http://publications.mfo.de:80 The DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material. Thu, 16 Apr 2026 23:42:17 GMT 2026-04-16T23:42:17Z http://publications.mfo.de/themes/Mirage2/images/apple-touch-icon.png http://publications.mfo.de:80 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. http://publications.mfo.de/handle/mfo/4404 Mini-Workshop: Algebraic Foliations: Analytic and Birational Viewpoint The main goal of the mini-workshop was starting strong collaborations between outstanding women in geometry with a broad spectrum of expertise. The focus was on the interplay of three topics in Geometry: analytic methods in Kähler geometry, Foliations and Cremona groups. More precisely, Foliation theory was a unifying theme. Wed, 01 Jan 2025 00:00:00 GMT http://publications.mfo.de/handle/mfo/4404 2025-01-01T00:00:00Z The main goal of the mini-workshop was starting strong collaborations between outstanding women in geometry with a broad spectrum of expertise. The focus was on the interplay of three topics in Geometry: analytic methods in Kähler geometry, Foliations and Cremona groups. More precisely, Foliation theory was a unifying theme. http://publications.mfo.de/handle/mfo/4403 Functional Inequalities: Geometric Calculus meets Stochastic Analysis Functional inequalities form a unifying theme across a wide spectrum of modern analysis, geometry, and probability. They encode deep geometric and analytic information - for instance through Poincaré, log-Sobolev, transportation, isoperimetric and curvature-dimension inequalities - and serve as crucial tools in the study of Markov semigroups, diffusion processes, metric measure spaces, and geometric flows. The workshop brought together researchers working in geometric analysis, stochastic analysis, and optimal transport in order to promote exchange of ideas and further strengthen the interaction between these rapidly developing fields. Substantial emphasis was placed on non-smooth or singular geometric structures, stochastic dynamics with degeneracies, and new bridges between discrete, fractal, and continuum settings. Wed, 01 Jan 2025 00:00:00 GMT http://publications.mfo.de/handle/mfo/4403 2025-01-01T00:00:00Z Functional inequalities form a unifying theme across a wide spectrum of modern analysis, geometry, and probability. They encode deep geometric and analytic information - for instance through Poincaré, log-Sobolev, transportation, isoperimetric and curvature-dimension inequalities - and serve as crucial tools in the study of Markov semigroups, diffusion processes, metric measure spaces, and geometric flows. The workshop brought together researchers working in geometric analysis, stochastic analysis, and optimal transport in order to promote exchange of ideas and further strengthen the interaction between these rapidly developing fields. Substantial emphasis was placed on non-smooth or singular geometric structures, stochastic dynamics with degeneracies, and new bridges between discrete, fractal, and continuum settings. http://publications.mfo.de/handle/mfo/4402 Recent Developments in SPDEs and BSDEs meet Harmonic and Functional Analysis The purpose of this workshop is the strengthening of the interaction between the fields of Stochastic Partial Differential Equations (SPDEs), Backward Stochastic Differential Equations (BSDEs), and Harmonic and Functional Analysis. We focus on the essential role of analytic techniques (including function spaces, weighted inequalities, and $A_p$-weights) in solving problems in critical stochastic settings. Special topics include SPDEs in critical spaces, regularization by noise, singular SPDEs, quadratic and nonlocal BSDEs. Wed, 01 Jan 2025 00:00:00 GMT http://publications.mfo.de/handle/mfo/4402 2025-01-01T00:00:00Z The purpose of this workshop is the strengthening of the interaction between the fields of Stochastic Partial Differential Equations (SPDEs), Backward Stochastic Differential Equations (BSDEs), and Harmonic and Functional Analysis. We focus on the essential role of analytic techniques (including function spaces, weighted inequalities, and $A_p$-weights) in solving problems in critical stochastic settings. Special topics include SPDEs in critical spaces, regularization by noise, singular SPDEs, quadratic and nonlocal BSDEs. http://publications.mfo.de/handle/mfo/4401 Arithmetic Statistics for Algebraic Objects The workshop focused on various directions of arithmetic statistics in algebra and number theory. These include statistical problems for random polynomials and varieties, probabilistic Galois theory, and counting and distribution problems for algebraic functions, algebraic number fields, elliptic curves, $L$-functions, as well as arithmetic problems in non-abelian settings (eg, arithmetic statistics for algebraic groups). Wed, 01 Jan 2025 00:00:00 GMT http://publications.mfo.de/handle/mfo/4401 2025-01-01T00:00:00Z The workshop focused on various directions of arithmetic statistics in algebra and number theory. These include statistical problems for random polynomials and varieties, probabilistic Galois theory, and counting and distribution problems for algebraic functions, algebraic number fields, elliptic curves, $L$-functions, as well as arithmetic problems in non-abelian settings (eg, arithmetic statistics for algebraic groups). http://publications.mfo.de/handle/mfo/4400 Analytic Number Theory Analytic number theory is a subject which continues to flourish and grow with several significant developments over the past few years making progress on some of the most famous open problems in mathematics. This workshop brought together world experts and young talent to discuss the various branches and recent developments in the subject. Wed, 01 Jan 2025 00:00:00 GMT http://publications.mfo.de/handle/mfo/4400 2025-01-01T00:00:00Z Analytic number theory is a subject which continues to flourish and grow with several significant developments over the past few years making progress on some of the most famous open problems in mathematics. This workshop brought together world experts and young talent to discuss the various branches and recent developments in the subject.