What, why, who?

Current morphometric methods that comprehensively measure shape cannot compare the disparate leaf shapes found in seed plants and are sensitive to processing...

In an age of proof abundance, what becomes scarce?

MAST90068

MAST30026

In this talk, I introduce the primary definition(s) of the course, along with some "low dimensional" examples from graph theory. Uploader: Jeremy Mann Duration:...

Here’s a mathematical situation that comes up a lot more often than is reasonable.

Synthetic Stone duality is an extension of homotopy type theory with four axioms. These axioms are strong enough to decide Bishop's omniscience principles.

Thorsten Altenkirch: ‘A constructive justification of Homotopy Type Theory’ My main thesis is that Homotopy Type Theory is a foundation of mathematics that is constructive in two ways: …

I'm working at Kerna Labs to make AI that makes mRNA-based medicines.

Posts about Homotopy Type Theory written by Christopher Brinkley

Attention is a computational primitive at the core of modern language models, allowing internal representations to reference and influence each other. It’s h...

Included as well are stripped-down versions (eg. definition-only; script-generated and doesn't necessarily make sense), example sheets, and the source code.

Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics.

How finding explicit strategies can mean biting off more than you can chew.

Course website: https://topos.site/poly-course Book: https://topos.site/poly-book.pdf Book suggestions document: https://docs.google.com/document/d/1qY5hLglgSW...

Tutorial given at Applied Category Theory 2023 https://act2023tutorials.netlify.app/ https://act2023.github.io/ Uploader: Applied Category Theory Duration: 106...

MIT Category Theory Seminar 2020/10/01 ©Spifong Speaker: David Jaz Myers Title: Paradigms of composition Abstract: Scientists and engineers manage the blist...

Découvrez l'histoire fascinante du groupe Bourbaki, ce collectif de mathématiciens qui a révolutionné la discipline au XXe siècle. Dans cette rencontre exceptio...

Uploader: xuan-gottfried YANG Duration: 3120s Views: 37543

Modal logic covers such areas of human knowledge as mathematics (especially, topology and graph theory), computer science, linguistics, artificial intelligence, and philosophy. In this post, we introduce the basic idea of modal logic, one of the most popular branches of mathematical logic.

This seminar series seeks to promote the learning and use of Category Theory by Machine Learning Researchers

Mathematics shouldn't survive logical errors—yet it does

Video lectures at MIT. See http://brendanfong.com/programmingcats.html Lecturers: Brendan Fong, Bartosz Milewski, David Spivak Summary: In this course we expl...

A set of lecture notes on differential geometry and theoretical fundamental physics, combining an introduction to traditional notions with an exposition of their formulation and refinement by higher geometry and extended prequantum field theory.

In this paper, we explore the universal properties underlying causal inference by formulating it in terms of a topos. More concretely, we introduce topos causal models (TCMs), a strict...

The Heidelberg Laureate Forum Foundation presents the HLF Laureate Portraits: Sir W. Timothy Gowers; Fields Medal, 1998. Interview recorded in 2019. In this s...

Chris Grossack's math blog and professional website.

Chris Grossack's math blog and professional website.

In this video we prove a version of Lawveres fixed point theorem that holds in Cartesian closed categories. It's a nice construction that specializes to results...

Sir William Timothy Gowers is a British mathematician and a Royal Society Research Professor at the Department of Pure Mathematics and Mathematical Statistics a...

0:09 Plans for Abel Prize funds incl. to Higher School of Economics 3:26 Importance of awards, prizes 4:46 The value of the multi-faceted threads of the Abel P...

David Bessis is a mathematician and the author of Mathematica: A Secret World of Intuition and Curiosity. We explore David's provocative claim that mathematical...

In 1890 Giuseppe Peano discovered a square-filling curve, and a year later David Hilbert published his variation.

In this post I would like to analyze the usual proof of Cantor's theorem and present an insightful reformulation of it which has applications outside set theory.

Alfred North Whitehead (1861–1947) was a British mathematician and philosopher best known for his work in mathematical logic and the philosophy of science.

An online resource for homotopy-coherent mathematics

The inaugural Natural Philosophy Symposium was held in Baltimore on May 29-31, 2025. It was sponsored by the Natural Philosophy Forum at Johns Hopkins (https://...

This presentation offers a heuristic proof (and simulations of a primordial soup) suggesting that life—or biological self-organization—is an inevitable and emer...

Taken from: Logic for CS, Shai Ben-David, U Waterloo Fall 2015 https://www.youtube.com/channel/UCg9V0y9_RxG7hg5GjcyS2OA Uploader: Adrian Apostol Duration: 4654s...

An introduction to the Twin Prime Conjecture and sieve methods, one of the most beautiful branches of modern number theory. ONLINE COURSE: I'm teaching an onl...

What does a mathematical theory consist of? This question was at the heart of the foundational dispute between the members of Bourbaki and practitioners of cate...

Prof. Richard Borcherds received a Fields medal in 1998. He is most famous for proving Monstrous Moonshine, a conjecture of John Conway and Simon Norton relatin...

Millennium Prize Problems Lecture 11/12/2025 Speaker: Pierre Deligne, Institute for Advanced Study Title: What is the Hodge conjecture? Abstract: The Hodge c...

first AGITTOC pseudolecture June 27, 2020 (see math216.wordpress.com for more on the AGITTOC experiment) Also a first stab at playing with this technology. Upl...

Grigori Perelman proved the Poincare conjecture and then refused a million dollar prize (the Millennium Prize). He is the only mathematician who has declined th...

All available speaker abstracts and slides can be found on our webpage - https://www.chapman.edu/scst/conferences-and-events/grothendieck-conference.aspx Upload...

My p-adic hat: https://www.bonfire.com/p-adic-hat-1/ My Patreon: https://patreon.com/K_Theory?utm_medium=unknown&utm_source=join_link&utm_campaign=creatorshare...

An axiomatic theory of truth is a deductive theory of truth as a primitive undefined predicate.

I am having a lot of trouble with the concept of Tarski's undefinability theorem as it relates to set theory. Tarski's undefinability theorem says that there is no formula $Tr$ on the natural numbers

This six-session course will introduce participants to thinking about physical interaction as communication, and hence thinking about physical systems as communicating agents.

