The Mathematics of Quantum Computers | Infinite Series PBS Infinite Series 2017-02-16 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiWhat is the math behind quantum computers? And why are quantum computers so amazing? Find out on this episode of Infinite Series. Tweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] comPrevious Episode - How the Infinite Explains the Finiteyoutube.com/watch?v=oBOZ2WroiVYWritten and Hosted by Kelsey Houston-EdwardsProduced by Rusty WardGraphics by Ray LuxMade by Kornhaber Brown (www.kornhaberbrown.com)Sources and further references:http://iqim.caltech.edu/outreach/For a great linear algebra tutorial, check out 3Blue1Brown's:youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_abSome of the people who wrote awesome programs that compute Goodstein Sequences:Donald HobsonDavid FrancoJeronimo Spuri MurguiaNathan SouffletIvan HopMax GoldsteinComments answered by Kelsey:Cybernetic Qualinautyoutube.com/watch?v=oBOZ2WroiVY&lc=z130wfx4gluhz55iy04cf15xcxmfe1q54hoEric Saumaryoutube.com/watch?v=oBOZ2WroiVY&lc=z122iz3rsu3vx3dgm04cj51riw3nxtegpbsBob Cyoutube.com/watch?v=oBOZ2WroiVY&lc=z12xtnujduinuvk2g23ye1fxxkrkzngog04
The End of An Infinite Series PBS Infinite Series 2018-05-17 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiThank you everyone. This show was a joy to produce and it was the audience that made it incredible. Gabe Perez-Giz@fizziksgabeTai-Danae Bradley@math3mahttp://www.math3ma.com/mathema/2015/2/1/a-math-blog-say-what
The Assassin Puzzle | Infinite Series PBS Infinite Series 2018-05-17 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiImagine you have a square-shaped room, and inside there is an assassin and a target. And suppose that any shot that the assassin takes can ricochet off the walls of the room, just like a ball on a billiard table. Is it possible to position a finite number of security guards inside the square so that they block every possible shot from the assassin to the target?Tweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] comSolution to the Assassin Puzzle:www.math3ma.com/mathema/2018/5/17/is-the-square-a-secure-polygonPrevious Episode:Instant Insanity Puzzleyoutube.com/watch?v=Lw1pF47N-0Q&t=1sLet’s walk through this puzzle a little more precisely. First, instead of thinking of a physical room with actual people inside, I really want you to think of a square in the xy plane. Pick any two points in the square, and let’s call one of those points A for “assassin,” and the other point T for “target.” Now a “shot” from the assassin is really just a ray emanating out of the point A which can, like a ball on a billiard table, bounce back and forth between the sides of the square. But unlike an actual game of pool, let’s assume the trajectory has constant speed and that it can bounce back and forth for forever! Written and Hosted by Tai-Danae BradleyProduced by Eric BrownGraphics by Matt RankinAssistant Editing and Sound Design by Mike Petrow and Linda Huang Made by Kornhaber Brown (www.kornhaberbrown.com)Special thanks to Roman Pinchuk for supporting us on our Converse level on Patreon. Along with thanks to Matthew O'Connor, Yana Chernobilsky, and John Hoffman who are supporting us on Patreon at the Identity level!And thanks to Mauricio Pacheco and Andrew Poelstra who are supporting us at the Lemma level!
Instant Insanity Puzzle | Infinite Series PBS Infinite Series 2018-04-26 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiImagine you have four cubes, whose faces are colored red, blue, yellow, and green. Can you stack these cubes so that each color appears exactly once on each of the four sides of the stack?Tweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfiniteseriesEmail us! pbsinfiniteseries [at] gmail [dot] comNow, the coloring on these cubes is very important. In other words: we’re only considering THESE four particular cubes. So, for example, cube 1 has exactly three red faces, and one yellow, green, and blue face, moreover, those colors appear on the particular faces as indicated. Similarly for the other cubes.Previous Episode:Defining Infinityyoutube.com/watch?v=VCksQ6g2yh0Written and Hosted by Tai-Danae BradleyGraphics by Matt RankinAssistant Editing and Sound Design by Mike Petrow and Linda Huang Made by Kornhaber Brown (www.kornhaberbrown.com)Print off a template for the cubes shown in the video!bit.ly/2HWYNaYResources:Introduction to Graph Theory by Robin J. Wilson amazon.com/Introduction-Graph-Theory-Robin-Wilson/dp/027372889X Special thanks to Roman Pinchuk for supporting us on our Converse level on Patreon. Along with thanks to Matthew O'Connor, Yana Chernobilsky, and John Hoffman who are supporting us on Patreon at the Identity level!And thanks to Mauricio Pacheco and Andrew Poelstra who is supporting us at the Lemma level!
Defining Infinity | Infinite Series PBS Infinite Series 2018-04-13 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiSet theory is supposed to be a foundation of all of mathematics. How does it handle infinity? Learn through active problem-solving at Brilliant: brilliant.org/InfiniteSeries Tweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] comPrevious Episode:Unraveling DNA with Rational Tanglesyoutube.com/watch?v=JXGyXtNsu14Earlier videos referenced around 1:00 are:Cantor's paradox: youtube.com/watch?v=TbeA1rhV0D0What are Numbers Made of? youtube.com/watch?v=S4zfmcTC5bMPeano axioms: youtube.com/watch?v=3gBoP8jZ1IsHiearchy of Infinities: youtube.com/watch?v=i7c2qz7sO0IVsauce How to Count Past Infinityyoutube.com/watch?v=SrU9YDoXE88Brilliant infinity wiki:brilliant.org/wiki/infinityBrilliant Number Theory course (with Exploring Infinity chapter):brilliant.org/courses/basic-number-theorySet theory arose in part to get a grip on infinity. Early “naive” versions were beset by apparent paradoxes and were superseded by axiomatic versions that used formal rules to demarcate "legal" mathematical statements from gibberish.Written and Hosted by Gabe Perez-Giz (@fizziksgabe)Produced by Rusty WardGraphics by Ray LuxAssistant Editing and Sound Design by Mike Petrow and Linda Huang Made by Kornhaber Brown (www.kornhaberbrown.com)Special thanks to Roman Pinchuk for supporting us on our Converse level on Patreon. Along with thanks to Matthew O'Connor, Yana Chernobilsky, and John Hoffman who are supporting us on Patreon at the Identity level!And thanks to Mauricio Pacheco and Andrew Poelstra who is supporting us at the Lemma level!
Unraveling DNA with Rational Tangles | Infinite Series PBS Infinite Series 2018-03-29 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiWhen you think about math, what do you think of? Numbers? Equations? Patterns maybe? How about… knots? As in, actual tangles and knots?Tweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] comPrevious Episode:How Big are All Infinities Combined? (Cantor’s Paradox)youtube.com/watch?v=TbeA1rhV0D0There is a special kind of mathematical tangle called a rational tangle, first defined by mathematician John Conway around 1970 which relates to biology and the study of DNA.Written and Hosted by Tai-Danae BradleyProduced by Rusty WardGraphics by Ray LuxAssistant Editing and Sound Design by Mike Petrow and Linda Huang Made by Kornhaber Brown (www.kornhaberbrown.com)Resources:Modeling protein-DNA complexes with tangles by Isabel Darcy(the tangle examples in today’s episode can be found here)sciencedirect.com/science/article/pii/S0898122107005718 Understanding Rational Tangles (Recreational Guide)mathteacherscircle.org/assets/session-materials/JTantonRationalTangles.pdf The Knot Book by Colin Adamsamazon.com/Knot-Book-Colin-Adams/dp/0821836781 Knot Theory and Its Applications by Kunio Murasugiamazon.com/Theory-Applications-Modern-Birkh%C3%A4user-Classics/dp/081764718X On the Classification of Rational Tangles by Louis Kauffman and Sofia Lambropoulouarxiv.org/pdf/math/0311499.pdf DNA Topology by Andrew Bates and Anthony Maxwellamazon.com/Topology-Oxford-Biosciences-Andrew-Bates/dp/0198506554 Proof of Conway’s Rational Tangle Theoremhttp://homepages.math.uic.edu/~kauffman/RTang.pdf The Shape of DNA (video with Mariel Vasquez)youtube.com/watch?v=AxxnziuL408 How DNA Unties its Own Knots (video on topoisomerase with Mariel Vasquez)youtube.com/watch?v=UkmQNbvlK8s Knots and Quantum Theory by Edward Wittenhttps://www.ias.edu/ideas/2011/witten-knots-quantum-theory Tangles, Physics, and Category Theoryhttp://math.ucr.edu/home/baez/tangles.html My Favorite Theorem Podcastkpknudson.com/my-favorite-theorem Topics in Knots and Algebra (Online Course at Bridgewater State University)youtube.com/watch?v=fpoGoAscqX4&list=PLL0ATV5XYF8BfT8CmmzKnfTlf3V9hQgj9 Special thanks to Roman Pinchuk for supporting us on our Converse level on Patreon. Along with thanks to Matthew O'Connor, Yana Chernobilsky, and John Hoffman who are supporting us on Patreon at the Identity level!And thanks to Mauricio Pacheco and Andrew Poelstra who is supporting us at the Lemma level!
