Browsing by Snapshot Mathematical Subject "Geometry and Topology"
Now showing items 2140 of 46

Higgs bundles without geometry
[SNAP2020008EN] (Mathematisches Forschungsinstitut Oberwolfach, 20200929)Higgs bundles appeared a few decades ago as solutions to certain equations from physics and have attracted much attention in geometry as well as other areas of mathematics and physics. Here, we take a very informal ... 
The Interaction of Curvature and Topology
[SNAP2019020EN] (Mathematisches Forschungsinstitut Oberwolfach, 20191218)In this snapshot we will outline the mathematical notion of curvature by means of comparison geometry. We will then try to address questions as the ways in which curvature might influence the topology of a space, and ... 
Jewellery from tessellations of hyperbolic space
[SNAP2022005EN] (Mathematisches Forschungsinstitut Oberwolfach, 20220602)In this snapshot, we will first give an introduction to hyperbolic geometry and we will then show how certain matrix groups of a numbertheoretic origin give rise to a large variety of interesting tessellations of 3dimensional ... 
Lagrangian mean curvature flow
[SNAP2021011EN] (Mathematisches Forschungsinstitut Oberwolfach, 20210916)Lagrangian mean curvature flow is a powerful tool in modern mathematics with connections to topics in analysis, geometry, topology and mathematical physics. I will describe some of the key aspects of Lagrangian mean curvature ... 
Matrixfaktorisierungen
[SNAP2014002DE] (Mathematisches Forschungsinstitut Oberwolfach, 2014)Im Folgenden soll ein kurzer Abriss des Themas Matrixfaktorisierungen gegeben werden. Wir werden darlegen, warum dieses recht simple Konzept zu erstaunlich tiefen mathematischen Gedankengängen führt und auch in der modernen ... 
Minimizing energy
[SNAP2015002EN] (Mathematisches Forschungsinstitut Oberwolfach, 2015)What is the most efficient way to fence land when you’ve only got so many metres of fence? Or, to put it differently, what is the largest area bounded by a simple closed planar curve of fixed length? We consider the answer ... 
Mixed volumes and mixed integrals
[SNAP2018014EN] (Mathematisches Forschungsinstitut Oberwolfach, 20181229)In recent years, mathematicians have developed new approaches to study convex sets: instead of considering convex sets themselves, they explore certain functions or measures that are related to them. Problems from ... 
The Periodic Tables of Algebraic Geometry
[SNAP2023002EN] (Mathematisches Forschungsinstitut Oberwolfach, 20230904)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 ... 
Polyhedra and commensurability
[SNAP2016009EN] (Mathematisches Forschungsinstitut Oberwolfach, 2016)This snapshot introduces the notion of commensurability of polyhedra. At its bottom, this concept can be developed from constructions with paper, scissors, and glue. Starting with an elementary example, we formalize it ... 
Positive Scalar Curvature and Applications
[SNAP2019004ENSNAP2019004ES] (Mathematisches Forschungsinstitut Oberwolfach, 20190425)[also available in Spanish] We introduce the idea of curvature, including how it developed historically, and focus on the scalar curvature of a manifold. A major current research topic involves understanding positive scalar ... 
Das Problem der Kugelpackung
[SNAP2016004DE] (Mathematisches Forschungsinstitut Oberwolfach, 2016)Wie würdest du Tennisbälle oder Orangen stapeln? Oder allgemeiner formuliert: Wie dicht lassen sich identische 3dimensionale Objekte überschneidungsfrei anordnen? Das Problem, welches auch Anwendungen in der digitalen ... 
Profinite groups
[SNAP2016014EN] (Mathematisches Forschungsinstitut Oberwolfach, 2016)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, ... 
Random permutations
[SNAP2019007EN] (Mathematisches Forschungsinstitut Oberwolfach, 20190712)100 people leave their hats at the door at a party and pick up a completely random hat when they leave. How likely is it that at least one of them will get back their own hat? If the hats carry name tags, how difficult ... 
Reflections on hyperbolic space
[SNAP2021007EN] (Mathematisches Forschungsinstitut Oberwolfach, 20210824)In school, we learn that the interior angles of any triangle sum up to pi. However, there exist spaces different from the usual Euclidean space in which this is not true. One of these spaces is the ''hyperbolic space'', ... 
Spaces of Riemannian metrics
[SNAP2017010ENSNAP2017010ES] (Mathematisches Forschungsinstitut Oberwolfach, 20171228)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 ... 
Swallowtail on the shore
[SNAP2014007EN] (Mathematisches Forschungsinstitut Oberwolfach, 2014)Platonic solids, Felix Klein, H.S.M. Coxeter and a flap of a swallowtail: The five Platonic solids tetrahedron, cube, octahedron, icosahedron and dodecahedron have always attracted much curiosity from mathematicians, not ... 
Topological Complexity, Robotics and Social Choice
[SNAP2018005EN] (Mathematisches Forschungsinstitut Oberwolfach, 20180810)Topological complexity is a number that measures how hard it is to plan motions (for robots, say) in terms of a particular space associated to the kind of motion to be planned. This is a burgeoning subject within the ... 
Topological recursion
[SNAP2018002EN] (Mathematisches Forschungsinstitut Oberwolfach, 20180305)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 ... 
Touching the transcendentals: tractional motion from the bir th of calculus to future perspectives
[SNAP2019013EN] (Mathematisches Forschungsinstitut Oberwolfach, 20191121)When the rigorous foundation of calculus was developed, it marked an epochal change in the approach of mathematicians to geometry. Tools from geometry had been one of the foundations of mathematics until the 17th century ... 
Tropical geometry
[SNAP2018007EN] (Mathematisches Forschungsinstitut Oberwolfach, 20180719)What kind of strange spaces hide behind the enigmatic name of tropical geometry? In the tropics, just as in other geometries, one of the simplest objects is a line. Therefore, we begin our exploration by considering tropical ...