http://publications.mfo.de/handle/mfo/4203 Tue, 07 Apr 2026 01:09:37 GMT 2026-04-07T01:09:37Z 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! http://publications.mfo.de/handle/mfo/4315 The Five Platonic Solids and their Connection to Root Systems Böhm, Sören Platonic solids have fascinated humans for thousands of years. In ancient times, they were associated with the elements fire, air, water, earth, and aether. These solids are completely symmetrical three-dimensional polyhedra. In this snapshot, it is first explained that there can only be five such polyhedra in the three-dimensional space. For this purpose, so-called Schläfli symbols and Coxeter graphs are introduced. More precisely, the (linear) Coxeter graphs correspond to the (linear) Schläfli symbols that, in turn, correspond exactly to the regular convex polyhedra. Through this one-to-one relationship, it is possible to classify the regular convex polytopes in any dimension by exploiting the classification of Coxeter graphs. Fri, 05 Sep 2025 00:00:00 GMT http://publications.mfo.de/handle/mfo/4315 2025-09-05T00:00:00Z Böhm, Sören Platonic solids have fascinated humans for thousands of years. In ancient times, they were associated with the elements fire, air, water, earth, and aether. These solids are completely symmetrical three-dimensional polyhedra. In this snapshot, it is first explained that there can only be five such polyhedra in the three-dimensional space. For this purpose, so-called Schläfli symbols and Coxeter graphs are introduced. More precisely, the (linear) Coxeter graphs correspond to the (linear) Schläfli symbols that, in turn, correspond exactly to the regular convex polyhedra. Through this one-to-one relationship, it is possible to classify the regular convex polytopes in any dimension by exploiting the classification of Coxeter graphs. http://publications.mfo.de/handle/mfo/4292 Five Ways to Spell ADE Kaufman, Dani The solutions to a surprising number of mathematical questions can be classified by the ADE Coxeter–Dynkin diagrams. This snapshot will show you a selection of these questions and how they correspond to the ADE Coxeter–Dynkin diagrams. Wed, 30 Jul 2025 00:00:00 GMT http://publications.mfo.de/handle/mfo/4292 2025-07-30T00:00:00Z Kaufman, Dani The solutions to a surprising number of mathematical questions can be classified by the ADE Coxeter–Dynkin diagrams. This snapshot will show you a selection of these questions and how they correspond to the ADE Coxeter–Dynkin diagrams. http://publications.mfo.de/handle/mfo/4232 Convex Polytopes and Linear Programs Joswig, Michael Convex polytopes are geometric objects that look deceptively simple. They occur everywhere in mathematics and have practical applications in everyday life – like organizing your grocery shopping list. In this snapshot, you get into contact with a long-standing, unsolved question in mathematics, which you can explore interactively. Tue, 01 Apr 2025 00:00:00 GMT http://publications.mfo.de/handle/mfo/4232 2025-04-01T00:00:00Z Joswig, Michael Convex polytopes are geometric objects that look deceptively simple. They occur everywhere in mathematics and have practical applications in everyday life – like organizing your grocery shopping list. In this snapshot, you get into contact with a long-standing, unsolved question in mathematics, which you can explore interactively. http://publications.mfo.de/handle/mfo/4204 Truncated Fusion Rules for Supergroups Heidersdorf, Thorsten In the '70s, physicists introduced a new type of symmetry – supersymmetry – to address some unresolved issues in particle physics models. Its mathematical foundations involve the representation theory of the associated symmetry groups, called supergroups. Our aim is to understand fusion rules, which describe how a combination of two physical systems can be broken down into more fundamental building blocks. Although the answer is largely unknown, we can get approximate answers in some cases. Mon, 03 Feb 2025 00:00:00 GMT http://publications.mfo.de/handle/mfo/4204 2025-02-03T00:00:00Z Heidersdorf, Thorsten In the '70s, physicists introduced a new type of symmetry – supersymmetry – to address some unresolved issues in particle physics models. Its mathematical foundations involve the representation theory of the associated symmetry groups, called supergroups. Our aim is to understand fusion rules, which describe how a combination of two physical systems can be broken down into more fundamental building blocks. Although the answer is largely unknown, we can get approximate answers in some cases.