Sandbox 2

Writing

Classify:
Reaction-diffusion systems
Cellular automata
Genetic algorithms
Dissipative structures

(How to correctly classify stuff described by PDEs vs. stuff described by ODEs?)

Nonlinear Dynamics

Phase transition, Phase transition classes
Critical exponent
Critical point
Percolation theory

(edit) Topics in Nonlinear Dynamics, Chaos, and Ergodic Theory (📁) Miscellaneous Time series analysis • Stability analysis • Population dynamics • Iterative map • Phase space • Attractor • Multistability • Bifurcation • Coupled map • Hopf decomposition

Statistical Mechanics

Ising model
Mean-field theory / Landau theory
Renormalization group
Scale decoupling and Coupled map lattice
Rigidity and Emergence and Order parameter and Correlation Length

(edit) Topics in Statistical Mechanics and Thermodynamics x (📁) Fundamentals Ensemble Theory • Canonical Ensemble • Microcanonical Ensemble • Grand Canonical Ensemble • Phase Space • Partition Function • Equiparition Theorem • Boltzmann Distribution • Entropy • Free energy (Gibbs, Helmholtz • Enthalpy • Thermodynamic potentials • Thermodynamic equilibrium • Laws of Thermodynamics Phase Transitions First Order Phase Transition • Second Order Phase Transition • Critical Exponents • Renormalization Group • RG flow • Scaling Theory • Universality Classes • Phase transition • Critical point • Order Parameter • Correlation Function • Universality • Hysteresis • Non-Equilibrium Phase Transitions • Kosterlitz-Thouless Transition • Symmetry Breaking • Ergodicity Breaking • Replica Symmetry Breaking Dynamical Systems Stochastic processes • Irreversible Process • Adiabatic process • Isothermal process • Onsager Relations • Linear response theory • Fluctuation-dissipation theorem • Brownian motion • Fokker-Planck equation • Langevin equation • Master equation • Hopf decomposition Disordered Systems Spin glass • Random field Ising model • Griffiths phase • Percolation theory • Localization Models Ising Model • Landau Theory • Curie-Weiss Model • Bragg-Williams Approximation • Spin glass • Random field Ising model • Quenched disorder • Annealed disorder • Griffiths phase • Percolation theory • Localization • Hopfield Networks • Attractor Networks Computational Methods Monte Carlo Simulations • Molecular Dynamics Simulations • Mean-Field Simulations • Percolation Algorithms • Tensor Networks • Path Integral Methods • Quantum Annealing • Simulated Annealing Advanced Topics Neural Mean Field Theory • Energy-Based Models • Statistical Mechanics of Deep Learning • Topological Mean-Field Theory • Order Parameter-Based Theory • Generalized Rigidity • Emergence in Statistical Mechanics • Anderson Localization • Jammed Systems • Jammey Transitions • Gibbs Measures • Metastable States • Glassy Dynamics • Lyapunov Functions • Synchronization Phenomena • Self-Organized Criticality • Complex Systems in Statistical Mechanics • Information Theory in Statistical Mechanics • Quantum Statistical Mechanics • Geometric Deep Learning • Topological Data Analysis in Statistical Mechanics • Statistical Field Theory • Variational Methods in Statistical Mechanics • Effective Field Theory • Coarse-Graining Methods • Stochastic Optimization • Bayesian Inference in Statistical Mechanics • Maximum Entropy Methods• Statistical Mechanics of Coupled Networks • Statistical Mechanics of Social Networks • Statistical Mechanics of Financial Markets • Statistical Mechanics of Epidemics • Statistical Mechanics of Traffic Flow • Statistical Mechanics of Ecosystems

Graphs and Networks

Centrality
Complex network
Neural gas model

Modeling networks as random graphs:

• Configuration model
• Generating function
• Rate equation

Random graph phenomena:

• Resilience
• Scale-free network
• Small world
• Cascades
• Random walk
• Trophic coherence

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Machine Learning

Feature learning
Autoencoder

Sheaf neural network (Spectral sheaf theory, Discourse sheaf)

Manifold learning
Spectral submanifold
Taken's theorem

(edit) Topics in Artificial Intelligence and Machine Learning (📁) Overview Supervised learning • Unsupervised learning • Universal Approximation Theorem • Neural Mean Field Theory Artificial Neural Networks Hopfield model • Boltzmann machine • Autoencoder • Transformer • Graph neural network • Sheaf neural network • Spiking neural network

Information Theory

Kolmogorov complexity, Kolmogorov's Invariance Theorem

Entropy
Transfer Entropy

Fisher information, a Hessian of Relative entropy

Algorithmic probability
Shannon's source coding theorem
Kraft-McMillan Inequality

