http://publications.mfo.de/handle/mfo/4366 Mon, 27 Apr 2026 11:01:38 GMT 2026-04-27T11:01:38Z 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.