I love the feeling of having a new way to think about the world. I especially love when there’s some vague idea that gets formalized into a concrete concept.

Platonism about mathematics (or mathematical platonism) is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices.

Proof theory is not an esoteric technical subject that was invented to support a formalist doctrine in the philosophy of mathematics; rather, it has been developed as an attempt to analyze aspects of mathematical experience and to isolate, possibly overcome, methodological problems in the foundations of mathematics.

This is a ~1 hour 30 minute talk + Q&A by Chris Fields (https://allencenter.tufts.edu/christopher-a-chris-fields-ph-d/) titled "From Experience to Math", given ...

This is a ~50-minute talk titled "Substrate-dependent mathematics hypothesis" by Olaf Witkowski (https://olafwitkowski.com/), presented for our Platonic Space s...

Teaching Tom Crawford a bit about my favorite subject -- Lie algebras. Check out Part 2: https://www.youtube.com/watch?v=ap7GZKCcgS8 🌟Support the channel🌟 Pat...

Links: - Patreon (Support the channel directly!): https://www.patreon.com/Asianometry - X: https://twitter.com/asianometry - Newsletter & Podcast (available thr...

How can a language model comprehend a million-token document without drowning in O(N²) attention cost? A statistical model revealing the success of block sparse attention through learned similarity gaps.

Gödel’s two incompleteness theorems are among the most important results in modern logic, and have deep implications for various issues.

We’ve all been there before: by the time you start graduate school in Princeton, you’ve already invented the Turing machine, pioneered the concept of computational universality, and pro…

On Thursday, I probably should've told you explicitly that I was compressing a whole math course into one lecture. On the one hand, that means I don't really expect you to have understood everything.

A global pandemic, apocalyptic fires, and the possible descent of the US into violent anarchy three days from now can do strange things to the soul. Bertrand Russell—and if he’d done no…

(Originally posted in December 2015: A dialogue between Ashley, a computer scientist who's never heard of Solomonoff's theory of inductive inference,…

> It can scarcely be denied that the supreme goal of all theory is to make the irreducible basic elements as simple and as few as possible without having to surrender the adequate representation of a…

Personal Blog

We all know how deep learning works.

[This work was performed as my final project for ARENA 5.0.] …

This is the completed article that Luke wrote the first half of. My thanks go to the following for reading, editing, and commenting; Luke Muehlhauser…

Hilbert's problems are 23 problems in mathematics published by German mathematician David Hilbert in 1900.

Ludwig Wittgenstein’s Philosophy of Mathematics is undoubtedly the most unknown and under-appreciated part of his philosophical opus.

Category theory is a beautiful and powerful field but it can feel impenetrable without the right entry point. This post hopes to serve as a sort of beginner's guide and reference.

Sequent Calculus is a way of doing logic that's very explicit and mechanical. It's used as an important system and notation for type theory and logic related to programming languages.

the quick and dirty of Lean

Over at Cosmic Variance, I learned that FQXi (the organization that paid for me to go to Iceland) sponsored an essay contest on “The Nature of Time”, and the submission deadline was las…

This is a series of lectures aimed at graduate students on the modern design of full-spectrum dependent type theories, such as the core calculi of proof assistants like Agda, Coq, and Lean.

Explores the geometric and trigonometric principles behind aiming and banking shots in pool, showing how players use a “diamond system” to compute angles and sp...

Walkthrough solution of a Japanese Math Olympiad problem requiring creative high-school level mathematics to determine an unknown value X.

Conference talk modifying Doom’s source to use intentionally incorrect π and trigonometric constants, demonstrating how breaking fundamental maths produces non-...

An educational deep-dive into the source coding theorem, entropy, arithmetic coding, and asymmetric numeral systems, illustrating how these information-theoreti...

Introduces an early, concrete application of category theory via algebraic topology, illustrating how categorical constructs map to homotopy concepts within top...

Survey-style overview of several famous open mathematical conjectures related to calculus and analysis, outlining what is known and why they remain unsolved.

An example-driven primer on categorical limits, building from sets and vector spaces to equalisers, fibre products, cones, and universal properties, aimed at ne...

The first lecture in a sheaf-theory series, defining presheaf stalks, sheafification, and exactness concepts such as kernels and images within a categorical fra...

Provides a rapid, physics-motivated introduction to geometric algebra, covering multivectors, grades, geometric products, and rotors as an extension of linear-a...

Live demonstration of the Lean interactive theorem prover, showing how formal logic rules are encoded, manipulated, and verified, and discussing its role in mat...

Clear technical explanation of the groundbreaking MIP*=RE complexity-theory result—valuable foundational content for theoretical computer scientists.

Provides an in-depth mathematical explanation of tensors, suitable for learners of linear algebra and theoretical physics.

Bartosz Milewski provides an intensive introduction to category theory with programming examples, fitting both educational and theoretical criteria.

Conference-style talk by Erik Meijer connecting category theory to interface-based design and Java 8 lambdas; valuable for programmers interested in theoretical...

Philip Wadler’s lecture introduces category theory concepts for programmers, bridging mathematics and software development.

Conference talk by Philip Wadler connecting category theory to programming; foundational material for programmers interested in type theory.

Educational explanation of imaginary (complex) numbers aimed at engineering students, providing foundational mathematical knowledge.

Filmed at the March 11, 2014 LispNYC meetup at Meetup HQ in NYC. ABOUT DATA COUNCIL: Data Council (https://www.datacouncil.ai/) is a community and conference ...

This is a gentle introduction to applied category theory – more about the history of the subject, what people are trying to do, and my own personal involvement ...

Description: Category theory and its applications Slides: No Slides Uploader: LambdaConf Duration: 3667s Views: 38695

Lesson 1 is concerned with defining the category of Abstract Sets and Arbitrary Mappings. We also define our first Limit and Co-Limit: The Terminal Object, and ...

Video lectures at MIT. See http://brendanfong.com/programmingcats.html Lecturers: Brendan Fong, Bartosz Milewski, David Spivak Summary: In this course we e...

Strong static typing detects a lot of bugs at compile time, so why would anyone prefer to program in JavaScript or Python? The main reason is that type systems ...

