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Submitted by IMAGINARY on
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 convex geometry become thereby accessible to analytic and probabilistic tools, and we can use these tools to make progress on very difficult open problems.
Submitted by IMAGINARY on
Sometimes 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.
Submitted by Emmanuelle Féau... on
Removable mirror rooms enabling to visualize spherical tesselations and platonic solids
Submitted by IMAGINARY on
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 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.
Submitted by IMAGINARY on
When you describe a physical process, for example, the weather on Earth, or an engineered system, such as a self-driving car, you typically have two sources of information. The first is a mathematical model, and the second is information obtained by collecting data. To make the best predictions for the weather, or most effectively operate the self-driving car, you want to use both sources of information. Data assimilation describes the mathematical, numerical and computational framework for doing just that.
Submitted by IMAGINARY on
Algorithms 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 comput- ing power. Translating and implementing algorithms on quantum computers is hard.
Submitted by IMAGINARY on
We present some examples of optimal transport problems and of applications to different sciences (logistics, economics, image processing, and a little bit of evolution equations) through the crazy story of an industrial dynasty regularly asking advice from an exotic mathematician.
Submitted by IMAGINARY on
The 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.
Submitted by IMAGINARY on
In 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.
Submitted by IMAGINARY on
Waves of diverse types surround us. Sound, light and other waves, such as microwaves, are crucial for speech, mobile phones, and other communication technologies. Elastic waves propagating through the Earth bounce through the Earth’s crust and enable us to “see” thousands of kilometres in depth. These propagating waves are highly oscillatory in time and space, and may scatter off obstacles or get “trapped” in cavities. Simulating these phenomena on computers is extremely important. However, the achievable speeds for accurate numerical modelling are low even on large modern computers.