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- ItemQuantum diffusion(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2015) Knowles, AnttiIf 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.
- ItemDeterminacy versus indeterminacy(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2020) Berg, ChristianCan a continuous function on an interval be uniquely determined if we know all the integrals of the function against the natural powers of the variable? Following Weierstrass and Stieltjes, we show that the answer is yes if the interval is finite, and no if the interval is infinite.
- ItemCounting self-avoiding walks on the hexagonal lattice(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Duminil-Copin, HugoIn 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.
- ItemSearching for structure in complex data: a modern statistical quest(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2021) Loh, Po-LingCurrent research in statistics has taken interesting new directions, as data collected from scientific studies has become increasingly complex. At first glance, the number of experiments conducted by a scientist must be fairly large in order for a statistician to draw correct conclusions based on noisy measurements of a large number of factors. However, statisticians may often uncover simpler structure in the data, enabling accurate statistical inference based on relatively few experiments. In this snapshot, we will introduce the concept of high-dimensional statistical estimation via optimization, and illustrate this principle using an example from medical imaging. We will also present several open questions which are actively being studied by researchers in statistics.
- ItemThe Algebraic Statistics of an Oberwolfach Workshop(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2018) Seigal, AnnaAlgebraic 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.
- ItemRandom matrix theory: Dyson Brownian motion(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2020) Finocchio, GianlucaThe theory of random matrices was introduced by John Wishart (1898–1956) in 1928. The theory was then developed within the field of nuclear physics from 1955 by Eugene Paul Wigner (1902–1995) and later by Freeman John Dyson, who were both concerned with the statistical description of heavy atoms and their electromagnetic properties. In this snapshot, we show how mathematical properties can have unexpected links to physical phenomenena. In particular, we show that the eigenvalues of some particular random matrices can mimic the electrostatic repulsion of the particles in a gas.
- ItemLimits of graph sequences(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Klimošová, TerezaGraphs 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.
- ItemVisual analysis of Spanish male mortality(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 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.
- ItemFokus-Erkennung bei Epilepsiepatienten mithilfe moderner Verfahren der Zeitreihenanalyse(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2016) Deistler, Manfred; Graef, AndreasViele 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.
- ItemStatistics and dynamical phenomena(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2014) Tong, HowellA 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.