Channel Title: Proof of Concept Channel Description: Mathematics for aspiring mathematicians. Created by Dr. Katherine Stange at the University of Colorado, Boulder. Learn about me under the "Community" tab, or here: https://math.katestange.net/ <video> Modular Arithmetic: Under the Hood Description: Modular arithmetic visually! For aspiring mathematicians already familiar with modular arithmetic, this video describes how to formalize the concept mathematically: to define the integers modulo n, to define the operations of addition and multiplication, and check that these are well-defined. This video is appropriate for an introduction to proof course, for undergraduate mathematics majors, or for the mathematically inclined, especially those interested in cryptography or number theory. This is part of a series on modular arithmetic: https://www.youtube.com/playlist?list=PLrm9Y---qlNyWBQEKBSrr_Uh16gEWhJzw For associated materials: https://proofofconcept.katestange.net/modular-arithmetic-under-the-hood/ </video> <video> The Intuition behind the Double-And-Add / Square-And-Multiply Algorithm (Also Just a Fun Puzzle!) Description: This video develops the Double-And-Add Algorithm entirely intuitively through a fun recreational math puzzle, so you will always be able to recreate it for yourself and use it when you need it. (Also known as Square-And-Multiply and occasionally as the Chanda Sutra Method.) The video requires that you are familiar with binary. Here's the tree I draw in the video: http://oeis.org/A232559 (In my video, there's a small error; I accidentally pruned out 27 and 29 and their descendants.) You can learn about its history as part of the Chanda Sutra 8.28-31 at Wikipedia: https://en.wikipedia.org/wiki/Sanskrit_prosody#Chandah_Sutra Click on my channel to find more instructional videos on modular arithmetic, number theory and cryptography. #cryptography #algorithms </video> <video> Modular Arithmetic: Multiplication in Motion Description: Modular arithmetic visually! We explore multiplication modulo n, and discover and prove that the multiplication-by-a map f(z) = az mod n is bijective if and only if a and n are coprime. We use a visualization tool called a "dynamical portrait." This treatment is inspired by Martin H. Weissman's beautiful book, An Illustrated Theory of Numbers. This video is appropriate for an introduction to proof course, for undergraduate mathematics majors, or for the mathematically inclined, especially those interested in cryptography or number theory. This is part of a series on modular arithmetic: https://www.youtube.com/playlist?list=PLrm9Y---qlNyWBQEKBSrr_Uh16gEWhJzw For associated materials: https://proofofconcept.katestange.net </video> <video> ADFGVX Cipher: Encryption and Decryption (OLD VERSION: SEE NEW VERSION LINK BELOW) Description: This video has been replaced with an updated video: visit https://youtu.be/T0xfKiU9Rr4 Encryption and Decryption of the World War One ADFGVX Cipher, through examples. </video> <video> The extended Euclidean algorithm in one simple idea Description: An intuitive explanation of the extended Euclidean algorithm as a simple modification of the Euclidean algorithm. This video is part of playlist on GCDs and the Euclidean algorithm: https://www.youtube.com/playlist?list=PLrm9Y---qlNxXccpwYQfllCrHRJWwMky- </video> <video> Modular Arithmetic: Addition in Motion Description: Modular arithmetic visually! We explore addition modulo n, and discover and prove the number of cycles and their sizes. We use a visualization tool called a "dynamical portrait." This treatment is inspired by Martin H. Weissman's beautiful book, An Illustrated Theory of Numbers. This video is appropriate for an introduction to proof course, for undergraduate mathematics majors, or for the mathematically inclined, especially those interested in cryptography or number theory. This is part of a series on modular arithmetic: https://www.youtube.com/playlist?list=PLrm9Y---qlNyWBQEKBSrr_Uh16gEWhJzw For associated materials: https://proofofconcept.katestange.net/modular-arithmetic-addition-in-motion/ </video> <video> The Euclidean Algorithm: How and Why, Visually Description: We explain the Euclidean algorithm to compute the gcd, using visual intuition. You'll never forget it once you see the how and why. Then we write it out formally and do an example. This is part of a playlist on GCDs and the Euclidean algorithm: https://www.youtube.com/playlist?list=PLrm9Y---qlNxXccpwYQfllCrHRJWwMky- </video> <video> Modular Arithmetic: