Analysis

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Touching the transcendentals: tractional motion from the birth of calculus to future perspectives

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When the rigorous foundation of calculus was developed, it marked an epochal change in the approach of mathematicians to geometry. Tools from geometry had been one of the foundations of mathematics until the 17th century but today, mainstream conception relegates geometry to be merely a tool of visualization. In this snapshot, however, we consider geometric and constructive components of calculus.

Limits of graph sequences

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Graphs are simple mathematical structures used to model a wide variety of real-life objects. With the rise of computers, the size of the graphs used for these models has grown enormously. The need to efficiently represent and study properties of extremely large graphs led to the development of the theory of graph limits.

On Logic, Choices and Games

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Can we always mathematically formalise our taste and preferences? We discuss how this has been done historically in the field of game theory, and how recent ideas from logic and computer science have brought an interesting twist to this beautiful theory.

Nonlinear Acoustics

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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.

Mixed volumes and mixed integrals

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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.

Geometry behind one of the Painlevé III differential equations

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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. 

Molecular quantum dynamics

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We provide a brief introduction to some basic ideas of Molecular Quantum Dynamics. We discuss the scope, strengths and main applications of this field of science. Finally, we also mention open problems of current interest in this exciting subject. 

Mathematische Modellierung von Krebswachstum

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Krebs ist eine der größten Herausforderungen der modernen Medizin. Der WHO zufolge starben 2012 weltweit 8,2 Millionen Menschen an Krebs. Bis heute sind dessen molekulare Mechanismen nur in Teilen verstanden, was eine erfolgreiche Behandlung erschwert. Mathematische Modellierung und Computersimulationen können helfen, die Mechanismen des Tumorwachstums besser zu verstehen. Sie eröffnen somit neue Chancen für zukünftige Behandlungsmethoden.

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