PBS Infinite Series
Proving Picks Theorem | Infinite Series
updated
Thank you everyone. This show was a joy to produce and it was the audience that made it incredible.
Gabe Perez-Giz
@fizziksgabe
Tai-Danae Bradley
@math3ma
http://www.math3ma.com/mathema/2015/2/1/a-math-blog-say-what
Imagine 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?
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Solution to the Assassin Puzzle:
www.math3ma.com/mathema/2018/5/17/is-the-square-a-secure-polygon
Previous Episode:
Instant Insanity Puzzle
youtube.com/watch?v=Lw1pF47N-0Q&t=1s
Let’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 Bradley
Produced by Eric Brown
Graphics by Matt Rankin
Assistant 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!
Imagine 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?
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Now, 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 Infinity
youtube.com/watch?v=VCksQ6g2yh0
Written and Hosted by Tai-Danae Bradley
Graphics by Matt Rankin
Assistant 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/2HWYNaY
Resources:
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!
Set 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! @pbsinfinite
Facebook: facebook.com/pbsinfinite series
Email us! pbsinfiniteseries [at] gmail [dot] com
Previous Episode:
Unraveling DNA with Rational Tangles
youtube.com/watch?v=JXGyXtNsu14
Earlier videos referenced around 1:00 are:
Cantor's paradox: youtube.com/watch?v=TbeA1rhV0D0
What are Numbers Made of? youtube.com/watch?v=S4zfmcTC5bM
Peano axioms: youtube.com/watch?v=3gBoP8jZ1Is
Hiearchy of Infinities: youtube.com/watch?v=i7c2qz7sO0I
Vsauce How to Count Past Infinity
youtube.com/watch?v=SrU9YDoXE88
Brilliant infinity wiki:
brilliant.org/wiki/infinity
Brilliant Number Theory course (with Exploring Infinity chapter):
brilliant.org/courses/basic-number-theory
Set 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 Ward
Graphics by Ray Lux
Assistant 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!
When 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! @pbsinfinite
Facebook: facebook.com/pbsinfinite series
Email us! pbsinfiniteseries [at] gmail [dot] com
Previous Episode:
How Big are All Infinities Combined? (Cantor’s Paradox)
youtube.com/watch?v=TbeA1rhV0D0
There 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 Bradley
Produced by Rusty Ward
Graphics by Ray Lux
Assistant 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 Adams
amazon.com/Knot-Book-Colin-Adams/dp/0821836781
Knot Theory and Its Applications by Kunio Murasugi
amazon.com/Theory-Applications-Modern-Birkh%C3%A4user-Classics/dp/081764718X
On the Classification of Rational Tangles by Louis Kauffman and Sofia Lambropoulou
arxiv.org/pdf/math/0311499.pdf
DNA Topology by Andrew Bates and Anthony Maxwell
amazon.com/Topology-Oxford-Biosciences-Andrew-Bates/dp/0198506554
Proof of Conway’s Rational Tangle Theorem
http://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 Witten
https://www.ias.edu/ideas/2011/witten-knots-quantum-theory
Tangles, Physics, and Category Theory
http://math.ucr.edu/home/baez/tangles.html
My Favorite Theorem Podcast
kpknudson.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!
Infinities 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! @pbsinfinite
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Email us! pbsinfiniteseries [at] gmail [dot] com
Previous Episode:
The Geometry of SET
youtube.com/watch?v=zurpOBPt4LI
Check out the solution to the Geometry of SET challenge problem right here: bit.ly/2Ggpw1d
Talking 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 Bradley
Produced by Rusty Ward
Graphics by Ray Lux
Assistant 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=UgzxzGjWJh8Pra57LDR4AaABAg
youtube.com/watch?v=zurpOBPt4LI&lc=Ugx2lMNH6inRYiTGFSx4AaABAg
youtube.com/watch?v=zurpOBPt4LI&lc=UgzA2HK0J9ytm01lUex4AaABAg
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!
In the card game SET, what is the maximum number of cards you can deal that might not contain a SET?
Tweet at us! @pbsinfinite
Facebook: facebook.com/pbsinfinite series
Email us! pbsinfiniteseries [at] gmail [dot] com
Resources
Official SET game instructions
setgame.com/sites/default/files/instructions/SET%20INSTRUCTIONS%20-%20ENGLISH.pdf
Simple SET Game Proof Stuns Mathematicians (Quanta Magazine)
quantamagazine.org/set-proof-stuns-mathematicians-20160531
The Problem with SET (NYTimes Puzzle)
nytimes.com/2016/08/22/crosswords/the-problem-with-set.html
Open Question: Best Bounds for Cap Sets (blog post by Terry Tao)
terrytao.wordpress.com/2007/02/23/open-question-best-bounds-for-cap-sets
SET and Group Theory by Pavel Etingof (see p. 13ff)
http://www-math.mit.edu/~etingof/groups.pdf
This 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/2Ggpw1d
Previous Episode:
What was Fermat’s “Marvelous" Proof?
youtube.com/watch?v=SsVl7_R2MvI
Let'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 Bradley
Produced by Rusty Ward
Graphics by Ray Lux
Assistant 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!
If Fermat had a little more room in his margin, what proof would he have written there?
