Theory

• 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.

• Linear Logic: Girard's Resource-Sensitive Logic, Exponential Modalities, and Linear Types in Rust · 2021-09-29

  A comprehensive exploration of linear logic's resource-conscious foundations, proof nets, the ! and ? modalities translating intuitionistic to linear, and how Rust's ownership system mirrors these ideas.

• Optimizing Distributed Consensus: Comparing Fast Paxos, Epaxos, And Multi Paxos In Wan Deployments With Latency Benchmarks · 2021-08-20

  A comprehensive technical exploration of optimizing distributed consensus, comparing Fast Paxos, Epaxos, and Multi Paxos in WAN deployments with latency benchmarks, covering key concepts, practical implementations, and real-world applications.

• Homotopy Type Theory: The Univalence Axiom, Higher Inductive Types, and ∞-Groupoids · 2021-08-11

  A deep dive into the univalent foundations of mathematics, where equality is homotopy, types are spaces, and the universe mirrors the ∞-groupoid of all ∞-groupoids.

• Category Theory for Programmers: Functors, Monads, and Natural Transformations · 2021-08-10

  A rigorous yet intuitive journey through the categorical structures that secretly power functional programming—from categories and functors to adjunctions and the monad-as-monoid correspondence.

• Integer Programming: Branch-and-Bound, Gomory Cuts, Lift-and-Project, and Solver Internals · 2020-02-23

  An inside look at integer programming algorithms—branch-and-bound, cutting planes, lift-and-project hierarchies—and how Gurobi and CPLEX solve NP-hard problems.

• Convex Optimization: Gradient Descent, Nesterov Acceleration, KKT Conditions, and the ML Stack · 2020-02-18

  A deep investigation of convex optimization—the engine of modern machine learning—from gradient descent and Nesterov momentum to KKT conditions and interior-point methods.

• Submodular Optimization: Diminishing Returns, the (1-1/e) Greedy Guarantee, and Machine Learning Applications · 2020-02-01

  A comprehensive study of submodular functions—the discrete analog of convexity—the greedy algorithm's optimal approximation, and applications in active learning and summarization.

• Matroid Theory: The Greedy Exchange Property, Matroid Intersection, and Applications in Spanning Trees and Matching · 2020-01-19

  A thorough exploration of matroid theory—the algebraic abstraction that explains why greedy algorithms work—matroid intersection, and their applications in combinatorial optimization.