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We describe a connection between quantum mechanics and nonlinear wave equations and highlight a few problems at the forefront of modern research in the intersection of these areas.
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Throughout the history of dynamical systems, instruments have been used to calculate and visualize (approximate) solutions of differential equations. Here we describe the approach of a group of physicists and engineers in the period 1948–1964, and we give ex- amples of the specific (analogue) mathematical in- struments they conceived and used. These examples also illustrate how their analogue culture and practices faced the advent of the digital computer, which appeared at that time as a new instrument, full of promises.
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Nonlinear acoustics has been a topic of research for more than 250 years. Driven by a wide range and a large number of highly relevant industrial and medical applications, this area has expanded enormously in the last few decades. Here, we would like to give a glimpse of the mathematical modeling techniques that are commonly employed to tackle problems in this area of research, with a selection of references for the interested reader to further their knowledge into this mathematically interesting field.
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100 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 re- turned 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.
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We introduce the idea of curvature, including how it developed historically, and focus on the scalar curvature of a manifold. A major current research topic involves understanding positive scalar curvature. We discuss why this is interesting and how it relates to general relativity.
Also available in Spanish.
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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.
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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.
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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.
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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.
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We introduce and outline a workshop concept called Science Spaces aimed at High School students, which combines mathematics and natural sciences with art to strengthen creativity, imagination, and perceptual abilities. The workshop guides the participants through the process of designing and implementing a public exhibition.