Kalman filter
Covariance intersection
Sensor fusion

(edit) Topics in Information Theory and Control Theory (📁) Entropy Shannon entropy • Conditional entropy • Mutual Information • Joint entropy • Relative entropy • Rényi entropy • Tsallis entropy • Hartley's measure • Transfer entropy Information Geometry Fisher information (metric) • Exponential family • Kullback–Leibler divergence • Bregman divergence • f-divergence • Statistical divergence • Amari's alpha-divergence • Information manifold • Statistical structure • Cramér-Rao bound Model Selection Maximum likelihood estimation • Parameter space • Likelihood function (log-likelihood) • Bayesian inference • Akaike Information Criterion (AIC) • Bayesian Information Criterion (BIC) • Cross-validation • Bootstrap method • Minimum description length Network Information Theory Multiuser information theory • Broadcast channels • Multiple-access channels • Interference channels • Relay channels • Network coding • Slepian-Wolf coding Algorithmic Information Theory Kolmogorov complexity • Kolmogorov invariance theorem • Algorithmic probability • Chaitin's constant • Solomonoff's theory of inductive inference • Universal Turing machine Quantum Information Theory Quantum entropy • Quantum mutual information • Quantum channel capacity • Quantum error correction • Quantum cryptography Coding Theory Source Coding Shannon's source coding theorem • Huffman coding • Arithmetic coding • LZW coding • Run-length encoding • Golomb coding • Kraft-McMillan Inequality Channel Coding Shannon's channel coding theorem • Error detection and correction • Hamming codes • Reed-Solomon codes • Turbo codes • Convolutional codes • LDPC codes • Viterbi algorithm State Spaces State variables • State-transition matrix • Controllability • Observability • Stability • Kalman filter • Luenberger observer • Covariance intersection • Sensor fusion Control Theory Classical Transfer functions • Block diagrams • PID controllers • Root locus • Bode plots • Nyquist stability criterion • Routh-Hurwitz stability criterion Optimal Pontryagin's maximum principle • Hamilton-Jacobi-Bellman equation • Linear-quadratic regulator (LQR) • Dynamic programming • Model-predictive control Robust H-infinity methods • Sliding mode control • Robust stability • Uncertainty modeling • Youla–Kucera parametrization Adaptive Model reference adaptive control (MRAC) • Self-tuning regulator • Adaptive observers • Gain scheduling Nonlinear Feedback linearization • Backstepping • Passivity-based control • Lyapunov stability • Nonlinear system identification Stochastic Wiener filter • Stochastic optimal control • Markov decision processes • Stochastic stability • Stochastic differential equations Networks Networked Control System Architectures • Time Delay Systems • Multi-Agent Networks Applications Process control (Robotics, Aerospace, Automotive) • Biological Systems

Self-organized criticality control

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Social models

Bounded confidence model

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Advanced Maths

Liquid tensor experiment

Applied Topology

Manifold
Simplicial complex
Euler characteristic
Homology
Exact sequence
Cohomology
Morse theory
Homotopy
Sheaf
Categorification

(edit) Topics in Topological Data Analysis (📁) Background Point-set topology • Algebraic topology Manifolds Manifold • Configuration space • Differentiability • Vector space (bundle) • Tangent space (bundle) • Cotangent space (bundle) • Vector field (Flow, Critical points) • Braid • Transversality • Whitney embedding theorem • Stratified space Complexes CW Complex • Δ-Complex • Regular Complex (Simplicial, Cubical, Vietoris-Rips, Čech, Witness, Flag, Strategy • Decision Task) • Nerve • Phylogenetic tree • Discretized graph configuration space • State complex Euler Characteristic Euler characteristic • Curvature • Nonvanishing vector field • Fixed point index • Tame topology • Euler calculus • Target enumeration • Fubini theorem • Euler integral transform • Intrinsic volume • Gaussian random field Exact Sequences Chain homotopy • Snake lemma • Good pair • Mayer-Vietoris sequence • 5-lemma • Cellular homology • Coverage • Degree • Sperner's Lemma • Borsuk-Ulam theorem • Short exact sequence • Lefschetz index • Nash equilibrium • Hex • Barcode • Space of natural images • Zigzag persistence Homology Simplicial homology • Künneth theorem • Coefficient • Singular homology • Reduced homology • Čech homology • Relative homology • Excision Theorem • Local homology • Relation homology • Functoriality • Inverse kinematics • Winding number • Brouwer fixed point theorem Cohomology Duality • Cochain complex • Cohomology • Universal coefficient theorem • Poincaré duality • Alexander duality • Helly's Theorem • Numerical Euler integration • Forms • De Rham cohomology • Cup product • Helicity Theorem • Laplacian • Hodge Theorem • Cohomological coordination Morse Theory Morse homology • Excursion set • Definable functional • Euler-Fourier transform • Stratified Morse theory • Conley index • Forcing theory • Attractor-repeller pairs • Floer homology • Symplectic form • Hamiltonian field • Arnold's Conjecture • Lefschetz-Hopf theorem • Stratified Lefschetz-Morse Index • Discrete Morse theory • LS category • Unimodal decomposition Homotopy Fundamental group • Van-Kampen theorem • Covering space • Knot • Braid • Seifert conjecture • Higher homotopy groups • Biaxial nematics • Hurewicz Theorem • Whitehead Theorem • Eilenberg-MacLane space • Topological social choice • Fiber bundle • Section • Path space • Farber lemma • Sectional category • Topological complexity • Fibration • Homotopy extension property • Model category • Homotopy type theory Sheaves Cellular sheaf • Local cohomology • Fiber homology • Sheaf cohomology • Global sections theorem • Linear sheaf • Flow sheaf • Network coding sheaf • Information flow theorem • Čech cohomology • Preseaf • Stalk • Gluing axiom • PDE sheaf • Operations on sheaves • Vietoris mapping theorem • Sheaf-theoretic Nyquist-Shannon • Fourier-Sato transform • Cosheaf • Bézier curve • Barcodes • Interval Indecomposables • Étale Space • Leray spectral sequence • Spectral sheaf theory • Sheaf neural network Categorification Category • Groupoid • Monoid • Morphism • Equalizer • Functor • Kan complex • Nerve • Clustering • Natural transformation • Cone • Interleaving Limit • Terminal object • Product • Pullback • Homology Colimit • Initial object • Coproduct • Pushout • Cohomology Topological sheaf • Sheaf operations • Derived functors

Misc (New Research)

Tangle machines

Neuroscience and Psychology

Cognitive effort, Stroop task
Corpus callosum
Salience network

Bouba-Kiki effect

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