The relationships among the properties of flexible shapes have fascinated mathematicians for centuries.

I've been collaborating on an exciting project for quite some time now, and today I'm happy to share it with you. There is a new topology book on the market! Topology: A Categorical Approach is a graduate-level textbook that presents basic topology from the modern perspective of category theory. Coauthored with Tyler Bryson and John Terilla, Topology is published through MIT Press and will be released on August 18, 2020. But you can pre-order on Amazon now!

Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: https://to.pbs.org/donateinfi What happens when you multiply shapes? This is part 2 of our episode on multiplying things that aren't numbers. You can check out part 1: The Multiplication Multiverse right here https://www.youtube.com/watch?v=H4I2C3Ts7_w Tweet at us! @pbsinfinite Facebook: facebook.com/pbsinfinite series Email us! pbsinfiniteseries [at] gmail [dot] com And discuss the episode further over on reddit at https://www.reddit.com/r/PBSInfiniteSeries/ Previous Episode The Multiplication Multiverse | Infinite Series https://www.youtube.com/watch?v=H4I2C3Ts7_w In our last episode, we talked about different properties of multiplication: associativity and commutativity are the most familiar, but they’re just two of many. We also saw it’s possible to multiply things that aren’t numbers, and in that case we may not have... associativity, for instance. But that’s not a bad thing. In fact, it’s a beautiful thing! References:: More on the associahedra: http://www.ams.org/samplings/feature-column/fcarc-associahedra http://www.claymath.org/library/academy/LectureNotes05/Lodaypaper.pdf https://arxiv.org/pdf/math/0212126.pdf More on multiplying non-numbers: http://www.math3ma.com/mathema/2017/11/24/multiplying-non-numbers An introduction to operads: http://www.math3ma.com/mathema/2017/10/23/what-is-an-operad-part-1 http://www.math3ma.com/mathema/2017/10/30/what-is-an-operad-part-2 Some applications in math and physics: https://arxiv.org/abs/1202.3245 http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.22.2871&rep=rep1&type=pdf http://bookstore.ams.org/conm-227 http://www.springer.com/us/book/9780817647346 Richard Stanley’s book on the Catalan Numbers: https://www.amazon.com/Catalan-Numbers-Richard-P-Stanley/dp/1107427746 Written and Hosted by Tai-Danae Bradley Produced by Rusty Ward Graphics by Ray Lux Assistant Editing and Sound Design by Meah Denee Barrington Made by Kornhaber Brown (www.kornhaberbrown.com) Thanks to Matthew O'Connor and Yana Chernobilsky who are supporting us on Patreon at the Identity level! And thanks to Nicholas Rose and Mauricio Pacheco who are supporting us at the Lemma level!

Course notes for Pure Mathematics Topic D 2019. Contribute to DavidMichaelRoberts/AlgebraicTopology2019 development by creating an account on GitHub.

From [Adrien Vakili](vaki.li) General Topology Munkres -- Topology Chapter 1, sections 1-6 Chapter 2 Chapter 3, sections 1-2 Algebraic Topology Hatcher -- Algebraic Topology Provides good geometric intuition but avoids categorical language. Chapters 0, 1, 4 Strom -- Modern Classical Homotopy Theory More axiomatic/categorical treatment. Fomenko -- Visual Geometry and Topology For pictures.

I begin to feel that I can trust mathematics as a guiding beacon for how programming can be done well. This is why I feel strongly that simple mathematical constructs, like pure functions, monoids, etc., form a strong foundation of abstraction as opposed to the overly complicated, and often ad-hoc, design patterns we see in software engineering. ​ I spend a lot of my time trying to find new and creative ways to bring seemingly complex functional programming ideas down to earth and make them approachable to a wider audience. ​ but it does give us an opportunity to explore a strange and surprising result in computation and mathematics. It can help show that the connection between the two topics is perhaps deeper than we may first think.