How Big are All Infinities Combined? (Cantors Paradox) | Infinite Series PBS Infinite Series 2018-03-23 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiInfinities come in different sizes. There’s a whole tower of progressively larger "sizes of infinity". So what’s the right way to describe the size of the whole tower?Tweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] comPrevious Episode:The Geometry of SETyoutube.com/watch?v=zurpOBPt4LICheck out the solution to the Geometry of SET challenge problem right here: bit.ly/2Ggpw1dTalking about the sizes of infinite things is tricky in part because the word “infinite” is often used in two distinct ways -- to refer to the sets themselves, and also to refer to the *sizes* of those sets. In what follows, let’s try to keep as sharp a distinction as we can between infinite sets and infinite set *sizes*, because doing so will let me highlight an especially paradoxical feature about infinite sizing that I don't think gets enough coverage. The technical term for a “size”, infinite or otherwise, is “cardinality”, and I should probably use a term like “numerousness” or “numerosity” rather than “size” because the idea it tries to generalize is the notion of "how many". Still, I’m going to say “size” a lot in this episode just because it’s easier.Written and Hosted by Tai-Denae BradleyProduced by Rusty WardGraphics by Ray LuxAssistant Editing and Sound Design by Mike Petrow and Linda Huang Made by Kornhaber Brown (www.kornhaberbrown.com)Thanks to Peleg Shilo, Anurag Bishnoi, and Yael Dillies for your comments on last week's episode:youtube.com/watch?v=zurpOBPt4LI&lc=UgzxzGjWJh8Pra57LDR4AaABAgyoutube.com/watch?v=zurpOBPt4LI&lc=Ugx2lMNH6inRYiTGFSx4AaABAgyoutube.com/watch?v=zurpOBPt4LI&lc=UgzA2HK0J9ytm01lUex4AaABAgSpecial thanks to Roman Pinchuk for supporting us on our Converse level on Patreon. Along with thanks to Matthew O'Connor, Yana Chernobilsky, and John Hoffman who are supporting us on Patreon at the Identity level!And thanks to Mauricio Pacheco and Andrew Poelstra who are supporting us at the Lemma level!
The Geometry of SET | Infinite Series PBS Infinite Series 2018-03-15 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiIn the card game SET, what is the maximum number of cards you can deal that might not contain a SET?Tweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] comResourcesOfficial SET game instructionssetgame.com/sites/default/files/instructions/SET%20INSTRUCTIONS%20-%20ENGLISH.pdfSimple SET Game Proof Stuns Mathematicians (Quanta Magazine)quantamagazine.org/set-proof-stuns-mathematicians-20160531The Problem with SET (NYTimes Puzzle)nytimes.com/2016/08/22/crosswords/the-problem-with-set.htmlOpen Question: Best Bounds for Cap Sets (blog post by Terry Tao)terrytao.wordpress.com/2007/02/23/open-question-best-bounds-for-cap-setsSET and Group Theory by Pavel Etingof (see p. 13ff)http://www-math.mit.edu/~etingof/groups.pdfThis episodes challenge question is: Among the 9 cards shown in this episode what is the maximum number of them that may not contain a SET? Can you rephrase this question in an equivalent yet geometric way and then answer it using the hint - SETs correspond to lines in the Z/3Z grid?Email your answers to pbsinfiniteseries@gmail.com with the subject line "SET Challenge" along with your proof. A random winner will be selected among the submissions to win a PBS Digital t-shirt.(Spoiler Alert!) Here's the solution to the SET challenge problem: bit.ly/2Ggpw1dPrevious Episode:What was Fermat’s “Marvelous" Proof?youtube.com/watch?v=SsVl7_R2MvILet's talk about the card game SET. To play, you start with a deck of cards, each of which has a certain number of shapes in different colors and shadings. You deal out 12 cards and start looking for a SET---a collection of 3 cards that have either all the same or all different patterns. Now, once you deal those 12 cards, it’s possible that there might not be a SET among them. When that happens, you just deal out 3 more cards. And… in some cases, there still might not be a SET. So… you can add 3 more cards. And this begs the question: What is the maximum number of cards you can deal that might not contain a SET? Written and Hosted by Tai-Danae BradleyProduced by Rusty WardGraphics by Ray LuxAssistant Editing and Sound Design by Mike Petrow and Linda Huang Made by Kornhaber Brown (www.kornhaberbrown.com)Special thanks to Roman Pinchuk for supporting us on our Converse level on Patreon. Along with thanks to Matthew O'Connor, Yana Chernobilsky, and John Hoffman who are supporting us on Patreon at the Identity level!And thanks to Mauricio Pacheco and Andrew Poelstra who are supporting us at the Lemma level!
What was Fermat’s “Marvelous Proof? | Infinite Series PBS Infinite Series 2018-03-08 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiIf Fermat had a little more room in his margin, what proof would he have written there?Tweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] comResources:Contemporary Abstract Algebra by Joseph Gallianamazon.com/Contemporary-Abstract-Algebra-Joseph-Gallian/dp/1133599702Standard Definitions in Ring Theory by Keith Conradhttp://www.math.uconn.edu/~kconrad/blurbs/ringtheory/ringdefs.pdf Rings and First Examples (online course by Prof. Matthew Salomone)youtube.com/watch?v=h4UCMd8dyiMFermat's Enigma by Simon Singhamazon.com/Fermats-Enigma-Greatest-Mathematical-Problem/dp/0385493622Who was first to differentiate between prime and irreducible elements? (StackExchange)hsm.stackexchange.com/questions/3754/who-was-first-to-differentiate-between-prime-and-irreducible-elementsPrevious Episodes:What Does It Mean to be a Number?youtube.com/watch?v=3gBoP8jZ1IsWhat are Numbers Made of?youtube.com/watch?v=S4zfmcTC5bMGabe's references from the comments:Blog post about the Peano axioms and construction of natural numbers by Robert Low:http://robjlow.blogspot.co.uk/2018/01/whats-number-1-naturally.htmlRecommended by a viewer for connections to formulation of numbers in computer science:https://softwarefoundations.cis.upenn.edu/In 1637, Pierre de Fermat claimed to have the proof to his famous conjecture, but, as the story goes, it was too large to write in the margin of his book. Yet even after Andrew Wiles’s proof more than 300 years later, we’re still left wondering: what proof did Fermat have in mind? The mystery surrounding Fermat’s last theorem may have to do with the way we understand prime numbers. You all know what prime numbers are. An integer greater than 1 is called prime if it has exactly two factors: 1 and itself. In other words, p is prime if whenever you write p as a product of two integers, then one of those integers turns out to be 1. In fact, this definition works for negative integers, too. We simply incorporate -1. But the prime numbers satisfy another definition that maybe you haven’t thought about: An integer p is prime if, whenever p divides a product of two integers, then p divides at least one of those two integers.Written and Hosted by Tai-Danae BradleyProduced by Rusty WardGraphics by Ray LuxAssistant Editing and Sound Design by Mike Petrow and Linda Huang Made by Kornhaber Brown (www.kornhaberbrown.com)Special thanks to Roman Pinchuk for supporting us on our Converse level on Patreon. Along with thanks to Matthew O'Connor, Yana Chernobilsky, and John Hoffman who are supporting us on Patreon at the Identity level!And thanks to Mauricio Pacheco who are supporting us at the Lemma level!
What are Numbers Made of? | Infinite Series PBS Infinite Series 2018-03-01 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiIn the physical world, many seemingly basic things turn out to be built from even more basic things. Molecules are made of atoms, atoms are made of protons, neutrons, and electrons. So what are numbers made of?Check out the previous episode to find out What It Means to be a Numberyoutube.com/watch?v=3gBoP8jZ1IsAnd to see Gabe's solution to the Torus Clock Challenge check out:http://bit.ly/pbs_clock_challengeTweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] comPrevious Episode:What Does it Mean to Be A Number?youtube.com/watch?v=3gBoP8jZ1IsTorus Clock Challenge:youtube.com/watch?v=KZT5hrYOERsHow to Divide by "Zero"youtube.com/watch?v=uxpowBoPieQ&t=3sBlog post about the Peano axioms and construction of natural numbers by Robert Low:http://robjlow.blogspot.co.uk/2018/01/whats-number-1-naturally.htmlRecommended by a viewer for connections to formulation of numbers in computer science:https://softwarefoundations.cis.upenn.edu/Any set N and function S that meet these conditions will *behave* , respectively, like the natural numbers and the operation "next" or "successor". You can even define operations that fully mimic run-of-the-mill addition and multiplication in terms of any suitable S, regardless of the details of how S works . In this sense, the Peano axioms distill numberhood down to its bare essentials.Written and Hosted by Gabe Perez-GizProduced by Rusty WardGraphics by Ray LuxAssistant Editing and Sound Design by Mike Petrow and Linda Huang Made by Kornhaber Brown (www.kornhaberbrown.com)Special thanks to Roman Pinchuk for supporting us on our Converse level on Patreon. Along with thanks to Matthew O'Connor, Yana Chernobilsky, and John Hoffman who are supporting us on Patreon at the Identity level!And thanks to Mauricio Pacheco who are supporting us at the Lemma level!
What Does It Mean to Be a Number? (The Peano Axioms) | Infinite Series PBS Infinite Series 2018-02-27 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiIf you needed to tell someone what numbers are and how they work, without using the notion of number in your answer, could you do it?Tweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] comPrevious Episodes:Telling Time on a Torusyoutu.be/KZT5hrYOERsCrisis in the Foundation of Mathematicsyoutube.com/watch?v=KTUVdXI2vngHow to Divide by "Zero"youtube.com/watch?v=uxpowBoPieQBeyond the Golden Ratioyoutube.com/watch?v=MIxvZ6jwTuAAre the natural numbers fundamental, or can they be constructed from more basic ingredients? It turns out that you can capture the essence of numberhood in a small set of axioms, analogous to Euclid’s axioms in geometry. They will allow us to build a set N that will behave just like the natural numbers without ever explicitly mentioning numbers or counting or arithmetic as we do so. These axioms were first published in 1889, more or less in their modern form, by Giuseppe Peano, building on and integrating earlier work by Peirce and Dedekind.Written and Hosted by Gabe Perez-GizProduced by Rusty WardGraphics by Ray LuxAssistant Editing and Sound Design by Mike Petrow and Linda HuangMade by Kornhaber Brown (www.kornhaberbrown.com)Special thanks to Roman Pinchuk for supporting us on our Converse level on Patreon. Along with thanks to Matthew O'Connor, Yana Chernobilsky, and John Hoffman who are supporting us on Patreon at the Identity level!And thanks to Mauricio Pacheco who is supporting us at the Lemma level!
