http://publications.mfo.de/handle/mfo/1336 Fri, 17 Apr 2026 11:25:05 GMT 2026-04-17T11:25:05Z http://publications.mfo.de/handle/mfo/1400 Mixed volumes and mixed integrals Rotem, Liran 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. 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. Sat, 29 Dec 2018 00:00:00 GMT http://publications.mfo.de/handle/mfo/1400 2018-12-29T00:00:00Z Rotem, Liran 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. 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. http://publications.mfo.de/handle/mfo/1396 Estimating the volume of a convex body Baldin, Nicolai 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. Sun, 30 Dec 2018 00:00:00 GMT http://publications.mfo.de/handle/mfo/1396 2018-12-30T00:00:00Z Baldin, Nicolai 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. http://publications.mfo.de/handle/mfo/1384 Topological Complexity, Robotics and Social Choice Carrasquel, José; Lupton, Gregory; Oprea, John 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. Fri, 10 Aug 2018 00:00:00 GMT http://publications.mfo.de/handle/mfo/1384 2018-08-10T00:00:00Z Carrasquel, José Lupton, Gregory Oprea, John 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. http://publications.mfo.de/handle/mfo/1381 A short story on optimal transport and its many applications Santambrogio, Filippo 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. Wed, 08 Aug 2018 00:00:00 GMT http://publications.mfo.de/handle/mfo/1381 2018-08-08T00:00:00Z Santambrogio, Filippo 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. http://publications.mfo.de/handle/mfo/1380 Number theory in quantum computing Schönnenbeck, Sebastian 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 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. Tue, 07 Aug 2018 00:00:00 GMT http://publications.mfo.de/handle/mfo/1380 2018-08-07T00:00:00Z Schönnenbeck, Sebastian 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 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. http://publications.mfo.de/handle/mfo/1378 Tropical geometry Brugallé, Erwan; Itenberg, Ilia; Shaw, Kristin; Viro, Oleg What kind of strange spaces hide behind the enigmatic name of tropical geometry? In the tropics, just as in other geometries, one of the simplest objects is a line. Therefore, we begin our exploration by considering tropical lines. Afterwards, we take a look at tropical arithmetic and algebra, and describe how to define tropical curves using tropical polynomials. Thu, 19 Jul 2018 00:00:00 GMT http://publications.mfo.de/handle/mfo/1378 2018-07-19T00:00:00Z Brugallé, Erwan Itenberg, Ilia Shaw, Kristin Viro, Oleg What kind of strange spaces hide behind the enigmatic name of tropical geometry? In the tropics, just as in other geometries, one of the simplest objects is a line. Therefore, we begin our exploration by considering tropical lines. Afterwards, we take a look at tropical arithmetic and algebra, and describe how to define tropical curves using tropical polynomials. http://publications.mfo.de/handle/mfo/1375 Data assimilation: mathematics for merging models and data Morzfeld, Matthias; Reich, Sebastian 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. Tue, 10 Jul 2018 00:00:00 GMT http://publications.mfo.de/handle/mfo/1375 2018-07-10T00:00:00Z Morzfeld, Matthias Reich, Sebastian 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. http://publications.mfo.de/handle/mfo/1370 Fast Solvers for Highly Oscillatory Problems Barnett, Alex 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. Our snapshot will introduce the reader to recent progress in designing algorithms that allow for much more rapid solutions. Tue, 26 Jun 2018 00:00:00 GMT http://publications.mfo.de/handle/mfo/1370 2018-06-26T00:00:00Z Barnett, Alex 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. Our snapshot will introduce the reader to recent progress in designing algorithms that allow for much more rapid solutions. http://publications.mfo.de/handle/mfo/1367 Geometry behind one of the Painlevé III differential equations Hertling, Claus 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. Wed, 20 Jun 2018 00:00:00 GMT http://publications.mfo.de/handle/mfo/1367 2018-06-20T00:00:00Z Hertling, Claus 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. http://publications.mfo.de/handle/mfo/1365 The codimension Lerario, Antonio 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. Tue, 19 Jun 2018 00:00:00 GMT http://publications.mfo.de/handle/mfo/1365 2018-06-19T00:00:00Z Lerario, Antonio 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.