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- 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.
- 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.
- 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.
- 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.
- ItemMixed volumes and mixed integrals(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2018) Rotem, LiranIn 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 convex geometry become thereby accessible to analytic and probabilistic tools, and we can use these tools to make progress on very difficult open problems. We discuss in this Snapshot such a functional extension of some “volumes” which measure how “big” a set is. We recall the construction of “intrinsic volumes”, discuss the fundamental inequalities between them, and explain the functional extensions of these results.
- ItemTopological recursion(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2018) Sułkowski, PiotrIn 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.
- ItemComputing with symmetries(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2018) Roney-Dougal, Colva M.Group theory is the study of symmetry, and has many applications both within and outside mathematics. In this snapshot, we give a brief introduction to symmetries, and how to compute with them.
- ItemNumber theory in quantum computing(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2018) Schönnenbeck, SebastianAlgorithms are mathematical procedures developed to solve a problem. When encoded on a computer, algorithms must be "translated" to a series of simple steps, each of which the computer knows how to do. This task is relatively easy to do on a classical computer and we witness the benefits of this success in our everyday life. Quantum mechanics, the physical theory of the very small, promises to enable completely novel architectures of our machines, which will provide specific tasks with higher computing power. Translating and implementing algorithms on quantum computers is hard. However, we will show that solutions to this problem can be found and yield surprising applications to number theory.
- ItemThe Algebraic Statistics of an Oberwolfach Workshop(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2018) Seigal, AnnaAlgebraic Statistics builds on the idea that statistical models can be understood via polynomials. Many statistical models are parameterized by polynomials in the model parameters; others are described implicitly by polynomial equalities and inequalities. We explore the connection between algebra and statistics for some small statistical models.
- ItemThe mathematics of aquatic locomotion(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2018) Tucsnak, MariusAquatic locomotion is a self-propelled motion through a liquid medium. It can be of biological nature (fish, microorganisms,. . .) or performed by robotic swimmers. This snapshot aims to introduce the reader to some of the challenges raised by the mathematical modelling of aquatic locomotion, even in seemingly very simple cases.