Snapshots of Modern Mathematics from Oberwolfach
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Browsing Snapshots of Modern Mathematics from Oberwolfach by Subject "Geometry and Topology"
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- ItemAperiodic Order and Spectral Properties(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2017) Baake, Michael; Damanik, David; Grimm, UwePeriodic 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, materials science, and physics. Let us catch a glimpse into the inherent beauty of aperiodic order!
- ItemArrangements of lines(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2014) Harbourne, Brian; Szemberg, TomaszWe discuss certain open problems in the context of arrangements of lines in the plane.
- ItemBilliards and flat surfaces(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2015) Davis, DianaBilliards, the study of a ball bouncing around on a table, is a rich area of current mathematical research. We discuss questions and results on billiards, and on the related topic of flat surfaces.
- ItemBiological shape analysis with geometric statistics and learning(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2022) Utpala, Saiteja; Miolane, NinaThe advances in biomedical imaging techniques have enabled us to access the 3D shapes of a variety of structures: organs, cells, proteins. Since biological shapes are related to physiological functions, shape data may hold the key to unlocking outstanding mysteries in biomedicine. This snapshot introduces the mathematical framework of geometric statistics and learning and its applications to biomedicine.
- ItemCharacterizations of intrinsic volumes on convex bodies and convex functions(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2022) Mussnig, FabianIf we want to express the size of a two-dimensional shape with a number, then we usually think about its area or circumference. But what makes these quantities so special? We give an answer to this question in terms of classical mathematical results. We also take a look at applications and new generalizations to the setting of functions.
- ItemClosed geodesics on surfaces(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2022) Dozier, BenjaminWe consider surfaces of three types: the sphere, the torus, and many-holed tori. These surfaces naturally admit geometries of positive, zero, and negative curvature, respectively. It is interesting to study straight line paths, known as geodesics, in these geometries. We discuss the issue of counting closed geodesics; this is particularly rich for hyperbolic (negatively curved) surfaces.
- ItemClosed geodesics on surfaces and Riemannian manifolds(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2017) Radeschi, MarcoGeodesics 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.
- ItemThe codimension(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2018) Lerario, AntonioIn 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.
- ItemConfiguration spaces and braid groups(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Jiménez Rolland, Rita; Xicoténcatl, Miguel A.In this snapshot we introduce configuration spaces and explain how a mathematician studies their ‘shape’. This will lead us to consider paths of configurations and braid groups, and to explore how algebraic properties of these groups determine features of the spaces.
- ItemDescribing distance: from the plane to spectral triples(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2021) Arici, Francesca; Mesland, BramGeometry draws its power from the abstract structures that govern the shapes found in the real world. These abstractions often provide deeper insights into the underlying mathematical objects. In this snapshot, we give a glimpse into how certain “curved spaces” called manifolds can be better understood by looking at the (complex) differentiable functions they admit.
- ItemEstimating the volume of a convex body(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2018) Baldin, NicolaiSometimes the volume of a convex body needs to be estimated, if we cannot calculate it analytically. We explain how statistics can be used not only to approximate the volume of the convex body, but also its shape.
- ItemExpander graphs and where to find them(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Khukhro, AnaGraphs are mathematical objects composed of a collection of “dots” called vertices, some of which are joined by lines called edges. Graphs are ideal for visually representing relations between things, and mathematical properties of graphs can provide an insight into real-life phenomena. One interesting property is how connected a graph is, in the sense of how easy it is to move between the vertices along the edges. The topic dealt with here is the construction of particularly well-connected graphs, and whether or not such graphs can happily exist in worlds similar to ours.
- ItemFelder und Räume: Symmetrie und Lokalität in Mathematik und theoretischen Wissenschaften(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2023) Saberi, IngmarWir werden einige grundlegende Ideen der Eichtheorie und der dazugehörigen Differentialtopologie erkunden. Damit kann sich die Leserin ein Bild des Modulraums flacher Zusammenhänge machen und ihn mit den physikalisch motivierten Ideen dahinter in Beziehung bringen. Den Begriffen von Symmetrien und Feldern gehen wir gründlich nach. Außerdem werfen wir einen flüchtigen Blick auf unendliche Symmetrie in zwei Dimensionen und auf vor kurzem entdeckte Verallgemeinerungen.
- ItemA few shades of interpolation(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2017) Szpond, JustynaThe 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.
- ItemFinite geometries: pure mathematics close to applications(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2021) Storme, LeoThe 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”.
- ItemFootballs and donuts in four dimensions(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2016) Klee, StevenIn this snapshot, we explore connections between the mathematical areas of counting and geometry by studying objects called simplicial complexes. We begin by exploring many familiar objects in our three dimensional world and then discuss the ways one may generalize these ideas into higher dimensions.
- ItemFrom Betti numbers to ℓ²-Betti numbers(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2020) Kammeyer, Holger; Sauer, RomanWe provide a leisurely introduction to ℓ²-Betti numbers, which are topological invariants, by relating them to their much older cousins, Betti numbers. In the end we present an open research problem about ℓ²-Betti numbers.
- ItemFrom computer algorithms to quantum field theory: an introduction to operads(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2015) Krähmer, UlrichAn operad is an abstract mathematical tool encoding operations on specific mathematical structures. It finds applications in many areas of mathematics and related fields. This snapshot explains the concept of an operad and of an algebra over an operad, with a view towards a conjecture formulated by the mathematician Pierre Deligne. Deligne’s (by now proven) conjecture also gives deep inights into mathematical physics.
- ItemFrom the dollar game to the Riemann-Roch Theorem(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2021) Lamboglia, Sara; Ulirsch, MartinWhat is the dollar game? What can you do to win it? Can you always win it? In this snapshot you will find answers to these questions as well as several of the mathematical surprises that lurk in the background, including a new perspective on a century-old theorem.
- ItemGeometry behind one of the Painlevé III differential equations(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2018) Hertling, ClausThe Painlevé equations are second order differential equations, which were first studied more than 100 years ago. Nowadays they arise in many areas in mathematics and mathematical physics. This snapshot discusses the solutions of one of the Painlevé equations and presents old results on the asymptotics at two singular points and new results on the global behavior.
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