Science Slam: Andrew Conner

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04/05/2018 Added

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Andrew Conner, mathematics

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[00:00:02.917]My name's Andrew.
[00:00:03.950]I'm not a scientist, I'm a mathematician.
[00:00:06.218]And usually when I say that I study math
[00:00:07.558]that gets kind of the same reaction
[00:00:08.749]as if I just said that I punched your grandma.
[00:00:10.532](audience laughs) So if that's you,
[00:00:12.460]if you're math phobic,
[00:00:13.551]then you know, no worries, tune me out.
[00:00:14.692]It's a ten minute talk, just wait for the next guy.
[00:00:17.164]But if you've got a little bit more patience then
[00:00:18.970]you know I don't do science,
[00:00:20.881]even my math doesn't really have any applications.
[00:00:22.906]How exactly do I make this career out of
[00:00:24.531]something that people just hate so bitterly.
[00:00:26.948]Right, like what do I even see in it?
[00:00:28.723]And I can show you.
[00:00:30.570]So back in undergrad when I was a young warthog
[00:00:33.567]you know I thought that I wanted, I got really interested
[00:00:35.580]in this field of math that's called number theory,
[00:00:37.545]which is the study of positive whole numbers.
[00:00:40.297]So that's, from now on, all my numbers
[00:00:41.994]are positive and whole.
[00:00:43.236]The positive whole numbers,
[00:00:44.091]especially the prime numbers.
[00:00:45.603]But I went to this small liberal arts school
[00:00:47.100]and it didn't have a class in number theory
[00:00:49.318]so you know I thought, "Aw, man, no problem."
[00:00:51.144]I went to the library and I dug up
[00:00:52.491]this number theory textbook, went off to the second floor,
[00:00:54.708]sat down and cracked it open to page one.
[00:00:56.492]Right, 'cause optimal learning strategy,
[00:00:58.311]sit by yourself, talk to no one.
[00:01:00.556](audience laughs) Write that down, it's a tip.
[00:01:03.502]But you know, despite that right,
[00:01:04.465]the first few pages went pretty well.
[00:01:05.964]It was stuff that you know I had sort of
[00:01:07.234]picked up along the way as a math major.
[00:01:08.636]Right, like I know what a prime is.
[00:01:11.241]I know how many primes there are.
[00:01:14.062]I know what we do with primes.
[00:01:18.087]Uh-oh.
[00:01:20.542]Oh no.
[00:01:40.150]Okay, let's try this.
[00:01:42.061]Yeah, that's what I meant.
[00:01:43.822]There we go.
[00:01:45.183]So you know, none of that stuff was new to me.
[00:01:46.932]But I only made it to page five
[00:01:49.799]before I absolutely just ran headlong
[00:01:51.515]into this crazy wonderful beautiful problem
[00:01:53.693]that I had no idea, never heard of before
[00:01:55.668]and before I share it to you
[00:01:57.463]kinda needs a little backup, or background.
[00:02:01.022]The primes are absolutely just totally a mystery.
[00:02:05.072]We just don't know.
[00:02:06.711]There's some that we can say,
[00:02:08.052]especially just like global big large scale
[00:02:10.221]patterns that we do understand,
[00:02:11.747]but there's a lot of things that we don't.
[00:02:13.195]For example, like this handy dandy Happy Birthday balloon.
[00:02:16.759]If I hold it like this and take off this orange clip,
[00:02:20.276]I can tell you that the air is gonna go that way.
[00:02:23.255]But there air in this balloon is bouncing around
[00:02:25.388]at you know 70 degrees Fahrenheit, right?
[00:02:27.007]If you asked me, "Andrew, what's going to happen
[00:02:28.440]to this particular air molecule right here?
[00:02:30.149]Which direction is it heading?"
[00:02:31.858]Pfft, I don't know.
[00:02:33.668]I don't know, they're bouncing around, right?
[00:02:35.004]I can tell you that on average
[00:02:35.971]the air is going that way.
[00:02:36.840]I cannot tell you where any molecule is going.
[00:02:39.500]And the primes are the same way,
[00:02:40.413]that I can say big level things but I
[00:02:41.568]absolutely can't tell you small level.
[00:02:44.436]So here's a big level pattern.
[00:02:45.982]The primes get rarer as they get larger.
[00:02:48.907]Out of the first hundred primes,
[00:02:50.631]the first hundred numbers, there's 25 primes.
[00:02:53.754]Out of this hundred numbers, there's only 12.
[00:02:59.115]But despite that, right, this is a true fact,
[00:03:01.114]the primes do get rarer as they get larger.
[00:03:02.970]If you hand me a big number it's a very
[00:03:04.386]small probability that it's going to be prime.
[00:03:07.003]And the gaps between them grow, and grow, and grow,
[00:03:08.744]except sometimes not.
[00:03:12.001]Those are as close as two prime numbers can get, right?
[00:03:13.998]You can't be even and a prime
[00:03:15.106]unless you're specifically two.
