Mathematics
27 posts in category “Mathematics”.
The Johnson-Lindenstrauss Lemma and the Geometry of High-Dimensional Data
2025-09-05Explore the surprising geometry of high-dimensional spaces: the Johnson-Lindenstrauss lemma showing that random projections preserve pairwise distances, the concentration phenomena that make it work, and its profound applications in nearest-neighbor search, compressed sensing, and machine learning.
Kolmogorov Complexity and Algorithmic Information Theory: The Deepest Measure of Information
2025-07-30Dive into algorithmic information theory: Kolmogorov complexity as the ultimate measure of information content, its relationship to randomness (Martin-Löf tests), the incompressibility method for proving lower bounds, and the philosophical implications for science and mathematics.
Spectral Graph Theory: How Eigenvalues Reveal the Hidden Structure of Graphs
2025-06-12Explore how the eigenvalues and eigenvectors of graph matrices — adjacency, Laplacian, normalized Laplacian — encode fundamental graph properties: connectivity, expansion, mixing time, clustering structure, and more.
The Fast Fourier Transform: From Cooley-Tukey to Modern Signal Processing and Fast Multiplication
2025-04-12Master the FFT from first principles: the Cooley-Tukey algorithm as recursive divide-and-conquer, the underlying group theory, modern variants for arbitrary sizes, and applications from polynomial multiplication to GPU signal processing.
Shannon's Information Theory from First Principles: Entropy, Channel Capacity, and the Fundamental Limits of Communication
2025-03-05Build Shannon's information theory from the ground up: entropy as a measure of uncertainty, source coding theorem, channel capacity, and the noisy-channel coding theorem that established the theoretical limits of reliable communication.
Analytic Combinatorics: The Symbolic Method, Generating Functions, and Average-Case Algorithm Analysis
2022-11-09A rigorous exploration of analytic combinatorics—the symbolic method for deriving generating functions, singularity analysis, saddle-point asymptotics, and applications to average-case analysis of algorithms and random structures.
Additive Combinatorics: Szemerédi's Theorem, Sumset Inequalities, and Applications in Property Testing
2022-10-25A rigorous exploration of additive combinatorics—Szemerédi's theorem on arithmetic progressions, Plünnecke-Ruzsa inequalities, the Balog-Szemerédi-Gowers theorem, and their applications in property testing and pseudorandomness.
Combinatorial Designs and Coding Theory: Block Designs, Steiner Systems, and Finite Geometry
2022-10-10An exploration of combinatorial design theory—block designs, Steiner systems, finite projective planes—and their deep connections to error-correcting codes and experimental design.
Tropical Geometry: Algorithmic Applications in Optimization, Phylogenetics, and Deep Learning
2022-09-15A rigorous exploration of tropical geometry—the min-plus semiring, tropical varieties, and the unexpected connections between algebraic geometry and combinatorial algorithms.
Algebraic Geometry in Computer Science: Gröbner Bases, the Nullstellensatz, and Applications in Cryptography and Coding Theory
2022-08-21A rigorous exploration of how algebraic geometry—Gröbner bases, Hilbert's Nullstellensatz, and elliptic curves—powers modern cryptography, error-correcting codes, and complexity theory.
Information Geometry: Statistical Manifolds, the Fisher Information Metric, and Natural Gradient Descent
2022-07-12A rigorous journey through information geometry—the Riemannian geometry of statistical models, the Fisher metric as the unique invariant metric, natural gradient, and the dually flat structure of exponential families.
Online Learning: Regret Minimization, the Multiplicative Weights Algorithm, and Adversarial Bandits
2022-04-15A rigorous treatment of online learning—regret minimization, multiplicative weights, EXP3 for adversarial bandits, and the deep connections to game theory and boosting.
Statistical Learning Theory: PAC Learning, VC Dimension, and the Bias-Complexity Tradeoff
2022-03-31A rigorous development of statistical learning theory—the PAC framework, VC dimension and Sauer's lemma, the fundamental theorem, Rademacher complexity, and the mathematical limits of learning from data.
Large Deviations Theory: Cramér's Theorem, Importance Sampling, and Rare Event Simulation
2022-02-13A rigorous exploration of large deviations—the theory of exponentially rare events—from Cramér's theorem to Sanov's theorem, and their application to importance sampling for reliable networks.
Renewal Theory for Computer Science: The Renewal Equation, Key Renewal Theorem, and Applications in Cache Analysis and Failure Recovery
2022-02-12A rigorous journey through renewal theory—the mathematics of recurring events—from the renewal equation and key renewal theorem to applications in garbage collection, cache replacement, and fault-tolerant system analysis.
Stochastic Processes for Computer Science: Poisson, Brownian Motion, Queueing and Reliability
2022-02-12A rigorous treatment of continuous-time stochastic processes—Poisson processes, CTMCs, Brownian motion with the reflection principle—and their applications in queueing theory, reliability engineering, and network performance.
Markov Chains for Computer Science: MCMC, Mixing Times, and Randomized Algorithms
2022-01-31A rigorous treatment of Markov chains from a computer science perspective—Metropolis-Hastings, coupling bounds, spectral gaps, and the role of rapid mixing in modern randomized algorithms.
Abstract Interpretation: The Cousot Framework, Galois Connections, and Sound Static Analysis by Construction
2022-01-20A rigorous exploration of abstract interpretation—Patrick and Radhia Cousot's unifying framework for static program analysis, from Galois connections to widening operators and the soundness proofs that guarantee analysis correctness.
Domain Theory: Scott's D∞ Construction, Denotational Semantics, and the Mathematics of Recursive Types
2022-01-15A rigorous exploration of domain theory—Scott's D∞ construction, continuous lattices, the fixpoint theorem, and how domains provide the mathematical foundation for denotational semantics of programming languages.
Process Calculi: Milner's CCS, the π-Calculus, Bisimulation, and Session Types for Protocol Correctness
2022-01-10A rigorous exploration of process calculi—from CCS to the π-calculus, the theory of bisimulation, and the Curry-Howard line connecting session types to linear logic.
Separation Logic: The Frame Rule, Separating Conjunction, and Concurrent Verification
2022-01-01An exploration of separation logic—O'Hearn and Reynolds's revolutionary extension of Hoare logic for local reasoning about mutable state, the frame rule, and concurrent separation logic.
Abstract Interpretation: Cousot's Galois Connection Framework, Widening/Narrowing, and Sound Static Analysis by Construction
2021-12-29A deep exploration of abstract interpretation—the mathematical theory of sound approximation that underpins every modern static analyzer, from the Astrée system to the Rust borrow checker.
Domain Theory: Scott's D∞ Construction, Solving Recursive Domain Equations, and the Foundations of Denotational Semantics
2021-12-19An in-depth exploration of domain theory—Scott's construction of a universal domain D∞ isomorphic to its own function space, continuous lattices, and how these ideas gave birth to denotational semantics.
Game Semantics: Fully Abstract Models of PCF, AJM Games, and Strategies as Sheaves
2021-09-30A rigorous exploration of game semantics—the technique that cracked the full abstraction problem for PCF by modeling computation as dialogue between Player and Opponent.