Tweet at us! @pbsinfinite
Facebook: facebook.com/pbsinfinite series
Email us! pbsinfiniteseries [at] gmail [dot] com
Resources:
Contemporary Abstract Algebra by Joseph Gallian
amazon.com/Contemporary-Abstract-Algebra-Joseph-Gallian/dp/1133599702
Standard Definitions in Ring Theory by Keith Conrad
http://www.math.uconn.edu/~kconrad/blurbs/ringtheory/ringdefs.pdf
Rings and First Examples (online course by Prof. Matthew Salomone)
youtube.com/watch?v=h4UCMd8dyiM
Fermat's Enigma by Simon Singh
amazon.com/Fermats-Enigma-Greatest-Mathematical-Problem/dp/0385493622
Who was first to differentiate between prime and irreducible elements? (StackExchange)
hsm.stackexchange.com/questions/3754/who-was-first-to-differentiate-between-prime-and-irreducible-elements
Previous Episodes:
What Does It Mean to be a Number?
youtube.com/watch?v=3gBoP8jZ1Is
What are Numbers Made of?
youtube.com/watch?v=S4zfmcTC5bM
Gabe'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.html
Recommended 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 Bradley
Produced by Rusty Ward
Graphics by Ray Lux
Assistant 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!
In 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 Number
youtube.com/watch?v=3gBoP8jZ1Is
And to see Gabe's solution to the Torus Clock Challenge check out:
http://bit.ly/pbs_clock_challenge
Tweet at us! @pbsinfinite
Facebook: facebook.com/pbsinfinite series
Email us! pbsinfiniteseries [at] gmail [dot] com
Previous Episode:
What Does it Mean to Be A Number?
youtube.com/watch?v=3gBoP8jZ1Is
Torus Clock Challenge:
youtube.com/watch?v=KZT5hrYOERs
How to Divide by "Zero"
youtube.com/watch?v=uxpowBoPieQ&t=3s
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.html
Recommended 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-Giz
Produced by Rusty Ward
Graphics by Ray Lux
Assistant 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!
If 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! @pbsinfinite
Facebook: facebook.com/pbsinfinite series
Email us! pbsinfiniteseries [at] gmail [dot] com
Previous Episodes:
Telling Time on a Torus
youtu.be/KZT5hrYOERs
Crisis in the Foundation of Mathematics
youtube.com/watch?v=KTUVdXI2vng
How to Divide by "Zero"
youtube.com/watch?v=uxpowBoPieQ
Beyond the Golden Ratio
youtube.com/watch?v=MIxvZ6jwTuA
Are 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-Giz
Produced by Rusty Ward
Graphics by Ray Lux
Assistant 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 is supporting us at the Lemma level!
What 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! @pbsinfinite
Facebook: facebook.com/pbsinfinite series
Email us! pbsinfiniteseries [at] gmail [dot] com
Previous Episode:
How to Divide by Zero
youtube.com/watch?v=uxpowBoPieQ
Some 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-Giz
Produced by Rusty Ward
Graphics by Ray Lux
Assistant 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!
What happens when you divide things that aren’t numbers?
Tweet at us! @pbsinfinite
Facebook: facebook.com/pbsinfinite series
Email us! pbsinfiniteseries [at] gmail [dot] com
RESOURCES
Visual Group Theory by Nathan Carter amazon.com/Visual-Group-Theory-Problem-Book/dp/088385757X
Geek37 - The silver ratio and the octagon
youtube.com/watch?v=o-6kWnYfdVk
Polygons, Diagonals, and the Bronze Mean by Antonia Redondo
Buitrago
link.springer.com/content/pdf/10.1007/s00004-007-0046-x.pdfl
Previous Episodes:
Beyond the Golden Ratio
youtube.com/watch?v=MIxvZ6jwTuA&t=600s
The Multiplication Multiverse
youtube.com/watch?v=H4I2C3Ts7_w
The Mathematics of Diffie Hillman Key Exchange
youtube.com/watch?v=ESPT_36pUFc
How to Break Cryptography
youtube.com/watch?v=12Q3Mrh03Gk&t=130s&list=PLa6IE8XPP_glwNKmFfl2tEL0b7E9D0WRr&index=42
We 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 Bradley
Produced by Rusty Ward
Graphics by Ray Lux
Assistant 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!
You 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/InfiniteSeriesOpenProblem
Tweet at us! @pbsinfinite
Facebook: facebook.com/pbsinfinite series
Email us! pbsinfiniteseries [at] gmail [dot] com
RESOURCES
Polygons, Diagonals, and the Bronze Mean by Antonia Redondo
Buitrago
link.springer.com/content/pdf/10.1007/s00004-007-0046-x.pdf
Previous Episode
Proving Brouwer's Fixed Point Theorem
youtu.be/djaSbHKK5yc
Cut 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-Giz
Produced by Rusty Ward
Graphics by Ray Lux
Assistant 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!
There is a proof for Brouwer's Fixed Point Theorem that uses a bridge - or portal - between geometry and algebra.
Tweet at us! @pbsinfinite
Facebook: facebook.com/pbsinfinite series
Email us! pbsinfiniteseries [at] gmail [dot] com
Previous Episode
The Mathematics of Diffie-Hellman Key Exchange
youtube.com/watch?v=ESPT_36pUFc&feature=youtu.be
Analogous 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 Bradley
Produced by Rusty Ward
Graphics by Ray Lux
Assistant Editing and Sound Design by Mike Petrow and Meah Denee Barrington
Made by Kornhaber Brown (www.kornhaberbrown.com)
REFERENCES
The 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-proof
Algebraic Topology by Allen Hatcher, page 31:
https://www.math.cornell.edu/~hatcher/AT/AT.pdf
A 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/0201627280
To 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-1
VSauce - Fixed Points
youtube.com/watch?v=csInNn6pfT4
Congratulations to Robby Weintraub for being the winner of the Topology vs "a" Topology Challenge Question. youtube.com/watch?v=tdOaMOcxY7U
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!
