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Deciding which mall, hospital or school is closest to us is a problem we face everyday. It even comes on holidays with us, when we optimize our plans to make sure that we have enough time to visit all the attractions we want to see. In this article, we show how concepts from metric algebraic geometry help us to rise to this task while planning a weekend trip to the Black Forest.
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Geproci sets arise from applying the perspective of inverse scattering problems to algebraic geometry. Analogous to the reconstruction of an object from multiple X-ray images, we aim at a classification of sets with certain algebraic properties under multiple projections.
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To understand our world, we classify things. A famous example is the periodic table of elements, which describes the properties of all known chemical elements and gives us a classification of the building blocks we can use in physics, chemistry, and biology. In mathematics, and algebraic geometry in particular, there are many instances of similar periodic tables”, describing fundamental classification results. We will go on a tour of some of these.
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The research field of finite geometries investigates structures with a finite number of objects. Classical examples include vector spaces, projective spaces, and affine spaces over finite fields. Although many of these structures are studied for their geometrical importance, they are also of great interest in other, more applied domains of mathematics. In this snapshot, finite vector spaces are introduced. We discuss the geometrical concept of partial t-spreads together with its implications for the “packing problem” and a recent application in the existence of “cooling codes”.
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In school, we learn that the interior angles of any triangle sum up to π. However, there exist spaces different from the usual Euclidean space in which this is not true. One of these spaces is the “hyperbolic space”, which has another geometry than the classical Euclidean geometry. In this snapshot, we consider the geometry of hyperbolic polytopes, for example polygons, how they tile hyperbolic space, and how reflections along the faces of polytopes give rise to important mathematical structures. The classification of these structures is an open area of research.
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In this snapshot, we give a survey of some problems in the study of rational points on higher genus curves, discussing questions ranging from the era of the ancient Greeks to a few posed by mathematicians of the 20th century. To answer these questions, we describe a selection of techniques in modern number theory that can be used to determine the set of rational points on a curve.
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In this snapshot, we explain two important mathematical concepts (representation and degeneration) in elementary terms. We will focus on the simplest meaningful examples, and motivate both concepts by study of symmetry.
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In this snapshot, we will first give an introduction to hyperbolic geometry and we will then show how certain matrix groups of a number-theoretic origin give rise to a large variety of interesting tessellations of 3-dimensional hyperbolic space. Many of the building blocks of these tessellations exhibit beautiful symmetry and have inspired the design of 3D printed jewellery.
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The Monster finite simple group is almost unimaginably large, with about 8×1053 elements in it. Trying to understand such an immense object requires both theory and computer programs. In this snapshot, we discuss finite groups, representations, and finally Brauer trees, which offer some new understanding of this vast and intricate structure.
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I am going to describe the Robinson–Schensted algorithm which transforms a permutation of the numbers from 1 to n into a pair of combinatorial objects called “standard Young tableaux”. I will then say a little bit about a few of the fascinating properties of this transformation, and how it connects to current research.