Atlas / Across domains
Crossings
Each domain in the atlas is surveyed on its own, but ideas do not stay put. A crossing is a use written on one field's page that lands in another domain: geometry becoming gravity, physics revealing the double helix. Solid lines land in a field this atlas has surveyed. Dashed lines reach ground not yet mapped.
Linked crossings · 73
Mathematics → Physics
Algebraic Topology → Topological Matter
Topological phases of matter
Condensed matter physics
Bayesian Statistics → Physical Cosmology
Weighing the universe
Cosmology
Calculus → Classical Mechanics
The laws of motion are differential equations
Physics
Combinatorial Optimisation → Statistical Mechanics
Ground states of spin glasses
Statistical physics
Complex Analysis → Quantum Mechanics
Quantum mechanics is written in complex numbers
Quantum physics
Differential Equations → Classical Mechanics
Every law of motion is a differential equation
Mechanics
Dynamical Systems → Classical Mechanics
Low-energy routes through the solar system
Spaceflight
Enumerative Combinatorics → Statistical Mechanics
Entropy is a count
Statistical physics
Ergodic Theory → Statistical Mechanics
Why thermodynamics works
Statistical physics
Geometric Topology → Physical Cosmology
What shape is the universe?
Cosmology
Geometric Topology → Quantum Information
Computing with braids
Quantum computing
Graph Theory → Electromagnetism
Kirchhoff's laws are graph theory
Electrical engineering
Information Theory → Statistical Mechanics
Entropy is missing information
Statistical physics
Monte Carlo Methods → Statistical Mechanics
Simulating matter
Statistical physics
Non-Euclidean Geometry → Special Relativity
The geometry of velocities
Special relativity
Numerical Analysis → Classical Mechanics
The return of Halley's comet
Astronomy
Numerical Linear Algebra → Quantum Mechanics
Energy levels as eigenvalues
Quantum physics
Numerical Methods for PDEs → Classical Mechanics
Numerical weather prediction
Weather and climate
Numerical Methods for PDEs → General Relativity
Templates for LIGO
Gravitational waves
Probability Theory → Kinetic Theory of Gases
Brownian motion and atoms
Statistical physics
Representation Theory → Quantum Mechanics
Atomic spectra and spin
Quantum physics
Representation Theory → Particle Physics
The Eightfold Way
Particle physics
Riemannian Geometry → General Relativity
Gravity is curvature, and satellites notice
Relativity and GPS
Statistical Learning Theory → Phase Transitions
Learning phases of matter
Condensed matter physics
Stochastic Processes → Kinetic Theory of Gases
Newton's law with noise
Physics
Mathematics → Biology
Algebraic Geometry → Evolutionary Biology
Algebraic statistics for evolutionary trees
Evolutionary biology
Bayesian Statistics → Evolutionary Biology
Bayesian family trees
Evolution
Chaos Theory → Population Ecology
Booms and crashes without outside causes
Ecology
Computational Complexity → Genomics
Assembling genomes: an easy path instead of a hard one
Genomics
Continuous Optimisation → Molecular Biology
Predicting protein structures
Structural biology
Enumerative Combinatorics → Evolutionary Biology
Why the tree of life can't be found by brute force
Phylogenetics
Fourier Analysis → Molecular Biology
Crystallography is a Fourier transform
Structural biology
Geometric Topology → Molecular Biology
Knotted DNA
Molecular biology
Euler's bridges, in DNA
Genome assembly
Information Theory → Molecular Biology
Reading information in DNA
Molecular biology
Monte Carlo Methods → Evolutionary Biology
Bayesian family trees of species
Evolution
Probability Theory → Population Genetics
Genetic drift is a random walk
Population genetics
Statistical Inference → Population Genetics
Splitting variation into its causes
Genetics
Statistical Inference → Genomics
A million tests at once
Genomics
Statistical Learning Theory → Molecular Biology
Predicting protein shapes
Structural biology
Stochastic Processes → Cell Biology
Noisy genes
Cell biology
Physics → Mathematics
Crystallography → Euclidean Geometry
Penrose tilings and quasicrystals
Geometry
Magnetism → Combinatorial Optimisation
Simulated annealing
Optimisation
Quantum Field Theory → Geometric Topology
Knot invariants from quantum fields
Mathematics
Quantum Information → Public-Key Cryptography
Breaking and remaking public-key cryptography
Cryptography
Radioactivity → Probability Theory
Decay follows the Poisson law
Statistics
Topological Matter → Algebraic Topology
A periodic table of topological matter
Topology
Physics → Biology
Crystallography → Biochemistry
A library of protein structures
Structural biology
Electromagnetism → Molecular Biology
X-ray crystallography reveals the molecules of life
Structural biology
Exoplanetary Science → Evolutionary Biology
Is Earth's life unusual?
