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- ItemEstimating the volume of a convex body(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2018) Baldin, NicolaiSometimes 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.
- ItemDomino tilings of the Aztec diamond(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2015) Rué, JuanjoImagine 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 dominoes? One specific paper cutout can be mathematically described as the so-called Aztec Diamond, and a way to cover it with dominoes is a domino tiling. In this snapshot we revisit some of the seminal combinatorial ideas used to enumerate the number of domino tilings of the Aztec Diamond. The existing connection with the study of the so-called alternating-sign matrices is also explored.
- ItemRandom permutations(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Betz, Volker100 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.
- ItemThe Kadison-Singer problem(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2014) Valette, AlainIn 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.
- 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.
- 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.
- 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.