Maria de Maeztu Annual Workshop | Arnau Padrol

e [Music] so this talk is about combinatorial combinatorial polytop Theory so let me start by telling you what I mean by a polytop so in this talk at least a polop will be the convex Hull of Finly many points okay so you have a finite set of points say in R to the D and you take the convex Hull and such a polytop can always be described as the intersection of Finly many uh half spaces when this intersection is bounded okay if you take only one half space then it's not it's not a polytop and these are examples of polytop this is probably the simplest example it's called a Simplex okay you take a uh some a finally independent points you take the convex H and you get a Simplex or a more like famous family of polytopes are the platonic solids uh so there are these are these five uh polytopes in R3 and and the way I drew them there's some symmetry this is given by polarity it's a simple geometric operation that will send this octahedron to the cube and what it does it it sends vertices which are the faces of Dimension zero two facets which are the F faces of codimension one so here you see this octahedron has six vertices and this Cube has six facets okay and here we have eight facets and here we will have eight vertices and in this case in general you send faces of Dimension K to faces of Dimension D minus one minus K so if you do the numbers you'll see that edges go to edges and that's why they have the same number of edges okay and the Simplex here in the middle it's self B there it will appear often this word simplicial simplicial polytopes are those where all the fa faces which are not the polytop itself are simplies like this one this one and this one simplicial poops are polar to simple poops let me just say the definition of being simple is that it's polar it's it's simplicial um in some sense most polytopes are simplicial because if you take a random set of points for anything any reasonable like definition of random and you take the convex H then you'll get a simplicial poly right it's it's like it's when you per Ser slightly the vertices you won't have four points in a plane so you you will never have faces which have more than three vertices for example have three okay but in the same sense most polytopes are simple because if you take a random set of house spaces and you take the intersection of of a random set of hard spaces that if you do some small perturbation you always get a simple polytop and this is important in some sense because we are counting and it always depends on how you look at the problem to to say what's like what's most right and I will mostly be centered in things that are vertex centered and not facet Center okay and if you want to do the other picture you just have to polarize okay so this is combinatorial uh so these were were polytop and let me tell you what's combinatorial about them and that's the phase structure so let me Define a phase as the intersection of the polytop with hyperplane that does not cut the polytop that does not go through the interior of the polytop okay so we'll have vertices when this intersection is of Dimension zero edges when this intersection is of Dimension One and in this case which we a threedimensional polto we have facets when the intersection is of Dimension two um if the hyperplane doesn't intersect the polytop at all then we'll have the empty set and I would like like call say that this is a phase and I want also want to say which is not compatible with my definition but I also want to say think in general that the whole polytope is a is a phase and the reason why I want to present it this way is that if I order the faces Now by inclusion this partially ordered set is a ltis okay and that's uh why what we call the face latice of the polytop and this is what's combinatorial about the polytopes for me when I look at the polytop I don't see or I don't own only see a geometric object I see a combinatorial object because I only see its face lates okay and this again important when you are counting because of course there are infinitely many polytop okay any subset of points give you a polytop when you take the convex H but I want to consider that up to some equivalence relation and my equivalence relation is having the same phase ltis is this clear enough okay so we learned on in Alberto talk it's good to give some historical remarks uh so I'll follow the the pattern uh except that I'm very bad at this so I I decided to cheat and to take one of the of the sources okay so this is groom bound's book on convex polytopes if you go to the Google engram search and you look for convex polytop you see that this started moving when this book was publ published okay it's it's like the seminar book on on combinatorial polytop Theory like now there are many more books on the topic but this is really the like the the first one for modern combinatorial polop Theory and if you start uh at the preface it uh it starts like telling you slightly the story of combinatorially combinatorial polytop theory and he says that politops have been with us since Antiquity I mean actually like if you go to UK's elements the last book was just the construction of this platonic solids so it's something that really goes back then he talks about uh Oiler uh oil's famous theorem this is the one that says if you have a three-dimensional polytop and you count the vertices plus the number of facets you get the number of edges plus two and which is considered to be the first landmark of of combinatorial polop theory okay so this is my like summary of this preface it starts with Antiquity and then we have oil relation and then there were like some work he