Dans cet entretien, le mathématicien Etienne Ghys interroge ses pairs Terence Tao, Nalini Anantharaman et Timothy Gowers autour de la question de la preuve dans...

As I mentioned in the video, here are the links to the three math books that changed my life for the better: 1) Peter Selby and Steven L. Slavin's "Practical A...

When you hear that someone is "studying algebra". What comes to mind? Are they drilling through thousands of factorisation problems? Are they an undergraduate s...

To fully utilise the exciting category theory we've learnt so far, we need a way to abstract definitions from a specific category and then be able to apply them...

In this video, I describe how all of the different theorems of multivariable calculus (the Fundamental Theorem of Line Integrals, Green's Theorem, Stokes' Theor...

By first outlining a mathematically rigorous definition of a category, we can embark on a fascinating journey through category theory with examples from mathema...

What did category theory ever do for us (functional programmers)? - An extreme pragmatic and un-academic approach. Examples are in Scala. Talk given at Scale b...

Category theory is a framework that unifies all of mathematics in an abstract and homogeneous language by extracting the essence of mathematical structures. The...

PL Virtual Meetup: https://www.meetup.com/Programming-Languages-Toronto-Meetup/ CtFP Textbook: https://github.com/hmemcpy/milewski-ctfp-pdf Github Repo: https:/...

http://cppnow.org — Presentation Slides, PDFs, Source Code and other presenter materials are available at: http://cppnow.org/history/2019/talks/ — Composition i...

In this video I will share something that will change the way you think about mathematics forever. If you learn this one thing, you can then go on and learn mor...

P.S. I am looking for a job! If you can help me I would be very grateful. Message me on LinkedIn: www.linkedin.com/in/marc-maliar/ Uploader: Marc Maliar Duratio...

The provided audio is an explanation of sets in mathematics. It details ways to define sets, including roster notation, semantic descriptions, and set-builder n...

Euler's proof combines calculus and combinatorics in a remarkable way to prove that the primes never end! Euler's proof also reveals a deep connection between t...

In this Lambda World 2019 keynote, Emily Riehl discusses category theory and computational effects. Slides are available here: http://www.math.jhu.edu/~eriehl...

In this video, we explore basic concepts of Measure Theory and the Lebesgue Integral. We will learn about important theorems of Lebesgue Integration like the Mo...

This video is part of the “Real Analysis” series I am making. Thanks and enjoy the video! Real Analysis Playlist: https://www.youtube.com/playlist?list=PLDidd...

A peek into the world of Riemann surfaces, and how complex analysis is algebra in disguise. Secure your privacy with Surfshark! Enter coupon code ALEPH for an e...

Algebraic geometry is often presented as the study of zeroes of polynomial equations. But it's really about something much deeper: the duality between abstract ...

Definition of category. Example categories. Isomorphisms and monomorphisms. Uploader: Alex Simpson Duration: 5241s Views: 10578

STEMerch Store: https://stemerch.com/ Support the Channel: https://www.patreon.com/zachstar PayPal(one time donation): https://www.paypal.me/ZachStarYT Versión...

Playlist: https://www.youtube.com/playlist?list=PLOROtRhtegr7DmeMyFxfKxsljAVsAn_X4 What is category theory? In this lecture we introduce categories, which incl...

Kurt Gödel showed that mathematical thinking cannot be captured in a formal axiomatic reasoning system. What does this deep result mean in practice? What are th...

Carl Gauss was a child prodigy who reinvented mathematics. Try https://brilliant.org/Newsthink/ for FREE for 30 days, and get 20% off your annual premium subscr...

Sign up with brilliant and get 20% off your annual subscription: https://brilliant.org/ZachStar/ STEMerch Store: https://stemerch.com/ Support the Channel: htt...

This lecture is part of an online course on algebraic topology. We define the fundamental group, calculate it for some easy examples (vector spaces and spheres...

Goal. Explaining basic concepts of algebraic topology in an intuitive way. This time. What is...homotopy? Or: The same shape!? Disclaimer. Nobody is perfec...

Lecture 1 of Algebraic Topology course by Pierre Albin. Uploader: Mat Neth Duration: 3821s Views: 146152

The first in a two-lecture series on our recently-announced Categorical Deep Learning framework (categoricaldeeplearning.com), given as a lecture for the Geomet...

#logic #prooftheory #modeltheory #goedel Access exclusive content on Patreon: https://www.patreon.com/user?u=86649007 All the way at the foundations of mathe...

An explanation of fractal dimension. Help fund future projects: https://www.patreon.com/3blue1brown An equally valuable form of support is to simply share some ...

We explore a counter-intuitive property of proof by induction, where one inequality can be proven by induction easily, but a seemingly easier to prove, weaker, ...

In this lecture from Sam Cohen’s 3rd year ‘Information Theory’ course, one of eight we are showing, Sam asks: how do we measure the amount of information we lea...

There's often a lot of emphasis in math on generalizing concepts beyond the domains where they were originally defined, but what are the limits of this process?...

Claude Shannon is the father of Information Theory. Try https://brilliant.org/Newsthink/ for FREE for 30 days, and get 20% off your annual premium subscription....

Continuous functions play a crucial role in various disciplines in math. We discuss the epsilon-delta criterion and formalize it in the programming language and...

Is category theory a mathematical theory? Or something more? In this brief presentation, educational designer Paul Dancstep shares an animated description of wh...

Yann LeCun, Meta, gives the AMS Josiah Willard Gibbs Lecture at the 2025 Joint Mathematics Meetings on “Mathematical Obstacles on the Way to Human-Level AI.” Th...

In this video I look back on my told topology assignments from 2017 and tear them apart. Submit your garbage proofs to: ktheorytutoring@gmail.com Uploader: K-T...

In this video I RIP in viewers GARBAGE proofs. If you want your proofs roasted, or you want 1-on-1 tutoring, please email me at ktheorytutoring@gmail.com Intr...

Time to go to the next level! Uploader: Reclaiming Curiosity Duration: 3695s Views: 505

The principle of Propositions as Types links logic to computation. At first sight it appears to be a simple coincidence---almost a pun---but it turns out to be ...

