Snapshots of Modern Mathematics from Oberwolfach

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(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2022) Dozier, Benjamin

We consider surfaces of three types: the sphere, the torus, and many-holed tori. These surfaces naturally admit geometries of positive, zero, and negative curvature, respectively. It is interesting to study straight line paths, known as geodesics, in these geometries. We discuss the issue of counting closed geodesics; this is particularly rich for hyperbolic (negatively curved) surfaces.

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Patterns and Waves in Theory, Experiment, and Application

(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2023) Bramburger, Jason J.

In this snapshot of modern mathematics we describe some of the most prevalent waves and patterns that can arise in mathematical models and which are used to describe a number of biological, chemical, physical, and social processes. We begin by focussing on two types of patterns that do not change in time: space-filling patterns and localized patterns. We then discuss two types of waves that evolve predictably as time goes on: spreading waves and rotating waves. All our examples are motivated with real-world applications and we highlight some of the main lines of research that mathematicians pursue to better understand them.

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A tale of three curves

(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2022) Balakrishnan, Jennifer S.

In this snapshot, we give a survey of some problems in the study of rational points on higher genus curves, discussing questions ranging from the era of the ancient Greeks to a few posed by mathematicians of the 20th century. To answer these questions, we describe a selection of techniques in modern number theory that can be used to determine the set of rational points on a curve.

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Route planning for bacteria

(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2022) Hellmuth, Kathrin; Klingenberg, Christian

Bacteria have been fascinating biologists since their discovery in the late 17th century. By analysing their movements, mathematical models have been developed as a tool to understand their behaviour. However, adapting these models to real situations can be challenging, because the model coefficients cannot be observed directly. In this snapshot, we study this question mathematically and explain how the idea of “route planning” can be used to determine these model coefficients.

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Characterizations of intrinsic volumes on convex bodies and convex functions

(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2022) Mussnig, Fabian

If we want to express the size of a two-dimensional shape with a number, then we usually think about its area or circumference. But what makes these quantities so special? We give an answer to this question in terms of classical mathematical results. We also take a look at applications and new generalizations to the setting of functions.

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What is pattern?

(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2022) Baake, Michael; Grimm, Uwe; Moody, Robert V.

Pattern is ubiquitous and seems totally familiar. Yet if we ask what it is, we find a bewildering collection of answers. Here we suggest that there is a common thread, and it revolves around dynamics.

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Voronoi Cells: Or How to Find the Nearest Bakery

(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2024) Hess, Sarah; Torres, Angélica; van der Eyden, Mirte

Deciding which mall, hospital or school is closest to us is a problem we face everyday. It even comes on holidays with us, when we optimize our plans to make sure that we have enough time to visit all the attractions we want to see. In this article, we show how concepts from metric algebraic geometry help us to rise to this task while planning a weekend trip to the Black Forest.

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Representations and degenerations

(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2022) Dumanski, Ilya; Kiritchenko, Valentina

In this snapshot, we explain two important mathematical concepts (representation and degeneration) in elementary terms. We will focus on the simplest meaningful examples, and motivate both concepts by study of symmetry.

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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.

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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.

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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.

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Geproci Sets: a New Perspective in Algebraic Geometry

(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2023) Chiantini, Luca; Harbourne, Brian

Geproci sets arise from applying the perspective of inverse scattering problems to algebraic geometry. Analogous to the reconstruction of an object from multiple X-ray images, we aim at a classification of sets with certain algebraic properties under multiple projections.

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The Geometry of Fair Division

(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2023) Frick, Florian

How can we fairly divide a necklace with various types of beads? We use this problem as a motivating example to explain how geometry naturally appears in solutions of non-geometric problems. The strategy we develop to solve this problem has been used in several other contexts.

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Algebras and Quantum Games

(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2023) Paulsen, Vern I.

Everyone loves a good game, but when the players can access the counterintuitive world of quantum mechanics, watch out!

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4 = 2 × 2, or the Power of Even Integers in Fourier Analysis

(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2023) Negro, Giuseppe; Oliveira e Silva, Diogo

We describe how simple observations related to vectors of length 1 recently led to the proof of an important mathematical fact: the sharp Stein–Tomas inequality from Fourier restriction theory, a pillar of modern harmonic analysis with surprising applications to number theory and geometric measure theory.

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Waves and Incidences

(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2024) Yung, Po-Lam

The wave equation in Euclidean spaces describes many natural phenomena such as sound, light, or water waves. We explore how its solutions are related to the geometric problem of how long thin cylinders can intersect each other and discuss a related open problem.

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Felder und Räume: Symmetrie und Lokalität in Mathematik und theoretischen Wissenschaften

(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2023) Saberi, Ingmar

Wir werden einige grundlegende Ideen der Eichtheorie und der dazugehörigen Differentialtopologie erkunden. Damit kann sich die Leserin ein Bild des Modulraums flacher Zusammenhänge machen und ihn mit den physikalisch motivierten Ideen dahinter in Beziehung bringen. Den Begriffen von Symmetrien und Feldern gehen wir gründlich nach. Außerdem werfen wir einen flüchtigen Blick auf unendliche Symmetrie in zwei Dimensionen und auf vor kurzem entdeckte Verallgemeinerungen.

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The Periodic Tables of Algebraic Geometry

(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2023) Belmans, Pieter

To understand our world, we classify things. A famous example is the periodic table of elements, which describes the properties of all known chemical elements and gives us a classification of the building blocks we can use in physics, chemistry, and biology. In mathematics, and algebraic geometry in particular, there are many instances of similar “periodic tables”, describing fundamental classification results. We will go on a tour of some of these.

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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.

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Jewellery from tessellations of hyperbolic space

(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2022) Gangl, Herbert

In this snapshot, we will first give an introduction to hyperbolic geometry and we will then show how certain matrix groups of a number-theoretic origin give rise to a large variety of interesting tessellations of 3-dimensional hyperbolic space. Many of the building blocks of these tessellations exhibit beautiful symmetry and have inspired the design of 3D printed jewellery.

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