Symmetric 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-Q4ldHi56mYsBuOg2Qw
Tweet at us! @pbsinfinite
Facebook: facebook.com/pbsinfinite series
Email us! pbsinfiniteseries [at] gmail [dot] com
Previous Episode
Topology vs. “a” Topology
youtube.com/watch?v=tdOaMOcxY7U&t=13s
Symmetric 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-Giz
Produced by Rusty Ward
Graphics by Ray Lux
Assistant 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!
What exactly is a topological space? Learn through active problem-solving at Brilliant: brilliant.org/InfiniteSeries
Tweet at us! @pbsinfinite
Facebook: facebook.com/pbsinfinite series
Email us! pbsinfiniteseries [at] gmail [dot] com
Previous Episode
This Video was Not Encrypted with RSA | Infinite Series
youtube.com/watch?v=4Tb1q8dSIlI
We’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. Vickers
books.google.com/books/about/Topology_Via_Logic.html?id=peXBNAEACAAJ
Topology by Munkres
amazon.com/Topology-2nd-James-Munkres/dp/0131816292
For the solution to the challenge question check out:
http://www.math3ma.com/mathema/2017/12/21/topology-vs-a-topology
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Produced by Rusty Ward
Graphics by Ray Lux
Assistant Editing and Sound Design by Meah Denee Barrington and Emma Dessau
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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!
Learn through active problem-solving at Brilliant: brilliant.org/InfiniteSeries
Last episode we discussed Symmetric cryptography youtube.com/watch?v=NOs34_-eREk Here we break down Asymmetric crypto and more.
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Previous Episode
(Almost) Unbreakable Crypto | Infinite Series
youtube.com/watch?v=NOs34_-eREk
How To Break Cryptography
youtube.com/watch?v=12Q3Mrh03Gk&list=PLa6IE8XPP_gnot4uwqn7BeRJoZcaEsG1D&index=2
Last 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-Giz
Produced by Rusty Ward
Graphics by Ray Lux
Assistant Editing and Sound Design by Mike Petrow and Meah Denee Barrington
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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!
Despite 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_t6FwNsqr3WWoL6dFqG9w
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Previous Episode
Associahedra: The Shapes of Multiplication | Infinite Series
youtube.com/watch?v=N7wNWQ4aTLQ
In 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-Giz
Produced by Rusty Ward
Graphics by Ray Lux
Assistant 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!
What happens when you multiply shapes? This is part 2 of our episode on multiplying things that aren't numbers. You can check out part 1: The Multiplication Multiverse right here youtube.com/watch?v=H4I2C3Ts7_w
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And discuss the episode further over on reddit at reddit.com/r/PBSInfiniteSeries
Previous Episode
The Multiplication Multiverse | Infinite Series
youtube.com/watch?v=H4I2C3Ts7_w
In our last episode, we talked about different properties of multiplication: associativity and commutativity are the most familiar, but they’re just two of many. We also saw it’s possible to multiply things that aren’t numbers, and in that case we may not have... associativity, for instance. But that’s not a bad thing. In fact, it’s a beautiful thing!
References::
More on the associahedra:
http://www.ams.org/samplings/feature-column/fcarc-associahedra
http://www.claymath.org/library/academy/LectureNotes05/Lodaypaper.pdf
arxiv.org/pdf/math/0212126.pdf
More on multiplying non-numbers:
http://www.math3ma.com/mathema/2017/11/24/multiplying-non-numbers
An introduction to operads:
http://www.math3ma.com/mathema/2017/10/23/what-is-an-operad-part-1
http://www.math3ma.com/mathema/2017/10/30/what-is-an-operad-part-2
Some applications in math and physics:
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 Bradley
Produced by Rusty Ward
Graphics by Ray Lux
Assistant Editing and Sound Design by Meah Denee Barrington
Made by Kornhaber Brown (www.kornhaberbrown.com)
Thanks to Matthew O'Connor and Yana Chernobilsky who are supporting us on Patreon at the Identity level!
And thanks to Nicholas Rose and Mauricio Pacheco who are supporting us at the Lemma level!
What happens if you multiply things that aren’t numbers? And what happens if that multiplication is not associative?
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Previous Episode
The Heat Equation + Special Announcement | Infinite Series
youtube.com/watch?v=NHucpzbD600
Multiplication 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.pdf
Written and Hosted by Tai-Danae Bradley
Produced by Rusty Ward
Graphics by Ray Lux
Assistant Editing and Sound Design by Mike Petrow
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!
What is the heat equation? And find out who the two new hosts of Infinite Series are!
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Previous Episode
Hilbert's 15th Problem: Schubert Calculus | Infinite Series
youtube.com/watch?v=U8sq3BplCfI
Imagine 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 Graph
youtube.com/watch?v=twBcpxrWm5E
Normal Distribution Graph
http://www.statisticshowto.com/wp-content/uploads/2013/09/normal-distribution-probability.jpg
3D Plot Heat Graph
mathworks.com/help/examples/matlab/win64/ThreeDPlotsGSExample_01.png
Written and Hosted by Kelsey Houston-Edwards
Produced by Rusty Ward
Graphics by Ray Lux
Assistant Editing and Sound Design by Mike Petrow
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!
Get 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.