Smooth lesson from my smooth brain about... CHANNEL LINKS 🗞️ Substack — https://verynormal.substack.com ☕ Buy me a Ko-fi! — https://ko-fi.com/verynormal USEFU...

In this experiment, I took a statement in universal algebra that a collaborator of mine (Bruno Le Floch) on the Equational Theories Project had written a one-pa...

Following on from the previous video at https://www.youtube.com/watch?v=cyyR7j2ChCI, I now attempt to formalize a different proof of the same assertion using th...

DON'T memorize this – understand it. Matrix multiplication isn’t just some weird set of rules. It's a carefully designed system that fits real-world problems pe...

The tensor product of vector spaces (or modules over a ring) can be difficult to understand at first because it's not obvious how calculations can be done with ...

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Course: https://www.udemy.com/course/introduction-to-power-system-analysis/?couponCode=KELVIN ✅ If you want to support me to make more frequent videos, consider...

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Terence Tao is widely considered to be one of the greatest mathematicians in history. He won the Fields Medal and the Breakthrough Prize in Mathematics, and has...

Make a donation to Closer To Truth to help us continue exploring the world's deepest questions without the need for paywalls: https://shorturl.at/OnyRq Conside...

Make a donation to Closer To Truth to help us continue exploring the world's deepest questions without the need for paywalls: https://shorturl.at/OnyRq “Power ...

IT’S ALL ABOUT MATH! An ongoing series hosted by The Department of Mathematics of the University of Toronto How playing games led to more numbers than anybody ...

at Napoleonic hall of Brera palace Uploader: Università degli Studi dell'Insubria Duration: 1871s Views: 14453

80,000 Hours want to help you find a fulfilling career that makes a positive difference in the world: https://80000Hours.org/bensyversen How a numerical table ...

How the deformation mapping and the deformation gradient are used to mathematically describe deformation - with many visual examples. This is the first video o...

Speaker, institute & title 1) Alex Alberts, Purdue University, Information field theory for solving Bayesian inverse problems Uploader: CRUNCH Group: Home of M...

Goal. I would like to tell you a bit about my favorite theorems, ideas or concepts in mathematics and why I like them so much. This time. What is...homotopy o...

In this Presidential Lecture, Terence Tao will survey historical and recent developments in the use of machines in mathematics. He will also speculate on the fu...

Our last AI PhD grad student feature was Shunyu Yao, who happened to focus on Language Agents for his thesis and immediately went to work on them for OpenAI. Ou...

Abstract: I will try to share a glimpse of this strange unification of many different ideas. This talk is aimed at a general audience, and no particular backgro...

Type theory is one of the central ideas in theoretical computer science and formal linguistics. But what is it, where did it come from, and how does it work? We...

Algebraic geometry seminar Department of Pure Mathematics University of Waterloo September 22nd, 2016 Following the notes of Ravi Vakil, available at http://mat...

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I'm sure we've all heard of the Yoneda Lemma before, but why does it get so much hype in category theory when the result itself is quite elementary to prove? W...

This video’s sponsor Brilliant is a great way to learn more. You can try Brilliant for free for thirty days by visiting https://brilliant.org/WelchLabs and the...

In this video, we generalize Euclidean vector space to obtain Hilbert spaces. In the process, we come across Bessel's inequality and Parseval's identity. The th...

In this video, we aim to explore the full geometric meaning of the second derivative test through the lens of linear algebra. We'll uncover the power of the spe...

Jacobian matrix and determinant are very important in multivariable calculus, but to understand them, we first need to rethink what derivatives and integrals me...

To learn more about various areas of Group Theory: https://en.wikipedia.org/wiki/Group_theory Galois Theory article in Encyclopedia of Mathematics: https://en...

Advait Shinde discusses the history of the theory of computation, delving into axiomatic thinking, Peano axioms, Turing Machines, Lambda Calculus, the Y Combina...

The Lambda Calculus is a tiny symbol manipulation system which suffices to compute anything Turing-computable. Thanks to this expressive power, LC is woven into...

This video has a list of books, videos, and exercises that goes through the undergrad pure mathematics curriculum from start to finish. --- REAL ANALYSIS Boo...

3Blue1Brown visually proves and contextualizes the Central Limit Theorem and its importance in probability and data analysis.

Clarifies entropy as a measure of information—not disorder—linking thermodynamic and Shannon definitions via microstate counting.

Stephen Wolfram discusses his computational approach to the second law of thermodynamics, entropy growth, and implications for AI governance.

Seminar proves the self-regularizing sparsity of nonparametric MLEs for mixture models, yielding logarithmic bounds on component count.

Wide-ranging conversation with Po-Shen Loh about competitive mathematics, combinatorics, and effective strategies for learning and teaching math.

Recreates Feynman’s geometric proof of Keplerian orbits, demonstrating why planetary motion forms ellipses using elementary mechanics.

Explores why linear algebra underpins diverse scientific computations, illustrating its unifying power across optimization and physics.

Visual geometric interpretation of Bayes’ theorem, illustrating belief updates and prior-posterior relationships in probabilistic inference.

Get 30% off unlimited access to Ground News, giving you full coverage of breaking news and allowing you to navigate media bias seamlessly 👉 https://www.ground.n...

Talk uses geometric algebra to show why no unique general vector-vector product exists in 3-D, highlighting dot, cross, and outer products.

The full report (PDF): http://math.fau.edu/yiu/Oldwebsites/MPS2010/TerenceTao1984.pdf Terence did note in his answers that questions 6 and 8 (A & E) at 2:18 can...

(*) Among current mathematicians, many people regard Professor Terence Tao as the world's finest... Opinions on such things vary, of course. Professor Tao kindl...

Euler's Number, e, is one of the most prominent constants in mathematics and exponential functions are some of the most important in maths. In this video: we ta...

Remember when I used a video with a coconut in the thumbnail to drive a stake through the heart of mathematical structure? Today, in this introduction to the ba...

This is the most information-dense introduction to group theory you'll see on this website. If you're a computer scientist like me and have always wondered what...

Playlist: https://www.youtube.com/playlist?list=PLOROtRhtegr7DmeMyFxfKxsljAVsAn_X4 When are two shapes the "same"? Topics covered include deformation retract, ...