Telling Time on a Torus | Infinite Series PBS Infinite Series 2018-02-15 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiWhat shape do you most associate with a standard analog clock? Your reflex answer might be a circle, but a more natural answer is actually a torus. Surprised? Then stick around.Tweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] comPrevious Episode:How to Divide by Zeroyoutube.com/watch?v=uxpowBoPieQSome configurations of a clock, like the hour hand at 3 with the minute hand at 12, represent "valid" times of day -- if the hands sweep around continuously at their usual steady rates, this configuration will actually happen every 12 hours, at precisely 3 o'clock.Written and Hosted by Gabe Perez-GizProduced by Rusty WardGraphics by Ray LuxAssistant Editing and Sound Design by Mike Petrow and Meah Denee Barrington Made by Kornhaber Brown (www.kornhaberbrown.com)Special thanks to Roman Pinchuk for supporting us on our Converse level on Patreon. Along with thanks to Matthew O'Connor, Yana Chernobilsky, and John Hoffman who are supporting us on Patreon at the Identity level!And thanks to Nicholas Rose, Jason Hise, Thomas Scheer, Marting Sergio H. Faester, CSS, and Mauricio Pacheco who are supporting us at the Lemma level!
How to Divide by Zero | Infinite Series PBS Infinite Series 2018-02-01 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiWhat happens when you divide things that aren’t numbers?Tweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] comRESOURCESVisual Group Theory by Nathan Carter amazon.com/Visual-Group-Theory-Problem-Book/dp/088385757X Geek37 - The silver ratio and the octagonyoutube.com/watch?v=o-6kWnYfdVkPolygons, Diagonals, and the Bronze Mean by Antonia RedondoBuitragolink.springer.com/content/pdf/10.1007/s00004-007-0046-x.pdflPrevious Episodes:Beyond the Golden Ratioyoutube.com/watch?v=MIxvZ6jwTuA&t=600sThe Multiplication Multiverseyoutube.com/watch?v=H4I2C3Ts7_wThe Mathematics of Diffie Hillman Key Exchangeyoutube.com/watch?v=ESPT_36pUFcHow to Break Cryptographyyoutube.com/watch?v=12Q3Mrh03Gk&t=130s&list=PLa6IE8XPP_glwNKmFfl2tEL0b7E9D0WRr&index=42We all know you can’t divide by the number zero. But in some sense the notion of “dividing by zero” appears every time you use modular arithmetic! The structures that underlie this “modding business” are called equivalence relations and quotient sets. And that’s what I’d like to dive into today.Written and Hosted by Tai-Danae BradleyProduced by Rusty WardGraphics by Ray LuxAssistant Editing and Sound Design by Mike Petrow and Meah Denee Barrington Made by Kornhaber Brown (www.kornhaberbrown.com)Special thanks to Roman Pinchuk for supporting us on our Converse level on Patreon. Along with thanks to Matthew O'Connor, Yana Chernobilsky, and John Hoffman who are supporting us on Patreon at the Identity level!And thanks to Nicholas Rose and Mauricio Pacheco for supporting us at the Lemma level!
Beyond the Golden Ratio | Infinite Series PBS Infinite Series 2018-01-26 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiYou know the Golden Ratio, but what is the Silver Ratio? Learn through active problem-solving at Brilliant: brilliant.org/InfiniteSeries Dive into more open problem solving right here brilliant.org/InfiniteSeriesOpenProblemTweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] comRESOURCESPolygons, Diagonals, and the Bronze Mean by Antonia RedondoBuitragolink.springer.com/content/pdf/10.1007/s00004-007-0046-x.pdfPrevious EpisodeProving Brouwer's Fixed Point Theorem youtu.be/djaSbHKK5ycCut a line segment into unequal pieces of lengths a and b such that the ratio a to b is the same as the ratio (a + b) to a --- that is, so that big over medium equals medium over small. This is how you construct the golden ratio Phi. If a rectangle has an aspect ratio of Phi, you can subdivide it forever into a square and another golden rectangle, and make fun logarithmic spirals by connecting the corners.Written and Hosted by Gabe Perez-GizProduced by Rusty WardGraphics by Ray LuxAssistant Editing and Sound Design by Mike Petrow and Meah Denee Barrington Made by Kornhaber Brown (www.kornhaberbrown.com)Special thanks to Roman Pinchuk for supporting us on our Converse level on Patreon. Along with thanks to Matthew O'Connor, Yana Chernobilsky, and John Hoffman who are supporting us on Patreon at the Identity level!And thanks to Nicholas Rose, Jason Hise, Thomas Scheer, Marting Sergio H. Faester, CSS, and Mauricio Pacheco who are supporting us at the Lemma level!
Proving Brouwers Fixed Point Theorem | Infinite Series PBS Infinite Series 2018-01-18 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiThere is a proof for Brouwer's Fixed Point Theorem that uses a bridge - or portal - between geometry and algebra. Tweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] comPrevious EpisodeThe Mathematics of Diffie-Hellman Key Exchangeyoutube.com/watch?v=ESPT_36pUFc&feature=youtu.beAnalogous to the relationship between geometry and algebra, there is a mathematical “portal” from a looser version of geometry -- topology -- to a more “sophisticated” version of algebra. This portal can take problems that are very difficult to solve topologically, and recast them in an algebraic light, where the answers may become easier.Written and Hosted by Tai-Danae BradleyProduced by Rusty WardGraphics by Ray LuxAssistant Editing and Sound Design by Mike Petrow and Meah Denee Barrington Made by Kornhaber Brown (www.kornhaberbrown.com)REFERENCESThe functor in today’s episode is called “the fundamental group.” To learn more about the fundamental group and the proof of Brouwer’s Fixed Point Theorem, check out:Brouwer's Fixed Point Theorem (Proof) on Math3ma:http://www.math3ma.com/mathema/2018/1/18/brouwers-fixed-point-theorem-proofAlgebraic Topology by Allen Hatcher, page 31:https://www.math.cornell.edu/~hatcher/AT/AT.pdfA Concise Course in Algebraic Topology by Peter May, page 10: https://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf Category Theory in Context by Emily Riehl, page 15: http://www.math.jhu.edu/~eriehl/context.pdf To learn more about algebraic topology, check out:Algebraic Topology: An Introduction by W. S. Massey amazon.com/Algebraic-Topology-Introduction-Graduate-Mathematics/dp/0387902716 Elements of Algebraic Topology by J. Munkres amazon.com/Elements-Algebraic-Topology-James-Munkres/dp/0201627280To learn more about category theory and functors, check out:“What is category theory, anyway?” http://www.math3ma.com/mathema/2017/1/17/what-is-category-theory-anyway “What is a functor?” http://www.math3ma.com/mathema/2017/1/31/what-is-a-functor-part-1VSauce - Fixed Pointsyoutube.com/watch?v=csInNn6pfT4Congratulations to Robby Weintraub for being the winner of the Topology vs "a" Topology Challenge Question. youtube.com/watch?v=tdOaMOcxY7USpecial thanks to Roman Pinchuk for supporting us on our Converse level on Patreon. Along with thanks to Matthew O'Connor, Yana Chernobilsky, and John Hoffman who are supporting us on Patreon at the Identity level!And thanks to Nicholas Rose, Jason Hise, Thomas Scheer, Marting Sergio H. Faester, CSS, and Mauricio Pacheco who are supporting us at the Lemma level!
The Mathematics of Diffie-Hellman Key Exchange | Infinite Series PBS Infinite Series 2018-01-11 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiSymmetric keys are essential to encrypting messages. How can two people share the same key without someone else getting a hold of it? Upfront asymmetric encryption is one way, but another is Diffie-Hellman key exchange. This is part 3 in our Cryptography 101 series. Check out the playlist here for parts 1 & 2: youtube.com/watch?v=NOs34_-eREk&list=PLa6IE8XPP_gmVt-Q4ldHi56mYsBuOg2QwTweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] comPrevious EpisodeTopology vs. “a” Topology youtube.com/watch?v=tdOaMOcxY7U&t=13sSymmetric single-key encryption schemes have become the workhorses of secure communication for a good reason. They’re fast and practically bulletproof… once two parties like Alice and Bob have a single shared key in hand. And that’s the challenge -- they can’t use symmetric key encryption to share the original symmetric key, so how do they get started?Written and Hosted by Gabe Perez-GizProduced by Rusty WardGraphics by Ray LuxAssistant Editing and Sound Design by Mike Petrow and Meah Denee Barrington Made by Kornhaber Brown (www.kornhaberbrown.com)Thanks to Matthew O'Connor, Yana Chernobilsky, and John Hoffman who are supporting us on Patreon at the Identity level!And thanks to Nicholas Rose, Jason Hise, Thomas Scheer, Marting Sergio H. Faester, CSS, and Mauricio Pacheco who are supporting us at the Lemma level!
Topology vs a Topology | Infinite Series PBS Infinite Series 2017-12-21 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiWhat exactly is a topological space? Learn through active problem-solving at Brilliant: brilliant.org/InfiniteSeries Tweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] comPrevious EpisodeThis Video was Not Encrypted with RSA | Infinite Seriesyoutube.com/watch?v=4Tb1q8dSIlIWe’ve talked about topology here on Infinite Series. It’s the branch of math where we study properties of shapes that are preserved no matter how you bend, twist, stretch or deform them. And you’ve probably come across some cool examples of these shapes - or topological spaces - like spheres and tori, Mobius bands and Klein bottles. In this episode we discuss the the 3 axioms that underlie all of topology. If you want to dive deeper after watching this episode check out the links below:References:Topology Via Logic by S. Vickersbooks.google.com/books/about/Topology_Via_Logic.html?id=peXBNAEACAAJTopology by Munkresamazon.com/Topology-2nd-James-Munkres/dp/0131816292For the solution to the challenge question check out:http://www.math3ma.com/mathema/2017/12/21/topology-vs-a-topology Written and Hosted by Tai-Dinae BradleyProduced by Rusty WardGraphics by Ray LuxAssistant Editing and Sound Design by Meah Denee Barrington and Emma DessauMade by Kornhaber Brown (www.kornhaberbrown.com)Thanks to Matthew O'Connor, Yana Chernobilsky and John Hofmann 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!