In Motion Description: Modular arithmetic visually! We use a visualization tool called a "dynamical portrait." We explore addition and multiplication modulo n, and discover and prove the portrait is made of cycles if and only if the function (f(z) = z+a mod n or f(z) = az mod n) is bijective. This treatment is inspired by Martin H. Weissman's beautiful book, An Illustrated Theory of Numbers. This video is appropriate for an introduction to proof course, for undergraduate mathematics majors, or for the mathematically inclined, especially those interested in cryptography or number theory. This is part of a series on modular arithmetic: https://www.youtube.com/playlist?list=PLrm9Y---qlNyWBQEKBSrr_Uh16gEWhJzw For associated materials: https://proofofconcept.katestange.net/modular-arithmetic-in-motion/ </video> <video> Studying Apollonian Circle Packings using Group Theory Description: This video is in response to a colleague who asked for short videos about how we use group theory in our research. One of my interests is Apollonian circle packings. Some links: Wikipedia: https://en.wikipedia.org/wiki/Apollonian_gasket An article in New Scientist by Dana Mackenzie: https://www.americanscientist.org/article/a-tisket-a-tasket-an-apollonian-gasket A good math paper to start: https://arxiv.org/abs/math/0009113 The main result I mention at the end, in the direction of the local-global conjecture, is due to Bourgain and Kontorovich: https://link.springer.com/article/10.1007/s00222-013-0475-y </video> <video> Cryptanalysis of Vigenere cipher: not just how, but why it works Description: The Vigenere cipher, dating from the 1500's, was still used during the US civil war. We introduce the cipher and explain a standard method of cryptanalysis based on frequency analysis and the geometry of vectors. We focus on visual intuition to explain why it works. The only background needed is some familiarity with vectors and probabilities. For more on this method: _Introduction_to_Cryptography_with_Coding_Theory_ by Wade Trappe and Lawrence C. Washington, Second Edition, Section 2.3. Here the method is exactly as in the video, because this is the text I teach my course from. _Secret_History:_The_Story_of_Cryptology_ by Craig P. Bauer, Section 3.3. Here the method is done with more mathematical sophistication (so it looks slightly different). This is a very valuable and detailed textbook for anyone interested in old ciphers and their cryptanalysis. I also have some cryptanalysis tools here you can use to implement the method (which I do with my students in the classroom): https://crypto.katestange.net/cryptanalysis-tools/ </video> <video> Lehmer Factor Stencils: A paper factoring machine before computers Description: In 1929, Derrick N. Lehmer published a set of paper stencils used to factor large numbers by hand before the advent of computers. We explain the math behind the stencils, which includes modular arithmetic, quadratic residues, and continued fractions, including my favourite mathematical visualization for continued fractions. *VIDEO CORRECTION*: I made a copying error when setting up the recurrences and doing out the computations (so some intermediate values are incorrect). I'm so sorry about this! See the PDF User's Guide below for full details and correction of this issue. *DO IT YOURSELF!* PDF user's guide: https://math.colorado.edu/~kstange/stencils/stencil-users-guide.pdf Accompanying Python & SVG files to make your own: https://github.com/katestange/lehmerfactorstencils Original stencils: Derrick N. Lehmer, Factor Stencils, Washington, DC: Carnegie Institution of Washington, 1929. - Worldcat link: https://www.worldcat.org/title/factor-stencils-by-derrick-norman-lehmer/oclc/504616344&referer=brief_results - Smithsonian Link: https://americanhistory.si.edu/collections/search/object/nmah_1214102 Updated on Hollerith cards: D. N. Lehmer, Factor Stencils, rev. John D. Elder, Washington, D.C.: Carnegie Institution of Washington, 1939. - Worldcat Link: https://www.worldcat.org/title/factor-stencils/oclc/9739317&referer=brief_results - Smithsonian Link: https://americanhistory.si.edu/collections/search/object/nmah_1214103 Lehmer's essay on his factoring machine that I quote from in the video: D. N. Lehmer, Hunting Big Game in the Theory of Numbers, Scripta Mathematica, September, 1932. http://ed-thelen.org/comp-hist/Lehmer-NS03.html A great video about homemade factor stencils by Chris Staecker: https://www.youtube.com/watch?v=Fwn_6uqmeqw Videos on modular