Patterns and waves in theory, experiment, and application

Snapshots of modern mathematics from Oberwolfach

Patterns and waves in theory, experiment, and application

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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Mathematical subjects

Analysis
Numerics and Scientific Computing

Author(s)

Jason J. Bramburger

License

DOI (Digital Object Identifier)

10.14760/SNAP-2023-001-EN

Download PDF

snapshot-2023-001.pdf

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snapshots: overview

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Cutoff phenomenon: Surprising behaviour in card shuffling and other Markov chains
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Algebras and quantum games
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Felder und Räume: Symmetrie und Lokalität in Mathematik und theoretischen Wissenschaften
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The periodic tables of algebraic geometry
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Patterns and waves in theory, experiment, and application
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Closed geodesics on surfaces
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Route planning for bacteria
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Characterizations of intrinsic volumes on convex bodies and convex functions
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A tale of three curves
Michael Baake, Uwe Grimm, Robert V. Moody
What is Pattern?
Saiteja Utpala, Nina Miolane
Biological Shape Analysis with Geometric Statistics and Learning
Ilya Dumanski, Valentina Kiritchenko
Representations and degenerations
Jonas Latz, Björn Sprungk
Solving inverse problems with Bayes’ theorem
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Jewellery from tessellations of hyperbolic space
Chr istina Frederick, Yunan Yang
Seeing through rock with help from optimal transport
David A. Craven
Searching for the Monster in the Trees
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The Robinson–Schensted algorithm
Franca Hoffmann, Sara Merino-Aceituno
Emergence in biology and social sciences
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Lagrangian mean curvature flow
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Finite geometries: pure mathematics close to applications
Francesca Arici, Bram Mesland
Describing distance: from the plane to spectral triples
Martijn Caspers
Quantum symmetry
Fadoua Balabdaoui, Yulia Kulagina
The Enigma behind the Good–Turing formula
Anna Haensch
Reflections on hyperbolic space
Isaac Goldbring
Ultrafilter methods in combinatorics
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Zopfgruppen, die Yang–Baxter-Gleichung und Unterfaktoren
Baptiste Rognerud
Invitation to quiver representation and Catalan combinatorics
Po-Ling Loh
Searching for structure in complex data: A modern statistical quest
David Kerr
C∗-algebras: structure and classification
Sara Lamboglia, Martin Ulirsch
From the dollar game to the Riemann-Roch Theorem
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Higgs bundles without geometry
Behrend Heeren, Martin Rumpf
Shape space - a paradigm for character animation in computer graphics
Joshua Zahl
Rotating needles, vibrating strings, and Fourier summation
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Quantum symmetry
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Determinacy versus indeterminacy
Jayadev S. Athreya, David Aulicino
Vertex-to-Self Trajectories on the Platonic Solids
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Random matrix theory: Dyson Brownian motion
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From Betti numbers to l2-Betti numbers
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Is it possible to predict the far future before the near future is known accurately?
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The Interaction of Curvature and Topology
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A surprising connection between quantum mechanics and shallow water waves
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Formation Control and Rigidity Theory
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Mixed-dimensional models for real-world applications
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Limits of graph sequences
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On Logic, Choices and Games
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Nonlinear Acoustics
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Random permutations
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Counting self-avoiding walks on the hexagonal lattice
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Algebra, matrices, and computers
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Positive Scalar Curvature and Applications
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Diophantine equations and why they are hard
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On radial basis functions