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Previous Episode
Crisis in the Foundation of Mathematics | Infinite Series
youtube.com/watch?v=KTUVdXI2vng
In 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.pdf
3264 and All That -- David Eisenbud and Joe Harris
The 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-Edwards
Produced by Rusty Ward
Graphics by Ray Lux
Assistant Editing and Sound Design by Mike Petrow
Made by Kornhaber Brown (www.kornhaberbrown.com)
Special Thanks to Allen Knutson and Balázs Elek
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!
What 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.
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Previous Episode
How to generate Pseudorandom Numbers? | Infinite Series
youtube.com/watch?v=C82JyCmtKWg
Mathematics 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.pdf
Philosophy of mathematics (Selected readings) edited by Paul Benacerraf and Hilary Putnam
Written and Hosted by Kelsey Houston-Edwards
Produced by Rusty Ward
Graphics by Ray Lux
Assistant Editing and Sound Design by Mike Petrow
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!
What 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?
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Previous Episode
How many Cops to catch a Robber? | Infinite Series
youtube.com/watch?v=fXvN-pF76-E
Computers 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-Edwards
Produced by Rusty Ward
Graphics by Ray Lux
Assistant Editing and Sound Design by Mike Petrow
Made by Kornhaber Brown (www.kornhaberbrown.com)
Special Thanks to Alex Townsend
Big 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!
Last 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?
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Previous Episode
The Cops and Robbers Theorem | Infinite Series
youtube.com/watch?v=9mJEu-j1KT0
Cops 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-Edwards
Produced by Rusty Ward
Graphics by Ray Lux
Assistant Editing and Sound Design by Mike Petrow
Made 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.pdf
What is Cop Number? -Anthony Bonato
http://www.math.ryerson.ca/~abonato/papers/whatis_copnumber_new.pdf
The Game of Cops and Robbers on Graph - Anthony Bonato and Richard Nowakowski
Anthony Bonato -- "What is... Cops and Robbers"
http://www.ams.org/notices/201208/rtx120801100p.pdf
Special Thanks to Anthony Bonato and Brendan Sullivan
Big 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!
Can a cop catch a robber? There's some surprising and compelling graph theory that go into answering that question.
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Previous Episode
Higher Dimensional Tic Tac Toe | Infinite Series
youtube.com/watch?v=FwJZa-helig&t=17s
Cops 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-Edwards
Produced by Rusty Ward
Graphics by Ray Lux
Assistant Editing and Sound Design by Mike Petrow
Made 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.pdf
The Game of Cops and Robbers on Graphs - Anthony Bonato and Richard Nowakowski
Anthony Bonato -- "What is... Cops and Robbers" http://www.ams.org/notices/201208/rtx120801100p.pdf
Brendan W. Sullivan, Nikolas Townsend, Mikayla Werzanski - “An Introduction to Lazy Cops and Robbers on Graphs,” to appear in College Mathematics Journal in 2017
Brendan W. Sullivan, Nikolas Townsend, Mikayla Werzanski "The 3x3 rooks graph is the unique smallest graph with lazy cop number 3" -- arxiv.org/abs/1606.08485
Special Thanks to Anthony Bonato and Brendan Sullivan
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!
Regular 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?
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Previous Episode
How the Axiom of Choice Gives Sizeless Sets | Infinite Series
youtube.com/watch?v=hcRZadc5KpI
The 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 Math
Written and Hosted by Kelsey Houston-Edwards
Produced by Rusty Ward
Graphics by Ray Lux
Assistant Editing and Sound Design by Mike Petrow
Made 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.pdf
Golomb/Hales paper: http://library.msri.org/books/Book42/files/golomb.pdf
youtube.com/watch?v=p1YzYLzRwtk
http://www.austms.org.au/Gazette/2005/Jul05/mathellaneous.pdf
Special Thanks: Benjamin Houston-Edwards and Nathan Kaplan
Mathologer Video:
youtube.com/watch?v=aDOP0XynAzA
Special 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!
Does 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.
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Previous Episodes
Your Brain as Math - Part 1
youtube.com/watch?v=M0M3srBoTkY
Simplicial Complexes - Your Brain as Math Part 2
youtube.com/watch?v=rlI1KOo1gp4
Your Mind Is Eight-Dimensional - Your Brain as Math Part 3
youtube.com/watch?v=akgU8nRNIp0
In 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-Edwards
Produced by Rusty Ward
Graphics by Ray Lux
Assistant Editing and Sound Design by Mike Petrow
Made by Kornhaber Brown (www.kornhaberbrown.com)
Resources:
https://math.vanderbilt.edu/schectex/ccc/choice.html
https://plato.stanford.edu/entries/axiom-choice/
http://www.math.kth.se/matstat/gru/godis/nonmeas.pdf
Vsauce
youtube.com/watch?v=s86-Z-CbaHA
Special Thanks: Lian Smythe and James Barnes
Thanks 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!
How 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=rlI1KOo1gp4
Tweet at us! @pbsinfinite
Facebook: facebook.com/pbsinfinite series
Email us! pbsinfiniteseries [at] gmail [dot] com
Previous Episode
Simplicial Complexes - Your Brain as Math - Part 2
youtube.com/watch?v=rlI1KOo1gp4
Last 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-Edwards
Produced by Rusty Ward
Graphics by Ray Lux
Assistant Editing and Sound Design by Mike Petrow
Made by Kornhaber Brown (www.kornhaberbrown.com)
Resources:
Cliques of Neurons Bound into Cavities Provide a Missing Link between Structure and Function
http://journal.frontiersin.org/article/10.3389/fncom.2017.00048/full
The Blue Brain Project:
http://bluebrain.epfl.ch/
Barcodes: The Persistent Topology of Data
https://www.math.upenn.edu/~ghrist/preprints/barcodes.pdf
Network Neuroscience
http://www.nature.com/neuro/journal/v20/n3/full/nn.4502.html
Algebraic Topology
https://www.math.cornell.edu/~hatcher/AT/ATpage.html
Special 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!
