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- ItemMatrixfaktorisierungen(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2014) Lerche, WolfgangIm 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 theoretischen Physik wichtige Anwendungen hat.
- ItemSpaces of Riemannian metrics(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2017) Bustamante, Mauricio; Kordaß, Jan-BernhardRiemannian 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.
- ItemRandom permutations(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Betz, Volker100 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 is it to arrange for all hats to be returned to their owner? These classical questions of probability theory can be answered relatively easily. But if a geometric component is added, answering the same questions immediately becomes very hard, and little is known about them. We present some of the open questions and give an overview of what current research can say about them.
- 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.
- 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.
- ItemMinimizing energy(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2015) Breiner, ChristineWhat 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 to this question and others like it, making note of recent results in the same spirit.
- ItemHiggs bundles without geometry(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2020) Rayan, Steven; Schaposnik, Laura P.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 stroll through some aspects of linear algebra that anticipate the deeper structure in the moduli space of Higgs bundles.
- ItemTopological Complexity, Robotics and Social Choice(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2018) Carrasquel, José; Lupton, Gregory; Oprea, JohnTopological 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 wider area of Applied Algebraic Topology. Surprisingly, the same mathematics gives insight into the question of creating social choice functions, which may be viewed as algorithms for making decisions by artificial intelligences.
- 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.
- 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.