 thoughts about mathematics and an enormous influence on mathematics in general. He passed away last August, and after he died I was asked to give a colloquium talk about his work at the University of Maryland and since then I've been studying some of his work and his thoughts about mathematics and that's what I'd like to present at this time. So let me briefly describe and give a biography of him. So Thursday he was born here in Washington D.C. in 1946 and grew up in Wheaton, Maryland. He went to college in New College in Sarasota, Florida and then in 1972 earned a PhD from the University of California Berkeley. At this point he was really revolutionizing mathematics in a big way and described a little bit of this later in the talk. Two years later in 1974 he was appointed a full professor at Princeton a good deal of charade at the age of 28. I met him after I was an undergraduate at Princeton and at Princeton University I had a tradition of encouraging undergraduate research and the senior thesis is a very big thing there and I was looking for something to work on for a senior thesis where the professor said, well we just hired this new guy that might be an interesting person to work with and they were right and I don't know. Started working with him my last two years as an undergraduate and wrote my senior thesis with him. He encouraged me to go to graduate school at Berkeley which is what I did. I ended up working with this thesis advisor, Mo Hirsch and when I finished he and Dennis Sullivan spent the academic year in 1981 of post-doc. So after a self-imposed exile to Berkeley I worked with him my last two years as an undergraduate at post-doc. After that in 1982 he was awarded a PhD. Somewhat later he had always been very much involved in experimentation in mathematics. I remember when I first met him I went to his office in Princeton in 1975 and he had this little calcomp plotter growing some incredible pictures and asked what it was and he said, will you take him to the irrational slow coagulation on a torus and apply the virus for us p-function? It makes a lot of sense now that he had always been involved in and was aware that technology could be used to visualize geometric phenomena and around 1990 he was instrumental in setting up the Geometry Center in Minnesota. So this was, I don't know if these things exist anymore but it was a Science and Technology Center and this was the one NSF-funded Science and Technology Center in mathematics. Later it unfortunately closed around 1995. Around that in 1991 he moved to California back to Berkeley as in 1993 he became the director at ISRI and a few years later he moved to the University of California, Davis and a few years later he moved to... I mentioned the Fields Medal that he won a number of other awards such as the Bevelum Prize in Geometry and last year, about a year ago, it was like meetings, 2012 it was more... So in this talk I'd like to describe some of his mathematical contributions I think that the, more than just the theorems that he proved and the ideals that he generated their whole approach to mathematics influenced and I, in researching this talk I was led to look at a very interesting paper that I recommended in 1994 in the full of the AMS article about on progress and proof and some of it is explaining what he thinks we're doing as mathematicians and this was in response to a controversial article by Jackie and Quinn about theoretical mathematics and lack of rigor and the harms of lack of rigor and one paragraph is directed to Thurston and his lack of rigor and part of the article is to say that the questions that they're raising are not really the right questions and what we're doing as mathematics and the first part of the article describes what it is and his idea is what we're doing as mathematicians and he says that the, there's just a statement how do mathematicians advance the human understanding of mathematics I think it's very interesting reading this and I recommend it the second part of the article is somewhat autobiographical and it talks about his experience in the areas that he studied so I'm not going to be able to describe in a short talk all the different contributions that he's made and I'm not confident to do that anyway but what I would like to talk about in sort of the historical order I'll tell you what I'm not going to talk about the first subject that he worked on was falliation theory answered the main questions that no one was even close to looking at this was done maybe up until by 1975 and I was given work that awarded him to a professorship in Princeton I was reading the papers when I started working with him that it didn't seem like anybody else was reading and the papers aren't that easy to read and they were sort of the end of the story and he addresses this in his article that people were congratulating for killing off the field that he had solved these problems that nobody else had thought about might be true and rather than trying to digest the ideas they left the field and eventually he had no one to talk to about the subject he left the field and remarks that more people knew about the subject of falliation in 1975 than they do now probably even more true now and I think this was maybe an interesting reflection on the self-destructive aspects of our profession the successes to kill the field and comments in the article how people were congratulating killing the field destroying jobs well he was more successful at creating jobs in the next two areas and so when I started working with him in 1975 he was giving a course on surface morphisms and two-dimensional topology and as a student interested in topology at the time people said well everything's known in one dimension and two dimensions nobody knows anything that's three dimensions but everything was enough there's a very successful theory of classifying