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  • What am I ever going to need this?

  • I'm looking at your screen shot and I think the answer is never you are never gonna need this.

  • I'm Professor Moon Duchin comma mathematician today, I'm here to answer any and all math questions on twitter.

  • This is math support at records, fluorescent says what is an algorithm?

  • Keep hearing this word?

  • Hmm the way you spelled algorithm like it has rhythm in it.

  • I like it.

  • I'm going to keep it a mathematician.

  • What we mean by algorithm is just any clear set of rules of procedure for doing something.

  • The word comes from ninth century Baghdad where al who worries me?

  • His name became algorithm but he also gave us the word that became algebra.

  • He was just interested in building up the science of manipulating what we would think of as equations.

  • Usually when people say algorithm, they made something more computerese.

  • Right?

  • So usually when we have a computer program we think of the underlying set of instructions as an algorithm given some inputs.

  • It's going to tell you kind of how to make a decision.

  • If an algorithm is just like a precise procedure for doing something, then an example is a procedure that's so precise that a computer can do it at llama Lord 10 91 asks how bofa did the Mayans developed the concept of zero?

  • Everybody's got a zero in the sense that everybody's got the concept of nothing.

  • The math concept of zero is kind of the idea that nothing is a number.

  • The heart of it is how do different cultures incorporate zero as a number.

  • I don't know much about the mayan example particularly but you can see different cultures wrestling with.

  • Is it a number?

  • What makes it number three Math is decided kind of collectively is that it is useful to think about it as a number because you can do arithmetic to it.

  • So it deserves to be called the number at jess peacock says how can math be misused or abused because the reputation of math is just being like plain, right or wrong and also being really hard.

  • It gives mathematicians a certain kind of authority and you can definitely see that being abused and this is true more and more now that data science is kind of taking over the world but the flip side of that is that math is being used and used well about five years ago I got obsessed with redistricting and gerrymandering and trying to think about how you could use math models to better and fairer redistricting.

  • Ancient ancient math was being used.

  • You just close your eyes and do random redistricting.

  • You're not going to get something that's very good for minorities And now that's become much clearer because of these mathematical models and when you know that you can fix it.

  • And I think that's an example of math being used to kind of move the needle in a direction.

  • That's pretty good at chris ex chris explain news that is hard to say analytic valley girl.

  • I honestly have no idea what math research looks like and all I'm envisioning is a dude with a mid atlantic accent narrating over footage of guys in lab coats looking at shapes and like a number four on a whiteboard.

  • There's this fatal error at the center of your account.

  • The white board, like no mathematicians are fairly united on this point of disdaining whiteboards together.

  • So we really like these beautiful things called chalkboards and we especially like this beautiful fetish object, japanese chalk and then when you write, it's really smooth.

  • The things that are fun about this, like the colors are really vivid and also it erases well which matters.

  • You just feel that much smarter when you're using good chalk.

  • One thing I would say about math research that probably is a little known is how collaborative it is.

  • Typical math papers have multiple authors and we're just working together all the time.

  • It's kind of fun to look back at the paper correspondence of mathematicians from like 100 years ago who are actually putting all this like cool math into letters and sending them back and forth.

  • We've done this really good job of packaging math to teach it and so that it looks like it's all done and clean and neat but math research is like messy and creative and original and new and you're trying to figure out how things work and how to put them together in new ways.

  • It looks nothing like the math in school, which is sort of a much polished up after the fact finished product version of something that's actually like out there and messy and weird.

  • So Dylan john kemp says serious question, that sounds like it's not a serious question for mathematicians, scientists and engineers.

  • Do people use imaginary numbers to build real things?

  • Yes, they do.

  • You can't do much without them.

  • In particular, equation solving requires these things.

  • They got called imaginary at some point because just people didn't know what to do with them.

  • There were these concepts that you needed to be able to handle and manipulate but people didn't know whether they count as numbers.

  • No pun intended.

  • Here's the usual number line that you're comfortable with.

  • 012 and so on real numbers over here and then just give me this number up here and call it I that gives me a building block to get anywhere.

  • So now I come out here, this will be like three plus two.

  • I so I is now the building block that can get me anywhere in space.

  • Yes, every bridge and every spaceship and all the rest.

  • Like you better hope someone could handle imaginary numbers well at let Clara Vinny it says hashtag movie errors that bugged me the seventh equation down on the third chalkboard in a beautiful mind was erroneously shown with two extra variables and an incomplete constant boy that requires some zooming, I will say though for me and lots of mathematicians watching the math in movies is a really great sport.

  • So what's going on here is I see a bunch of sums, I see some partial derivatives.

  • This movie about john nash who is actually famous for a bunch of things in math world.

  • One of them is like game theory, ideas and economics, but I do not think that's what's on the board here.

  • If I had to guess, I think what he's doing is earlier.

  • Very important work of his.

  • Um this is like Nash embedding theorems I think.

  • So this is like fancy geometry, you can't tell because it looks like a bunch of sums and squiggles.

  • You're missing the part of the board that defines the terms.

  • So um do I agree with Jk Vinnie that stuff is missing from the bottom row.

  • I don't think that I do.