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Snake graphs, perfect matchings and continued fractions
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Estimating the volume of a convex body
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Mixed volumes and mixed integrals
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A short story on optimal transport and its many applications
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Number theory in quantum computing
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Data assimilation: mathematics for merging models and data
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Geometry behind one of the Painlevé III differential equations
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The codimension
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The mathematics of aquatic locomotion
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Tropical geometry
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Fast Solvers for Highly Oscillatory Problems
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Computing with symmetries
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Topological recursion
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The Algebraic Statistics of an Oberwolfach Workshop
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Solving quadratic equations in many variables
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Mathematics plays a key role in scientific computing
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Spaces of Riemannian metrics
Shaula Fiorelli Vilmart, Gilles Vilmart
Computing the long term evolution of the solar system with geometric numerical integrators
Justin Solomon
Computational Optimal Transport
Justyna Szpond
A few shades of interpolation
George Hagedorn, Caroline Lasser
Molecular quantum dynamics
Marco Radeschi
Closed geodesics on surfaces and Riemannian manifolds
Christian Engwer, Markus Knappitsch
Mathematische Modellierung von Krebswachstum
Michael Baake, David Damanik, Uwe Grimm
Aperiodic Order and Spectral Properties
Lukas Pottmeyer
News on quadratic polynomials
Felix Günther
Winkeltreue zahlt sich aus
Jacob Bedrossian
Towards a Mathematical Theory of Turbulence in Fluids
Laurent Bartholdi
Profinite groups
Dietmar Gallistl
The adaptive finite element method
Steven Klee
Footballs and donuts in four dimensions
Nikolai Nowaczyk
The Willmore Conjecture
Lior Bary-Soroker
Prime Tuples in Function Fields
Rafael Guglielmetti, Matthieu Jacquemet
Polyhedra and commensurability
Manfred Deistler, Andreas Graef
Fokus-Erkennung bei Epilepsiepatienten mithilfe moderner Verfahren der Zeitreihenanalyse
Robert Altmann, Jan Heiland
Wie steuert man einen Kran?
Anthony T Patera, Karsten Urban
High performance computing on smartphones
Eugenio Giannelli, Jay Taylor
Symmetry and characters of finite groups
Maria Dostert, Stefan Krupp, Jan Hendrick Rolfes
Das Problem der Kugelpackung
Tomasz Szemberg
On the containment problem
Éric Fusy
Random sampling of domino and lozenge tilings
Magnus Egerstedt
Swarming robots
Ulrich Krähmer
From computer algorithms to quantum field theory: an introduction to operads
Juanjo Rué
Domino tilings of the Aztec Diamond
Sebastian Funk
The mystery of sleeping sickness – why does it keep waking up?
Antti Knowles
Quantum diffusion
Amanda Swan, Albert Murtha
Modelling the spread of brain tumours
J. Marron
Visual Analysis of Spanish Male Mortality
Christopher Sangwin
Curriculum development in university mathematics: where mathematicians and education collide
Jürg Kramer, Anna-Maria Pippich
Special values of zeta functions and areas of triangles
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Valentina Kiritchenko, Evgeny Smirnov, Vladlen Timorin
Ideas of Newton-Okounkov bodies
Ben Schweizer
Darcy’s law and groundwater flow modelling
Raluca Eftimie
Modelling communication and movement: from cells to animals and humans
Tom Solomon
Chaos and chaotic fluid mixing
Thorsten Holm
Friezes and tilings
George Willis
Zero-dimensional symmetry
Christine Breiner
Minimizing energy
Diana Davis
Billiards and flat surfaces
Peter Benner, Hermann Mena, René Schneider
Drugs, herbicides, and numerical simulation
Gregory Knese
Operator theory and the singular value decomposition
Alain Valette
The Kadison-Singer problem
Ragnar-Olaf Buchweitz, Eleonore Faber
Swallowtail on the shore
Howell Tong
Statistics and dynamical phenomena
Brian Harbourne, Thomas Szemberg
Arrangements of lines
Bruce Reznick
What does ">" really mean?
Harald Helfgott
The ternary Goldbach Problem
Wolfgang Lerche
Matrixfaktorisierungen
John E. McCarthy
Dirichlet Series

Mathematical subjects

Algebra and Number Theory
Analysis
Didactics and Education
Discrete Mathematics and Foundations
Geometry and Topology
Numerics and Scientific Computing
Probability Theory and Statistics

Connections to other fields

Chemistry and Earth Science
Computer Science
Engineering and Technology
Finance
Humanities and Social Sciences
Life Science
Physics
Reflections on Mathematics

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