manifolds in higher dimensions so first we showed that in dimensions one and two there's actually a very rich structure that hadn't really been addressed until 1997 so I'll say something about the theory of surface dipmorphisms which really led to an explosion relating two-dimensional topology and dynamical systems Riemann's surface theory and type-morph theory the following year I mentioned three and gave a course on three manifolds and in particular geometric structures on three manifolds and the most interesting geometric structures and these were things that he had been looking at since he was an undergraduate a new college there were three manifolds that had a hyperbolic geometry on them a hyperbolic three manifolds at the beginning of the course he described a notion of geometric structure which is locally homogeneous geometric structure these were the study of which really began to bear a spawn in the 1930s and these were ways of putting a geometry on a topological manifold where the coordinate charts map into a certain geometry like a hyperbolic geometry or euclidean geometry and as you go from one coordinate system to the next one system of local coordinates to overlapping local coordinates then the coordinate change is given by an automorphism of the geometry and this is the context and he said that there were going to be eight geometries which would be useful for studying three manifolds and conjectured that every three manifolds would be canonically decomposed into geometric pieces and I think maybe this most significant mathematical contribution would be the geometry geometricization of the injection which was recently proved to be a curl and I think it's maybe significant that the most significant achievement is actually the injection it's the open-endedness of our subject okay so these are the topics that I will discuss in this talk but they're just a small amount of number of things that I'm using so as I mentioned the theory of surfaces in three manifolds and hyperbolic geometry were very closely related to Riemann surfaces and modulated Riemann surfaces which is called typular theory about the same time as this was in the late 70s there are some remarkable developments in homomorphic dynamical systems and some of the ideas that were used in geometries are geometries in three manifolds and applications and analogs in complex dynamics I mentioned also that at the same time in the late 70s he was working with millen around a one-dimensional in topology the most important invariant of a two-dimensional space or three-dimensional space is its fundamental group and the universal covering space of a manifold looks like the fundamental group manifolds compact and the subject of combinatorial group theory say groups given by generators and relations acquired a new geometric flavor by through the work of Thurston and Gromov and the subject was renamed or renovated which is now called geometric group theory to which Thurston had a major influence before the geometry center represented an effort to institutionalize and really develop them visualization tools in a significant way at the geometry center they produced two videotapes which I highly recommend classics because I don't think anything like that will be produced now you can find them on YouTube Thurston really liked puns and the first one is called knot knot and it describes the hyperbolic geometry of a knot a knot is an embedded circle in the three-spaces and so the confine of a knot is what's knot in the knot and is the subject of this videotape it's a little bit strange because it doesn't really have it's unclear what the audience is so it starts at a very low level and in about 20 minutes into the talk you're gliding through a tessellation of hyperbolic three-spaces and somewhere it accelerates it's well worth looking at it was followed by another videotape also on YouTube called outside in which animates any version of the sphere that he came up with amazing picture of how you take the sphere and turn it inside out through a regular homotopy of inversions in three-spaces his efforts in visualization I remember I got involved in visualization after many years of post-op involvement I had been studying geometric structures on manifolds and in particular projective structures and really wanted to know what they looked like I realized at the time that it was going to be a big time sink to work since I had never done a computer programming before and it turned out to be a big time sink but I think it was very useful but I remember a discussion with one of my colleagues who said, oh this is very interesting they saw these pictures of things on the boundary of complex hyperbolic space but it's not really mathematics and it wasn't really rigorous mathematics but it certainly developed an intuition and it communicated some mathematical ideas and I think it really is mathematics but it wasn't a traditional theorem or proof and people would ask, well does it help you prove theorems? I never really was sure if that was the case or not because you don't really have anything to test against but it certainly developed intuition and it was a form of communicating mathematics and I think Thurston's realization that these are valid forms of mathematical expression was very much related to his interest in education and outreach about the time that he was involved in the Geometry Center he had developed a course with several other people called Geometry and the Imagination named after Albert and Conebossa by that name and as far as teaching and disseminating mathematics I might mention what I want the relationship of mathematics and art and last summer I went to the one day at the Bridges Conference in Tannisland and saw lots of interesting examples of things related to hyperbolic geometry and things made with three dimensional printers pretty amazing