2015, Knowles, Antti

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 something that is never observed in the real world. Such diffusive and irreversible behaviour is ubiquitous in nature. Nevertheless, the fundamental equations that describe the motion of individual particles – Newton’s and Schrödinger’s equations – are reversible in time: a film depicting the motion of just a few particles looks as realistic when played forwards as when played backwards. In this snapshot, we discuss how one may try to understand the origin of diffusion starting from the fundamental laws of quantum mechanics.

2019, Klimošová, Tereza

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.

2014, Tong, Howell

A friend of mine, an expert in statistical genomics, told me the following story: At a dinner party, an attractive lady asked him, "What do you do for a living?" He replied, "I model." As my friend is a handsome man, the lady did not question his statement and continued, "What do you model?" "Genes." She then looked at him up and down and said, "Mh, you must be very much in demand." "Yes, very much so, especially after I helped discover a new culprit gene for a common childhood disease." The lady looked puzzled. In this snapshot, I will give you an insight into Statistics, the field that fascinated my friend (and myself) so much. I will concentrate on phenomena that change over time, in other words, dynamical events.

2014, Valette, Alain

In quantum mechanics, unlike in classical mechanics, one cannot make precise predictions about how a system will behave. Instead, one is concerned with mere probabilities. Consequently, it is a very important task to determine the basic probabilities associated with a given system. In this snapshot we will present a recent uniqueness result concerning these probabilities.

2019, Duminil-Copin, Hugo

In how many ways can you go for a walk along a lattice grid in such a way that you never meet your own trail? In this snapshot, we describe some combinatorial and statistical aspects of these so-called self-avoiding walks. In particular, we discuss a recent result concerning the number of self-avoiding walks on the hexagonal (“honeycomb”) lattice. In the last part, we briefly hint at the connection to the geometry of long random self-avoiding walks.

2015, Marron, J. S.

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 data decomposition both reflects expected historical events and reveals some perhaps less expected trends in mortality over the years.

2019, Betz, Volker

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 returned 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. We present some of the open questions and give an overview of what current research can say about them.

2018, Seigal, Anna

Algebraic Statistics builds on the idea that statistical models can be understood via polynomials. Many statistical models are parameterized by polynomials in the model parameters; others are described implicitly by polynomial equalities and inequalities. We explore the connection between algebra and statistics for some small statistical models.

2016, Deistler, Manfred, Graef, Andreas

Viele epileptische Anfälle entstehen in einer begrenzten Region im Gehirn, dem sogenannten Anfallsursprung. Eine chirurgische Entfernung dieser Region kann in vielen Fällen zu Anfallsfreiheit führen. Aus diesem Grund ist die Frage nach der Lokalisation des Anfallsursprungs aus EEG-Aufzeichnungen wichtig. Wir beschreiben hier ein Verfahren zur Lokalisation des Anfallsursprungs mittels Zeitreihenanalyse, das auf der Schätzung von Spektren im EEG beruht.

2018, 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.