The codimension
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In this snapshot we discuss the notion of codimension, which is, in a sense, “dual” to the notion of dimension and is useful when studying the relative position of one object insider another one.
Geometry and Topology
mathemathical subject image:
Spaces of Riemannian metrics
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Riemannian metrics endow smooth manifolds such as surfaces with intrinsic geometric properties, for example with curvature. They also allow us to measure quantities like distances, angles and volumes. These are the notions we use to characterize the “shape” of a manifold. The space of Riemannian metrics is a mathematical object that encodes the many possible ways in which we can geometrically deform the shape of a manifold.
Also available in Spanish.
Spaces of Riemannian metrics
Submitted by IMAGINARY on
Riemannian metrics endow smooth manifolds such as surfaces with intrinsic geometric properties, for example with curvature. They also allow us to measure quantities like distances, angles and volumes. These are the notions we use to characterize the “shape” of a manifold. The space of Riemannian metrics is a mathematical object that encodes the many possible ways in which we can geometrically deform the shape of a manifold.
Topological recursion
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In this snapshot we present the concept of topological recursion – a new, surprisingly powerful formalism at the border of mathematics and physics, which has been actively developed within the last decade. After introducing necessary ingredients – expectation values, random matrices, quantum theories, recursion relations, and topology – we explain how they get combined together in one unifying picture.
A few shades of interpolation
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The topic of this snapshot is interpolation. In the ordinary sense, interpolation means to insert something of a different nature into something else. In mathematics, interpolation means constructing new data points from given data points. The new points usually lie in between the already known points. The purpose of this snapshot is to introduce a particular type of interpolation, namely, polynomial interpolation. This will be explained starting from basic ideas that go back to the ancient Babylonians and Greeks, and will arrive at subjects of current research activity.
Closed geodesics on surfaces and Riemannian manifolds
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Geodesics are special paths in surfaces and so-called Riemannian manifolds which connect close points in the shortest way. Closed geodesics are geodesics which go back to where they started. In this snapshot we talk about these special paths, and the efforts to find closed geodesics.
Aperiodic Order and Spectral Properties
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Periodic structures like a typical tiled kitchen floor or the arrangement of carbon atoms in a diamond crystal certainly possess a high degree of order. But what is order without periodicity? In this snapshot, we are going to explore highly ordered structures that are substantially nonperiodic, or aperiodic. As we construct such structures, we will discover surprising connections to various branches of mathematics, ma- terials science, and physics. Let us catch a glimpse into the inherent beauty of aperiodic order!
Winkeltreue zahlt sich aus
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Nicht nur Seefahrerinnen, auch Computergrafikerinnen und Physikerinnen wissen Winkeltreue zu schätzen. Doch beschränkte Rechenkapazitäten und Vereinfachungen in theoretischen Modellen erfordern es, winkeltreue Abbildungen nur mit einer überschaubaren Datenmenge zu beschreiben. Entsprechende Theorien werden in der diskreten Mathematik untersucht. Im Folgenden lade ich Sie auf eine Reise in die faszinierende Welt der winkeltreuen Abbildungen ein.
Math Creations - A Math-Art Competition
Submitted by Bianca Violet on
We describe a math-art contest for new creative ideas and their realizations based on modern mathematical concepts currently worked on in two Berlin-based collaborative research centers. By combining mathematics with art and design the power, the fun, and the beauty of this interdisciplinary connection is shown.
Profinite groups
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Profinite objects are mathematical constructions used to collect, in a uniform manner, facts about infinitely many finite objects. We shall review recent progress in the theory of profinite groups, due to Nikolov and Segal, and its implications for finite groups.