mentions like goosi Steiner Sylvester Kay mobus gmany and tight and says however about the turn of the century there was a decline in the interest of uh convex poops and he gives two causes and they suit very well my the first one suits very well my talk because because the first cause is that there was this problem enumerating the combinatorial types of convex polytopes which has very hard even in dimension three and people decided that actually working on combinatorial polop theory was hopelessly hard so they gave up okay the second reason was that other mathematicians didn't see this as a serial Topic in general okay and I'm happy that this has changed and this has one of the proofs is that we have a new combinatorics group at UB and I'm very happy about this that this is like the first time that you know like nowadays goric is not anymore uh seen as a as a let's say lesser part of mathematics okay so we continue with this history and like you know like it starts the people start working and then they just hit this problem which is counting probably cting poopes and uh they just give up okay so let me do a parenthesis I will continue with groom bounds book later but let me do a parenthesis at counting polytopes so that just so that we are clear on what like what's the problem that that we want to talk about um so I I already mentioned this okay we will want to count up to some equivalence relation and this equivalence relation will be the combinatorial type okay okay so we want to say that two poops are equivalent if they have the same pH ltis H so like well in in the plane which is where I can do the pictures this is not extremely interesting okay they are only polygons so I want to say that all quadrilaterals are equivalent and they are not equivalent to a pentagon this is somehow has some extra hardness which are symmetries which again you don't see them in the polygons but when you are counting you will have to take into account the symmetries of the face ltis if you want to count things to be different uh so actually it's much easier or at least like you you separate like two problems which one is considering Symmetry and the other one is considering counting polytopes when what you count instead is labeled combinatorial types and this is when you have the vertices of your polytop you give them numbers okay let's say you have n vertices you give them n vertices so for me these two polytopes have the same labeled combinatorial type because if you look at the faces here I have an h12 and here I have an h12 and here I have an H2 three H2 3 and so on but I will consider this one to be different from this combinatorial type because here I don't have the edge one okay so during the whole talk when I'm talking about counting poops I'm actually talking about uh counting labeled uh po poops labeled combinatorial types in general you have a factor of n factorial from one to the other but if you want to really look like it's not exactly n factorial and if you want to mod out this then it's a lot of work okay uh because you have to understand the symmetries of the politops most poops don't have any but uh nevertheless uh so anyway this is what what we're going to do um so let's start let's start counting okay so in dimension zero there's only one polytope okay which is a vertex a point it has one vertex in dimension one there's only one polytop which is a segment which has two vertices and in dimension two if you look at the unlabelled combinatorial type you have one for any n n n to me so just if I forget n is always the number of vertices and D is the dimension okay so for every n that is at least three you have one pogon with n sides and if you want to count with my rules you have to take labeling into account okay so actually you have one way of labeling a triangle you have three ways of of of labeling a a a square and in general you have n minus one factorial div divided by two ways of of labeling a polygon okay and this is because actually polygons do have symmetries have the dihedral group but in general in higher Dimensions you won't have this symmetri okay and let me do a parenthesis here um which is strongly related to counting polytop but it's not counting poops which is counting triangulations uh so a triangulation will be a subdivision in this case of a polygon into triangles and I want to count triangulations is just for the sake of these parentheses the rules are different so now the polygon the vertices are fixed and we are only looking in the way of subdividing it and we can count how many ways are there of triangulating a triangle there's one two ways of triangulating a square five ways of triangulating the Pentagon 14 ways triangulating the hexagon and and so on and I wanted to do this parenthesis because these are the Catalan numbers so um but the there's nothing cat about them uh they are called Catalan because some Belgian guy called Jin Catalan studied them uh he did not invent them uh it seems like the first people who really like found the the like actually the formula for the Catalan numbers was uh Oiler and sner there's uh some nice pieces of of let's say uh uh I don't know what historic like history of math that tell you actually on the one hand who found these Catalan numbers which was not Catalan and then somebody also tried to see why do we call them the Catalan numbers so exactly where who was the first people person who like started like saying these are this number are the C numbers uh aart it's like you know Olan numbers it's not enough information because ier did too many things so that's uh why I think the Catal numbers stick uh so anyway the catalon number the nth catalon number is the