What 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=M0M3srBoTkY
Check out Part 3 here: youtube.com/watch?v=akgU8nRNIp0
Tweet at us! @pbsinfinite
Facebook: facebook.com/pbsinfinite series
Email us! pbsinfiniteseries [at] gmail [dot] com
Previous Episode
Your Brain as Math - Part 1
youtu.be/M0M3srBoTkY
Network Mathematics and Rival Factions
youtube.com/watch?v=qEKNFOaGQcc
Last 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-Edwards
Produced by Rusty Ward
Graphics by Ray Lux
Assistant Editing and Sound Design by Mike Petrow
Made by Kornhaber Brown (www.kornhaberbrown.com)
Resources:
Cliques of Neurons Bound into Cavities Provide a Missing Link between Structure and Function
http://journal.frontiersin.org/article/10.3389/fncom.2017.00048/full
The Blue Brain Project:
http://bluebrain.epfl.ch/
Barcodes: The Persistent Topology of Data
https://www.math.upenn.edu/~ghrist/preprints/barcodes.pdf
Network Neuroscience
http://www.nature.com/neuro/journal/v20/n3/full/nn.4502.html
Algebraic Topology
https://www.math.cornell.edu/~hatcher/AT/ATpage.html
Special 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!
What 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=rlI1KOo1gp4
Check out Part 3 here: youtube.com/watch?v=akgU8nRNIp0
Tweet at us! @pbsinfinite
Facebook: facebook.com/pbsinfinite series
Email us! pbsinfiniteseries [at] gmail [dot] com
Previous Episode
Stochastic Supertasks
youtube.com/watch?v=Sdp_V0L99sw
In 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-Edwards
Produced by Rusty Ward
Graphics by Ray Lux
Assistant Editing and Sound Design by Mike Petrow
Made by Kornhaber Brown (www.kornhaberbrown.com)
Resources:
Cliques of Neurons Bound into Cavities Provide a Missing Link between Structure and Function
http://journal.frontiersin.org/article/10.3389/fncom.2017.00048/full
The Blue Brain Project
http://bluebrain.epfl.ch/
Barcodes: The Persistent Topology of Data
https://www.math.upenn.edu/~ghrist/preprints/barcodes.pdf
Network Neuroscience
http://www.nature.com/neuro/journal/v20/n3/full/nn.4502.html
Algebraic Topology
https://www.math.cornell.edu/~hatcher/AT/ATpage.html
Special thanks to Kathryn Hess and Florian Frick!
Comments answered by Kelsey:
Vriskanon
youtube.com/watch?v=Sdp_V0L99sw&lc=z22ic1ehnlqhi3g4g04t1aokgo4wjichzrqe5sqovhw4rk0h00410
Challenge Winner : Florence B
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!
Supertasks 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.
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Email us! pbsinfiniteseries [at] gmail [dot] com
Previous Episode
The Honeybcomcs of 4-Dimensional Bees
youtube.com/watch?v=X8jOxEGVyPo
Vsauce - Supertasks
youtube.com/watch?v=ffUnNaQTfZE
Written and Hosted by Kelsey Houston-Edwards
Produced by Rusty Ward
Graphics by Ray Lux
Assistant Editing and Sound Design by Mike Petrow
Made by Kornhaber Brown (www.kornhaberbrown.com)
Resources:
http://julio.staff.ipb.ac.id/files/2015/02/Ross_8th_ed_English.pdf -- Pg 46
Special 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:
Lowtech
youtube.com/watch?v=X8jOxEGVyPo&lc=z235h5yquwukfhh2nacdp431mqnc3ul0fvs4cy1tqpdw03c010c
Daniel Shapiro
youtube.com/watch?v=X8jOxEGVyPo&lc=z235h5yquwukfhh2nacdp431mqnc3ul0fvs4cy1tqpdw03c010c.1501799589673862
Professor Politics
youtube.com/watch?v=X8jOxEGVyPo&lc=z23xhrzyrzmxeraluacdp43aqfqbpbddfp3g4myyjclw03c010c
Alex 24757
youtube.com/watch?v=X8jOxEGVyPo&lc=z23fvxw5zmq1vzvxa04t1aokgodujwhjcmcr4pmcbu5lbk0h00410
Be 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: infinite
The 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=17024143
en.wikipedia.org/wiki/Weaire%E2%80%93Phelan_structure#/media/File:Irregular_dodecahedron.png
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Previous Episode
Why Computers are Bad at Algebra
youtube.com/watch?v=pQs_wx8eoQ8
Why 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-Edwards
Produced by Rusty Ward
Graphics by Ray Lux
Assistant Editing and Sound Design by Mike Petrow
Made by Kornhaber Brown (www.kornhaberbrown.com)
Resources:
Nature paper http://www.nature.com/news/how-honeycombs-can-build-themselves-1.13398#/b1
Hales’ proof of honeycomb conjecture: arxiv.org/pdf/math/9906042.pdf
Older article on honeycomb conjecture http://www.ams.org/journals/bull/1964-70-04/S0002-9904-1964-11155-1/S0002-9904-1964-11155-1.pdf
Overview of proof of honeycomb conjecture http://www.maa.org/frank-morgans-math-chat-hales-proves-hexagonal-honeycomb-conjecture
http://www.npr.org/sections/krulwich/2013/05/13/183704091/what-is-it-about-bees-and-hexagons
Kelvin -- http://soft-matter.seas.harvard.edu/images/1/17/Kelvin_Cell.pdf
http://www.slate.com/articles/health_and_science/science/2015/07/hexagons_are_the_most_scientifically_efficient_packing_shape_as_bee_honeycomb.html
The answer lies in the weirdness of floating-point numbers and the computer's perception of a number line.