This Video was Not Encrypted with RSA | Infinite Series PBS Infinite Series 2017-12-15 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiLearn through active problem-solving at Brilliant: brilliant.org/InfiniteSeriesLast episode we discussed Symmetric cryptography youtube.com/watch?v=NOs34_-eREk Here we break down Asymmetric crypto and more. Tweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] comPrevious Episode(Almost) Unbreakable Crypto | Infinite Seriesyoutube.com/watch?v=NOs34_-eREkHow To Break Cryptographyyoutube.com/watch?v=12Q3Mrh03Gk&list=PLa6IE8XPP_gnot4uwqn7BeRJoZcaEsG1D&index=2Last time, we discussed symmetric encryption protocols, which rely on a user-supplied number called "the key" to drive an algorithm that scrambles messages. Since anything encrypted with a given key can only be decrypted with the same key, Alice and Bob can exchange secure messages once they agree on a key. But what if Alice and Bob are strangers who can only communicate over a channel monitored by eavesdroppers like Eve? How do they agree on a secret key in the first place?Written and Hosted by Gabe Perez-GizProduced by Rusty WardGraphics by Ray LuxAssistant Editing and Sound Design by Mike Petrow and 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!
(Almost) Unbreakable Crypto | Infinite Series PBS Infinite Series 2017-12-12 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiDespite what many believe, the essence of encryption isn’t really about factoring or prime numbers. So what is it about? Thanks to Vanessa Hill for playing the part of our evil hacker! Be sure to check out Braincraft youtube.com/channel/UCt_t6FwNsqr3WWoL6dFqG9wTweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] comPrevious EpisodeAssociahedra: The Shapes of Multiplication | Infinite Seriesyoutube.com/watch?v=N7wNWQ4aTLQIn previous episodes, Kelsey explained how you could crack RSA encryption with an algorithm capable of quickly factoring ginormous numbers into primes. That might give you the impression that fast factoring algorithms would compromise all digital encryption. But not so -- for instance, YouTube's encryption of this video would be unaffected. And that's because the essence of encryption isn’t really about factoring or prime numbers per se. So what is it about? Written and Hosted by Gabe Perez-GizProduced by Rusty WardGraphics by Ray LuxAssistant Editing and Sound Design by Mike Petrow and 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!
Associahedra: The Shapes of Multiplication | Infinite Series PBS Infinite Series 2017-11-30 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiWhat 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 youtube.com/watch?v=H4I2C3Ts7_wTweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] comAnd discuss the episode further over on reddit at reddit.com/r/PBSInfiniteSeriesPrevious EpisodeThe Multiplication Multiverse | Infinite Seriesyoutube.com/watch?v=H4I2C3Ts7_wIn 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 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-1http://www.math3ma.com/mathema/2017/10/30/what-is-an-operad-part-2 Some applications in math and physics: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:amazon.com/Catalan-Numbers-Richard-P-Stanley/dp/1107427746 Written and Hosted by Tai-Danae BradleyProduced by Rusty WardGraphics by Ray LuxAssistant Editing and Sound Design by Meah Denee BarringtonMade 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!
The Multiplication Multiverse | Infinite Series PBS Infinite Series 2017-11-23 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiWhat happens if you multiply things that aren’t numbers? And what happens if that multiplication is not associative?Tweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] comPrevious EpisodeThe Heat Equation + Special Announcement | Infinite Seriesyoutube.com/watch?v=NHucpzbD600Multiplication of numbers is an associative property and we can make sense of “multiplication” between things that aren’t numbers but that’s not considered as associativity. And since we’re talking about associativity, you might wonder about that other property of real numbers: You know: when multiplying two numbers, swapping their order doesn’t change the answer. This property is called commutativity. But keep in mind: it’s a very special property to have! Not everything in life is commutative. For example, getting dressed in the morning... putting on your socks and then your shoes is NOT the same as first putting on your shoes and then your socks. Link to Resources:The Fundamental Group: ‘Loop concatenation’http://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/Landau.pdfWritten and Hosted by Tai-Danae BradleyProduced by Rusty WardGraphics by Ray LuxAssistant Editing and Sound Design by Mike PetrowMade 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!
The Heat Equation + Special Announcement! | Infinite Series PBS Infinite Series 2017-11-17 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiWhat is the heat equation? And find out who the two new hosts of Infinite Series are!Tweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] comPrevious EpisodeHilbert's 15th Problem: Schubert Calculus | Infinite Seriesyoutube.com/watch?v=U8sq3BplCfIImagine a giant -- endless, infinite -- sheet of metal. For mathematical convenience, let’s overlay a grid. Now, imagine you light a very tiny and very powerful match directly underneath the origin, the point (0,0). This super special match instantaneously creates infinite heat at the origin, and zero heat everywhere else. Immediately remove the match and watch the heat diffuse along the surface, radiating out from the origin. Links to Resources:Distribution of Heat Graphyoutube.com/watch?v=twBcpxrWm5ENormal Distribution Graphhttp://www.statisticshowto.com/wp-content/uploads/2013/09/normal-distribution-probability.jpg3D Plot Heat Graphmathworks.com/help/examples/matlab/win64/ThreeDPlotsGSExample_01.pngWritten and Hosted by Kelsey Houston-EdwardsProduced by Rusty WardGraphics by Ray LuxAssistant Editing and Sound Design by Mike PetrowMade 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!
Hilberts 15th Problem: Schubert Calculus | Infinite Series PBS Infinite Series 2017-11-10 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiGet 2 months of Curiosity Stream free by going to www.curiositystream.com/infinite and signing up with the promo code "infinite." It's said that Hermann Schubert performed the mathematical equivalent of "landing a jumbo jet blindfolded." Find out why. Tweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] comPrevious EpisodeCrisis in the Foundation of Mathematics | Infinite Seriesyoutube.com/watch?v=KTUVdXI2vngIn the late 1800s, a mathematician, Hermann Schubert, computed all sorts of wild enumerative geometry problems, like the number of twisted cubics tangent to 12 quadrics -- which is apparently 5,819,539,783,680. And maybe that exact number doesn’t seem particularly important -- but the fact that Schubert was able to figure it out it is pretty amazing. Schubert Calculus, Kleiman and Laksov -- jstor.org/tc/accept?origin=/stable/pdf/2317421.pdf3264 and All That -- David Eisenbud and Joe HarrisThe Honeycomb Model of GLn(C) Tensor Products II -- Knutson, Tao, Woodward -- THE HONEYCOMB MODEL OF GLn(C) TENSOR PRODUCTS II:Written and Hosted by Kelsey Houston-EdwardsProduced by Rusty WardGraphics by Ray LuxAssistant Editing and Sound Design by Mike PetrowMade by Kornhaber Brown (www.kornhaberbrown.com)Special Thanks to Allen Knutson and Balázs ElekThanks 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!
Crisis in the Foundation of Mathematics | Infinite Series PBS Infinite Series 2017-10-19 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiWhat if the foundation that all of mathematics is built upon isn't as firm as we thought it was?Note: The natural numbers sometimes include zero and sometimes don't -- it depends on how you define it. Within logic, zero is always included as a natural number.Correction - The image shown at 8:15 is of Netwon's Principia and not Russell and Whitehead's Principia Mathematica as was intended.Tweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] comPrevious EpisodeHow to generate Pseudorandom Numbers? | Infinite Seriesyoutube.com/watch?v=C82JyCmtKWgMathematics is cumulative -- it builds on itself. That’s part of why you take math courses in a fairly prescribed order. To learn about matrices - big blocks of numbers - and the procedure for multiplying matrices, you need to know about numbers. Matrices are defined in terms of - in other words, constructed from - more fundamental objects: numbers.References::Probability website mentioned in comments: http://people.ischool.berkeley.edu/~nick/aaronson-oracle/index.html https://plato.stanford.edu/entries/philosophy-mathematics/https://plato.stanford.edu/entries/logicism/Ernst Snapper :: maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/1980/0025570x.di021111.02p0048m.pdfPhilosophy of mathematics (Selected readings) edited by Paul Benacerraf and Hilary PutnamWritten and Hosted by Kelsey Houston-EdwardsProduced by Rusty WardGraphics by Ray LuxAssistant Editing and Sound Design by Mike PetrowMade 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!
How to Generate Pseudorandom Numbers | Infinite Series PBS Infinite Series 2017-10-13 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiWhat is a the difference between a random and a pseudorandom number? And what can pseudo random numbers allow us to do that random numbers can't?Tweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] comPrevious EpisodeHow many Cops to catch a Robber? | Infinite Seriesyoutube.com/watch?v=fXvN-pF76-EComputers need to have access to random numbers. They’re used to encrypt information, deal cards in your game of virtual solitaire, simulate unknown variables -- like in weather prediction and airplane scheduling, and so much more. But How can a computer possibly produce a random number?Written and Hosted by Kelsey Houston-EdwardsProduced by Rusty WardGraphics by Ray LuxAssistant Editing and Sound Design by Mike PetrowMade by Kornhaber Brown (www.kornhaberbrown.com)Special Thanks to Alex TownsendBig 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!
How Many Cops to Catch a Robber? | Infinite Series PBS Infinite Series 2017-10-06 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiLast episode, we used graph theory to figure out how a cop could catch a robber. But what happens when we introduce multiple cops? What happens if you have "lazy" cops or "drunk" robbers?Tweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] comPrevious EpisodeThe Cops and Robbers Theorem | Infinite Seriesyoutube.com/watch?v=9mJEu-j1KT0Cops and Robbers is played on a finite and connected graph - meaning that any two vertices are joined by a path of edges. The game begins by placing a cop and a robber each on a single vertex; we say it “occupies” that vertex. They alternate moving along the edges, from a vertex to neighboring vertex. Or, on any given turn, the player can choose to not move -- to stay where they are. We’ll assume that the cop always goes first. If, eventually, the cop lands on the robber’s vertex, the game is over -- we say that the game is a “win” for the cop. But, if the robber can avoid the cop indefinitely, we say that the game is a win for the robber. Written and Hosted by Kelsey Houston-EdwardsProduced by Rusty WardGraphics by Ray LuxAssistant Editing and Sound Design by Mike PetrowMade by Kornhaber Brown (www.kornhaberbrown.com)Resources:M. Aigner and M. Fromme -- A Game of Cops and Robbers:https://www.math.ucdavis.edu/~erikslivken/classes/2016_spring_180/aigner%20fromme.pdfWhat is Cop Number? -Anthony Bonatohttp://www.math.ryerson.ca/~abonato/papers/whatis_copnumber_new.pdfThe Game of Cops and Robbers on Graph - Anthony Bonato and Richard NowakowskiAnthony Bonato -- "What is... Cops and Robbers" http://www.ams.org/notices/201208/rtx120801100p.pdfSpecial Thanks to Anthony Bonato and Brendan SullivanBig 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!