number of way of ways of triangulation of triangulating a labeled n plus2 Gun and uh I'll start with the the segment I also say that they they have one way of triangulating the segment and since I have slightly more like the privilege of being senior is that I have like a 15 month minutes I have the time to give you a proof so let's count to the Catalan numbers and here's the proof you take your polygon and I want to count the triangulations uh and you know the edge n + one one will be attached to a triangle that will go to some vertex and then we have like two independent triangulations one of the left side and one of the right side okay and I'll do that for every possible neighbor of this Edge and this gives this recursive formula for the catalon numbers okay you have to sum for every I the triangulations of what you have here times the triangulations of what you have here and now you can take the generating function so this is a formal power Series where uh the coefficient of Z to the N is the n catalon number and this recursive function uh tells you translates into this equation for the generating function uh which you can solve and you get this nice formula which is one of the formulas handwritten in the in the wall of the CRM and now you can go check is if this one is the one that has the mistake or not okay so let's continue with the groom bound okay so we are at the moment where politops are not interesting anymore in general except for some people that are either like maybe on the boundary of polytop theory like alexandrov coxeter minkovski with one notable exception which is er Stein okay uh so he did a lot of work on on polop Theory during this period and in particular he made a theorem which is nowadays known as St theorem so let me tell you about St theorem and what he says is if you want to understand a three-dimensional polytope uh you can understand it combinator the following way so you can consider the graph which is made by the vertices and the edges of the of your three polytop okay so take any polytop so you take your a polytop and you consider the vertices of your polytop as the vertices of your graph and the edges of the polop and as the edges of your graph and you have a graph and you want to characterize those graphs okay he says actually it's relatively easy the graph of a polytop will be a planer graph meaning that you can draw it into the plane you see it just put a light bulb inside your polytop it will cast The Shadow on the sphere and then you can project the sphere to the plane so you have just drawn the graph of your poletop onto the plane and it will be three connected meaning that when you leave two vertices out you can still uh connect any two vertices of the graph and this is not hard to prove so this direction is actually quite easy to prove the down Direction and what he did is to prove the converse which is that if you have any three connected plan or graph then you have a convex polytop with that graph uh if you join this Pol this theorem that that characterizes graphs of polytopes with the fact that if you know the graph of a polytop of a threedimensional polytop then you know the whole face latis this is very this is only for three-dimensional polytop but if you know the graph of a threedimensional polytope you can't tell who is a phase because that's at are actually non disconnected non disconnecting Circles of the gra just again not hard to prove what this tells you is that actually like somehow the combinatorial types of three-dimensional polytopes is something very combinatorial I'm not talking about geometry I I only have to enumerate planer graphs that are three connected if I want to enumerate threedimensional po I don't have to worry whether there are coordinates that make something convex uh with this with this structure okay St did that for us and this is I this is like actually I I want to like highlight this this is what makes a huge difference between three-dimensional polytop Theory and higher dimensional polytop Theory okay this is no longer through through in in high Dimensions uh in particular for example if you have anything that looks like a polytop but it's not convex at all but like you say topologically let's say you have a a simplicial complex of Dimension two that isomorphic to a sphere then you can realize that uh as a convex as the boundary of a convex three threedimensional polytop okay so which means that every simplical two sphere is polytop again these are things that will no longer be true in higher Dimensions uh and thanks to this theorem now enumerating threedimensional polytop which is actually the very hard problem that gr alluded to this is essentially solved nowadays it's is very combinatorial because you only need to enumerate graphs with certain properties okay and I mean to tell you how precise this this is already in 1962 I mean there's some technicalities which are again always the problem is symmetry so want to kill symmetries but if you like if you root your polytop so that you are killing some symmetries and you're only looking at simplicial polytop that he's not even giv asymptotic approximation this is a a precise formula this is an exact number okay for any so this problem is so this is the simplicial case if you don't do the simplicial fa case there's again a precise formula uh we know anything okay except how to deal with the symmetries which are very very rare but if you want the exact exact exact number you need to know how these symmetries play so that that's the only thing that people are still working and and have not managed to solve but otherwise like you know like as everything is known asymptotically about uh poops in dimension