Simon Donaldson: 2015 Breakthrough Prize in Mathematics Symposium

our next speaker will be Samuel Donna's son old Simon Donaldson from the Simon Center at Stony Brook in the morning we heard about a little bit about his work in four dimensional geometry and now he's going to tell us about dimensions seven and eight thank you so the the the subject I need to speak about is not something which I'm in a sense really a specialist in and when we get to the more detailed results later in the talk I'll be talking further understand about work of other people but this is I think a very interesting the fascinating area which I have got some kind of research involvement in and we were asked to be preparing these talks to speak about an area where the prospects for progress over the next five years or so and I felt this is an area where that might that might might be appropriate so I hoped I hope to if possible explain to you in this talk that there are interesting and hard questions in this topic because serving exceptional Helana me and what I probably can't really if we'd probably have no real firm grounds for believing but if a person can't really convey so much is a is a belief that actually we may see a substantial advance in our understanding of these problems over that kind of time period so there'll be two introductory sections in the talk beginning with one jewel so what I'm talking about here mathematical objects which are sort of special precious rare and which in a sense you completely understand a little hole in your in your hand as you'll see as you'll see from examples so an example of what I mean classical example would be the the platonic solids regular polyhedra the polyhedra space with the maximal possible symmetry where of course we have a small list of five known to antiquity and there are no this is all you get you don't get any anymore looking at three-dimensional regular polyhedra another example somewhat that's following on from Jacob Lewis talk is to consider speaking number systems where technically what I mean here is what we call normed division algebra so things satisfying mmm suitable set of axioms so there of course we have we have the real numbers the number line familiar with then the the complement the complex numbers think of as the the Argan plane and a complex number is specified by a a pair of real numbers and so that's something that's sort of familiar then perhaps perhaps less familiar we could go on to a full dimensional number system discovered famously by Hamilton in 19th century they called the quaternions which we can think of it in different ways so one way better right and down as we write any term in terms of for real numbers we have a basis element of the identity I J and K where I J and K squared at all minus 1 and they satisfy this kind of cyclic simile which I excite like permutations also this identity holes so the difference is that these are not commutative IJ is not is minus J I or another description will be to think in terms of the quaternions has built up of a pair of complex numbers with a suitable multiplication law and we could take take we can single out I has been preferred from IJ and K and then right at equatorial is Z plus W times J with Zed and W as complex numbers well third description making things more familiar in terms of for the vector calculus we think of a quaternion as given by one real number the coefficient of the identity and the vector a three dimensional vector and in the multiplication on the the the vector part is given by a combination of the cross products of vectors and the dot product as little as you if you because the identity multiplies in the obvious way so this this law tells you how to define the multiplication for any pair of quaternions so this forms a four dimensional number system in the sense that we want to talk about the difference is that it's not that multiplication is not commutative but otherwise you have many of the familiar properties and then you can go one step further to also discovered it in the 19th century we call the opto nians an eighth dimensional number system she went we went Chum actually write down the the definition of multiplication but one way of thinking about it is in terms of an opto nyan is given by a pair of quaternions they have eight coefficients in total with a certain with a certain law you could write down a for howdy how you multiply these things so they what's what you lose when you go to this stage is that you also lose associativity multiplication is not associative but we do get if we if we we take the imaginary quaternions so we can speak and write any quaternion as a mult of the identity plus complimentary piece so they play the imaginary quaternions just the complementary pieces in just the same way as before if we restrict the we take the imaginary component of the product in the quaternions then we get a cross product on r7 which is analogous there were many analogous properties to the ordinary cross product of vectors in r3 well we could write down a formula but the sake of this talk there's no real need to do that ok so this is an example of saying this this is a complete list of these norm division algebras there's no other famous old theorem that there's no other normed these particular no other number systems in this sense with a suitable precise definition of what we mean this is connected to another famous some classification result in mathematics the classification of us will compact Li groups essentially continuous symmetry groups continuous ways in which you have continuous symmetries of an object there are there are there are three families essentially corresponding to matrices with other real complex of quaternion entries and then there are some exceptional groups which are connected to the these octo nians Estonians are associative if you can't just do your linear algebra code in automatic