  • Sorry Vinnie at a D H S Jag club asks questions without using numbers and without using a search engine.

  • Do you know how to explain what pi is in words you sort of need pie or something like it to, to talk about any measurements of circles, Everything you want to describe about round things.

  • You need pie to make it precise circumference, surface area area, volume um anything that relates length to other measurements on circles needs pie, but here's a fun one.

  • So what if you took four and you subtracted four thirds and then you added back for fifth and then subtracted 4/7 and so on.

  • So it turns out that if you kept going forever, this actually equals pi, they don't teach you this in school.

  • So this is what's called a power series.

  • And it's it's pretty much like all the originators of calculus were kind of thinking this way about these like infinite sums.

  • So that's another way to think about it.

  • If you are allergic to circles because you're the only one bro why did math?

  • People have to invent infinity?

  • Because it is so convenient.

  • It completes us.

  • Um Could we do math without infinity?

  • The fact that the numbers go on forever.

  • 1234 dot dot dot It would be pretty hard to do math without the dot dot dots.

  • In other words, without the idea of things that go on forever.

  • We kind of need that but we maybe didn't have to create a symbol for it and creating arithmetic around it and create like a geometry for it where there's like a point of infinity.

  • That was optional.

  • But it's pretty at the fill which Alex, what is the sexiest equation?

  • I'm going to show you an identity or a theorem that I love, I just think is really pretty and that I use a lot.

  • So this is about surfaces and the geometry of surfaces looks like this.

  • This is called Minsky's product regions theorem.

  • So this is a kind of almost equality that we really like in my kind of math, the picture that goes along with this theorem looks something like this.

  • You have a surface, you have some curves.

  • This is called a genius to surface.

  • It's like a double inner tube.

  • It's sort of like to hollow donuts kind of surgery together in the middle.

  • And so this is telling you what happens when you take some curves, like the ones that I've colored here and you squeeze them really thin.

  • So it's the thin part, 1st Set of Curves.

  • And it's telling you that um this looks just like what would happen if you like pinch them all the way off and cut open the surface there.

  • You'd get something simpler.

  • And a leftover part that is well understood at Absa says what if Blockchain is just a plot by math majors to convince governments VC funds and billionaires to give money to low level math research?

  • No, and here's how I know we're really bad at telling the world what we're doing.

  • And incidentally getting money for it.

  • Most people could tell you something about new physics ideas, chemistry, new biology, ideas from say the 20th century.

  • And most people probably think there aren't new things in math right?

  • There are breakthroughs in math all the time.

  • One of the breakthrough ideas from the 20th century is turns out there aren't three basic three dimensional geometries.

  • There are eight flat, like like a piece of paper round like a sphere.

  • And then the third one looks like a pringle.

  • It's this hyperbolic geometry or like saddle shape.

  • Another one is actually instead of a single Pringle, you passed to a stack of Pringles.

  • So like this so we call this H two cross are put these all together and you get a three dimensional geometry and then the last three are nil.

  • This guy over here sol which is a little bit like nil but it's hard to explain.

  • And then the last one which I kid you not is called sl two are twiddle.

  • Really?

  • That's what it's called.

  • Finally it was proved to like the community satisfaction what is now called the geometry?

  • Ization theorem.

  • The idea of how you can build stuff out of those eight kinds of of worlds.

  • It's just one example of the publicity mathematicians are failing to generate.

  • Did we invent Blockchain to like get money for ourselves?

  • No we did not at Riley?

  • Alonzo is geometric group theory.

  • Just an ability anthropology.

  • And then there's this like my absolute favorite part of this is the laughing crying emoji because Riley is just like cracking herself up here or Riley's I think really saying here has to do with just like how much things commute, right?

  • So you're used to A.

  • B equals B.

  • A.

  • That's when things commute.

  • And then you can sort of do math where that's not true anymore for like you know A B equals B.

  • A.

  • Times a new thing called C.

  • That's just not the math you learned in school.

  • Like what is this new thing and how do you understand it?

  • Well it turns out this is the math of this model here.

  • This is a model of what's called nil or nil potent geometry.

  • It's pretty cool as I rotate it.

  • You can probably see that there's some complexity here from some angles.

  • It looks one way from some angles.

  • You see different kinds of structure.

  • This is my favorite.

  • I love to think about this one.

  • A.

  • And B.

  • Are kind of moving horizontally and see is kind of moving up in this model.

  • So that really shows you something about what Riley's calling geometric group theory.

  • You start with just like the group theory of how to multiply things.

  • And it builds geometry for you like you know it's sort of stringing a bunch of words together and trying to make meaning out of them.

  • And I think that's the joke here.

  • And like all jokes when you try to explain it.

  • It sounds desperately unfunny at ruth Townsend Law question for mathematicians, why don't we solve maths problems in a particular order of operations?

  • E.

  • G.

  • Why multiplication first?

  • This is like asking in a chess game how come bishops move diagonally?

  • It's because over time those rules were developed and they produced a pretty good game.

  • I could make up a chess game where the bishops move differently.

  • But then it would be my burden to show that it's a good game.

  • We could do arithmetic differently and we do in math all the time.

  • We set up other number systems with other arithmetic.

  • So you just have to show that they have some internal consistency.