objects, you can see hyperbolic geometry dresses crochets that are made in hyperbolic geometry and it's probably easy to find these things by searching on the web and most recently, around 2010 working with the Japanese fashion designer applied ideas of geometrization and the eight geometries to fashion design and there's a nice article by Kelly Delpin in Math Horizons last summer that describes mathematics and fashion design some of these are some of the things that I will and won't talk about yes, so maybe I should emphasize the geometric group theory and visualization were really part of another major contribution that he made and that's the relationship of mathematics and computer science and the geometries which I'll say a little bit later in the talk all correspond to different kinds of combinatorial configurations and networks and there are some very interesting papers about using different kinds of geometries to model communications and in geometric group theory a group is a group given by generators of relations that's raised to a graph it's clearly a graph of a group and the combinatorial properties of that correspond to certain properties of the network and that's what I'm talking about computer science so let me say a little bit about how he first came on the scene in studying topology so algebraic topology maybe to be said to be done in the late 19th century worked upon karate and others and the techniques of algebraic topology led to certain invariance of spacelessness one would have the naive hope of using these invariance to classify spaces they have to homeomorphism so for example you might take a surface and then decompose it into pieces that were like triangles such a decomposition is called the triangulation and the simplest example so you'd have a triangulation so a manifold is a space that at every point it has a neighborhood which is homeomorphic to an Euclidean space that every point has Euclidean local coordinates and a triangulation would be a decomposition the simplices are very simple zero dimensional simplices are just points one dimensional simplices are line segments two dimensional simplices are three dimensional simplices would be tetrahedron etc and the simplest example of an algebraic topological invariant would be the order characteristic which in the case of the surface would be the number of vertices number of zero dimensional simplices minus the number of n-1 plus the symmetry for a closed orientable surface say if the surface has genus J and this is a complete invariant of homeomorphism well, the theory developed in the next hundred years and by 1970 it was known that these invariants generalizing the order characteristics led to invariants that could classify manifolds so the precise statement that these could classify closed, we assume that the fundamental truth is finite set of invariants that could classify these manifolds up to homeomorphism and this was the culmination of a great deal of work so very little was known at that time in dimensions three and four mentioned to it and settled much earlier things were very complicated because the fundamental group could be but if you assume that the fundamental group was a finite group then you wanted to know if that's were badly enough and had some very good algebraic colleagues that you could ask to compute various groups with one could effectively classify manifolds under these conditions and so this was sort of the situation of geometric policy around 1970 and so people wanted to who were looking for other subjects to to work on them more geometrically minded ones that often went into subjects like dynamic systems one of the major results that led to this huge success story in topology was the age cohortism fear of male which was enough to prove that the Tom Craig conjectured into the Bivenberger and I think around the time that Smale did that he thought that very interesting problems really were more in dynamical systems and trying to predict what he would have under an arbitrary homomorphism so in the 1960s there was an American school of dynamical systems developed and this was in some sense the context in which there were struggles sort of halfway between dynamical systems one example would be a vector field on a manifold so you might imagine at every point there's a direction that's specific and this gives rise to a flow an action of the real numbers by the homomorphism so you sort of take a point and you move it in the direction of the arrow and if you assume that this vector field has no zero then the trajectories fill up the amount so at every point it has a neighborhood which is filled up by these submetals in this case there's one dimensional and submetals are curves and this picture is what is a special case of a general object of geometry called a foliation so a foliation is a decomposition of a manifold into submetals just locally a manifold locally looks like Rn and the foliation locally looks like a decomposition of Rn and our co-dimension n-minus and these things exist in many natural situations and in particular some of the most interesting dynamical systems the Smale and the Nossoff and Sinai had studied generalizing the geodesic flows on negatives remodeling manifolds which hyperbolic manifolds in a special case which is very interesting in differential geometry one has a flat connection one gets a foliation degree complicated and interesting and so many questions that have been studied for manifolds could be then applied with a sort of dynamical aspect to study foliations and so the main thing when it's building the characteristic class theory was a classifying space the space that represented all possible kinds of topological behavior and the thing like the tangent bundle of a manifold could be understood in terms of maps into a classifying space so in this standard case this would correspond to