arithmetic, done visually: https://www.youtube.com/playlist?list=PLrm9Y---qlNyWBQEKBSrr_Uh16gEWhJzw *Chapters*: 00:00 intro 00:50 demo of the stencils 02:27 history 04:11 overview 04:34 difficulty of factoring 09:14 relationship to cryptography 09:50 Lehmer quote 11:40 sieving for primes 12:04 quadratic residues and modular arithmetic 14:23 sieving for primes with quadratic residues 15:37 how many stencils are needed 16:47 recurrence relations and how to find quadratic residues 17:29 geoboard demo and rubber band points 20:31 Farey subdivision 22:55 Continued fractions 24:45 summary and runtime *Thanks to*: Glen Whitney, of MOMATH, for help with paper cutter 3blue1brown and LeiosOS, for motivation (Summer of Math Exposition, SoME1 https://www.3blue1brown.com/blog/some1) Edmund Harriss and Steve Trettel, for collaboration on some lattice images Jonathan Wise, as always Image credits: Lehmer on piano – Konrad Jacobs – CC-BY-SA 2.0 DE https://commons.wikimedia.org/wiki/File:Derrick_Norman_Lehmer.jpg Lehmer’s machines – Marcin Whichery – CC-BY 3.0 https://commons.wikimedia.org/wiki/File:Computer_History_Museum_(2664736595).jpg RSA keyfob – Alexander Klink – CC-BY 3.0 https://commons.wikimedia.org/wiki/File:RSA_SecurID_SID800.jpg Farey subdivision – collaboration with Edmund Harriss and Steve Trettel If you like, subscribe and comment, it does encourage me :) Personal notes: This was my first real YouTube video project. I learned a lot of lessons, including that my voice is different on different days even with the same microphone setup! It was a fun adventure. I'm not going to be the next big YouTube star, but your comments and feedback make it feel worth the time and energy! #SummerOfMathExposition #SoME1 #3b1b Licensed under CC BY-SA 4.0 </video> <video> ADFGVX Cipher: Encryption and Decryption (Updated) Description: Encryption and Decryption of the World War One ADFGVX Cipher, through examples. This video replaces an earlier one with the same content, but has some improvements and corrections. #cryptography </video> <video> Rethinking the real line #SoME3 Description: We take a geometric approach to rational numbers, to rethink how to organize the real line. Along the way, we visualize Diophantine approximation and continued fractions. And your favourite number, pi. Much of the mathematics here is based on the following article: Series, C. The geometry of markoff numbers. The Mathematical Intelligencer 7, 20–29 (1985). https://doi.org/10.1007/BF03025802 A big thanks to the Summer of Math Exposition competition for the motivation to make this happen, and a big thanks to my audience for forgiving my video-editing non-skills. Some of the software used in creating this: Sage Mathematics Software, Manim, VPython, p5.js, Krita, Audacity, Kdenlive. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Music used in the video: Walk Through the Park -- TrackTribe George Street Shuffle -- Kevin MacLeod Quarter Mix -- Freedom Trail Studio Love Struck -- E's Jammy Jams George Street Shuffle by Kevin MacLeod is licensed under a Creative Commons Attribution 4.0 license. https://creativecommons.org/licenses/by/4.0/ Source: http://incompetech.com/music/royalty-free/index.html?isrc=USUAN1300035 Artist: http://incompetech.com/ </video> <video> The modular inverse via Gauss not Euclid Description: We demonstrate a lesser-known algorithm for taking the inverse of a residue modulo p, where p is prime. This algorithm doesn't depend on the extended Euclidean algorithm, so it can be learned independently. This is part of a larger series on modular arithmetic: https://www.youtube.com/playlist?list=PLrm9Y---qlNyWBQEKBSrr_Uh16gEWhJzw </video> <video> Greatest common divisor and least common multiple: building blocks Description: The intuition behind gcd and lcm, from the perspective of factorizations. For aspiring mathematicians already familiar with prime numbers as the building blocks of integers. Learn the intuition, the definitions, and gain some practice. This video is appropriate for high school, for an introduction to proof course, for undergraduate mathematics majors, or for the mathematically inclined, especially those interested in cryptography or number theory. </video>

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Mathematics for aspiring mathematicians. Created by Dr. Katherine Stange at the University of Colorado, Boulder. Learn about me under the "Community" tab, or here: https://math.katestange.net/

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