Mathematics

• The Johnson-Lindenstrauss Lemma and the Geometry of High-Dimensional Data · 2025-09-05

  Explore 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 · 2025-07-30

  Dive 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 · 2025-06-12

  Explore 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 · 2025-04-12

  Master 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 · 2025-03-05

  Build 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 · 2022-11-09

  A 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 · 2022-10-25

  A 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 · 2022-10-10

  An 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 · 2022-09-15

  A 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 · 2022-08-21

  A 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 · 2022-07-12

  A 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 · 2022-04-15

  A 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 · 2022-03-31

  A 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 · 2022-02-13

  A 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 · 2022-02-12

  A 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 · 2022-02-12

  A 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 · 2022-01-31

  A 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 · 2022-01-20

  A 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 · 2022-01-15

  A 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 · 2022-01-10

  A 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 · 2022-01-01

  An 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 · 2021-12-29

  A 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 · 2021-12-19

  An 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 · 2021-09-30

  A rigorous exploration of game semantics—the technique that cracked the full abstraction problem for PCF by modeling computation as dialogue between Player and Opponent.