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Previous Episode
Making Probability Mathematical
youtube.com/watch?v=-6HxjiW_KwA
Written and Hosted by Kelsey Houston-Edwards
Produced by Rusty Ward
Graphics by Ray Lux
Assistant Editing and Sound Design by Mike Petrow
Made by Kornhaber Brown (www.kornhaberbrown.com)
Resources:
Random ASCII
randomascii.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:
Neroox05
youtube.com/watch?v=-6HxjiW_KwA&lc=z132ehrixviktnumh23fgzmgfkv1dz0c1
Joshua Sherwin
youtube.com/watch?v=-6HxjiW_KwA&lc=z13xs5mplpn4d5dgc04cilyxbqr1ifxwd24
Joshua Hillerup
youtube.com/watch?v=-6HxjiW_KwA&lc=z12wc1iipuzsirj2h22ucfv5burmibno304
What 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.html
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Previous Episode
youtube.com/watch?v=qEKNFOaGQcc
Written and Hosted by Kelsey Houston-Edwards
Produced by Rusty Ward
Graphics by Ray Lux
Assistant Editing and Sound Design by Mike Petrow
Made by Kornhaber Brown (www.kornhaberbrown.com)
Resources and Special thanks:
terrytao.wordpress.com/2010/01/01/254a-notes-0-a-review-of-probability-theory
Kolmogorov - Foundations of the Theory of Probability
Ian Hacking - The Emergence of Probability
Throughout 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:
Zutaca
youtube.com/watch?v=qEKNFOaGQcc&lc=z13ky5eruxbiermgx04cdx1ztyjlxzfyavc0k
Comments answered by Kelsey:
Ja-Shwa Cardell
youtube.com/watch?v=qEKNFOaGQcc&lc=z12yhrgjdqm5wrdbv04cejno4t2icnmpy1c
The 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.
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Previous Episode
Arrow’s Impossibility Theorem
youtube.com/watch?v=AhVR7gFMKNg
Written and Hosted by Kelsey Houston-Edwards
Produced by Rusty Ward
Graphics by Ray Lux
Made 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.pdf
Antal, Krapivsky, and Redner :: http://physics.bu.edu/~redner/pubs/pdf/dresden.pdf
Steven Strogatz Lecture :: youtube.com/watch?v=P60sWUxluyk
Special Thanks: Steven Strogatz
Commonly, 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 Cat
youtube.com/watch?v=AhVR7gFMKNg&lc=z13lyxtxmom1tv51y23tzluompyzg5xkw
David de Kloet
youtube.com/watch?v=AhVR7gFMKNg&lc=z13lyxtxmom1tv51y23tzluompyzg5xkw.1498299129602777
Comments answered by Kelsey:
Edelopo
youtube.com/watch?v=AhVR7gFMKNg&lc=z12ftje5aybuxhro204chdn4tuqovxwjapk0k
The 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.
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Previous Episode
Voting Systems and the Condorcet Criterion
youtube.com/watch?v=HoAnYQZrNrQ
Written and Hosted by Kelsey Houston-Edwards
Produced by Rusty Ward
Graphics by Ray Lux
Made by Kornhaber Brown (www.kornhaberbrown.com)
Additional Resources
Networks, 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.pdf
Different 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 Richter
youtube.com/watch?v=HoAnYQZrNrQ&lc=z12bt1nabyievh4yg04chlvpdnisxnw5rx00k
Nat Tuck
youtube.com/watch?v=HoAnYQZrNrQ&lc=z12tetkx1wzocn2ue23wzdfg5sn2dhhh004
What 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/diyft53
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Previous Episode
Pantographs and the
youtube.com/watch?v=KYaCtHPCARc
Written and Hosted by Kelsey Houston-Edwards
Produced by Rusty Ward
Graphics by Ray Lux
Made 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 Methods
http://www.ams.org/samplings/feature-column/fcarc-voting-decision
The Mathematics of Voting
https://www.math.ku.edu/~jmartin/courses/math105-F11/Lectures/chapter1-part1.pdf
The Mathematics of Voting, Power and Sharing
http://web.math.princeton.edu/math_alive/6/Notes1.pdf
CGP Grey Voting Playlist
youtube.com/playlist?list=PLej2SlXPEd37YwwEY7mm0WyZ8cfB1TxXa
Comments answered by Kelsey:
FossilFighters101
youtube.com/watch?v=XOzhF3QoTCA&lc=z12fi1jgnle5wre0k22puzfojxnbzjirk04
Abi Gail
youtube.com/watch?v=XOzhF3QoTCA&lc=z13kin5rcqatih1i004cjdfwsofhi1hopgo
Lucas Hoffses
youtube.com/watch?v=XOzhF3QoTCA&lc=z13nilewrtbmfp24t04cgdlziqafvzbahkk0k
When 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.