Entropy, Cross-Entropy and KL-Divergence are often used in Machine Learning, in particular for training classifiers. In this short video, you will understand wh...

Goedel-Prover-V2: The Strongest Open-Source Theorem Prover to Date

Lifted from Mathoverflow: I think (almost) everyone agrees that Hartshorne's Algebraic Geometry is still the best. Then what might be the 2nd best? It can be a book, preprint, online lecture note,

Muon from first principles, what makes it different from other optimizers, and why it works so well.

Dependent type theory is a powerful and expressive language, allowing you to express complex mathematical assertions, write complex hardware and software specifications, and reason about both of these in a natural and uniform way.

The single most undervalued fact of linear algebra: matrices are graphs, and graphs are matrices

Last week we took an intuitive peek into the First Isomorphism Theorem as one example in our ongoing discussion on quotient groups.

Quantum entanglement is, as you know, a phrase that's jam-packed with meaning in physics. But what you might not know is that the linear algebra behind it is quite simple.

Yet it seems to me that the situation right now is that LtU has readers with very different backgrounds, among them many readers who haven't studied PL formally.

The origins of set theory can be traced back to a Bohemian priest, Bernhard Bolzano (1781-1848), who was a professor of religion at the University of Prague.

A re-construction of the fundamentals of programming as a small mathematical theory (PRISM) based on elementary set theory. Highlights: $\bullet$ Zero axioms. No properties are assumed, all are proved (from standard set theory). $\bullet$ A single concept covers specifications and programs. $\bullet$ Its definition only involves one relation and one set. $\bullet$ Everything proceeds from three operations: choice, composition and restriction. $\bullet$ These techniques suffice to derive the axioms of classic papers on the "laws of programming" as consequences and prove them mechanically. $\bullet$ The ordinary subset operator suffices to define both the notion of program correctness and the concepts of specialization and refinement. $\bullet$ From this basis, the theory deduces dozens of theorems characterizing important properties of programs and programming. $\bullet$ All these theorems have been mechanically verified (using Isabelle/HOL); the proofs are available in a public repository. This paper is a considerable extension and rewrite of an earlier contribution [arXiv:1507.00723]

The links below are to various freely (and legitimately!) available online mathematical resources for those interested in category theory at an elementary/intermediate level. There is supplementary page, introductory readings for philosophers, for reading suggestions for those looking for the most accessible routes into category theory and/or links to philosophical discussions. A gentle introduction? My Category … Category Theory: Lecture Notes and Online Books Read More »

The book can be ordered from amazon. com / co.

The unprecedented pace of machine learning research has lead to incredible advances, but also poses hard challenges. At present, the field lacks strong theoretical underpinnings, and many important achievements stem from ad hoc design choices which are hard to justify in principle and whose effectiveness often goes unexplained. Research debt is increasing and many papers are found not to be reproducible. This thesis is a survey that covers some recent work attempting to study machine learning categorically. Category theory is a branch of abstract mathematics that has found successful applications in many fields, both inside and outside mathematics. Acting as a lingua franca of mathematics and science, category theory might be able to give a unifying structure to the field of machine learning. This could solve some of the aforementioned problems. In this work, we mainly focus on the application of category theory to deep learning. Namely, we discuss the use of categorical optics to model gradient-based learning, the use of categorical algebras and integral transforms to link classical computer science to neural networks, the use of functors to link different layers of abstraction and preserve structure, and, finally, the use of string diagrams to provide detailed representations of neural network architectures.

Recently, I have stumbled upon certain articles and lecture videos that use category theory to explain certain aspects of machine learning or deep learning (e.g. Cats for AI and the paper An enriched

Databases have been studied category-theoretically for decades. The database schema---whose purpose is to arrange high-level conceptual entities---is generally modeled as a category or sketch. The data itself, often called an instance, is generally modeled as a set-valued functor, assigning to each conceptual entity a set of examples. While mathematically elegant, these categorical models have typically struggled with representing concrete data such as integers or strings. In the present work, we propose an extension of the set-valued functor model, making use of multisorted algebraic theories (a.k.a. Lawvere theories) to incorporate concrete data in a principled way. This also allows constraints and queries to make use of operations on data, such as multiplication or comparison of numbers, helping to bridge the gap between traditional databases and programming languages. We also show how all of the components of our model---including schemas, instances, change-of-schema functors, and queries - fit into a single double categorical structure called a proarrow equipment (a.k.a. framed bicategory).

The Cartesian closed categories have been shown by several authors to provide the right framework of the model theory of λ-calculus. The second author…

We present our position on the elusive quest for a general-purpose framework for specifying and studying deep learning architectures. Our opinion is that the key attempts made so far lack a coherent bridge between specifying constraints which models must satisfy and specifying their implementations. Focusing on building a such a bridge, we propose to apply category theory -- precisely, the universal algebra of monads valued in a 2-category of parametric maps -- as a single theory elegantly subsuming both of these flavours of neural network design. To defend our position, we show how this theory recovers constraints induced by geometric deep learning, as well as implementations of many architectures drawn from the diverse landscape of neural networks, such as RNNs. We also illustrate how the theory naturally encodes many standard constructs in computer science and automata theory.

Deep learning, despite its remarkable achievements, is still a young field. Like the early stages of many scientific disciplines, it is marked by the discovery of new phenomena, ad-hoc design decisions, and the lack of a uniform and compositional mathematical foundation. From the intricacies of the implementation of backpropagation, through a growing zoo of neural network architectures, to the new and poorly understood phenomena such as double descent, scaling laws or in-context learning, there are few unifying principles in deep learning. This thesis develops a novel mathematical foundation for deep learning based on the language of category theory. We develop a new framework that is a) end-to-end, b) unform, and c) not merely descriptive, but prescriptive, meaning it is amenable to direct implementation in programming languages with sufficient features. We also systematise many existing approaches, placing many existing constructions and concepts from the literature under the same umbrella. In Part I we identify and model two main properties of deep learning systems parametricity and bidirectionality by we expand on the previously defined construction of actegories and Para to study the former, and define weighted optics to study the latter. Combining them yields parametric weighted optics, a categorical model of artificial neural networks, and more. Part II justifies the abstractions from Part I, applying them to model backpropagation, architectures, and supervised learning. We provide a lens-theoretic axiomatisation of differentiation, covering not just smooth spaces, but discrete settings of boolean circuits as well. We survey existing, and develop new categorical models of neural network architectures. We formalise the notion of optimisers and lastly, combine all the existing concepts together, providing a uniform and compositional framework for supervised learning.