three and what happens in dimension four is that we have a phenomenon that will kill all these approaches and all the hopes of solving this which is universal faity so um this to present this I need to introduce the concept of realization space so now what I want you to imagine is I I choose one my combinatorial type my preferred combinatorial type say the cube and now I want all the geometric polytop with the same combinatorics okay so to parameterize this I can just take the coordinates of the vertices if the convex H have the face lates that I want then so you know like if I have a polytop with n vertices uh then in dimension D I can make a matrix of D * n with the coordinates of the of the of my polytop take the convex H if this has the combinatorial type I want I put it in my set okay and this is the realization space which is a subset of R to the N to the times D make sense okay so this is uh you know it's it's the opposite of modulized spaces in the sense that this is only one equivalence class of the modul space okay it's the one equivalence class of this combinatorial equivalent that that I'm talking so this is the realization space and what happen so realization spaces of threedimensional polytopes are nice if you have two things that look like octahedra you will be able to continuously go from one to the other and through the path you only will have octahedra okay this is no longer true in the dim menion four and and larger actually quite the opposite is true like imagine your worst nickmar you know like the ugliest thing you can imagine in the Roy like and and and this is like well there's some big words but imagine your like any semi-algebraic set there will be one polytop that encodes this semi-algebraic set in a sense that the realization space will be as ugly as your favorite algebraic set but semi-algebraic sets are quite flexible for particular they can be disconnected or they can be anything so that means that these realization spaces can be as ugly as any SE algebraic set which is uglier than what you want okay in particular it's very hard to decide whether a sem algebraic set is empty or not okay and this tells you that it's very hard to decide whether a polytop exists or not and what I mean by that is like if if I I don't want to go back to the slides but imagine you go to one of the first slides where there was a face lce okay so I I give you the problem I I give you the phas lce and you have to decide is this the phas lce of a polytope or not okay so this is the problem and there are some simple checks you can you can do okay uh I don't know like every phase of Dimension D has to have at least D+ one vertices but this problem is hard it's hopelessly hard hard you will not be able to solve it okay it's at least NP hard it's it's actually exist are complete which is to be equivalent to being able to solve systems of of polinomial inequalities okay so we will not be able to decide whether a phas ltis is the phas latis of a polytop or not so there's like no hope for combinatorial enumeration okay anything that we count can count uh if you are able to count you are able to decide whether it exists right and I'm telling you that this is hard so all what we can hope for is some approximation which would be better or worse but like exact enumeration is out of the out of the out of the game okay you have questions I'm I'm going on with the groom bound uh so we were left with uh despair um I mean well universality groom bound didn't talk about it because it was not discovered at that time um but uh he was uh we were at the time where poops were not interesting okay and then uh something happens and this something are computers okay uh and you know like people needed things that you can compute Pon and polytop actually turned out to be very useful for computers and actually all combinatorics had like regrowth okay and um and in particular for polytopes was the linear optimization was what was the a big motivation because you could sell your research saying that you were uh doing linear optimization or just you were doing linear optimization and that gave you interesting problems on poops okay uh and that was what was the reason for the rebirth of of korial polop theory and he mentions as one of the focal point points uh the phenomenon of neighborly uh poops so let me tell you what a neighborly polop is uh so yeah I didn't read everything but he talks about neighborly politops and the upper bound conjecture um uh so let me tell you about this um so neighborly polyon and I won't present you the upper bound conjecture but the upper bound theorem uh was proved some years later um actually the story is uh so in he mentions the upper bound conjecture by mskin what we have of modin is an abstract uh of a talk he gave where he doesn't present a conjecture but state man okay uh he says this uh we never saw a proof by but he also makes other statements that later turned out to be false so I think we can assume that like he did not have a proof he was just a so that's why it's called the upper one conector now for that we have a proof um so the problem here and now you have to go like beyond your three dimensional uh like mental picture um so let's say I give you a polytop with n vertices okay what's the largest number of edges you can hope for like I want you to give me an upper Bound for the number of edges of a let's say of a four-dimensional polytope edges is one the dional faces I give you an upper bound which is well best you can hope every pair of vertices are joined by an edge right so the upper bound the trivial upper bound is n choose two okay and let's continue what's the most you can hope for the number of two