[00:03:17.393]So primes that differ by two like that are called twins.
[00:03:20.910]And twin primes appear to persist.
[00:03:23.534]There are small ones, there are medium ones,
[00:03:25.430]there are big ones,
[00:03:27.305]and there are many, many, many more.
[00:03:30.935]Which is like nuts, right?
[00:03:32.025]How can, I literally just said that
[00:03:34.108]they get rarer, except no, they don't?
[00:03:36.334]They're close together, and far apart?
[00:03:38.979]Like, how, what?
[00:03:39.962]What does that even mean?
[00:03:41.259]But that's also kind of not a surprise
[00:03:43.343]with this physics metaphor that I'm gonna make.
[00:03:46.957]The primes, the individual ones
[00:03:48.469]are sort of pseudo randomly distributed right,
[00:03:50.323]you know, we can't predict them.
[00:03:51.992]So I have these lima beans
[00:03:53.524](beans hit the floor)
[00:03:55.293]and now they're pseudo randomly distributed on the ground.
[00:03:57.592](audience laughs)
[00:04:00.246]Some of them scattered, right?
[00:04:02.150]But some of them are close together.
[00:04:03.883]It's not a surprise that some of the beans piled up.
[00:04:05.950]And so it's not a surprise that some
[00:04:07.242]of the primes are only twins,
[00:04:08.880]only two away from each other.
[00:04:11.490]And this was first formally stated
[00:04:12.895]as the Twin Prime Conjecture in the mid 1800's,
[00:04:15.277]that the available evidence seems to suggest
[00:04:17.031]that that never stops happening,
[00:04:18.250]that there are infinitely many such twin primes.
[00:04:22.751]That's what I found on page five.
[00:04:26.378]It's so crazy, it's so cool but it was,
[00:04:28.648]you know everybody seemed to suggest you know,
[00:04:30.238]we'd find these primes and we'd keep finding them.
[00:04:31.878]It appears that it's true but nobody could prove it.
[00:04:34.004]It just sat around for over 150 years
[00:04:36.087]until this guy named Zhang in 2013
[00:04:38.254]absolutely came out of nowhere,
[00:04:39.988]hadn't published a paper in like a decade.
[00:04:41.898]He proved that there are infinitely many
[00:04:43.245]primes that different by N,
[00:04:45.398]where N is some number than he couldn't pin down
[00:04:47.379]but is was less than 70 million.
[00:04:48.778](audience laughs)
[00:04:51.418]It was like a joke, on yeah, oh big help, right?
[00:04:54.724]But he proved that theorems in this area are possible,
[00:04:58.697]like this is an attackable area and I can
[00:05:00.707]put a finite bound on it.
[00:05:02.713]And that was enough to get him a MacArthur Genius Grant,
[00:05:05.095]if you've heard of those.
[00:05:07.463]And it absolutely set the research community on fire.
[00:05:09.786]There's this huge collaborative team
[00:05:11.191]of mathematicians that within a year,
[00:05:12.974]they'd improved and found refinements
[00:05:15.407]in his arguments, that kind of stuff.
[00:05:16.911]There are infinitely many primes that differ by N.
[00:05:18.430]They got the 70 million down to 246.
[00:05:22.906]And then there's this other related
[00:05:24.162]conjecture in the same area that
[00:05:25.376]if we add that as one of our hypotheses,
[00:05:27.961]you can get it down to six.
[00:05:32.222]And if you would like to become world-famous
[00:05:34.413]and get a tenure track job at
[00:05:35.553]a university of your choice,
[00:05:36.657]replace the six by a two.
[00:05:42.397]But despite all this, this you know just
[00:05:44.547]amazing, wonderful, crazy problem,
[00:05:46.683]this book that I was reading casually mentioned
[00:05:48.931]in less than a single sentence that obviously
[00:05:51.241]triplet primes can't happen.
[00:05:54.272]There's you know three and five, that's a twin,
[00:05:56.382]but three, five, seven, those are triplets.
[00:05:59.013]Eleven, 13 that's a twin, but 11, 13, 15,
[00:06:02.005]not a triplet, 'cause 15 factors.
[00:06:06.389]What? That killed me.
[00:06:09.288]How can like, twin primes are just so impossibly difficult
[00:06:12.654]that if you make 70 million degrees of progress
[00:06:14.878]you get a MacArthur Genius Grant,
[00:06:16.516]and triplet primes are just so easy that
[00:06:19.267]he doesn't even work out the argument,
[00:06:20.504]he just says, "Oh yeah, sure, clear."
[00:06:22.551]How does that even happen?
[00:06:24.440]So I shut the book.
[00:06:26.008]I had to do a dot dot dot 'cause if
[00:06:28.146]talking to a scientifically literate audience
[00:06:29.530]would kind of give it away.
[00:06:30.363]But I shut the book and I tried to figure out
[00:06:32.760]what was that reason, what happened?
[00:06:35.384]But how do, if you're not familiar with
[00:06:36.741]mathematical style argument,
[00:06:38.647]what am a supposed to do to talk to you about prime numbers?