The Cops and Robbers Theorem | Infinite Series PBS Infinite Series 2017-09-28 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiCan a cop catch a robber? There's some surprising and compelling graph theory that go into answering that question.Tweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] comPrevious EpisodeHigher Dimensional Tic Tac Toe | Infinite Seriesyoutube.com/watch?v=FwJZa-helig&t=17sCops and Robbers is played on a finite and connected graph - meaning that any two vertices are joined by a path of edges. The game begins by placing a cop and a robber each on a single vertex; we say it “occupies” that vertex. They alternate moving along the edges, from a vertex to neighboring vertex. Or, on any given turn, the player can choose to not move -- to stay where they are. We’ll assume that the cop always goes first. If, eventually, the cop lands on the robber’s vertex, the game is over -- we say that the game is a “win” for the cop. But, if the robber can avoid the cop indefinitely, we say that the game is a win for the robber. Written and Hosted by Kelsey Houston-EdwardsProduced by Rusty WardGraphics by Ray LuxAssistant Editing and Sound Design by Mike PetrowMade by Kornhaber Brown (www.kornhaberbrown.com)Resources:M. Aigner and M. Fromme -- A Game of Cops and Robbers:https://www.math.ucdavis.edu/~erikslivken/classes/2016_spring_180/aigner%20fromme.pdfThe Game of Cops and Robbers on Graphs - Anthony Bonato and Richard NowakowskiAnthony Bonato -- "What is... Cops and Robbers" http://www.ams.org/notices/201208/rtx120801100p.pdfBrendan W. Sullivan, Nikolas Townsend, Mikayla Werzanski - “An Introduction to Lazy Cops and Robbers on Graphs,” to appear in College Mathematics Journal in 2017Brendan W. Sullivan, Nikolas Townsend, Mikayla Werzanski "The 3x3 rooks graph is the unique smallest graph with lazy cop number 3" -- arxiv.org/abs/1606.08485Special Thanks to Anthony Bonato and Brendan SullivanThanks 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!
Higher-Dimensional Tic-Tac-Toe | Infinite Series PBS Infinite Series 2017-09-21 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiRegular tic-tac-toe can get a bit boring -- if both players are playing optimally, it always ends in a draw. But what happens if you increase the width of the board? Or increase the dimension of the board? Or increase both?Tweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] comPrevious EpisodeHow the Axiom of Choice Gives Sizeless Sets | Infinite Seriesyoutube.com/watch?v=hcRZadc5KpIThe standard game of tic-tac-toe is too easy. How can we, as mathematicians, play with the combinatorics of tic-tac-toe? There are (at least) three easy ways to modify the game of tic-tac-toe: increase the width of the board - like *this* 5x5 board - increase the dimension of the board - like *this* 3x3x3 board - or increase both, like this 4x4x4 board.Challenge Winner of the How the Axiom of Choice Gives Sizeless Sets:For Your MathWritten and Hosted by Kelsey Houston-EdwardsProduced by Rusty WardGraphics by Ray LuxAssistant Editing and Sound Design by Mike PetrowMade by Kornhaber Brown (www.kornhaberbrown.com)Resources:Hales/Jewett paper: http://www.ams.org/journals/tran/1963-106-02/S0002-9947-1963-0143712-1/S0002-9947-1963-0143712-1.pdfGolomb/Hales paper: http://library.msri.org/books/Book42/files/golomb.pdfyoutube.com/watch?v=p1YzYLzRwtkhttp://www.austms.org.au/Gazette/2005/Jul05/mathellaneous.pdfSpecial Thanks: Benjamin Houston-Edwards and Nathan KaplanMathologer Video:youtube.com/watch?v=aDOP0XynAzASpecial thanks to Matthew O'Connor and Yana Chernobilsky who are supporting us on Patreon at the Identity level!And thanks to Mauricio Pacheco and Nicholas Rose who are supporting us on Patreon at the Lemma level!
How the Axiom of Choice Gives Sizeless Sets | Infinite Series PBS Infinite Series 2017-09-14 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiDoes every set - or collection of numbers - have a size: a length or a width? In other words, is it possible for a set to be sizeless? This in an updated version of our September 8th video. We found an error in our previous video and corrected it within this version. Tweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] comPrevious EpisodesYour Brain as Math - Part 1youtube.com/watch?v=M0M3srBoTkYSimplicial Complexes - Your Brain as Math Part 2youtube.com/watch?v=rlI1KOo1gp4Your Mind Is Eight-Dimensional - Your Brain as Math Part 3youtube.com/watch?v=akgU8nRNIp0In this episode, we look at creating sizeless sets which we call size the Lebesgue measure - it formalizes the notion of length in one dimension, area in two dimensions and volume in three dimensions.Written and Hosted by Kelsey Houston-EdwardsProduced by Rusty WardGraphics by Ray LuxAssistant Editing and Sound Design by Mike PetrowMade by Kornhaber Brown (www.kornhaberbrown.com)Resources:https://math.vanderbilt.edu/schectex/ccc/choice.htmlhttps://plato.stanford.edu/entries/axiom-choice/http://www.math.kth.se/matstat/gru/godis/nonmeas.pdfVsauceyoutube.com/watch?v=s86-Z-CbaHASpecial Thanks: Lian Smythe and James BarnesThanks to Mauricio Pacheco and Nicholas Rose who are supporting us at the Lemma level on Patreon!And thanks to Matthew O'Connor and Yana Chernobilsky who are supporting us at the Theorem Level on Patreon!
Your Mind Is Eight-Dimensional - Your Brain as Math Part 3 | Infinite Series PBS Infinite Series 2017-08-24 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiHow Algebraic topology can help us unlock the connection between neurological structure and function. This is Part 3 in our Your Brain as Math mini-series. Check out Part 1 here: youtube.com/watch?v=M0M3srBoTkY Check out Part 2 here: youtube.com/watch?v=rlI1KOo1gp4Tweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] comPrevious EpisodeSimplicial Complexes - Your Brain as Math - Part 2youtube.com/watch?v=rlI1KOo1gp4Last episode, we learned that your brain can be modeled as a simplicial complex. And algebraic topology can tell us the Betti numbers of that simplicial complex. Why is that helpful? Let’s find out.Written and Hosted by Kelsey Houston-EdwardsProduced by Rusty WardGraphics by Ray LuxAssistant Editing and Sound Design by Mike PetrowMade by Kornhaber Brown (www.kornhaberbrown.com)Resources:Cliques of Neurons Bound into Cavities Provide a Missing Link between Structure and Functionhttp://journal.frontiersin.org/article/10.3389/fncom.2017.00048/fullThe Blue Brain Project: http://bluebrain.epfl.ch/Barcodes: The Persistent Topology of Datahttps://www.math.upenn.edu/~ghrist/preprints/barcodes.pdfNetwork Neurosciencehttp://www.nature.com/neuro/journal/v20/n3/full/nn.4502.htmlAlgebraic Topologyhttps://www.math.cornell.edu/~hatcher/AT/ATpage.htmlSpecial thanks to Kathryn Hess and Florian Frick!Special thanks to Matthew O'Connor and Yana Chernobilsky who are supporting us on Patreon at the Identity level!And thanks to Mauricio Pacheco who is supporting us at the Lemma level!
Simplicial Complexes - Your Brain as Math Part 2 | Infinite Series PBS Infinite Series 2017-08-23 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiWhat is a Simplicial Complex and how can it help us decode the brain’s neurological structure? This is Part 2 in our Your Brain as Math mini-series. Check out Part 1 here: youtube.com/watch?v=M0M3srBoTkYCheck out Part 3 here: youtube.com/watch?v=akgU8nRNIp0Tweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] comPrevious EpisodeYour Brain as Math - Part 1youtu.be/M0M3srBoTkYNetwork Mathematics and Rival Factionsyoutube.com/watch?v=qEKNFOaGQccLast episode we saw that your neural network can be modeled as a graph, which -- we’ll show in this episode -- can be viewed as a higher-dimensional simplicial complex. So… what is a simplicial complex??Written and Hosted by Kelsey Houston-EdwardsProduced by Rusty WardGraphics by Ray LuxAssistant Editing and Sound Design by Mike PetrowMade by Kornhaber Brown (www.kornhaberbrown.com)Resources:Cliques of Neurons Bound into Cavities Provide a Missing Link between Structure and Functionhttp://journal.frontiersin.org/article/10.3389/fncom.2017.00048/fullThe Blue Brain Project: http://bluebrain.epfl.ch/Barcodes: The Persistent Topology of Datahttps://www.math.upenn.edu/~ghrist/preprints/barcodes.pdfNetwork Neurosciencehttp://www.nature.com/neuro/journal/v20/n3/full/nn.4502.htmlAlgebraic Topologyhttps://www.math.cornell.edu/~hatcher/AT/ATpage.htmlSpecial thanks to Kathryn Hess and Florian Frick!Special thanks to Matthew O'Connor and Yana Chernobilsky who are supporting us on Patreon at the Identity level!And thanks to Mauricio Pacheco who is supporting us at the Lemma level!