where you can't multiply matrices with octonal entries but some part of what you would do sort of works and so you do get some exceptional groups which are sort of vestige of algebra nanak algebra innocence so in particular for the for what they going to talk about we just need to focus on one of these exceptional lis groups called g2 which is the the autumn or foursome group of the imaginary quaternions with them actually octo nians with this cross-product so the invertible linear maps from our seven to our seven which preserves the this cross-product so that's turns out to be this exceptional 14 dimensional Lee group g2 so again there's a there's a famous theorem the classification of lis groups actually or you could do a complex version with semi simple groups to be ended up being the same thing but these are these are you get complete lists of all the all the possible groups familiar ones and a few sort of exotic exceptional things so now it comes up to differential geometry the essential notion we want to recall is that of a multiple technial at or ssin free connection but let's just say this is a this is a notion of parallel transport of vectors and lot through our purposes so that's the familiar really a case of this is when we're talking about working on on a sphere we want to have a rule for how we parallel transport or transport a tangent vector as we move along a path in the sphere for simplicity discuss considered geodesic paths that say great circles and the rule is that you just you keep you keep the tan that keep the angle fixed with your your great circle you're moving along so you take your vector here you keep you move along this great circle keeping a fixed angle now we change direction and come down this great circle so again we keep the sikh fixed angle and then come we can come back to where we came with we can come back to here again keeping the same angle but we get a different vector of a basic phenomena with parallel transport on a curved space is that you as you go around a path and come back to the starting point you'll you'll generally come back to come back with a different vector from the one you started with so in fact given up give them on these connections we get up to the Helana me group which is just given by all the transformations we get on tangent vector as we take a base point look at all the loops of that base point each loop gives us a transformation of tangent vectors by composing loops you see that there's things form a group and we so we get a we get a whole onami group of all possible transformations we get by parallel transport but a famous results in differential geometry infected muscle pair J this is a classification of again the list form will say the compact alanna me holonomy groups torsion free connections and there's a special class a very special class what's called symmetric spaces so well we'll leave those out those of those that completely you can quickly write those down apart from that then again there are there are three classical families corresponding to well the real numbers the complex numbers and the quaternions so the real numbers will be some generic case complex numbers would be what we correspond to having a complex manifold that's called a Kahler structures of a differential geometric structure compatible with the complex geometry and then there are slightly different quaternion it come variants of that idea but for this talk we want to focus not on these three sort of fairly obvious families all natural families but there are two special cases which arise connected to these exceptional groups and have a special phenomena and it's doing algebraic where the octo nians so there's one one cases in seven dimensions there's another case in eight dimensions that we went this is very much related to the 7-dimensional one but we won't actually talk about it any further just to say there are analogous questions everything I'm going to say in seven dimensions or analogous questions in eight dimensions what is this seven dimensional case we're looking at seven dimensional manifolds with a torsion free connection with holonomy this exceptional group g2 we're talking about so what does this mean it means we have a manifold on each tangent space we have a we have a cross-product the model algebraically on this standard model we have a torsion free connection aware of transporting tangent vectors along paths and this should be compatible with the cross product if you take two vectors and multiply them and transport the product is the same as transporting them and they're multiplying them so that's just what unwinding the definitions what this means alternatively you can think about this it's not quite obvious but you can think about this in terms of these structures are given by Romanian metrics on a seven manifold with a special special property a special class of Romanian metrics that's because any any Romanian metric has what's called the levy chavita connection associated with it and so the connections were looking at always arise in that way so we can think of another way frontier we're studying a particular branch of Romanian geometry so these structures these these g2 structures you might think of them as a sort of rather recondite area in differential geometry but they are I mean the other hand maybe were there what we're all made of maybe they're the structure of the universe there is the M theory which I know almost nothing about beyond what I've written but anyway these g to these g2 structures are the fundamental idea in this the idea would be that space-time is really 11 dimensional for visible dimensions and the sort of tiny dimensions given by a manifold with one of these g2 structures so there is there's a lot of mathematical motivation for studying these things because they're very kind of a few separate special beautiful things if classify everything you've only got