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Previous Episode
Dissecting Hypercubes with Pascal’s Triangle
youtube.com/watch?v=KYaCtHPCARc
Written and Hosted by Kelsey Houston-Edwards
Produced by Rusty Ward
Graphics by Ray Lux
Made 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.html
http://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/Mackevicius.pdf
http://www.math.toronto.edu/~drorbn/People/Eldar/thesis/default.htm
https://www.math.ucdavis.edu/~kapovich/EPR/link.pdf
Giant Drawing Machines by Nick Sayer
behance.net/gallery/18833343/Giant-Drawing-Machines
Pantographo Legal’s Cartoon Sketches
youtube.com/watch?v=Muic3yyhSv8
Thanks to Benjamin Houston-Edwards and Andrew Marshall
Comments answered by Kelsey:
NotaWalrus
youtube.com/watch?v=KYaCtHPCARc&lc=z12nit1j0za0wfqul22cin3qspjmi5hdp
Joel Nordstrom
youtube.com/watch?v=KYaCtHPCARc&lc=z12yhltz4sfhzlxa304cdtbz0vnsupmpahk
Rob Nicolaides
youtube.com/watch?v=KYaCtHPCARc&lc=z12yhltz4sfhzlxa304cdtbz0vnsupmpahk
What 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.”
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Previous Episode
Devil’s Staircase
youtube.com/watch?v=dQXVn7pFsVI
Written and Hosted by Kelsey Houston-Edwards
Produced by Rusty Ward
Graphics by Ray Lux
Made 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 Probability
http://www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/1992/0025570x.di021171.02p0016b.pdf
Challenge Winner:
Asthmen
youtube.com/watch?v=dQXVn7pFsVI&lc=z132hl040ozfc3q4i04cchci1suscbwg5ik
Comments answered by Kelsey:
Tehom
youtube.com/watch?v=dQXVn7pFsVI&lc=z133jzax0tansbqh504cedgq1mfvs1wjsyo0k
Badplayz
youtube.com/watch?v=dQXVn7pFsVI&lc=z12dchu4qtfwshkmv231elxqsnuudjt2y
Find 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/UC4K10PNjqgGLKA3lo5V8KdQ
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Previous Episode
Topology Riddles
youtube.com/watch?v=H8qwqGjOlSE
Written and Hosted by Kelsey Houston-Edwards
Produced by Rusty Ward
Graphics by Ray Lux
Made 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 Wieland
Honorable Mentions:
Felix Beichler
Andrew Weller
Comments answered by Kelsey:
Thomas Perry
youtube.com/watch?v=H8qwqGjOlSE&lc=z13ncngjhwy5iv5tp23lungjrsbxyup0.1494583661471832
Ralph Dratman
youtube.com/watch?v=H8qwqGjOlSE&lc=z13isv1g0krfg52ds235h1rbvqqtxndq204.1494681742451739
Can 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=NrDM6Ic2xMM
And special thanks to Danny Haymond Jr aka @slothman86 for for his incredible fan art!
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Previous Episode
Building an Infinite Bridge
youtube.com/watch?v=1_DOzuaBE84
The Heart of Mathematics
amazon.com/Heart-Mathematics-invitation-effective-thinking/dp/1931914419
Picture Hanging Article:: arxiv.org/pdf/1203.3602.pdf
Written and Hosted by Kelsey Houston-Edwards
Produced by Rusty Ward
Graphics by Ray Lux
Made 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
@Chalkface
Comments answered by Kelsey:
Louis JX
youtube.com/watch?v=1_DOzuaBE84&lc=z12lu5vx5sepxjemc22mhnualyfajxthv04
Erdmannelchen
youtube.com/watch?v=1_DOzuaBE84&lc=z132hrbbhxbpwvieu231wl3ytxmyznsu0
Using the harmonic series we can build an infinitely long bridge. It takes a very long time though. A faster method was discovered in 2009.
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Previous Episode
Hacking at Quantum Speed with Shor’s Algorithm
youtube.com/watch?v=wUwZZaI5u0c
Additional Resources:
Overhang article:: http://www.maa.org/sites/default/files/images/upload_library/22/Robbins/Patterson1.pdf
Maximum overhang article:: http://www.maa.org/sites/default/files/pdf/upload_library/22/Robbins/Patterson2.pdf
Physical limitations:: quantamagazine.org/20161117-overhang-insights-puzzle
http://mathworld.wolfram.com/BookStackingProblem.html
There’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-Edwards
Produced by Rusty Ward
Graphics by Ray Lux
Made by Kornhaber Brown (www.kornhaberbrown.com)
Comments answered by Kelsey:
MultiMurmaider
youtube.com/watch?v=wUwZZaI5u0c&lc=z12sergyhwjzv5zhx04cfj5akpqbhnjargc
Marhsall Webb
youtube.com/watch?v=wUwZZaI5u0c&lc=z13cwnehjmjgfdpfx23dzjwbczayvh41v
10TinsThought
youtube.com/watch?v=wUwZZaI5u0c&lc=z13cwnehjmjgfdpfx23dzjwbczayvh41v
Classical computers struggle to crack modern encryption. But quantum computers using Shor’s Algorithm make short work of RSA cryptography. Find out how.
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Previous Episode
How to Break Cryptography
youtube.com/watch?v=12Q3Mrh03Gk
The Mathematics Behind Quantum Computers
youtube.com/watch?v=IrbJYsep45E
Additional Resources:
Scott Aaronson's Blog (Great Intro to Shor's Alg.):: http://www.scottaaronson.com/blog/?p=208
Shor's Original Paper:: arxiv.org/abs/quant-ph/9508027v2
Lectures on Shor's Algorithm:: arxiv.org/pdf/quant-ph/0010034.pdf
Decrypting 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-Edwards
Produced by Rusty Ward
Graphics by Ray Lux
Made by Kornhaber Brown (www.kornhaberbrown.com)
Thanks to Spiros Michalakis for helpful discussions and feedback.