Your Brain as Math - Part 1 | Infinite Series PBS Infinite Series 2017-08-22 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiWhat does your brain look like when it's broken down mathematically? And what can this tell us? This is Part 1 in our Your Brain as Math mini-series. Check out Part 2 here: youtube.com/watch?v=rlI1KOo1gp4Check out Part 3 here: youtube.com/watch?v=akgU8nRNIp0Tweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] comPrevious EpisodeStochastic Supertasksyoutube.com/watch?v=Sdp_V0L99swIn order to dive deeper into an exciting topic, we're mixing up the format. Over the next three days, we’ll spend the next three episodes exploring an incredible application of seemingly purely-abstract mathematics: how algebraic topology can help us decode the connections among neurons in our brains, to help us understand their function.Written and Hosted by Kelsey Houston-EdwardsProduced by Rusty WardGraphics by Ray LuxAssistant Editing and Sound Design by Mike PetrowMade by Kornhaber Brown (www.kornhaberbrown.com)Resources:Cliques of Neurons Bound into Cavities Provide a Missing Link between Structure and Functionhttp://journal.frontiersin.org/article/10.3389/fncom.2017.00048/fullThe Blue Brain Project http://bluebrain.epfl.ch/Barcodes: The Persistent Topology of Datahttps://www.math.upenn.edu/~ghrist/preprints/barcodes.pdfNetwork Neurosciencehttp://www.nature.com/neuro/journal/v20/n3/full/nn.4502.htmlAlgebraic Topologyhttps://www.math.cornell.edu/~hatcher/AT/ATpage.htmlSpecial thanks to Kathryn Hess and Florian Frick!Comments answered by Kelsey:Vriskanon youtube.com/watch?v=Sdp_V0L99sw&lc=z22ic1ehnlqhi3g4g04t1aokgo4wjichzrqe5sqovhw4rk0h00410Challenge Winner : Florence BSpecial thanks to Matthew O'Connor and Yana Chernobilsky who are supporting us on Patreon at the Identity level!And thanks to Mauricio Pacheco who is supporting us at the Lemma level!
Stochastic Supertasks | Infinite Series PBS Infinite Series 2017-08-10 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiSupertasks allow you to accomplish an infinite number of tasks in a finite amount of time. Find out how these paradoxical feats get even stranger once randomness is introduced. Tweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] com Previous EpisodeThe Honeybcomcs of 4-Dimensional Beesyoutube.com/watch?v=X8jOxEGVyPo Vsauce - Supertasksyoutube.com/watch?v=ffUnNaQTfZE Written and Hosted by Kelsey Houston-EdwardsProduced by Rusty WardGraphics by Ray LuxAssistant Editing and Sound Design by Mike PetrowMade by Kornhaber Brown (www.kornhaberbrown.com) Resources:http://julio.staff.ipb.ac.id/files/2015/02/Ross_8th_ed_English.pdf -- Pg 46Special Thanks to John Pike What happens when you try to empty an urn full of infinite balls? It turns out that whether the vase is empty or full at the end of an infinite amount of time depends on what order you try to empty it in. Check out what happens when randomness and the Ross-Littlewood Paradox collide. Comments answered by Kelsey: Lowtechyoutube.com/watch?v=X8jOxEGVyPo&lc=z235h5yquwukfhh2nacdp431mqnc3ul0fvs4cy1tqpdw03c010c Daniel Shapiroyoutube.com/watch?v=X8jOxEGVyPo&lc=z235h5yquwukfhh2nacdp431mqnc3ul0fvs4cy1tqpdw03c010c.1501799589673862 Professor Politicsyoutube.com/watch?v=X8jOxEGVyPo&lc=z23xhrzyrzmxeraluacdp43aqfqbpbddfp3g4myyjclw03c010c Alex 24757youtube.com/watch?v=X8jOxEGVyPo&lc=z23fvxw5zmq1vzvxa04t1aokgodujwhjcmcr4pmcbu5lbk0h00410
The Honeycombs of 4-Dimensional Bees ft. Joe Hanson | Infinite Series PBS Infinite Series 2017-08-03 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiBe sure to check out It's OK to be Smart's video on nature's love of hexagons youtu.be/Pypd_yKGYpA And try CuriosityStream today: http://curiositystream.com/infinite Use the promo code: infiniteThe image of the 3D honeycomb sheet used at 7:33 and within the thumbnail image is a recolored/modified version of Andrew Kepert's "Tesselation of space using truncated octahedra." commons.wikimedia.org/wiki/File:Truncated_octahedra.jpg The original of this image is used again at 8:33 and 9:29.The images of the Weaire-Phelan Structure, the truncated Hexagonal Trapezohedron and the Pyritohedron at 9:14 were created by Tomruen, links below:commons.wikimedia.org/w/index.php?curid=10471229 commons.wikimedia.org/w/index.php?curid=17024143en.wikipedia.org/wiki/Weaire%E2%80%93Phelan_structure#/media/File:Irregular_dodecahedron.pngTweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] comPrevious EpisodeWhy Computers are Bad at Algebrayoutube.com/watch?v=pQs_wx8eoQ8Why is there a hexagonal structure in honeycombs? Why not squares? Or asymmetrical blobby shapes? In 36 B.C., the Roman scholar Marcus Terentius Varro wrote about two of the leading theories of the day. First: bees have six legs, so they must obviously prefer six-sided shapes. But that charming piece of numerology did not fool the geometers of day. They provided a second theory: Hexagons are the most efficient shape. Bees use wax to build the honeycombs -- and producing that wax expends bee energy. The ideal honeycomb structure is one that minimizes the amount of wax needed, while maximizing storage -- and the hexagonal structure does this best.Written and Hosted by Kelsey Houston-EdwardsProduced by Rusty WardGraphics by Ray LuxAssistant Editing and Sound Design by Mike PetrowMade by Kornhaber Brown (www.kornhaberbrown.com)Resources:Nature paper http://www.nature.com/news/how-honeycombs-can-build-themselves-1.13398#/b1Hales’ proof of honeycomb conjecture: arxiv.org/pdf/math/9906042.pdfOlder article on honeycomb conjecture http://www.ams.org/journals/bull/1964-70-04/S0002-9904-1964-11155-1/S0002-9904-1964-11155-1.pdfOverview of proof of honeycomb conjecture http://www.maa.org/frank-morgans-math-chat-hales-proves-hexagonal-honeycomb-conjecturehttp://www.npr.org/sections/krulwich/2013/05/13/183704091/what-is-it-about-bees-and-hexagonsKelvin -- http://soft-matter.seas.harvard.edu/images/1/17/Kelvin_Cell.pdfhttp://www.slate.com/articles/health_and_science/science/2015/07/hexagons_are_the_most_scientifically_efficient_packing_shape_as_bee_honeycomb.html
Why Computers are Bad at Algebra | Infinite Series PBS Infinite Series 2017-07-21 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiThe answer lies in the weirdness of floating-point numbers and the computer's perception of a number line. Tweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] com Previous EpisodeMaking Probability Mathematicalyoutube.com/watch?v=-6HxjiW_KwA Written and Hosted by Kelsey Houston-EdwardsProduced by Rusty WardGraphics by Ray LuxAssistant Editing and Sound Design by Mike PetrowMade by Kornhaber Brown (www.kornhaberbrown.com) Resources:Random ASCIIrandomascii.wordpress.com/2012/02/25/comparing-floating-point-numbers-2012-edition Special thanks to Professor Alex Townsend In 1994, Intel recalled - to the tune of $475 million - an early model of their Pentium processor after they discovered it was making arithmetic errors. Arithmetic mistakes - like the one Intel’s Pentium processors were making - are often rooted in computer’s unusual version of the real number line. Comments answered by Kelsey: Neroox05youtube.com/watch?v=-6HxjiW_KwA&lc=z132ehrixviktnumh23fgzmgfkv1dz0c1 Joshua Sherwinyoutube.com/watch?v=-6HxjiW_KwA&lc=z13xs5mplpn4d5dgc04cilyxbqr1ifxwd24 Joshua Hillerupyoutube.com/watch?v=-6HxjiW_KwA&lc=z12wc1iipuzsirj2h22ucfv5burmibno304
Making Probability Mathematical | Infinite Series PBS Infinite Series 2017-07-13 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiWhat happened when a gambler asked for help from a mathematician? The formal study of Probability. Go to http://squarespace.com/infiniteseries and use code “INFINITE” for 10% off your first order. Find out the players probability of winning based on their current score (Link referenced at 2:24):http://mathforum.org/isaac/problems/prob1.htmlTweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] comPrevious Episodeyoutube.com/watch?v=qEKNFOaGQccWritten and Hosted by Kelsey Houston-EdwardsProduced by Rusty WardGraphics by Ray LuxAssistant Editing and Sound Design by Mike PetrowMade by Kornhaber Brown (www.kornhaberbrown.com)Resources and Special thanks:terrytao.wordpress.com/2010/01/01/254a-notes-0-a-review-of-probability-theoryKolmogorov - Foundations of the Theory of ProbabilityIan Hacking - The Emergence of ProbabilityThroughout much of human history, people consciously and intentionally produced randomness. They frequently used dice - or dice-shaped animal bones and other random objects - to gamble, for entertainment, predict the future and communicate with deities. Despite all this engagement with controlled random processes, people didn’t really think of probability in mathematical terms prior to 1600. All of the ingredients were there -- people had rigorous theories of geometry and algebra, and the ability to rig a game of dice would have certainly provided an incentive to study probability -- but, there’s very little evidence that they thought about randomness in mathematical terms.Challenge Winner:Zutacayoutube.com/watch?v=qEKNFOaGQcc&lc=z13ky5eruxbiermgx04cdx1ztyjlxzfyavc0kComments answered by Kelsey:Ja-Shwa Cardellyoutube.com/watch?v=qEKNFOaGQcc&lc=z12yhrgjdqm5wrdbv04cejno4t2icnmpy1c