these possibilities but there is also a motivation from physics because possibly you're studying the structure of the universe so very maybe not but never we don't really so a second section this is some shorter this is also very general so but I want talk about is partial differential equations well I mean I think most of math petitions so I'm going to I'm going to be telling you what you will know so hung says everything everything in the world almost everything be safe can be described by a partial differential equations we so for example the oldest case would be Newtonian gravity and I thought I should write down an actual differential equation so that's fine put that that would be the terms of the gravitational Matter density and the gravitational potential is this is the Laplace operator of course or Maxwell's equations or the Schrodinger equation or fluid mechanics we possibly maybe maybe there's some control possible reasons to doubt whether this is actually a precise model of a fluid and fluid dynamics but well closer to what we are we can talk about that superbee I'm thinking about the minimal surface equation the surface the equation which disks which determines the shape of a soap film as illustrated so I mean we can take soap spanning some kind of wire it takes up some kind of shape what shape is determined by solving a partial differential equation I'm telling you what you all know so it's that these things are the kind of opposite of jewels they're everywhere everything is a partial differential equation it be possible to save the thing about partial differential equations because they're everything can be there must be but don't mean conversely if we if we have a partial differential equation which just comes from somewhere not necessarily from any kind of real-world application we can study it the solutions may do something interesting say something that's okay make sense we there's a whole world in this differential equation even though it doesn't even though it's not necessarily actually a a real world so a whole new world to explore so what does that this obvious remarks have to do with the first section the point is that finding one of these g2 structures on a seven dimensional manifold is solving a PDE I'm not going to write it down if you set it all up maybe you find you have to solver a PDF or something you can you can set it up as a PDF or a to form and then what you would get would be a kind of nonlinear version of the Poisson equation you get a posh definition equation whose linearization would involve this Laplace operator but there be lots of nonlinear terms which you could hardly even write down and no explicit way not easy to write down in a very explicit way but they're also there's not just these structures themselves they're also interesting things we can look at interesting geometry we can look at in a g2 manifold particular than a particular kind with something called associative sub manifolds which they're special forms of minimal sub manifolds so high dimensional generalizations with the minimal surfaces they're just they just define by saying they're tangent space is closed under the cross-product operation what we're supposed to have it also some in similar vein there are special solutions of the yang-mills equations which are nonlinear versions of Maxwell's equations also involving the notion of parallel transport but not just a tangent vectors but of more general objects so so we have all the moves we twist study well to study the g2 structures themselves and these kind of auxiliary objects in one of these g2 manifolds and all of these things are solving partial differential equations it's the point I'm making in this section so that's our kind of preliminary that's the introduction what are the prospects in what could be actually we'd like to know about these things what what can we say more one thing to say is that the topology in question is sort of well understood so I've interpreted a listen in good cases as well understood if you recall we just think about the topology of surfaces they are interval surfaces we can make these by taking the connected sum of copies of the torus this connected sum operation as indicated here and if we restrict to the case of it's called two connected seven dimensional manifolds which which are is an interesting it's a subclass but interesting subclass for these purposes there's a similar topological statement whether but our building blocks with whether it be braless the product of s3 at times s4 but actually you need to allow a kind of a twisted product in the same way as if you were doing non-orientable surfaces you might want to allow climb bothell summons which are can twist adverse of a Taurus these are fact well technically people to know this is what involving the pond tree argan class of the seven manifolds anyway at least in in in any case of interest you can write down you have a complete control of the topology of the manifolds you want to you want to consider and what is the state of knowledge that there are large this many thousands long of cases where these structures have been constructed initially if we work of Dominic Joyce in the 1990s a later kind of construction of Alexei Kovalev which was generalized more recently by corty Haskins Pass Feeny and Nordstrom so the upshot is you get huge numbers of examples in which from which various phenomena can be displayed such as some for two connected examples but there's no kind of systematic existence I think these examples with the way they all come from school gluing constructions so where you can take two things that you know with some of the simpler geometry glue them together to get an approximate solution of the equations and then used some sort of analysis argument to deform to an exact solution so ingénue ingenuity people produce examples but there's no kind of systematic understanding so let's