We give an introduction to logic tailored for algebraists, explaining how proofs in linear logic can be viewed as algorithms for constructing morphisms in symmetric closed monoidal categories with additional structure. This is made explicit by showing how to represent proofs in linear logic as linear maps between vector spaces. The interesting part of this vector space semantics is based on the cofree cocommutative coalgebra of Sweedler.

If you cannot find proofs, talk about them. Robert Reckhow with his advsior Stephen Cook famously started the formal study of the complexity of proofs with their 1979 paper. They were interested in…

Richard Wesley Hamming (February 11, 1915 – January 7, 1998) was an American mathematician whose work had many implications for computer engineering and telecommunications.

There are many books designed to introduce category theory to either a mathematical audience or a computer science audience. In this book, our audience is the broader scientific community. We attempt to show that category theory can be applied throughout the sciences as a framework for modeling phenomena and communicating results. In order to target the scientific audience, this book is example-based rather than proof-based. For example, monoids are framed in terms of agents acting on objects, sheaves are introduced with primary examples coming from geography, and colored operads are discussed in terms of their ability to model self-similarity. A new version with solutions to exercises will be available through MIT Press.

First my question: How much category theory should someone studying algebraic topology generally know? Motivation: I am taking my first graduate course in algebraic topology next semester, and,...

Okay, you wanna know what a topos is? First I'll give you a hand-wavy vague explanation, then an actual definition, then a few consequences of this definition, and then some examples.

Inspired by Whitehead and Russell's monumental Principia Mathematica, the Metamath Proof Explorer has over 26,000 completely worked out proofs in its main sections (and over 41,000 counting "mathboxes", which are annexes where contributors can develop additional topics), starting from the very foundation that mathematics is built on and eventually arriving at familiar mathematical facts and beyond.

Abstract page for arXiv paper 1803.05316: Seven Sketches in Compositionality: An Invitation to Applied Category Theory

Cambridge Core - Programming Languages and Applied Logic - An Invitation to Applied Category Theory

Shannon's mathematical theory of communication defines fundamental limits on how much information can be transmitted between the different components of any man-made or biological system. This paper is an informal but rigorous introduction to the main ideas implicit in Shannon's theory. An annotated reading list is provided for further reading.

1) Foundational math. 2) Classical machine learning. 3) Deep learning. 4) Cutting-edge machine learning.

We are Numerical Recipes, one of the oldest continuously operating sites on the Internet.

The post discusses how to compute the minimum number of AND and OR operators needed for Boolean functions with five variables. It describes the author's program that efficiently calculates this minimum for various functions while also improving algorithms for speed. The findings contribute to understanding the complexity of Boolean functions and their representations.

Sampling can help estimate the percentage of items with a specific trait accurately. The number of samples taken greatly affects the accuracy of the estimate. To get precise estimates, all items must have an equal chance of being selected during sampling.

Data compression involves modeling and coding to reduce the size of data files. Modern compressors typically use arithmetic coding for efficient compression. Algorithms like Huffman coding and run-length encoding are commonly used to achieve better compression results.

The text discusses Light's mistakes in using the Death Note and how they led to his de-anonymization by L. Light's errors, such as revealing his precise killing methods and using confidential police information, significantly reduced his anonymity. The text also explores strategies Light could have employed to better protect his anonymity while using the Death Note.

I'm sorry, but there is no content provided for me to summarize. If you provide me with the specific content or information you would like summarized, I would be happy to help.

This text introduces a book on linear algebra with chapters covering vectors, dot products, matrix operations, and more. It aims to help readers understand fundamental concepts and tools in linear algebra through clear explanations and examples. The book includes topics such as Gaussian elimination, determinants, rank, and eigenvalues.

The paper extends communication theory by considering noise in the channel, savings from message structure, and channel capacity. It discusses entropy, coding efficiency, channel capacity, noisy channels, equivocation, and optimal information transmission techniques. Examples and theorems are provided to explain the concepts of encoding, channel capacity, and noise in communication systems.

Anonymity on the internet is fragile, with each piece of information reducing anonymity. Revealing multiple bits of personal information can jeopardize anonymity, but deliberate disinformation can help regain some anonymity. To protect anonymity, it's best to minimize information disclosure.

This text discusses properties of vector spaces and matrices, particularly focusing on bases and eigenvalues. It establishes that any linearly independent system of vectors can be completed to form a basis in a finite-dimensional vector space. Additionally, it explains that operators in inner product spaces have an upper triangular matrix representation under certain conditions.

The Human Knowledge Compression Contest measures intelligence through data compression ratios. Better compression leads to better prediction and understanding, showcasing a link between compression and artificial intelligence. The contest aims to raise awareness of the relationship between compression and intelligence, encouraging the development of improved compressors.

The text discusses the complex relationship between Ludwig Wittgenstein and his peers, particularly Bertrand Russell. Wittgenstein's philosophical ideas and personal struggles are highlighted, showing the challenges he faced in expressing his thoughts and finding understanding from others. Despite his brilliance, Wittgenstein's life was marked by loneliness and inner turmoil, making it difficult for him to fully convey his philosophical insights.

The email exchange discusses the concept of negative entropy and its implications in mathematics and thermodynamics. Sungchul Ji questions the validity of negative entropy based on the Third Law of Thermodynamics. Arturo Tozzi argues for the existence of negative entropy in certain cases and relates it to information theory and free energy.