Network Mathematics and Rival Factions | Infinite Series PBS Infinite Series 2017-06-29 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiThe theory of social networks allows us to mathematically model and analyze the relationships between governments, organizations and even the rival factions warring on Game of Thrones. Tweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] comPrevious EpisodeArrow’s Impossibility Theoremyoutube.com/watch?v=AhVR7gFMKNgWritten and Hosted by Kelsey Houston-EdwardsProduced by Rusty WardGraphics by Ray LuxMade by Kornhaber Brown (www.kornhaberbrown.com)Resources and Special thanks:Network, Crowds and Markets, by David Easley and John Kleinberg :: https://www.cs.cornell.edu/home/kleinber/networks-book/Cartwright and Harary :: http://snap.stanford.edu/class/cs224w-readings/cartwright56balance.pdfAntal, Krapivsky, and Redner :: http://physics.bu.edu/~redner/pubs/pdf/dresden.pdfSteven Strogatz Lecture :: youtube.com/watch?v=P60sWUxluykSpecial Thanks: Steven StrogatzCommonly, in the field of social network analysis, one uses a graph - also called a network - where the vertices, or nodes, represent individuals and the edges represent something about the relationships or interactions between individuals. These networks might represent Facebook friendships, or help us understand the spread of disease.This episode focuses on one model of a social network that encodes whether relationships are positive or negative -- in other words, if they’re friendly or hostile -- and the notion of structural balance. Challenge Winners:Cantor’s Catyoutube.com/watch?v=AhVR7gFMKNg&lc=z13lyxtxmom1tv51y23tzluompyzg5xkwDavid de Kloetyoutube.com/watch?v=AhVR7gFMKNg&lc=z13lyxtxmom1tv51y23tzluompyzg5xkw.1498299129602777Comments answered by Kelsey:Edelopoyoutube.com/watch?v=AhVR7gFMKNg&lc=z12ftje5aybuxhro204chdn4tuqovxwjapk0k
Arrows Impossibility Theorem | Infinite Series PBS Infinite Series 2017-06-22 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiThe bizarre Arrow’s Impossibility Theorem, or Arrow’s Paradox, shows a counterintuitive relationship between fair voting procedures and dictatorships. Start your free trial with Squarespace at http://squarespace.com/infiniteseries and enter offer code “infinite” to get 10% off your first purchase. Tweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] com Previous EpisodeVoting Systems and the Condorcet Criterionyoutube.com/watch?v=HoAnYQZrNrQ Written and Hosted by Kelsey Houston-EdwardsProduced by Rusty WardGraphics by Ray LuxMade by Kornhaber Brown (www.kornhaberbrown.com) Additional ResourcesNetworks, Crowds and Markets:: https://www.cs.cornell.edu/home/kleinber/networks-book/Original Paper by Kenneth Arrow:: web.archive.org/web/20110720090207/http://gatton.uky.edu/Faculty/hoytw/751/articles/arrow.pdfDifferent voting systems can produce radically different election results, so it’s important to ensure the voting system we’re using has certain properties - that it fairly represents the opinions of the electorates. The impressively counterintuitive Arrow’s Impossibility Theorem demonstrates that this is much harder than you might think.Thanks: Ben Houston-Edwards and Iian Smythe Comments answered by Kelsey: Johan Richteryoutube.com/watch?v=HoAnYQZrNrQ&lc=z12bt1nabyievh4yg04chlvpdnisxnw5rx00k Nat Tuckyoutube.com/watch?v=HoAnYQZrNrQ&lc=z12tetkx1wzocn2ue23wzdfg5sn2dhhh004
Voting Systems and the Condorcet Paradox | Infinite Series PBS Infinite Series 2017-06-15 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiWhat is the best voting system? Voting seems relatively straightforward, yet four of the most widely used voting systems can produce four completely different winners. Get 10% off a custom domain and email address by going to hover.com/InfiniteSeries *Correction: The ballots at 1:20 were labeled incorrectly. At 1:20 the top ballot should read 1 Green, 2 Blue and 3 Purple and the bottom ballot should read 2 Green, 3 Blue and 1 Purple. Thank you to Hoarder who first noted this.*Correction: What's stated is the converse of the Condorcet Criterion. Oops - Stating conditionals can be tricky! For more details, see: reddit.com/r/math/comments/6hh9sb/voting_systems_and_the_condorcet_paradox_infinite/diyft53Tweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] com Previous EpisodePantographs and the youtube.com/watch?v=KYaCtHPCARc Written and Hosted by Kelsey Houston-EdwardsProduced by Rusty WardGraphics by Ray LuxMade by Kornhaber Brown (www.kornhaberbrown.com) With access to a complete set of ranked ballots - which means we know every person’s opinions - it seems like a clear winner should emerge. But it doesn’t. The outcome of the election depends critically on what process you use to convert all those individual’s preferences into a group preference. Further Resources: Voting and Election Decision Methodshttp://www.ams.org/samplings/feature-column/fcarc-voting-decision The Mathematics of Votinghttps://www.math.ku.edu/~jmartin/courses/math105-F11/Lectures/chapter1-part1.pdf The Mathematics of Voting, Power and Sharinghttp://web.math.princeton.edu/math_alive/6/Notes1.pdf CGP Grey Voting Playlistyoutube.com/playlist?list=PLej2SlXPEd37YwwEY7mm0WyZ8cfB1TxXa Comments answered by Kelsey: FossilFighters101youtube.com/watch?v=XOzhF3QoTCA&lc=z12fi1jgnle5wre0k22puzfojxnbzjirk04 Abi Gailyoutube.com/watch?v=XOzhF3QoTCA&lc=z13kin5rcqatih1i004cjdfwsofhi1hopgo Lucas Hoffsesyoutube.com/watch?v=XOzhF3QoTCA&lc=z13nilewrtbmfp24t04cgdlziqafvzbahkk0k
Pantographs and the Geometry of Complex Functions | Infinite Series PBS Infinite Series 2017-06-08 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiWhen we mathematically abstract a 400 year old drawing tool called the Pantograph, we find that they have an incredible relationship with complex functions. Start your free trial with Squarespace at http://squarespace.com/infiniteseries and enter offer code “infinite” to get 10% off your first purchase. Tweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] com Previous EpisodeDissecting Hypercubes with Pascal’s Triangleyoutube.com/watch?v=KYaCtHPCARc Written and Hosted by Kelsey Houston-EdwardsProduced by Rusty WardGraphics by Ray LuxMade by Kornhaber Brown (www.kornhaberbrown.com) Linkages, pantographs and the weird and beautiful geometry of complex functions. Further Resources:http://www.math.cornell.edu/~mec/Winter2009/Marshall/TOC.htmlhttp://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/Mackevicius.pdfhttp://www.math.toronto.edu/~drorbn/People/Eldar/thesis/default.htmhttps://www.math.ucdavis.edu/~kapovich/EPR/link.pdf Giant Drawing Machines by Nick Sayerbehance.net/gallery/18833343/Giant-Drawing-Machines Pantographo Legal’s Cartoon Sketchesyoutube.com/watch?v=Muic3yyhSv8 Thanks to Benjamin Houston-Edwards and Andrew MarshallComments answered by Kelsey: NotaWalrusyoutube.com/watch?v=KYaCtHPCARc&lc=z12nit1j0za0wfqul22cin3qspjmi5hdp Joel Nordstromyoutube.com/watch?v=KYaCtHPCARc&lc=z12yhltz4sfhzlxa304cdtbz0vnsupmpahk Rob Nicolaidesyoutube.com/watch?v=KYaCtHPCARc&lc=z12yhltz4sfhzlxa304cdtbz0vnsupmpahk
Dissecting Hypercubes with Pascals Triangle | Infinite Series PBS Infinite Series 2017-06-01 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiWhat does the inside of a tesseract look like? Pascal’s Triangle can tell us. Start your 60 day free trial of CuriosityStream by going to curiositystream.com/infinite and using the promo code “infinite.”Tweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] comPrevious EpisodeDevil’s Staircase youtube.com/watch?v=dQXVn7pFsVIWritten and Hosted by Kelsey Houston-EdwardsProduced by Rusty WardGraphics by Ray LuxMade by Kornhaber Brown (www.kornhaberbrown.com)What shape is formed by taking a diagonal slice of a 4-dimensional cube? Or, a 10-dimensional cube? It turns out that a very familiar mathematical object - Pascal’s triangle - can help us answer this question.Further Resources:Cube Slices, Pictoral Triangles and Probabilityhttp://www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/1992/0025570x.di021171.02p0016b.pdfChallenge Winner:Asthmenyoutube.com/watch?v=dQXVn7pFsVI&lc=z132hl040ozfc3q4i04cchci1suscbwg5ikComments answered by Kelsey:Tehomyoutube.com/watch?v=dQXVn7pFsVI&lc=z133jzax0tansbqh504cedgq1mfvs1wjsyo0kBadplayzyoutube.com/watch?v=dQXVn7pFsVI&lc=z12dchu4qtfwshkmv231elxqsnuudjt2y
The Devils Staircase | Infinite Series PBS Infinite Series 2017-05-19 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiFind out why Cantor’s Function is nicknamed the Devil’s Staircase. Try Skillshare at http://skl.sh/Infinite2 And check out the brand new PBS Digital Series Above the Noise youtube.com/channel/UC4K10PNjqgGLKA3lo5V8KdQTweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] comPrevious EpisodeTopology Riddlesyoutube.com/watch?v=H8qwqGjOlSEWritten and Hosted by Kelsey Houston-EdwardsProduced by Rusty WardGraphics by Ray LuxMade by Kornhaber Brown (www.kornhaberbrown.com)Cantor’s Function, also known as the Devil’s Staircase, is exceptionally strange. After mapping out Cantor’s Function, named after Georg Cantor, we find that its derivative disappears almost everywhere.Challenge Winner:Justin WielandHonorable Mentions:Felix BeichlerAndrew WellerComments answered by Kelsey:Thomas Perryyoutube.com/watch?v=H8qwqGjOlSE&lc=z13ncngjhwy5iv5tp23lungjrsbxyup0.1494583661471832Ralph Dratmanyoutube.com/watch?v=H8qwqGjOlSE&lc=z13isv1g0krfg52ds235h1rbvqqtxndq204.1494681742451739