a gross bond using the Gauss map where the tangent plane to a manifold lives well, Hefliger found a similar classifying space for foliations called the gamma and a foliation would have a classifying map which would be in the end of this very large space and the invariance the analog of characteristics of manifold were characteristics of foliations and there were some various examples of this constructed Thurston's first remarkable achievement was in studying foliations of the three sphere it's not immediately obvious that the three sphere can be foliated when divided up into surfaces that are all parallel if you know how to do it if you're down then it's challenging and he showed that there were so many foliations of the three sphere and his construction used hyperbolic geometry in an essential way he showed there were so many foliations of the three sphere that the third Hohmann-Tulbe group which was a group constructed out of maps from three spheres of the Hefliger classifying space so this is like a gross monion I think it is a nice infinite dimensional space should have to build out of lots of cells and this group which is usually going to be a finite regenerative helium group has a surjective homomorphism under the additive group of real numbers so it's really enormous so this result which he proved as a graduate student attracted a lot of attention his thesis concerned foliations of three manifolds also I won't say too much about it except that it has I think the most useless bibliographic reference that I can think of he references in his thesis W. Thurston Personal Communication so it's not particularly useful unless you happen to be there when you're talking to himself and I'm not sure I recommend this graduate students writing thesis so they don't get any ideas and anyway shortly thereafter he as a post-authent the institution for advanced study he showed the theorem through the following theorem I want I want to give the full generality just stated in two-dimension one so those of you who have a closed n-manifold now if you have a co-dimension one foliation if you foliate a manifold by hyper-surfaces of sub-manifolds of dimension n-1 then at every point you can take the normal line you give the manifold metric in that talk about the normal line and you pass to a covering space assume that that normal line field is orientable and get a vector field and so you get a vector field so the Euler characteristic has to be 0 if it has a co-dimension one if it has a the decomposed into n-1 dimensional sub-manifolds in a nice standard way first to show that the complex is true that this mild is actually sufficient up to that time many people were constructing examples of foliations of manifolds using things like open book decompositions and very beautiful ways of exploiting symmetry and very special constructions that didn't look like we would deal with the arbitrary n-dimensional manifold but Thurston showed that any manifold means relatively weak hypothesis did have this kind of structure let me say a little bit about the idea of a proof he with an assumption like this there isn't a whole lot to work with so he says trying to like the manifold divided up into simplices like we were discussing before and when you divide it up into simplices you can well, look at the foliation on the simplex so in the paper he draws this amazing picture a big bunch of leaves, foliation he actually puts a tree with branches, a trunk to illustrate the leaves and the important thing is to make sure that the leaves are all transverse to the faces of the simplex so he then has to adjust the foliation in the simplex so that it matches up on the boundary so the Heffler structures, the maps into the classifying space, they would correspond to foliation that it would have certain singularities so the first step is to make sure that it's actually a real foliation near the boundary which is not hard to do and then you get the singularities inside that you want to get rid of so he sort of projects that so he called the first step civilization something this picture of the tree with lots of leaves he shows how to civilize and then you sort of push it out to the boundary and that's what he called inflation and the main element in this paper is the civilization leads to inflation so this is the paper on foliation which would be that model if I could give this theorem well I think from colleagues working in the subject at that time people were intimidated by him and he left they started moving away from the field the papers that he wrote in this period I don't think there have been too many papers following up and it always struck me as being a bit strange that amazing ideas that solved problems that nobody was even close to seem to have gotten lost and he addresses that in 1994 so that destroyed jobs jobs were created in his later work on two manifolds three manifolds involved in this subject he was interested he was interested in a simple problem earthiness a sense of humor for Karen through all of this I didn't prepare any this talks on very high tech but if you look in the previous issue of the bulletin of the AMS there's a photograph and I'll just pass this around of a mural it's been in Heaven's Hall in Berkeley made before up until maybe seven years ago or so on the seventh floor there was a big mural that was painted of a really long simple closed curve on this surface and the discussion that he gave was that you imagine transforming the surface this is a disc with three holes removed the curve isn't by taking these two discs and moving them around followed by these two discs and so this first curve does it this curve that I drew doesn't get affected by that but then when you wrap it around like that then you get a longer curve you can see that on the last page when you imagine iterating this procedure and the fact that the curve