Jacob Lurie: 2015 Breakthrough Prize in Mathematics Symposium

so our next speaker is going to be a Jacob lorry from Harvard and about a few years ago I was giving a cloak him talk there and it's very fortunate for mathematics that Jacob was the audience I described in the course of my talk a certain conjecture of Andrey Vey from the 1950s which I thought deserved to be much more widely known and then the next morning Jacob came up to me and he says oh I thought about that last night and I think I know how to solve that and it was it turned out in the end it took some time for the details to work out but I was a sort of remarkable synthesis of applying ideas from geometry to a problem that came out of number theory and so it's really ideal example of the topic of what he's going to talk to us about which is analogy and abstraction in mathematics thank you very much so if you ask somebody on the street what mathematics is about they'll probably tell you that mathematics is about numbers when I was in school we learned that there are all different kinds of numbers there's counting numbers like 1 2 3 there are the integers where you also have negative numbers there's the rational numbers or fractions there's the real numbers where you include irrational numbers like pi and the square root of 2 and then when you got really advanced they were complex numbers where you also had a square root of negative 1 so each of these number systems is an enlargement of the previous one and for the most part the reason to consider these enlargements is that they make your life easier so one way that you can articulate this is to say that each of these enlargements allows you to solve equations that you couldn't solve before so if you want to solve X plus 4 equals 3 you need negative numbers for x equals 3 you need fractions X to the 4th equals 3 you need irrational numbers and so forth but this comes at a cost which is with each of these enlargements the kind of numbers that you're thinking about are in some sense more abstract I know what it means to have 3 apples and it's not so clear to me what it means to have a square root of negative 3 apples so this is something that has bothered mathematicians even throughout history so for example you can see that even in our terminology so some numbers we call real numbers and the square root of -1 that's an imaginary number so this terminology goes back to Descartes and it highlights a sort of discomfort that he had with allowing quantities like the square root of negative 1 and even 200 years later you can find this statement of Gauss Gauss is someone who certainly knew what complex numbers were he knew how to work with them he could prove all kinds of great theorems with them and about them but there was still something there that bothered him and even a little later than that there's a famous quote attributed to chronic or God made the integers all else is the work of man so this in other words by increasing considering these more and more abstract number systems were somehow getting away from the core activity of mathematics which is to study concrete things like the integers so these attitudes have very much shifted through the years and it's been a gradual shift but one inflection point that you can point to is the emergence of abstract algebra in the early 20th century so the notion more or less that is in use today of commutative ring was introduced by a mean auteur in the 1920s so a commutative ring is a collection of things that you can add subtract and multiply and addition and multiplication are required to satisfy some rules which are probably familiar to you like addition should be commutative here's a list of all the rules that they need to satisfy actually Emme notaire didn't require that there was a unit for multiplication nowadays we usually do so this is a definition the notion of a commutative ring it's we're sending him for a number system and I've already given you lots of examples the integers the rational numbers the real numbers the complex numbers those are all commutative rings and there are lots more examples so let me give you one right now though modular arithmetic so fix some positive integer N and consider the collection of numbers 0 through n minus 1 you can make this collection of numbers into a commutative ring I need to tell you how to add and how to multiply well what you do is just take two numbers that are in that range and add or multiply normally well you'll get an an integer which might not fall into that range but if it doesn't what you do is divide by n and take the remainder and that remainder will be back in the range of numbers 0 through n minus 1 so this is an important example of a commutative ring it has a name it's called it's denoted by this symbol here red Z mod n so anytime someone in mathematics introduces something new but particularly if it's an abstraction you legitimate question is to ask what purpose does this serve why should this be one of the things that mathematics is about so let me give you a couple of arguments so why should you care about other number systems well let's suppose that you care about the integers one answer is that by thinking about other number systems you can learn new things about the integers so here's this very simple example here's an equation which has no integer solutions so how could you know it has no integer solutions there's infinitely many possibilities for X infinitely many possibilities for why you can't possibly check them all but you could make a simple observation which is that the left-hand side of the equation is always an odd number and the right-hand side of the equation is always an even number so they can never be the same another way of phrasing that is you can check that