just started if we had if we just if you just gave me a seven manifold chances are you'd have no idea whether it have one of these you have no idea one way or the other but they have one of these g2 structures but if you did happen to know it had one then another interesting problem arises that these structures come in fine dimensional family or moduli a similar to that key now i guess for the tight mother's face of a riemann surface so there's a whole manifold t parameterizing these structures there's also a period map which takes this Maps T to the ecology of your manifold which is a local and homey amorphous or diffeomorphism so locally the moduli space is identified with the third ecology of the manifold by this period construction though is a bit more technical you can put together the metric and the cross product so the metric you identify the tangent space with its dual so you can regard the cross product as a three form the condition that the distortion free condition says a closed three form and then this period map is simply to take the class of this closed three form in the Durham kamala G for those who know so it's really easy to prove this gives a local complete local invariant of the structure as we make deformations but again we go beyond that then there are the basic questions that we asked which we young for know almost a little about is this is this modulized space connected could there be two completely different components is this is this period map globally and embedding how can we supposing it was what would the image be in in the three-dimensional ecology so these are the basic questions which said essentially nothing really is known about in any general way what kind of mathematics is relevant to understanding these questions certainly the convergence theory in Romanian geometry is very relevant if we imagine we wanted to understand the boundary of this image of this domain piece so we want to take a sequence of these structures and say what could happen in the limit speaking so we can think of this in terms of Romanian metrics as I said so we'd like me a situation like we'd like to understand what can we say about the limit of this sequence of Romanian manifolds and there's a whole theory of such questions one could start with the notion of a Gromov hausdorff limit which is a very coarse notion and and then bets go on to refine that to consider g2 space not manifolds but g2 space with singularities and they're out there are results in this direction of the cheetah holding and churn so certainly some things are known but nothing essentially nothing in detail is known about the singularities that would arise that would enable them to get any kind of precise answer to our questions about what the boundary of this period domain is a present there's a large gap between we do know something what we'd like to know was kind of a big gap similarly these these some these other objects who say the associative South manifolds and these special solutions of the yang-mills equation they they're interesting in their own right but they could also shed light on these questions that we've raised about understanding the moduli but one direction we could hope to and define it have an enumerative theory of these things so to find a notion of counting associative sub manifolds i've seen a given homology class say to a right to arrive at a number that will be an invariant under continuous deformations of our g2 structure so this this would be similar if you could do this this will be similar in spirit to the well-known cases of gromov-witten invariants in symplectic manifolds where one counts whole morphic curves or of the same with the invariants of four manifolds defined by counting yang-mills instant tongs or cyber britain solutions or similar things or in terms of this periods one who thinks opposing sound observation that we take the the volume of undies associative sub manifolds it's just given by the the integral of the three four five so it's entirely entirely homological object is given by the pairing of this this period point with the homology class of our associative sub manifold so if we knew if we had a supposing we have our jeetu structure and we know we have an associative sub manifold in some homology class supposing we knew as as we deformed our structure we always had an associative sub manifold in this homology class then we would know that we always had to satisfy this constraint that the pairing in homology between the the period point and this homology class and thing would have to be positive by this observation because our associates have sub manifold has got a positive volume and we'd expect that the picture we'd expected as we approach the boundary of P or we might have a case where this would go down to zero because our associative cycle is of shrinking down to a point so as if potentially all of these things cuz have tied together we might start to understand what's going on but I say interesting phenomena I mean so I'm saying this is that this is a kind of a picture a dream you might say that we're gonna help many people have but but interesting phenomena and very large difficulties arise basically we don't understand the PDE well enough to be able to actually do any of this rigorously or even to know whether it's a well-founded idea whether even one should expect to be doing this in twenty years time or when there are fundamental obstacles to doing this maybe it's too naive for example one kind of phenomena this is a work of 30 of Hannah's nordstrom won't find a phenomena that occurs is that so anything we have a feeling we have a pair of associative sub manifolds so those are there's are three-dimensional things in a seven dimensional manifold so we don't expect that they intersect but if we want to consider families if we have a one of have a one parameter family of these structures things they'll move