshop clarified central challenges in the field, identified key open problems, and initiated new collaborations spanning geometry, probability, and related areas. Wed, 01 Jan 2025 00:00:00 GMT http://publications.mfo.de/handle/mfo/4406 2025-01-01T00:00:00Z The mini-workshop brought together researchers from hyperbolic geometry and stochastic geometry with the aim of advancing the emerging field of hyperbolic stochastic geometry. It focused on understanding how negative curvature fundamentally influences the behaviour of random geometric models. Particular emphasis was placed on limit theorems, phase transitions, and scaling phenomena that differ substantially from those observed in Euclidean settings. The program combined survey lectures, research presentations, and discussion sessions to link geometric methods with probabilistic techniques tailored to hyperbolic spaces. As a result, the workshop clarified central challenges in the field, identified key open problems, and initiated new collaborations spanning geometry, probability, and related areas. http://publications.mfo.de/handle/mfo/4405 Mini-Workshop: Approximation of Manifold-Valued Functions The approximation of unknown functions from scattered, possibly high-dimensional data is central to many scientific applications. Advances in data acquisition have driven the need for flexible nonlinear models, including manifold-valued functions. Approximating and learning such functions differs fundamentally from classical linear methods and requires tools from numerical analysis, linear algebra, and differential geometry. This interdisciplinary framework has applications ranging from data science and machine learning to numerical PDEs and quantum chemistry. This mini-workshop brings together researchers developing constructive approximation methods for manifold-valued functions, their theory, and applications. Wed, 01 Jan 2025 00:00:00 GMT http://publications.mfo.de/handle/mfo/4405 2025-01-01T00:00:00Z The approximation of unknown functions from scattered, possibly high-dimensional data is central to many scientific applications. Advances in data acquisition have driven the need for flexible nonlinear models, including manifold-valued functions. Approximating and learning such functions differs fundamentally from classical linear methods and requires tools from numerical analysis, linear algebra, and differential geometry. This interdisciplinary framework has applications ranging from data science and machine learning to numerical PDEs and quantum chemistry. This mini-workshop brings together researchers developing constructive approximation methods for manifold-valued functions, their theory, and applications.