http://publications.mfo.de/handle/mfo/20 The snapshot project is designed to promote the understanding and appreciation of modern mathematics and mathematical research in the general public world-wide. It is part of the project "Oberwolfach meets IMAGINARY“, supported by the Klaus Tschira Foundation. Fri, 10 Apr 2026 13:36:54 GMT 2026-04-10T13:36:54Z http://publications.mfo.de:80/bitstream/id/3670a104-7004-438d-a11b-98e3fffbf777/ http://publications.mfo.de/handle/mfo/20 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/4391 Triangulations in Geometry: from Ptolemy to Teichmüller Pressland, Matthew Ptolemy's theorem is a classical result from ancient Greek mathematics, concerning the lengths of sides and diagonals of a polygon drawn in a circle. In this snapshot, I will explain why this theorem is still important today through its role in Teichmüller theory, a subject which seeks to describe all possible ''shapes'' of a surface with boundary. Fri, 27 Feb 2026 00:00:00 GMT http://publications.mfo.de/handle/mfo/4391 2026-02-27T00:00:00Z Pressland, Matthew Ptolemy's theorem is a classical result from ancient Greek mathematics, concerning the lengths of sides and diagonals of a polygon drawn in a circle. In this snapshot, I will explain why this theorem is still important today through its role in Teichmüller theory, a subject which seeks to describe all possible ''shapes'' of a surface with boundary. http://publications.mfo.de/handle/mfo/4388 Alternating Sign Matrix Bijections: Marvelous, Mysterious, Missing Striker, Jessica A bijection transforms one type of mathematical object into another. Such transformations provide new perspectives on these objects, revealing surprising properties and uncovering new mysteries. We discuss bijections from alternating sign matrices to other objects in mathematics and physics and recent progress in the search for a missing bijection. Tue, 24 Feb 2026 00:00:00 GMT http://publications.mfo.de/handle/mfo/4388 2026-02-24T00:00:00Z Striker, Jessica A bijection transforms one type of mathematical object into another. Such transformations provide new perspectives on these objects, revealing surprising properties and uncovering new mysteries. We discuss bijections from alternating sign matrices to other objects in mathematics and physics and recent progress in the search for a missing bijection. http://publications.mfo.de/handle/mfo/4387 Secure File Sharing and Cayley Graphs McKemmie, Eilidh Have you ever wondered how your computer knows it can trust certain downloads but not others? This snapshot describes some security concerns and one algebraic way of dealing with them. We'll see an interesting procedure that uses a very difficult problem in algebra to provide security, and discuss some of the procedure's important properties. Mon, 23 Feb 2026 00:00:00 GMT http://publications.mfo.de/handle/mfo/4387 2026-02-23T00:00:00Z McKemmie, Eilidh Have you ever wondered how your computer knows it can trust certain downloads but not others? This snapshot describes some security concerns and one algebraic way of dealing with them. We'll see an interesting procedure that uses a very difficult problem in algebra to provide security, and discuss some of the procedure's important properties. http://publications.mfo.de/handle/mfo/4386 Fracture Mechanics: a Nonlocal Approach Buczkowski, Nicole; Foss, Mikil; Radu, Petronela Nonlocal models consider interactions over a range of distances, not just at a single point. In this snapshot, we give a short introduction to nonlocal modeling, explain how it differs from its local counterpart, and present an application: fracture mechanics. Fri, 20 Feb 2026 00:00:00 GMT http://publications.mfo.de/handle/mfo/4386 2026-02-20T00:00:00Z Buczkowski, Nicole Foss, Mikil Radu, Petronela Nonlocal models consider interactions over a range of distances, not just at a single point. In this snapshot, we give a short introduction to nonlocal modeling, explain how it differs from its local counterpart, and present an application: fracture mechanics. http://publications.mfo.de/handle/mfo/4356 Why Oscillation Counts: Diophantine Approximation, Geometry, and the Fourier Transform Srivastava, Rajula Is it possible to approximate arbitrary points in space by vectors with rational coordinates, with which we, and computers, feel much more comfortable? If yes, can we approximate those points arbitrarily close? In this snapshot, we explore how the geometric configuration of these points influences the answers to these questions. Further, we delve into the closely related problem of counting rational vectors near surfaces. The unlikely tool which helps us in this endeavour is