The text discusses the challenges and complexities of measuring and quantifying information, particularly in terms of storage capacity, compression, and entropy. It explores various examples, such as genome information, human sensory capabilities, and the information content of objects like water molecules and black holes. The relationship between information, entropy, and physical properties is also highlighted.

t-SNE plots can be useful for visualizing high-dimensional data, but they can also be misleading if not interpreted correctly. The technique creates 2D "maps" of data with many dimensions, but these images can be misread. The perplexity parameter, which balances attention between local and global aspects of the data, has a significant impact on the resulting plots. Different perplexity values may be needed to capture different aspects of the data. t-SNE plots can equalize cluster sizes and distort distances between clusters, making it difficult to interpret relative sizes and distances. It's important to recognize random noise and avoid misinterpreting it as meaningful patterns. t-SNE plots can show some shapes accurately, but local effects and clumping can also affect the interpretation. For topological information, multiple plots at different perplexities may be required. Overall, using t-SNE effectively requires understanding its behavior and limitations.

Landauer's principle is a physical principle that establishes the minimum energy consumption of computation. It states that irreversible changes in information stored in a computer dissipate a minimum amount of heat to the surroundings. The principle was proposed by Rolf Landauer in 1961 and states that the minimum energy needed to erase one bit of information is proportional to the temperature at which the system is operating. While the principle is widely accepted, it has faced challenges in recent years. However, it has been shown that Landauer's principle can be derived from the second law of thermodynamics and the entropy change associated with information gain.

Bremermann's limit is a maximum rate of computation that can be achieved in a self-contained system in the material universe. It is based on Einstein's mass-energy equivalency and the Heisenberg uncertainty principle. This limit has implications for designing cryptographic algorithms, as it can determine the minimum size of encryption keys needed to create an uncrackable algorithm. The limit has also been analyzed in relation to the maximum rate at which a system with energy spread can evolve into an orthogonal state.

The Bekenstein bound is an upper limit on the entropy or information that can be contained within a given finite region of space with a finite amount of energy. It implies that the information of a physical system must be finite if the region of space and energy are finite. The bound was derived from arguments involving black holes and has implications for thermodynamics and general relativity. It can be proven in the framework of quantum field theory and has applications in various fields, such as black hole thermodynamics and the study of human brains.

The content provided is the table of contents for a book titled "Numerical Recipes: The Art of Scientific Computing, Third Edition." It includes various topics such as linear algebra, interpolation and extrapolation, integration of functions, evaluation of functions, special functions, random numbers, sorting and selection, root finding and nonlinear sets of equations, minimization or maximization of functions, eigensystems, and more.

—————SOURCES———————————————————————— Percolation – Béla Bollobás and Oliver Riordan Cambridge University Press, New York, 2006. Sixty Years of Percolation – Hugo Duminil-Copin https://www.ihes.fr/~duminil/publi/2018ICM.pdf Percolation – Geoffrey Grimmett volume 321 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, second edition, 1999. —————NOTES————————————————————————— Note at 10:42 – The uniqueness of the infinite cluster is known for the d-dimenional lattice since the works of Aizenman, Kesten and Newman - [Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation (1987)] and Burton and Keane - [Density and uniqueness in percolation (1989)]. It does not hold in general: when the graph in question is a regular tree for example, there are always infinitely many clusters during the supercritical phase. The two last results shown here are only known for site percolation (in which vertices are open or closed instead of edges) in the triangular lattice, where a scaling limit for the boundaries of critical clusters was proved to exist (more on that in the third note). It is believed that these results are universal, that is, valid in great generality for planar percolation processes near criticality. The third result is from an appendix by Gábor Pete in the paper [Scaling limits for the threshold window: When does a monotone Boolean function flip its outcome? (2017)] by Ahlberg and Steif. Consider an n by n box, and the event where there exists a left-right crossing of said box. Recall the uniform coupling from the video: intuitively, the result is saying that the point at which this crossing emerges in the uniform coupling is with high probability inside an interval of size n^{-3/4} around 1/2. The fourth result is saying that the average size of the cluster of the origin (or any other given point) goes to infinity as we let p approach the critical parameter like a specific power of the distance between p and p_c. This power is called a critical exponent. The existence of these exponents was proved by Smirnov and Werner in the paper [Critical exponents for two-dimensional percolation (2001)]. Note at 10:52 – Hugo Duminil-Copin has several major contributions to the study of processes arising in statistical physics, including Bernoulli percolation. Among his works on Ising and Ising-like processes we can cite [Random Currents and Continuity of Ising Model’s Spontaneous Magnetization (2015)] with Aizenman and Sidoravicius and [Sharp phase transition for the random-cluster and Potts models via decision trees (2019)] with Raoufi and Tassion. Note at 12:38 – In the triangular lattice site percolation, Stanislav Smirnov proved the conformal invariance of crossing probabilities at criticality (see https://www.unige.ch/~smirnov/papers/icmp-final.pdf for an overview), which led to the proof of the existence of scaling limits of exploration curves as Schramm–Loewner evolution processes. See [Critical percolation in the plane (2009)] by Smirnov. This provided a deep understanding of the critical phase in the triangular lattice site percolation, which to this day is not extended to the square lattice. Note at 17:52 – It is not at all obvious that the probability of being connected to infinity is continuous above criticality. This result can be proved in the d-dimenional hypercubic lattices using the uniqueness of the infinite cluster, and more generally it was proved for transitive graphs (intuitively, graphs in which all vertices look the same) by Häggström, Peres and Schonmann in [Percolation on transitive graphs as a coalescent process: Relentless merging followed by simultaneous uniqueness (1999)]. —————SECTIONS——————————————————————— 0:00 Introduction 1:37 Definition – Bernoulli Percolation 5:23 Definition – Uniform Coupling 7:56 Exploration – High-Resolution Square Grid 9:40 Exploration – Questions and Kesten's Theorem 10:58 Exploration – Ising Model 11:54 Exploration – Critical Percolation 12:50 Exploration – Three-Dimensional Cubic Lattice and Beyond 14:13 Proof – Theorem Statement 15:14 Proof – Simplifications 16:29 Proof – Definition of Critical Parameter 18:41 Proof – Critical Parameter is Greater Than Zero 20:44 Proof – Duality Definition 21:56 Proof – Critical Parameter is Less Than One 25:16 Proof – Summary and Idea for Kesten's Theorem 26:11 Conclusion —————CREDITS———————————————————————— Caio Alves – writing, 3D animation Aranka Hrušková – writing, clarinet Vilas Winstein – writing, 2D animation, editing, voice-over Special thanks to Anisah Awad, Gábor Pete, Jyotsna Sreenivasan, Angie Zavala This video is an entry in the second Summer of Mathematics Exposition (#SoME2) The photographs used in this video are licensed under the Creative Commons Attribution-ShareAlike license: https://creativecommons.org/licenses/by-sa/4.0/deed.en Uploader: Spectral Collective Duration: 1612s Views: 455517

This is a collection of (mostly) pen-and-paper exercises in machine learning. The exercises are on the following topics: linear algebra, optimisation, directed graphical models, undirected graphical models, expressive power of graphical models, factor graphs and message passing, inference for hidden Markov models, model-based learning (including ICA and unnormalised models), sampling and Monte-Carlo integration, and variational inference.

