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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.

Computing the long term evolution of the solar system with geometric numerical integrators

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Simulating the dynamics of the Sun–Earth–Moon sys- tem with a standard algorithm yields a dramatically wrong solution, predicting that the Moon is ejected from its orbit. In contrast, a well chosen algorithm with the same initial data yields the correct behavior. We explain the main ideas of how the evolution of the solar system can be computed over long times by taking advantage of so-called geometric numerical methods. Short sample codes are provided for the Sun–Earth–Moon system. 

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. 

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. 

Mathematics plays a key role in scientific computing

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I attended a very interesting workshop at the research center MFO in Oberwolfach on “Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws”. The title sounds a bit technical, but in plain language we could say: The theme is to survey recent research concerning how mathematics is used to study numerical algorithms involving a special class of equations.

Prony’s method: an old trick for new problems

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In 1795, French mathematician Gaspard de Prony invented an ingenious trick to solve a recovery problem, aiming at reconstructing functions from their values at given points, which arose from a specific application in physical chemistry. His technique became later useful in many different areas, such as signal processing, and it relates to the concept of sparsity that gained a lot of well-deserved attention recently. Prony’s contribution, therefore, has developed into a very modern mathematical concept. 

The Algebraic Statistics of an Oberwolfach Workshop

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Algebraic 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.

Solving quadratic equations in many variables

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Fields are number systems in which every linear equation has a solution, such as the set of all rational numbers Q or the set of all real numbers R. All fields have the same properties in relation with systems of linear equations, but quadratic equations behave differently from field to field. Is there a field in which every quadratic equation in five variables has a solution, but some quadratic equation in four variables has no solution? The answer is in this snapshot. 

Computational Optimal Transport

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Optimal transport is the mathematical discipline of matching supply to demand while minimizing shipping costs. This matching problem becomes extremely challenging as the quantity of supply and demand points increases; modern applications must cope with thousands or millions of these at a time. Here, we introduce the computational optimal transport problem and summarize recent ideas for achieving new heights in efficiency and scalability. 

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