doesn't intersect itself is preserved under this transformation under this homomorphism if it doesn't intersect itself and it's getting longer and longer then it's going to be following strands that are almost nearly parallel and in the limit what does it look like? in the limit it looks like a foliation having understood foliation so well and understanding some of the phenomenon occurring in an also dynamical systems he was led to develop a space 1975-76 this was the space of major foliation which was a kind of completion of the set of simple closed curves and now when I say simple I mean a curve that doesn't intersect itself the point is that simple closed curves can be quite complicated because as an example like this shows that they can wrap around many many times but they need to be very tightly constrained if they don't cross themselves and for example the simple closed curves on the torus these can be determined by their homology class they correspond to relatively prime pairs of integers and relatively prime pairs of integers of course correspond to rational numbers and how do you complete a continuous object completing the rational numbers while you take the real numbers and so the general picture that Burschen gets is you take a simple closed curve corresponding to a rational number p over q in reduced form is obtained by taking a curve on the square so the torus will be a plane with an integer lattice and then take a curve that has slope p over q and it will be a curve that goes around in the x direction p times or q times and in the y direction p times the p and q relatively prime the first curve you take will be simple and with the completion we'll let p and q be rational numbers approaching an irrational number that will correspond to a fulliation of the torus the square is identified by lines of that arbitrary irrational slope and I won't go into the details but Burschen showed that a similar picture existed for surfaces of higher genus and developed a space on which the morphism group of the surface active and this gave a normal form for a surface homomorphism group of dif-demorphisms of the surface homomorphisms of the surface Mondoisotope component in this space is a very important mathematical object called the mapping class group the structure group of the polynomial algebraic properties of the mapping class group are very important in studying dimension 3 and dimension 4 and using this theory he showed that gave a normal form for the element of the mapping class group his interest turned or began to concentrate in dimension 3 anyone that can know examples of what 3-manipals look like and I think this is the most lasting intellectual contribution picture of 3-manipals that he developed in the late 70's has turned out to be an accurate one and many of the conjectures that were made in the late 70's have since been proved in the work of many mathematicians in 1976 he just wanted to see what interesting examples would be so not compliments for one another example you might get would be from reflection groups and so say a little bit about what happened in the fall of 1976 you might take a polygon or a polyhedron and then look at say a group generated by reflections in the faces so for example you could take an equilateral triangle and start reflecting the equilateral triangle on its side and then you get a tiling and imagine doing the same thing in higher dimensions and imagine doing the same thing in other geometries such as spherical geometry so for example if you take a sphere you can chop it up into 8 spherical geometry and this corresponds to the 8 octets R3 and then get a finite group of reflection generated by the sides and in hyperbolic space you have even more freedom to build a polygon so in order to formulate conjectures one really needs a good supply of examples to test them against and this is a good way of doing that and he wanted to talk about a more general object which is going to be a manifold to the manifold quotient by one of these groups so you might imagine taking the plane its quotient by the group generated by reflections a fundamental domain is the triangle but the boundary and the vertices of the triangle have special significance so here you have it looks like a half plane and here it looks like a sector of 60 degrees and so these are sort of like manifolds that are folded up and when it developed a theory a folded up manifold so in the course of about a week people were talking about manifold debts to make sure that they were knew that they were talking about manifold debts rather than manifolds that they would fold it up manifold and that became somewhat objectionable so he then proposed that a new name a full name I don't know if that was much of an improvement and eventually after the enough disgruntlement it was decided to open it up since this is a democracy and we had a vote and the winner of the name was Horvathol which has now become quite standard this is a manifold which is locally in orbit space and the geometrization picture that he had maybe started with understanding some of these examples so let me close with the statement of what geometrization is and the picture of three manifolds the dimension two we have the Euler character it's a single term so we have Euler character plus two and the two sphere has euclidean geometry spherical next value of the Euler character is the Euler and the Taurus and the Taurus is represented by a square that's on opposite sides identified and that is right so we're euclidean geometry on the Taurus and then once the Euler character is negative you get much more veridation you get surfaces of higher genus and they all have hyperbolic geometry 19th century and so the geometrization of two manifolds is into these three three flavors well three manifolds are more complicated and another threat which Thurston united was the