this equation has no solutions in the ring Z mod 2 where you don't have to check infinitely many possibilities you just have to check two values for X and two values for Y so that's a simple example let me give you a little bit more sophisticated one there's a famous theorem of form ah not his most famous theorem but uh for most two squares theorem which asserts that every prime number of the form for n plus 1 can be written as a sum of two squares now there's many different proofs of this but there's one that is commonly taught in undergraduate number theory courses at universities and that proof proceeds by studying a particular commutative ring that the ring of Gaussian integers ring of consisting of numbers of the form X plus iy where x and y are integers by studying this ring you can learn new concrete things about the usual integers so this is one argument let me give you a related argument sometimes you can maybe you don't learn something new about familiar things like the integers but you can look at facts that you already knew in a new way so let me give you an example of that I'm going to give you two facts one of which is a very concrete statement about the integers the fundamental theorem of arithmetic every positive integer factors uniquely as a product of prime numbers so this is a concrete fact about integers and it has a cousin which is a more abstract fact about number systems so any commutative ring with finitely many elements also has a unique factorization as a product of local rings so rather than tell you what that statement means exactly let me just tell you how it plays out in the one example that I've already given you so if you have some positive integer n then Z mod n is a commutative ring with finitely many elements and it factors as a product and that factorization reflects the prime factorization of n so this statement goes by the name of the Chinese remainder theorem concretely it says for example that if you want to know the remainder when a number is divided by six it suffice us to know the remainder when it's divided by two and the remainder when it's divided by three so let me give you another kind of argument getting a little bit less conservative one reason that you might study other number system is that they're interesting in their own right we can find new and interesting questions that we would not have thought to ask before so let me give you an example of that so a field in mathematics is a commutative ring where not only can you add subtract and multiply you can also divide and field's what finitely many elements are very well understood so there's a finite field with n elements if and only if n is a prime power and in that case the field is unique so finite fields are very fundamental objects in many areas of mathematics you'll see them all over the place in number theory and algebraic geometry in discrete mathematics some parts of theoretical computer science they're very interesting objects and we would not have known to be on the lookout for them if we hadn't been open-minded about considering other number systems things that behave like the usual numbers integers rationals and so forth but maybe we're a little bit different I want to give you one more argument which really I think is the most convincing one there are just so many examples so once you know what to be on the lookout for these things are everywhere so I've given you some examples already familiar number systems like the integers rational numbers and so forth I told you about modular arithmetic and finite fields if you're a number theorist well there's number fields and their rings of integers if you study linear algebra there's matrix rings if you're willing to put up with non commutativity polynomials form rings coordinate rings in algebraic and analytic geometry and so on and so forth I'm going to stop there not because I've run out of examples probably every mathematician in the audience could come up here and add a few more so why should we care about this notion of commutative ring well this is just a basic pattern that shows up everywhere in mathematics that you have an addition and a multiplication that satisfy that list of rules that amino Tarot down so examples are just so common that we need to give this notion a name and it has become part of the common language that all mathematicians have in common so I just want to highlight a sort of shift that has occurred when we start thinking about commutative rings so if I give you a commutative ring R there's all kinds of questions that you might ask so mathematical questions like do you have multiplicative inverses in other words can you divide or is there something like unique factorization in R or are there zero divisors in R you could go on and on there's all kinds of questions that mathematicians can ask about number system and I want to contrast this with another kind of question which I would say is it's not really a mathematical question like ontological questions are these things really there what do they really mean now this belongs to a set of questions which while they might be interesting that they're not questions that mathematics equips us with the tools to say anything about so I'm telling you this story give it I want to give this to you as an example of one process that plays out again and again in mathematics that we start life interested in some particular topic like theory of numbers and after time passes well we're still interested in numbers but we're also interested in things that behave like numbers or number systems we're interested in more abstract entities and we have many good reasons for being interested in them so I'd like to describe another example of this paradigm in mathematics one which is much more