2015
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From computer algorithms to quantum field theory: an introduction to operads
(Mathematisches Forschungsinstitut Oberwolfach, 2015)An operad is an abstract mathematical tool encoding operations on specific mathematical structures. It finds applications in many areas of mathematics and related fields. This snapshot explains the concept of an operad and ... 
Domino tilings of the Aztec diamond
(Mathematisches Forschungsinstitut Oberwolfach, 2015)Imagine you have a cutout from a piece of squared paper and a pile of dominoes, each of which can cover exactly two squares of the squared paper. How many different ways are there to cover the entire paper cutout with ... 
The mystery of sleeping sickness – why does it keep waking up?
(Mathematisches Forschungsinstitut Oberwolfach, 2015)Sleeping sickness is a neglected tropical disease that affects rural populations in Africa. Deadly when untreated, it is being targeted for elimination through case finding and treatment. Yet, fundamental questions about ... 
Quantum diffusion
(Mathematisches Forschungsinstitut Oberwolfach, 2015)If you place a drop of ink into a glass of water, the ink will slowly dissipate into the surrounding water until it is perfectly mixed. If you record your experiment with a camera and play the film backwards, you will see ... 
Modelling the spread of brain tumours
(Mathematisches Forschungsinstitut Oberwolfach, 2015)The study of mathematical biology attempts to use mathematical models to draw useful conclusions about biological systems. Here, we consider the modelling of brain tumour spread with the ultimate goal of improving treatment ... 
Visual analysis of Spanish male mortality
(Mathematisches Forschungsinstitut Oberwolfach, 2015)Statistical visualization uses graphical methods to gain insights from data. Here we show how a technique called principal component analysis is used to analyze mortality in Spain over about the last hundred years. This ... 
Curriculum development in university mathematics: where mathematicians and education collide
(Mathematisches Forschungsinstitut Oberwolfach, 2015)This snapshot looks at educational aspects of the design of curricula in mathematics. In particular, we examine choices textbook authors have made when introducing the concept of the completness of the real numbers. Can ... 
Special values of zeta functions and areas of triangles
(Mathematisches Forschungsinstitut Oberwolfach, 2015)In this snapshot we give a glimpse of the interplay of special values of zeta functions and volumes of triangles. Special values of zeta functions and their generalizations arise in the computation of volumes of moduli ... 
How to choose a winner: the mathematics of social choice
(Mathematisches Forschungsinstitut Oberwolfach, 2015)Suppose a group of individuals wish to choose among several options, for example electing one of several candidates to a political office or choosing the best contestant in a skating competition. The group might ask: what ... 
Ideas of NewtonOkounkov bodies
(Mathematisches Forschungsinstitut Oberwolfach, 2015)In this snapshot, we will consider the problem of finding the number of solutions to a given system of polynomial equations. This question leads to the theory of Newton polytopes and NewtonOkounkov bodies of which we will ... 
Darcy's law and groundwater flow modelling
(Mathematisches Forschungsinstitut Oberwolfach, 2015)Formulations of natural phenomena are derived, sometimes, from experimentation and observation. Mathematical methods can be applied to expand on these formulations, and develop them into better models. In the year 1856, ... 
Modeling communication and movement: from cells to animals and humans
(Mathematisches Forschungsinstitut Oberwolfach, 2015)Communication forms the basis of biological interactions. While the use of a single communication mechanism (for example visual communication) by a species is quite well understood, in nature the majority of species ... 
Chaos and chaotic fluid mixing
(Mathematisches Forschungsinstitut Oberwolfach, 2015)Very simple mathematical equations can give rise to surprisingly complicated, chaotic dynamics, with behavior that is sensitive to small deviations in the initial conditions. We illustrate this with a single recurrence ... 
Friezes and tilings
(Mathematisches Forschungsinstitut Oberwolfach, 2015)Friezes have occured as architectural ornaments for many centuries. In this snapshot, we consider the mathematical analogue of friezes as introduced in the 1970s by Conway and Coxeter. Recently, infinite versions of such ... 
Zerodimensional symmetry
(Mathematisches Forschungsinstitut Oberwolfach, 2015)This snapshot is about zerodimensional symmetry. Thanks to recent discoveries we now understand such symmetry better than previously imagined possible. While still far from complete, a picture of zerodimensional symmetry ... 
Minimizing energy
(Mathematisches Forschungsinstitut Oberwolfach, 2015)What is the most efficient way to fence land when you’ve only got so many metres of fence? Or, to put it differently, what is the largest area bounded by a simple closed planar curve of fixed length? We consider the answer ... 
Billiards and flat surfaces
(Mathematisches Forschungsinstitut Oberwolfach, 2015)Billiards, the study of a ball bouncing around on a table, is a rich area of current mathematical research. We discuss questions and results on billiards, and on the related topic of flat surfaces.