existing three dimensional topology it was known that every three manifold could be decomposed along two spheres so as you can see two spheres in Tauri are special among surfaces so you can imagine taking some three dimensional manifold and then connecting it up to another three manifold inside that two would be a two dimensional sphere this was a notion of connected sum you remove one from another and then glue the components together and Knazer and Milner showed that every three manifold could be uniquely decomposed as a connected sum so it makes sense to look at ones that can't be decomposed much later in fact in the 1970s it was shown that there was a similar decomposition along Tauri so this is the work of Jacob Shale and Neil Hansen J decomposition which is a decomposition along Tauri condition on the Taurus so first of all once you decompose along the Taurus there are lots of ways of filling it in with a two sphere boundary there's only one way to do it but for a Taurus there are lots of ways of filling it in with a solid Taurus so we want to look at three manifolds that are not necessarily closed but the boundary is a union of Tauri and then once you decompose along Tauri make sure that this was in a certain sense the geometrization conjecture of Thurston conjecture of Michaelon remaining pieces so it's more complicated than in Dimension 2 where every surface has a geometric structure one of these three types of geometry now you have to decompose it along a very specific way into various pieces in three dimensions it's more complicated than two dimensions here there are only three geometries in Dimension 3 there are eight geometries just list them three sphere with a spherical geometry and the Poincare conjecture can be stated that every three manifold that should have a spherical structure that has a finite fundamental loop is a quotient of the three sphere so this is the setting for Poincare conjecture there are also three manifolds that look like two manifolds like S2 process 1 this does a different geometry these are sort of forms that have some positive curvature the ones that correspond including a manifold potions are three by lattices and then potions of those there are only finitely many topological types grouped by your Biberlauch in the early 20th century there are some twisted versions from manifolds to potions by Milford Milford and Lee groups also some rather interesting three manifolds that are obtained by taking a hyperbolic automorphism of the torus an integer matrix that has real distinct eigenvalues look at the corresponding map of the torus to itself and then take a suspension of that an interesting geometry solved geometry basically well understood and then you have hyperbolic three space I'll put that down here because the picture should really look like this but most three manifolds should be hyperbolic then there were two geometries that have some hyperbolic two dimensional geometry such as the geometry of products and then you can have another geometry but it's not it's a twisted product which has been under a good model for that but these are the eight ways of putting geometry on three so this is the picture that he developed I think it's fair to say that nobody really saw this pattern and he addresses this in the article and proving this, he improved many special cases of the geometrization conjecture, involved bringing in techniques from all kinds of different mathematics dynamical systems from three-mapple topology number theory came in at certain points type known theory, inclinin groups which amazing equals quasi-conformable mapings which would become a very well developed analytic machine play a crucial role in his proof but the general picture was that the three-mapple had this pattern okay so I'll stop there but I should encourage you to look on the web to see how these eight geometries and the geometrization have applications to fashion this up I'll finish with a little story about myself every time I look at everything he wrote oh I know that, oh I was thinking about that because Thurston influenced Bill and Bill had somewhat of an influence on me and I kept thinking, Bill could tell me about this why didn't I learn this before but the weirdest contribution or where it affected my life is I took my parents to the Philadelphia flower show many many years ago when I was at the University of Pennsylvania and as I walk in they had this futuristic one and there it was playing on a big huge screen well back then it was a bunch of little screens not not and then I went to buy later on that that year I went to buy a washer dryer and they had somewhere they had a big screen TV there too local appliance store and lo and behold not not was playing on that too so it was just haunting me haunting me I didn't really like the movie and I watched it all the time we wouldn't want to watch it all it was dinner anyway are there any quick questions otherwise we'll have to talk to them afterwards because it's a little late maybe a quick comment a very big outreach program and I don't know if you could because I know that while he was director at MSRI Howard and I think that was when we established the best link with MSRI yeah and he came here himself and gave talk and not only that he joined us to a party at Leslie's exactly and definitely his talk is something memorable as you said he used his computer you would think he's a pure mathematician so he would not be working with computers but he does very nice pictures so if you can add a little bit more about his outreach program that not really that familiar with that I know the person I know that he had done a lot at MSRI but also I went to West Chester University which is a pretty small little place and I went there because Thurston went there to give a talk and he would go a lot of places where you wouldn't necessarily expect to see