[00:06:40.849]Am I, you know take a hundred primes
[00:06:42.343]and give 50 of them a placebo?
[00:06:44.361]What, how do I even take a sample of prime numbers?
[00:06:48.104]They're are infinitely many of them,
[00:06:49.280]how can I even put a dent in infinity?
[00:06:52.032]The idea of experimentation literally
[00:06:53.623]doesn't even apply to this question.
[00:06:57.281]But none of that dumb stuff is going to stop me.
[00:06:59.402](audience laughs)
[00:07:00.604]And we can do it right now.
[00:07:03.200]So here's something that we can say,
[00:07:04.164]despite the fact they're we're having
[00:07:05.345]to stare down infinity and we can't take a sample.
[00:07:07.711]Here's a true fact.
[00:07:09.350]If you divide any positive whole number by three,
[00:07:12.609]you either get remainder zero,
[00:07:14.170]remainder one, or remainder two.
[00:07:16.409]You can't get remainder three because
[00:07:17.285]that means that you're not done dividing, right?
[00:07:20.004]And here's the idea I had,
[00:07:20.842]sitting at this graffiti-covered desk,
[00:07:22.556]staring at the back of the theater building.
[00:07:24.923]Is that I can rephrase this using algebra.
[00:07:28.675]The number that leaves remainder zero when
[00:07:30.171]you divide by three is a multiple of three.
[00:07:33.380]It's three times something.
[00:07:35.462]And like we usually do, I've chosen to
[00:07:37.097]represent "something" with the letter X.
[00:07:39.985]And that means that if you leave remainder one,
[00:07:41.161]you're three X plus one.
[00:07:42.851]And if you leave remainder two, you're three X plus two.
[00:07:47.140]And once we have this the variable defined in hand
[00:07:49.436]and we can say every number fits in exactly
[00:07:51.557]one of these buckets,
[00:07:52.722]then it's really easy to see why there's no triplets.
[00:07:56.685]One of them has to be a multiple of three.
[00:08:01.135]Let me show you right now.
[00:08:03.149]So it's possible that our first triplet
[00:08:04.615]is a multiple of three itself right,
[00:08:06.030]that's one of our choices?
[00:08:07.904]In which case, that's the form that
[00:08:09.495]the other two triplets take.
[00:08:10.564]But who cares, 'cause the first one is already not prime.
[00:08:14.702]It's also a possibility that our second triplet
[00:08:16.756]leaves remainder one.
[00:08:18.612]In which case, look at the other two triplets,
[00:08:20.349]specifically the middle one.
[00:08:22.482]I can factor a three out of that.
[00:08:24.349]Or as I like to call it, not prime.
[00:08:27.580]Also it's a possibility that our first triple
[00:08:29.326]leaves remainder two.
[00:08:31.011]In which case, check out the last triplet.
[00:08:33.918]I can factor a three out of that one, which makes it-
[00:08:37.341][Audience Members] Not prime.
[00:08:38.382]Not prime.
[00:08:40.291]And that's it, we're done.
[00:08:43.201]Every single possible number ever
[00:08:44.978]fits in exactly one of those categories,
[00:08:46.594]and every one of those categories is a problem.
[00:08:49.747]There are no triplet primes except for, you know the dumb
[00:08:52.286]trivial starting one, three, five, seven.
[00:08:54.241]We did it.
[00:08:58.924]I felt just so well, pleased with myself,
[00:09:02.829](audience laughs) shouldn't put a you know,
[00:09:03.895]good veneer on it, but I just felt so you know,
[00:09:07.038]this massive historical problem that was so
[00:09:09.728]difficult and had all these books written about it right,
[00:09:12.122]I'd taken it and brought it to my own doorstep
[00:09:14.289]and I'd done part of it myself.
[00:09:16.864]And now we've done it.
[00:09:19.066]And if you personally, each one of you,
[00:09:21.536]if you want to do something just
[00:09:23.303]startlingly beautiful in math,
[00:09:25.086]you don't need a lab, you don't need pipettes,
[00:09:27.703]you don't need equipment, you need no machines,
[00:09:30.248]you don't need an IRB, nothing.
[00:09:32.624](audience laughs)
[00:09:34.591]All you need is pen, paper, and an idea,
[00:09:37.857]and the pen and paper are optional.
[00:09:41.825]So let me leave you with some questions.
[00:09:44.667]I just told you that you can do math,
[00:09:46.073]so here's some math to do.
[00:09:47.441]What happened to one, three, five?
[00:09:50.430]I said three, five, seven was the only triplet.
[00:09:54.130]Also another question, are triplets like
[00:09:56.073]the 11, 13, 15's of the world?
[00:09:58.227]What about the 11, 13, 17's?
[00:10:00.794]Any obviously reasons why, or for or against those?
[00:10:06.189]Also one final question,
[00:10:07.886]what happens on page six?
[00:10:09.654](audience laughs)
[00:10:12.799](applause)

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Science Slam: Andrew Conner

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