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In this chapter, the authors discuss probability theory and information theory. Probability theory is a mathematical framework for representing uncertain statements and is used in artificial intelligence for reasoning. Information theory, on the other hand, quantifies the amount of uncertainty in a probability distribution. The chapter explains various concepts, such as probability mass functions for discrete variables and probability density functions for continuous variables. It also introduces key ideas from information theory, such as entropy and mutual information. The authors provide examples and explanations to help readers understand these concepts.

Linear algebra is a fundamental topic in understanding and working with machine learning algorithms, especially deep learning algorithms. This chapter provides an introduction to scalars, vectors, matrices, and tensors, which are the key mathematical objects in linear algebra. It explains the concepts and notation used in linear algebra, such as matrix multiplication, transpose, identity and inverse matrices, and norms. The chapter also introduces special kinds of matrices and vectors, such as diagonal matrices, orthogonal matrices, and eigenvalues and eigenvectors. These concepts are important for analyzing and solving equations in machine learning.

—————SOURCES———————————————————————— Percolation – Béla Bollobás and Oliver Riordan Cambridge University Press, New York, 2006. Sixty Years of Percolation – Hugo Duminil-Copin https://www.ihes.fr/~duminil/publi/2018ICM.pdf Percolation – Geoffrey Grimmett volume 321 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, second edition, 1999. —————NOTES————————————————————————— Note at 10:42 – The uniqueness of the infinite cluster is known for the d-dimenional lattice since the works of Aizenman, Kesten and Newman - [Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation (1987)] and Burton and Keane - [Density and uniqueness in percolation (1989)]. It does not hold in general: when the graph in question is a regular tree for example, there are always infinitely many clusters during the supercritical phase. The two last results shown here are only known for site percolation (in which vertices are open or closed instead of edges) in the triangular lattice, where a scaling limit for the boundaries of critical clusters was proved to exist (more on that in the third note). It is believed that these results are universal, that is, valid in great generality for planar percolation processes near criticality. The third result is from an appendix by Gábor Pete in the paper [Scaling limits for the threshold window: When does a monotone Boolean function flip its outcome? (2017)] by Ahlberg and Steif. Consider an n by n box, and the event where there exists a left-right crossing of said box. Recall the uniform coupling from the video: intuitively, the result is saying that the point at which this crossing emerges in the uniform coupling is with high probability inside an interval of size n^{-3/4} around 1/2. The fourth result is saying that the average size of the cluster of the origin (or any other given point) goes to infinity as we let p approach the critical parameter like a specific power of the distance between p and p_c. This power is called a critical exponent. The existence of these exponents was proved by Smirnov and Werner in the paper [Critical exponents for two-dimensional percolation (2001)]. Note at 10:52 – Hugo Duminil-Copin has several major contributions to the study of processes arising in statistical physics, including Bernoulli percolation. Among his works on Ising and Ising-like processes we can cite [Random Currents and Continuity of Ising Model’s Spontaneous Magnetization (2015)] with Aizenman and Sidoravicius and [Sharp phase transition for the random-cluster and Potts models via decision trees (2019)] with Raoufi and Tassion. Note at 12:38 – In the triangular lattice site percolation, Stanislav Smirnov proved the conformal invariance of crossing probabilities at criticality (see https://www.unige.ch/~smirnov/papers/icmp-final.pdf for an overview), which led to the proof of the existence of scaling limits of exploration curves as Schramm–Loewner evolution processes. See [Critical percolation in the plane (2009)] by Smirnov. This provided a deep understanding of the critical phase in the triangular lattice site percolation, which to this day is not extended to the square lattice. Note at 17:52 – It is not at all obvious that the probability of being connected to infinity is continuous above criticality. This result can be proved in the d-dimenional hypercubic lattices using the uniqueness of the infinite cluster, and more generally it was proved for transitive graphs (intuitively, graphs in which all vertices look the same) by Häggström, Peres and Schonmann in [Percolation on transitive graphs as a coalescent process: Relentless merging followed by simultaneous uniqueness (1999)]. —————SECTIONS——————————————————————— 0:00 Introduction 1:37 Definition – Bernoulli Percolation 5:23 Definition – Uniform Coupling 7:56 Exploration – High-Resolution Square Grid 9:40 Exploration – Questions and Kesten's Theorem 10:58 Exploration – Ising Model 11:54 Exploration – Critical Percolation 12:50 Exploration – Three-Dimensional Cubic Lattice and Beyond 14:13 Proof – Theorem Statement 15:14 Proof – Simplifications 16:29 Proof – Definition of Critical Parameter 18:41 Proof – Critical Parameter is Greater Than Zero 20:44 Proof – Duality Definition 21:56 Proof – Critical Parameter is Less Than One 25:16 Proof – Summary and Idea for Kesten's Theorem 26:11 Conclusion —————CREDITS———————————————————————— Caio Alves – writing, 3D animation Aranka Hrušková – writing, clarinet Vilas Winstein – writing, 2D animation, editing, voice-over Special thanks to Anisah Awad, Gábor Pete, Jyotsna Sreenivasan, Angie Zavala This video is an entry in the second Summer of Mathematics Exposition (#SoME2) The photographs used in this video are licensed under the Creative Commons Attribution-ShareAlike license: https://creativecommons.org/licenses/by-sa/4.0/deed.en Uploader: Spectral Collective Duration: 1612s Views: 455517