proofs

The first lecture of MIT 6.042J Mathematics for Computer Science, taught by Tom Leighton, introducing mathematical proofs. It covers propositions, implication and basic proof methods.

WIRED has UCLA professor Amit Sahai explain zero-knowledge proofs to a child, a teen, a college student, a grad student and an expert. Each round goes deeper into the cryptography and why it matters.

A Google Tech Talk by Steve Weis that introduces modern cryptography and its use in practice and at Google. It covers proofs of security and security definitions and a special topic. HN commenters treat it as a talk on verifiable electronic voting.

David Christiansen gives a Strange Loop 2018 introduction to dependent types, where one language serves for both compile-time and run-time code and programs unite with proofs. It draws on his book The Little Typer.

3Blue1Brown walks through a very hard Putnam competition problem and its elegant solution, emphasizing the visual intuition and the thought process behind solving it.

The first lecture of Francis Su's Harvey Mudd real analysis course, covering the construction of the rational numbers. It begins a full lecture series that follows a standard analysis text.

The first lecture of N J Wildberger's History of Mathematics course, starting with Pythagoras' theorem and following Stillwell's text. It emphasizes the flow of ideas over proofs.

Vladimir Voevodsky's IAS lecture starts from Godel's second incompleteness theorem to explore whether first-order arithmetic could be inconsistent. It motivates his vision of univalent foundations and computer-checked proofs.

The first chapter of 3Blue1Brown's Essence of Linear Algebra series, introducing vectors with animated geometric visuals. It begins a series that builds intuition for linear algebra from the basics.

3Blue1Brown's animated visual introduction to the Fourier Transform, showing how a signal breaks into frequency components.

3Blue1Brown's first chapter on deep learning explains what neurons and layers are and the math underlying a basic neural network, using handwritten digit recognition with visual animations.

Freya Holmer explains how Bezier curves work, from the helper-line construction to their uses in animation, text rendering and shapes. It uses visuals from her Unity Shapes plugin and was a SoME1 contest winner.

Freya Holmer's 74-minute video essay on splines, covering Bezier curves, B-splines, derivatives and the different levels of continuity (C0, C1, C2), built with custom animation tools. She notes she did not cover non-uniform splines.

The full-length Numberphile interview with mathematician and hedge fund founder Jim Simons. He discusses his path from mathematics and codebreaking to founding Renaissance Technologies, his use of algorithms in markets, and his philanthropy.

The first lecture of Gilbert Strang's MIT 18.06 Linear Algebra course, covering the geometry of linear equations. It opens a full 35-lecture university course.

Apple security head Ivan Krstic gives a Black Hat 2016 talk on iOS security, covering HomeKit, Auto Unlock and iCloud Keychain. He explains key hierarchies and how data can stay safe even in adversarial clouds.

Tom Scott explains why electronic voting is a bad idea. Elections must be trusted by everyone and attacks on software scale in ways attacks on paper do not, so open source or blockchain fixes do not solve it.

3Blue1Brown explains the math and mechanics behind Bitcoin visually: ledgers, digital signatures, proof of work and the blockchain, building up a decentralized payment system from first principles.

A short Dyalog demonstration building Conway's Game of Life in a single line of APL, step by step. It shows how array operations replace loops.

Part 1 of a lecture by physicist Albert Bartlett on the exponential function and its consequences for population growth, resource use and energy. It argues that steady growth rates quickly lead to enormous quantities.

Microsoft's Brian Beckman gives an hour-long, whiteboard-style walk through monads, building from function composition and monoids using school-level math. It is aimed at practicing programmers rather than category theorists.

A SIGGRAPH 2019 course on geometric algebra. Charles Gunn gives an intro, then Steven De Keninck gives a programmer-focused course with live examples in ganja.js.

The first video of 3Blue1Brown's Essence of Calculus series, which builds toward integrals, derivatives, and the fundamental theorem of calculus from a simple geometry question. It aims to make viewers feel they could have invented calculus themselves.

Meredith Patterson and Sergey Bratus argue that networked software stays insecure because protocols and inputs are too complex to recognize safely. They propose using formal language theory and grammars to build secure, verifiable input handling.

The first lecture in Bartosz Milewski's category theory series for programmers. It covers the motivation and philosophy behind category theory and why composition matters for handling complexity.

Philip Wadler's Strange Loop talk on Propositions as Types, the link between logic and computation (the Curry-Howard correspondence). He traces its history through Church, Godel and Turing and shows how it shaped theorem provers and programming languages.

Numberphile visits Cliff Stoll, the man behind ACME Klein Bottles, and sees how he stores a thousand glass Klein bottles in a crawlspace under his house. A remote-controlled forklift fetches them.

A short explainer on the history and mechanics of public key cryptography and the Diffie-Hellman key exchange. It uses a mixing-paint-colors analogy to show how two parties can agree on a shared secret in public.

A five-minute preview of 3Blue1Brown's Essence of Linear Algebra series, which uses animation to build geometric intuition for topics from a standard linear algebra course.

An animated introduction to geometric algebra, presented as a language that can describe most of physics. It was made with manim and covers multivectors and how they generalize familiar tools like cross products and quaternions.

The first part of Welch Labs' series on imaginary numbers, presenting them as the natural extension of the number line into two dimensions. It covers the history and the math with strong visuals.

Moxie Marlinspike's BlackHat 2011 talk examines weaknesses of SSL and the certificate authority system and proposes Convergence, a decentralized notary approach with trust agility.

James Mickens of Harvard gives a funny lecture at HBS's Future Assembly arguing that blockchains are a very bad idea. He covers their trust, legal and practical shortcomings for real-world problems.

A 10-minute 3Blue1Brown chapter explaining the determinant as the factor by which a linear transformation scales area or volume. It builds the idea geometrically with animations before any formulas.

Dan Olson's roughly two-hour video essay critiques NFTs, cryptocurrency, and Web3, covering their technical, economic, and ideological flaws. It is heavily sourced and ends with the advice to ask whether a project needs crypto to work.

Veritasium's explainer on Godel's incompleteness theorems and the limits of mathematical proof, tracing set theory, Hilbert's program and the road to Turing and modern computing.

3Blue1Brown explains fractal dimension and shows why many fractals are not strictly self-similar. Self-similarity turns out to be a special case of the broader idea of fractional dimension.

Vsauce's roughly 24 minute explainer on the Banach-Tarski paradox, building from infinity and set theory to the decomposition of a sphere into pieces that form two spheres.

Tom Scott explains for Computerphile why electronic voting is a bad idea, arguing that paper ballots are transparent, auditable and hard to tamper with at scale. He covers why moving voting to phones, tablets or computers undermines trust and verifiability.

3Blue1Brown visualizes high-dimensional spheres to explain a surprising puzzle about how a sphere fits in a box as dimensions grow. It builds geometric intuition for counterintuitive properties of high-dimensional space.