Fourier analysis – the study of waves and oscillations! Tue, 16 Dec 2025 00:00:00 GMT http://publications.mfo.de/handle/mfo/4356 2025-12-16T00:00:00Z Srivastava, Rajula Is it possible to approximate arbitrary points in space by vectors with rational coordinates, with which we, and computers, feel much more comfortable? If yes, can we approximate those points arbitrarily close? In this snapshot, we explore how the geometric configuration of these points influences the answers to these questions. Further, we delve into the closely related problem of counting rational vectors near surfaces. The unlikely tool which helps us in this endeavour is Fourier analysis – the study of waves and oscillations! http://publications.mfo.de/handle/mfo/4355 Is there a Smooth Lattice Polytope which does not have the Integer Decomposition Property? Hofscheier, Johannes; Kasprzyk, Alexander We introduce Tadao Oda's famous question on lattice polytopes which was originally posed at Oberwolfach in 1997 and, although simple to state, has remained unanswered. The question is motivated by a discussion of the two-dimensional case – including a proof of Pick's Theorem, which elegantly relates the area of a lattice polygon to the number of lattice points it contains in its interior and on its boundary. Tue, 16 Dec 2025 00:00:00 GMT http://publications.mfo.de/handle/mfo/4355 2025-12-16T00:00:00Z Hofscheier, Johannes Kasprzyk, Alexander We introduce Tadao Oda's famous question on lattice polytopes which was originally posed at Oberwolfach in 1997 and, although simple to state, has remained unanswered. The question is motivated by a discussion of the two-dimensional case – including a proof of Pick's Theorem, which elegantly relates the area of a lattice polygon to the number of lattice points it contains in its interior and on its boundary. http://publications.mfo.de/handle/mfo/4347 Brackets, Trees and the Borromean Rings Jasso, Gustavo We describe some of the beautiful mathematical structures that arise from the study of the associativity equation. Our journey takes us from combinatorics to abstract algebra, with brief excursions through geometry and topology along the way. Mon, 08 Dec 2025 00:00:00 GMT http://publications.mfo.de/handle/mfo/4347 2025-12-08T00:00:00Z Jasso, Gustavo We describe some of the beautiful mathematical structures that arise from the study of the associativity equation. Our journey takes us from combinatorics to abstract algebra, with brief excursions through geometry and topology along the way. http://publications.mfo.de/handle/mfo/4345 Brauer's Problems: 60 Years of Legacy Rizo, Noelia; Schaeffer Fry, Mandi A. Richard Brauer (1901-1977) was a German-American mathematician who is regarded as the founder of a highly active mathematical area known as modular representation theory. This area grew from group theory, which can be thought of as the mathematical study of symmetries. In this snapshot, we hope to impress on the reader the legacy left by Brauer and celebrate the 60th anniversary of "Brauer's problems", a list of 43 conjectures and objectives suggested by Brauer in 1963. These problems inspired an entire branch within character theory, studying "local-global conjectures". Mon, 24 Nov 2025 00:00:00 GMT http://publications.mfo.de/handle/mfo/4345 2025-11-24T00:00:00Z Rizo, Noelia Schaeffer Fry, Mandi A. Richard Brauer (1901-1977) was a German-American mathematician who is regarded as the founder of a highly active mathematical area known as modular representation theory. This area grew from group theory, which can be thought of as the mathematical study of symmetries. In this snapshot, we hope to impress on the reader the legacy left by Brauer and celebrate the 60th anniversary of "Brauer's problems", a list of 43 conjectures and objectives suggested by Brauer in 1963. These problems inspired an entire branch within character theory, studying "local-global conjectures". http://publications.mfo.de/handle/mfo/4322 Trisections of Four-Dimensional Spaces Blackwell, Sarah This snapshot introduces the theory of trisections of smooth 4-manifolds, an area of exploration in low-dimensional topology aiming to make four-dimensional spaces more understandable. Along the way, we discuss the concepts of topology, dimension, manifolds, and more! Fri, 19 Sep 2025 00:00:00 GMT http://publications.mfo.de/handle/mfo/4322 2025-09-19T00:00:00Z Blackwell, Sarah This snapshot introduces the theory of trisections of smooth 4-manifolds, an area of exploration in low-dimensional topology aiming to make four-dimensional spaces more understandable. Along the way, we discuss the concepts of topology, dimension, manifolds, and more!