Snapshots of Modern Mathematics from Oberwolfach

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The Algebraic Statistics of an Oberwolfach Workshop
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 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.
Biological shape analysis with geometric statistics and learning
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2022) Utpala, Saiteja; Miolane, Nina
The advances in biomedical imaging techniques have enabled us to access the 3D shapes of a variety of structures: organs, cells, proteins. Since biological shapes are related to physiological functions, shape data may hold the key to unlocking outstanding mysteries in biomedicine. This snapshot introduces the mathematical framework of geometric statistics and learning and its applications to biomedicine.
Counting self-avoiding walks on the hexagonal lattice
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 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.
Cutoff Phenomenon: Surprising Behaviour in Card Shuffling and other Markov Chains
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2023) Baraquin, Isabelle; Lafrenière, Nadia; Schuh, Katharina
This snapshot compares two techniques of shuffling a deck of cards, asking how long it will take to shuffle the cards until a “well-mixed deck” is obtained. Surprisingly, the number of shuffles can be very different for very similar looking shuffling techniques.
Determinacy versus indeterminacy
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2020) Berg, Christian
Can 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.
Domino tilings of the Aztec diamond
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2015) Rué, Juanjo
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 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.
The Enigma behind the Good–Turing formula
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2021) Balabdaoui, Fadoua; Kulagina, Yulia
Finding the total number of species in a population based on a finite sample is a difficult but practically important problem. In this snapshot, we will attempt to shed light on how during World War II, two cryptanalysts, Irving J. Good and Alan M. Turing, discovered one of the most widely applied formulas in statistics. The formula estimates the probability of missing some of the species in a sample drawn from a heterogeneous population. We will provide some intuition behind the formula, show its wide range of applications, and give a few technical details.
Estimating the volume of a convex body
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 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.
Fokus-Erkennung bei Epilepsiepatienten mithilfe moderner Verfahren der Zeitreihenanalyse
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 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.
The Kadison-Singer problem
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 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.
Limits of graph sequences
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 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.
Quantum diffusion
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 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.
Random matrix theory: Dyson Brownian motion
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2020) Finocchio, Gianluca
The 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.
Random permutations
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 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.
Randomness is Natural - an Introduction to Regularisation by Noise
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2024) Djurdjevac, Ana; Elad Altman, Henri; Rosati, Tommaso
Differential equations make predictions on the future state of a system given the present. In order to get a sensible prediction, sometimes it is necessary to include randomness in differential equations, taking microscopic effects into account. Surprisingly, despite the presence of randomness, our probabilistic prediction of future states is stable with respect to changes in the surrounding environment, even if the original prediction was unstable. This snapshot will unveil the core mathematical mechanism underlying this "regularisation by noise" phenomenon.
Searching for structure in complex data: a modern statistical quest
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2021) Loh, Po-Ling
Current 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.
Solving inverse problems with Bayes' theorem
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2022) Latz, Jonas; Sprungk, Björn
The goal of inverse problems is to find an unknown parameter based on noisy data. Such problems appear in a wide range of applications including geophysics, medicine, and chemistry. One method of solving them is known as the Bayesian approach. In this approach, the unknown parameter is modelled as a random variable to reflect its uncertain value. Bayes' theorem is applied to update our knowledge given new information from noisy data.
Statistics and dynamical phenomena
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 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.
Topological recursion
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2018) Sułkowski, Piotr
In this snapshot we present the concept of topological recursion – a new, surprisingly powerful formalism at the border of mathematics and physics, which has been actively developed within the last decade. After introducing necessary ingredients – expectation values, random matrices, quantum theories, recursion relations, and topology – we explain how they get combined together in one unifying picture.
Visual 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.

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Browsing Snapshots of Modern Mathematics from Oberwolfach by Subject "Probability Theory and Statistics"

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