around so we expect that generically they might intersect in a one-parameter family so it's like this pitch shown in lower dimensions of a pair of a pair of curves in space which voted a certain a one-parameter family we can imagine changing from this picture to that picture changing a crossing like that so what happens as knowledge of its planes is that when when this when in in such a situation or happens is that when you drink this crossing a completely new associative SAP manifold is born in the sub in the homology class of the connected some of the indicated by this l0 here well it could be either a new thing is born or an old thing dies you can you can't really tell the fun but anyway something would the naive idea of counting the associative sub manifolds is not going to work because something drastic is going to change when you go through such a a crossing phenomena so this is the kind of thing the kind of new structure that you see by strictly by studying these differential equations which has to be taken into account in a putative theory along these lines so that's really all I have to say so I say a complete understanding of all these things a complete understanding of how you can take limits of these jitu structures or limits of families of associative sub manifolds all these things that really seems very a long way off because just as Brian White was explaining today but if we look at minimal sub manifolds in heiko dimension not Co dimension 1 then there's very little theory one can appeal to to understand these things but nevertheless I think although away it may not be possible to have a kind of a complete final word on these things in the immediate future I have a feeling that we're at a point where a lot of progress we made in the next five years in list in understanding watch understanding in more detail what should happen and that's getting a picture of what ought to be true and should be possible so I think it's an exciting subject to think about we have time for maybe one or two questions yeah does one of Miller's exotic seven spheres the 3-sphere bundle over the Forster does that appear in any way in these g2 well it does I over simplified in my account because when you take account of the differential topology you could you have that you could take the connected sum with an exotic sphere so there's a kind of a more refined classification of the smooth structure involving the z/28 of but and that has that has been completely analyzed by again Nordstrom and Crowley recently and an examples where you have interesting examples of having different smooth structures with different readers with the same total manifold with different smooth structures having g2 structures has been discovered so there are it does it does very much interact with those things but that's one of the one of the attractive things for me is that when I'm basically a low dimensional person so we always say oh we're high dimensions that follow Jesus everyone knows but but kind of all that beautiful topology is not often applied to actual problems so here is a situation where one can really apply the understanding from the 1960s and 70s of high dimensional manifold topology in very explicit examples and that's an attractive feature to my mind moduli spaces for different kinds of compare killer manifolds are given can be described in terms of the homogeneous projective varieties is now how does the moduli space for g2 structures well what do they look like the moduli spaces well that's what we don't know I mean this theory of periods is very much analogous to similar theories in algebra geometry but it's not it's somehow we don't have algebraic geometry to appeal to so there are no beyond what I said nothing is really known there's a local picture but nothing so the local picture would involve well just the gross Manion of was just it is just the whole if it's just very simple it's just a it is just this h stream but there's no kind of global understanding really you told us there are thousands of families of this do too many folds and they have all morals the same topology yeah and it's similar to what happens in in vocal obviously faults if you can see the cone if all transitions we have mild sweet conjectures it's in centuries only one model space need people speculated maybe different manifolds belong to the same family and one can make maybe some special progress one more space yeah that could be some analog of that guy I just maybe that's part of that a lot of this geometry is where we could say very analogous they're very analogous things in khalaby our geometry where we have a much better understanding because we have the algebra no but there are all transition that you go through non algebraic varieties if you get someone in Valle crossing happens in kind of non controlled way I don't know coma logically when to Lagrangian due to associative manifolds intersected and differential geometric information of topologically there's no control yes exactly it's no there's no way that we know is controlling nature if you have time for one more question we're know much all right well I said I couldn't really give you an intro I just feel that I just feel that there has been a lot of progress in the last few years I think I think it's partly pets because more people are working on maybe some these questions have been around many of them for 10 20 years but somehow they've come more into the limelight recently and so there has been a lot of work a lot of progress and I think so just maybe just weren't my hunch that that progress were going [Applause]

Channel: Breakthrough

Views: 9,971

Rating: 5 out of 5

Keywords: Simon Donaldson, Breakthrough Prize, Mathematics, State University Of New York At Stony Brook (College/University), Imperial College London (College/University)

Id: zpp_LGR5cWM

Length: 36min 12sec (2172 seconds)

Published: Thu Dec 04 2014