Astrobiology
Non-Equilibrium Physics → Molecular Biology
Measuring energy by pulling on RNA
Molecular biology
Nuclear Structure → Biochemistry
Radioactive tracers
Biochemistry
Phase Transitions → Molecular Biology
Cells organise themselves by phase separation
Cell biology
Quantum Mechanics → Molecular Biology
Chemical bonds and the shape of proteins
Structural biology
Radioactivity → Molecular Biology
Radioactive labels
Molecular biology
Radiometric Dating → Evolutionary Biology
Dating early humans
Human evolution
Radiometric Dating → Cell Biology
Birth-dating human cells
Cell biology
Solid-State Physics → Genomics
Reading DNA on a chip
Genome sequencing
Statistical Mechanics → Molecular Biology
Why proteins fold
Biophysics
Stellar Astrophysics → Evolutionary Biology
Enough time for evolution
Evolution
Thermodynamics → Molecular Biology
Living things feed on order
Life
Biology → Mathematics
Community Ecology → Probability Theory
Estimating how many species are unseen
Statistics
Computational Neuroscience → Computability Theory
Finite automata and regular expressions
Mathematics
Conservation Biology → Probability Theory
Counting tigers from camera traps
Statistics
Electrophysiology → Dynamical Systems
Excitable systems
Mathematics
Experimental Embryology → Differential Equations
Pattern formation as mathematics
Mathematics
Population Ecology → Differential Equations
The standard example of a nonlinear system
Mathematics
Population Ecology → Chaos Theory
Population models and the discovery of chaos
Mathematics
Population Genetics → Probability Theory
The invention of modern statistical method
Statistics
Stem Cell Biology → Probability Theory
Reprogramming as a random process
Mathematics
Biology → Physics
Developmental Genetics → Kinetic Theory of Gases
How precisely can a cell read a signal?
Physics
Ecosystem Ecology → Thermodynamics
Ecosystems as energy converters
Thermodynamics
Electrophysiology → Electromagnetism
The battery
Electricity
Unmapped · 21
Reaching past the survey
Uses that land in another domain, but in a part of it this atlas has not surveyed yet: medicine, crystallography, statistics. Each is a place a future field would connect.
Algebraic Geometry↗ Physics
Mirror symmetry and counting curves
Analytic Number Theory↗ Physics
Zeta zeros behave like quantum energy levels
Calculus↗ Biology
Modelling epidemics
Chaos Theory↗ Physics
Ensemble forecasts
Complex Dynamics↗ Physics
Why Feigenbaum's constant is universal
Computability Theory↗ Physics
An undecidable question in physics
Computational Complexity↗ Biology
Protein folding is NP-hard in simple models
Differential Equations↗ Biology
The SIR model
Differential Geometry of Surfaces↗ Biology
The shape of membranes and red blood cells
Dynamical Systems↗ Biology
Body clocks as limit cycles
Electromagnetism↗ Biology
Magnetic resonance imaging
Euclidean Geometry↗ Physics
The 230 symmetry groups of crystals
Fourier Analysis↗ Biology
Reconstructing MRI scans
Galactic Astronomy↗ Biology
Dung beetles steer by the Milky Way
Geometric Analysis↗ Physics
Calabi–Yau manifolds as hidden dimensions
Magnetism↗ Biology
Memories as low-energy states
Old Quantum Theory↗ Biology
The eye can count photons
Probabilistic Combinatorics↗ Biology
When an outbreak becomes an epidemic
Riemannian Geometry↗ Biology
Averaging diffusion tensors in brain scans
Special Relativity↗ Biology
PET scans count annihilating matter
Superconductivity↗ Biology
Listening to the brain's magnetic field