recent and is in my own field the field of topology so topology the study of shapes so an example is the sphere another familiar example is the torus or a donut pictured here these are both examples of two-dimensional shapes but we can study for example one dimensional shapes like the circle and we can study all kinds of higher dimensional shapes which I'm not going to try to show you in this slide so these are some familiar examples of shapes I want to describe one which is probably going to be unfamiliar so I'm going to describe an arts and crafts project that you can do at home so the first thing you're going to need to do is make yourself a mobius bat so you you tear off a strip of paper and you glue one end to the other introducing a single twist now there's a picture of one up there when after you do that don't throw the rest of the paper away because actually you're neat gonna need to make yourself another Mobius band and a third one and actually keep going you need infinitely many Mobius pants so now you're not done so I want you to look at the line that goes along the middle of the first Mobius bat and I want you to sew that on to the second Mobius band around the edge so I've indicated what to sew on to what with the dotted red lines in this picture now I call this an arts and crafts project that you can do at home this step you might have a little trouble doing at home if you live in three-dimensional space so the Mobius band is famous for having only one side which will make this somewhat of an engineering challenge to actually build but this is a mathematics talk I'm just giving you the specifications for how to build this shape and we'll leave the rest to the engineers so after you perform this operation we're not done I want you to do the same thing again take the middle of the second Mobius band and sew it on to the edge of the third and take the middle of the third Mobius band and sew it onto the edge of the fourth and so on and so forth so the end result is some kind of shape it's an object that topology allows us to study it's belongs to the menagerie of objects with which the subject is concerned so I'd like to show you a picture of this object but just like it's hard for you to build it at home it's hard for me to show you a picture of it on on the screen so let me just give you a very inaccurate picture of it so this construction the general construction of this type is called a telescope construction so here's a very impressionistic picture of what you get so in this picture each segment of the telescope represents one Mobius band and each one is attached to the next in a way that isn't really accurately reflected in the picture so like many sciences topology has a taxonomy of basic examples which get individual names and this is actually an important enough example that it has one of those special names so this shape goes by the name of k'sia join one half comma one now I want to there's a this shape is actually closely related to a much simpler shape called the circle which has a name in the same taxonomy that's just called K of Z comma one so these two shapes are closely related and the relationship can be stated informally by saying that the telescope is what you get when you take the circle and allow yourself to divide by two so what does that mean well if you take a piece of string and wrap it around the front end of the telescope so it goes once around if you were to then slide that down the telescope to the next segment you would find that it went around twice and if you were to slide it down once more you would find that it went around four times so that's a sense in which the telescope is somehow letting you divide things by two so why would you want to do this so let me close for you a little one-act play so here's a conversation which is fictional but this may be not so different from what you might expect to hear if you wandered into a mathematics department at some University during tea time so Alice asked Bob what he's up to and Bob says well he's thinking about the telescope and Alice asks why the telescope what's the point and Bob answers well he's not really interested in the telescope he's really interested in the circle he had some question about the circle that he wanted to answer but some something about the question was too hard so we thought he might simplify his life by asking the same question for the telescope instead the telescope you can divide by two and just like it might it's convenient to be able to rely it by two and working with ordinary numbers it can be convenient to be able to divide by two when thinking about topology so this general procedure of taking a shape and making a larger one where you can divide by two has a name it's called localization it's something you can do not only to the circle but to many other shapes as well or any other shape for that matter so it was already hard for me to show you a picture of what happened when you applied this procedure to the circle I'm not even going to attempt it for a more complicated shape let me just mention that it is something that you can do for more complicated shapes and there's nothing special about the number two you can take any shape and allow yourself to divide by three or allow yourself to divide by four or any other number that you like so if you allow yourself to divide by all positive integers well then you're entering into the realm of what's called rational homotopy theory and there's a slogan just goes back a long time which is rational homotopy theory is easy what this means is that there are lots of questions that you can ask in topology which are very hard to answer but if you take those questions and ask analogues of them in the setting where you're allowed denominators