Topology Riddles | Infinite Series PBS Infinite Series 2017-05-11 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiCan you turn your pants inside out without taking your feet off the ground? Start your free trial with Squarespace at http://squarespace.com/infiniteseries and enter offer code “infinite” to get 10% off your first purchase. And check out Vanessa Hill’s documentary Mutant Menu right here! youtube.com/watch?v=NrDM6Ic2xMMAnd special thanks to Danny Haymond Jr aka @slothman86 for for his incredible fan art!Tweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] comPrevious EpisodeBuilding an Infinite Bridgeyoutube.com/watch?v=1_DOzuaBE84The Heart of Mathematicsamazon.com/Heart-Mathematics-invitation-effective-thinking/dp/1931914419Picture Hanging Article:: arxiv.org/pdf/1203.3602.pdfWritten and Hosted by Kelsey Houston-EdwardsProduced by Rusty WardGraphics by Ray LuxMade by Kornhaber Brown (www.kornhaberbrown.com)Find out why Topology is often called Rubber Sheet Geometry and try to solve these counter-intuitive topological puzzles.Challenge Winners:Alan Chang@ChalkfaceComments answered by Kelsey:Louis JXyoutube.com/watch?v=1_DOzuaBE84&lc=z12lu5vx5sepxjemc22mhnualyfajxthv04Erdmannelchenyoutube.com/watch?v=1_DOzuaBE84&lc=z132hrbbhxbpwvieu231wl3ytxmyznsu0
Building an Infinite Bridge | Infinite Series PBS Infinite Series 2017-05-04 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiUsing the harmonic series we can build an infinitely long bridge. It takes a very long time though. A faster method was discovered in 2009.Tweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] comPrevious EpisodeHacking at Quantum Speed with Shor’s Algorithmyoutube.com/watch?v=wUwZZaI5u0cAdditional Resources:Overhang article:: http://www.maa.org/sites/default/files/images/upload_library/22/Robbins/Patterson1.pdfMaximum overhang article:: http://www.maa.org/sites/default/files/pdf/upload_library/22/Robbins/Patterson2.pdfPhysical limitations:: quantamagazine.org/20161117-overhang-insights-puzzlehttp://mathworld.wolfram.com/BookStackingProblem.htmlThere’s actually more than one way to build an infinitely long bridge. The traditional method is shown in many calculus classes but a super-fast method of building this bridge was demonstrated by two mathematicians in 2009.Written and Hosted by Kelsey Houston-EdwardsProduced by Rusty WardGraphics by Ray LuxMade by Kornhaber Brown (www.kornhaberbrown.com)Comments answered by Kelsey:MultiMurmaideryoutube.com/watch?v=wUwZZaI5u0c&lc=z12sergyhwjzv5zhx04cfj5akpqbhnjargcMarhsall Webbyoutube.com/watch?v=wUwZZaI5u0c&lc=z13cwnehjmjgfdpfx23dzjwbczayvh41v10TinsThoughtyoutube.com/watch?v=wUwZZaI5u0c&lc=z13cwnehjmjgfdpfx23dzjwbczayvh41v
Hacking at Quantum Speed with Shors Algorithm | Infinite Series PBS Infinite Series 2017-04-28 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiClassical computers struggle to crack modern encryption. But quantum computers using Shor’s Algorithm make short work of RSA cryptography. Find out how.Tweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] comPrevious EpisodeHow to Break Cryptographyyoutube.com/watch?v=12Q3Mrh03GkThe Mathematics Behind Quantum Computersyoutube.com/watch?v=IrbJYsep45EAdditional Resources:Scott Aaronson's Blog (Great Intro to Shor's Alg.):: http://www.scottaaronson.com/blog/?p=208Shor's Original Paper:: arxiv.org/abs/quant-ph/9508027v2Lectures on Shor's Algorithm:: arxiv.org/pdf/quant-ph/0010034.pdfDecrypting secure messages often involves attempting to find the factors that make up extremely large numbers. This process is too time consuming for classical computers but Shor’s Algorithm shows us how Quantum Computers can greatly expedite the process.Written and Hosted by Kelsey Houston-EdwardsProduced by Rusty WardGraphics by Ray LuxMade by Kornhaber Brown (www.kornhaberbrown.com)Thanks to Spiros Michalakis for helpful discussions and feedback.Comments answered by Kelsey:Neon Bullyoutube.com/watch?v=12Q3Mrh03Gk&lc=z135uxf5cxenutmxj04cc3swkvm4tpcrxikBhargav Ryoutube.com/watch?v=12Q3Mrh03Gk&lc=z13qjjioozbjdrqyz04cevdrtu3ti3y5sq40kBobCyoutube.com/watch?v=12Q3Mrh03Gk&lc=z12pjpzastylzz2qx04cjtc5jrq2y3yhmlk0k
How to Break Cryptography | Infinite Series PBS Infinite Series 2017-04-20 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiOnly 4 steps stand between you and the secrets hidden behind RSA cryptography. Find out how to crack the world’s most commonly used form of encryption.Tweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] comPrevious Episode:Can We Combine pi & e into a Rational Number?youtube.com/watch?v=bG7cCXqcJag&t=25sLinks to other resources:Shor's paper: arxiv.org/abs/quant-ph/9508027v2Lecture on Shor's Algorithm: arxiv.org/pdf/quant-ph/0010034.pdfBlog on Shor's algorithm: http://www.scottaaronson.com/blog/?p=208Video on RSA cryptography: youtube.com/watch?v=wXB-V_Keiu8Another video on RSA cryptography: youtube.com/watch?v=4zahvcJ9glgEuler's Big Idea: en.wikipedia.org/wiki/Euler%27s_theorem (I can find a non-wiki article, but I don't actually use this in the video. It's just where to learn more about the relevant math Euler did.)Written and Hosted by Kelsey Houston-EdwardsProduced by Rusty WardGraphics by Ray LuxMade by Kornhaber Brown (www.kornhaberbrown.com)Challenge Winner - Reddles37youtube.com/watch?v=bG7cCXqcJag&lc=z135cnmgxlbwch1ds233sbzgaojkivaz004Comments answered by Kelsey:Joel David Hamkinsyoutube.com/watch?v=bG7cCXqcJag&lc=z13zdpcwyk2ofhugh04cdh4agsr2whmbsmk0kPCreeper394youtube.com/watch?v=bG7cCXqcJag&lc=z135w324kw21j1qi104cdzvrpoixslmq1jw
Can We Combine pi & e to Make a Rational Number? | Infinite Series PBS Infinite Series 2017-04-13 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiCan you produce a rational number by exchanging infinitely many digits of pi and e?Tweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] comHierarchy of Infinitiesyoutube.com/watch?v=i7c2qz7sO0IPrevious Episode - Solving the Wolverine Problem using Graph Coloring!youtube.com/watch?v=59aljcwo7awLinks to other resources:Original Post from Math Overflowhttp://mathoverflow.net/questions/265310/if-i-exchange-infinitely-many-digits-of-pi-and-e-are-the-two-resulting-numSquare Root of 2 is Irrationalhttp://www.mathsisfun.com/numbers/square-root-2-irrational.htmlWritten and Hosted by Kelsey Houston-EdwardsProduced by Rusty WardGraphics by Ray LuxMade by Kornhaber Brown (www.kornhaberbrown.com)Thanks to Joel David Hampkins and Erin Carmody for helpful comments.Comments answered by Kelsey:Circuitrinosyoutube.com/watch?v=59aljcwo7aw&lc=z13mevgrmlzqh1xzc04chnc4vvegxfhj4x00kBrian Willettyoutube.com/watch?v=59aljcwo7aw&lc=z123tzuj5oiqudan522hdfdxfl2xttl01
Why I Love PBS PBS Infinite Series 2017-04-10 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiHere's how you can get involved: 1. Tell us what you love about public media using #IlovePBS2. Learn more about public media at http://pbs.org/value.3. Find your representative at govtrack.us/congress/members
Solving the Wolverine Problem with Graph Coloring | Infinite Series PBS Infinite Series 2017-04-06 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiAt one time, Wolverine served on four different superhero teams. How did he do it? He may have used graph coloring.Tweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] comGraph coloring may seem simple but the mathematics behind it is surprising difficult and it pops up in a number strange places. Kelsey discusses Sudoku, the Four Color Theorem, the Hadwiger Nelson Problem and how graph coloring can be used to schedule the most effective way to save the planet. Find out how math can defeat the combined might of Thanos, Magneto, Kang, Dormammu, Ultron, Apocalypse and Dr. Doom.Correction: At 3:58, "Four is the maximum number of required colors"Previous Episode - What is a Random Walk?youtube.com/watch?v=stgYW6M5o4kLinks to other resources:Sudoku: http://www.ams.org/notices/200706/tx070600708p.pdfGeneral Applications: http://web.math.princeton.edu/math_alive/5/Notes2.pdfPage on graph colorings: webdocs.cs.ualberta.ca/~joe/Coloring/index.htmlBook on general graph theory: http://diestel-graph-theory.comThe Four Color Theorem - Numberphile: youtube.com/watch?v=NgbK43jB4rQWritten and Hosted by Kelsey Houston-EdwardsProduced by Rusty WardGraphics by Ray LuxMade by Kornhaber Brown (www.kornhaberbrown.com)Comments answered by Kelsey:Ofir Davidyoutube.com/watch?v=stgYW6M5o4k&lc=z12vin2o2vn0dxvn522wsr5j0wexezd0jBertie Blueyoutube.com/watch?v=stgYW6M5o4k&lc=z22xhfxbgx3qsh2ujacdp4333fssxeq5w1z4avpprddw03c010cTehAvenger29youtube.com/watch?v=stgYW6M5o4k&lc=z12qtxqgqpyftrzmy22wyxl5gy3ssvcpf04
What is a Random Walk? | Infinite Series PBS Infinite Series 2017-03-23 | Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfiTo understand finance, search algorithms and even evolution you need to understand Random Walks. Tell PBS what types of shows you want to see at surveymonkey.com/r/pbsds2017 25 random participants in the survey will receive PBS t-shirts. Tweet at us! @pbsinfiniteFacebook: facebook.com/pbsinfinite seriesEmail us! pbsinfiniteseries [at] gmail [dot] comPrevious Episode - Proving Pick’s Theoremyoutube.com/watch?v=bYW1zOMCQnoMarkov Chainsyoutube.com/watch?v=63HHmjlh794Written and Hosted by Kelsey Houston-EdwardsProduced by Rusty WardGraphics by Ray LuxMade by Kornhaber Brown (www.kornhaberbrown.com)Random Walks are used in finance, computer science, psychology, biology and dozens of other scientific fields. They’re one of the most frequently used mathematical processes. So exactly what are Random Walks and how do they work?Comments answered by Kelsey:Petros Adamopoulosyoutube.com/watch?v=bYW1zOMCQno&lc=z12hcf1rayqazhrry04cdtbxknznfdvayto0kJonathan Castelloyoutube.com/watch?v=bYW1zOMCQno&lc=z13lslqjqzzyzbbew22dwlh54zj4tpo0jNiosusyoutube.com/watch?v=bYW1zOMCQno&lc=z13kwrpavz3igj32t230dnl4uun2hppow04
