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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.
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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.
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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.
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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.
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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!
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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.
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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.
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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”.
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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!
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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.