Comments answered by Kelsey:
Neon Bull
youtube.com/watch?v=12Q3Mrh03Gk&lc=z135uxf5cxenutmxj04cc3swkvm4tpcrxik
Bhargav R
youtube.com/watch?v=12Q3Mrh03Gk&lc=z13qjjioozbjdrqyz04cevdrtu3ti3y5sq40k
BobC
youtube.com/watch?v=12Q3Mrh03Gk&lc=z12pjpzastylzz2qx04cjtc5jrq2y3yhmlk0k
Only 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.
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Previous Episode:
Can We Combine pi & e into a Rational Number?
youtube.com/watch?v=bG7cCXqcJag&t=25s
Links to other resources:
Shor's paper: arxiv.org/abs/quant-ph/9508027v2
Lecture on Shor's Algorithm: arxiv.org/pdf/quant-ph/0010034.pdf
Blog on Shor's algorithm: http://www.scottaaronson.com/blog/?p=208
Video on RSA cryptography: youtube.com/watch?v=wXB-V_Keiu8
Another video on RSA cryptography: youtube.com/watch?v=4zahvcJ9glg
Euler'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-Edwards
Produced by Rusty Ward
Graphics by Ray Lux
Made by Kornhaber Brown (www.kornhaberbrown.com)
Challenge Winner - Reddles37
youtube.com/watch?v=bG7cCXqcJag&lc=z135cnmgxlbwch1ds233sbzgaojkivaz004
Comments answered by Kelsey:
Joel David Hamkins
youtube.com/watch?v=bG7cCXqcJag&lc=z13zdpcwyk2ofhugh04cdh4agsr2whmbsmk0k
PCreeper394
youtube.com/watch?v=bG7cCXqcJag&lc=z135w324kw21j1qi104cdzvrpoixslmq1jw
Can you produce a rational number by exchanging infinitely many digits of pi and e?
Tweet at us! @pbsinfinite
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Email us! pbsinfiniteseries [at] gmail [dot] com
Hierarchy of Infinities
youtube.com/watch?v=i7c2qz7sO0I
Previous Episode - Solving the Wolverine Problem using Graph Coloring!
youtube.com/watch?v=59aljcwo7aw
Links to other resources:
Original Post from Math Overflow
http://mathoverflow.net/questions/265310/if-i-exchange-infinitely-many-digits-of-pi-and-e-are-the-two-resulting-num
Square Root of 2 is Irrational
http://www.mathsisfun.com/numbers/square-root-2-irrational.html
Written and Hosted by Kelsey Houston-Edwards
Produced by Rusty Ward
Graphics by Ray Lux
Made by Kornhaber Brown (www.kornhaberbrown.com)
Thanks to Joel David Hampkins and Erin Carmody for helpful comments.
Comments answered by Kelsey:
Circuitrinos
youtube.com/watch?v=59aljcwo7aw&lc=z13mevgrmlzqh1xzc04chnc4vvegxfhj4x00k
Brian Willett
youtube.com/watch?v=59aljcwo7aw&lc=z123tzuj5oiqudan522hdfdxfl2xttl01
Here's how you can get involved:
1. Tell us what you love about public media using #IlovePBS
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At one time, Wolverine served on four different superhero teams. How did he do it? He may have used graph coloring.
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Graph 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=stgYW6M5o4k
Links to other resources:
Sudoku: http://www.ams.org/notices/200706/tx070600708p.pdf
General Applications: http://web.math.princeton.edu/math_alive/5/Notes2.pdf
Page on graph colorings: webdocs.cs.ualberta.ca/~joe/Coloring/index.html
Book on general graph theory: http://diestel-graph-theory.com
The Four Color Theorem - Numberphile: youtube.com/watch?v=NgbK43jB4rQ
Written and Hosted by Kelsey Houston-Edwards
Produced by Rusty Ward
Graphics by Ray Lux
Made by Kornhaber Brown (www.kornhaberbrown.com)
Comments answered by Kelsey:
Ofir David
youtube.com/watch?v=stgYW6M5o4k&lc=z12vin2o2vn0dxvn522wsr5j0wexezd0j
Bertie Blue
youtube.com/watch?v=stgYW6M5o4k&lc=z22xhfxbgx3qsh2ujacdp4333fssxeq5w1z4avpprddw03c010c
TehAvenger29
youtube.com/watch?v=stgYW6M5o4k&lc=z12qtxqgqpyftrzmy22wyxl5gy3ssvcpf04
To 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.
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Previous Episode - Proving Pick’s Theorem
youtube.com/watch?v=bYW1zOMCQno
Markov Chains
youtube.com/watch?v=63HHmjlh794
Written and Hosted by Kelsey Houston-Edwards
Produced by Rusty Ward
Graphics by Ray Lux
Made 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 Adamopoulos
youtube.com/watch?v=bYW1zOMCQno&lc=z12hcf1rayqazhrry04cdtbxknznfdvayto0k
Jonathan Castello
youtube.com/watch?v=bYW1zOMCQno&lc=z13lslqjqzzyzbbew22dwlh54zj4tpo0j
Niosus
youtube.com/watch?v=bYW1zOMCQno&lc=z13kwrpavz3igj32t230dnl4uun2hppow04