where you're allowed to divide by any number you like then these questions become much easier to answer so this was made precise in the work of Quillin and Sullivan who gave two independently gave concrete procedures for taking questions of topology in rational homotopy theory and converting them into purely algebraic questions which often can be addressed concretely so the analogy that I want to highlight here is that taking the integers and enlarging them to add fractions enlarging them to the rational numbers it's analogous to taking topology and messing with it in the way that I just described passing two rational homotopy theory where you have also allowed denominators so these are both procedures which in some sense they take you into some more abstract realm but they take you into a more abstract realm where the mathematical questions that you meet are easier to answer and this is give you another more elaborate analogy the integers are two other commutative rings as topology is to the study of other homotopy theories or exotic homotopy theories so just like there's this notion of commutative ring which is a kind of mathematical structure that is like the integers in certain important respects there's addition and multiplication that satisfies the familiar rules but might differ from it in other respects well there are mathematical disciplines where you can study structures which behave like the theory of shapes in some important respects and might differ from it in some other respects and this is a sort of theme that has developed in the study of topology or algebraic topology over the last 50 years and once again whenever someone introduces an abstraction a legitimate question is what is the point why should we study exotic homotopy theories and really all the same answers that I gave you earlier are applicable here as well so for example one reason to study exotic homotopy theories is that they give us tools which we can use to answer questions about things that we already care about answer to answer existing questions in topology another example is that they're interesting in their own right there are all sorts of interesting questions and phenomena that appear in the theory of exotic these exotic whole motoki theories and we wouldn't have known about these questions we wouldn't have thought to ask them unless we introduced the appropriate definitions but I think the strongest argument is again these things once you know where to look for them once you know to be looking for them you see them everywhere so what are some examples of exotic homotopy theories well there's the theory of topology itself the study of shapes that's maybe not an exotic one I just told you something about rational homotopy theory where you simplify your life by allowing denominators somehow there are other simplifications theory of stable homotopy theory there are other variants of topology so the list of examples I've given you so far these are all mathematical disciplines that were introduced in order to have applications to classical to questions of topology but there's also a whole host of other examples which for which the intentions are different so for example home illogical algebra the theory of chain complexes and derived categories this is a something which is more algebraic in nature but it really falls into the realm of these exotic homotopy theories since Quillen it's been understood that if you pretty much take any sort of algebraic structure in mathematics at all commutative rings groups the algebra all sorts of things that mathematicians like to think about there's an Associated home otoki theory if you let yourself think about simplicial algebra of the appropriate type so these our last two examples are sort of algebraic in nature and there are also homotopy theories that involve a mixture of topological and algebraic considerations like the theory of structured ring spectra which is well a particular interest of mine and let me stop there not because I've run out of examples but again this list could go on and on so I just wanted to bring this to you as an example of well something that apology is about nowadays or algebraic topology specifically topology began as the study of shapes and of course we're still interested in shapes but we're also now interested more and more in exotic homotopy theories we're interested in taking tools from that originated in order to attack topological questions and repurposing them for applications in other areas of mathematics like abstract algebra thank you very much so we have any questions Jacob no okay it's all I'll ask a question so you mentioned introducing rational homotopy theory to simplify problems but what about the reverse that if you simplify the question but you still want to return back to the original question well that's of course you don't you lose information so but for example suppose that you were wondering if some construction was possible and suppose you were to find out that you couldn't perform that construction even in rational homotopy theory then you've you've just saved yourself some effort and there are also techniques for you know addressing the question of how you get back to the integral story from the rational questions okay what's [Applause]

Channel: Breakthrough

Views: 52,783

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Keywords: Jacob Lurie, Mathematics, Harvard university, Breakthrough Prize

Id: rUuECLebqtU

Length: 25min 19sec (1519 seconds)

Published: Thu Dec 04 2014

Reddit Comments

[u/G-Brain]

For those interested in this K(Z, 1) and K(Z[1/2], 1) business, see Eilenberg–MacLane spaces.

Does anyone have a reference about localization in this context, and about the "Möbius telescope" K(Z[1/2], 1) in particular?