dimensional faces where if you are lucky uh you have like triangles and every three vertices will make your Tri a triangle so you will have and choose three okay and the question is well the first is can you do that and if you can have imagine you can have the and just too many edges how far can you push it can you do it for all Dimensions well the answer is you can do it up to Dimension and half okay so a polytope is called neighborly if up to Dimension and half you have all possible uh K dimensional faces so for every K up to D Hales you have that any subset of size K is a phase um uh so these are neighborly politops and these actually so they clearly maximize the number of faces in low Dimensions but they actually do also they all have the same number of faces and they also maximize for higher dimensions and if you're curious the maximum number of faces is uh of order and to the dves okay maximum number of facets okay and this is the upper bound theorem that tells you what is the maximum number of facets or of Faces in every Dimension that a polytop in dimension D with n vertices can have if you want to construct one such polytop that has the maximum number of faces um uh which is neighborly you can do it this way so take what we known as the moment curve which is the curve t t² t to the 3 T 4 Etc ET and take end points in this curve and you take the convex H you'll get a polytop which will always have the same combinatorial type it's called a cyclic polytop and it has the property of being neighborly and of maximizing the number of faces okay and what actually what mskin said was that this uh polytop the cyclic polytop would maximize the number of faces in every Dimension which is true and then he said and this is the only polytop that has this property which is false there are many more polytop that have the property of being neighborly okay let me uh actually it was so why did I put two names here well McMullen proved this uh the upper bound conjecture in 1970s so shortly after the Brun B's book but actually something much stronger is true that was proved by Stanley uh which is that this is also true if you get if you only take something which is topological so a simplicial sphere that has so a simplicial complex homeomorphic to a sphere that has n vertices in and it's a D minus one sphere it also has this uh number of faces maximum okay so if you want to have a simplicial complex that isomorphic to a sphere and that has n vertices you will not have more than n to the Daves uh many simplies uh and this is also very uh a very important result not only because of the result but because he used uh commutative algebra and algebraic geometry and and this is the birth of uh combinatorial commutative algebra uh so now that this is like a a very important field uh eventually this Le led to Jun who's uh Fields medal but it Go all goes back to this paper by standing um okay corollary uh can do our first bound because we know that we don't have more than a certain number of facets let's call M and I have n vertices how many combinatorial types do I have well every combinatorial type is given by how do I like like spread the vertices into the facets okay so I have M facet Let each facet choose its D vertices okay and this gives you one trivial upper Bound for the number of combinatorial types which is of order n to the N to the thehab okay we said this was valid for spheres so this is a a valid upper Bound for the number of simplicial spheres okay this was uh first observed by C but it's really uh a trivial um up or bound okay so here it's saying this is the number of facets and every facet will be a subset of D vertices so this is the number of possible subset of D ver subsets of D vertices so here is every facet let let you let it choose its subset of the vertices this is a very coarse upper bound is it too is it so coarse and the answer is no okay it's actually not so far from the truth I mean okay when when you're working with such big numbers this not so far from the to truth it's it's huge okay so this is two to the N to the DS and this is n to the nend to the dhabs but the difference between these two numbers it's huge okay it's a it's a log n in the exponent but yet uh I mean it's reasonably small because it's only a log n in the exponent okay but I mean these things sometimes when when you work on this you you lose perspective you say you know like well like every tiny Improvement you have to produce many more you know like so so how do you do this you do a construction for many simplicial spheres okay if you want to do some more but if you want to improve like you have to like do many many many more then like a small Improvement they won't they won't appear in the in these in these numbers okay so magnitudes uh so this is what we know uh for simplicial spheres there's two dates because like we have like one for OD Dimension and one for even Dimension there was like something about this rounded down and we know that we don't have more poops than spheres because every polytop like if we stay in the simplicial case every polytop is a simplicial sphere the boundary but of course we don't know if it's far from these or okay uh and it's far okay so this is not so many poops so there was a first paper that the title tells you everything need to tell you in this section which is there are asymptotically far fewer politops than we thought okay then this was slightly improved later and this is what I'm setting and this is an upper Bound for the number of the dimensional polto with n vertices okay which is I I like to draw it this way n factorial to the D squ okay this is very far from what we have here which is two to the end to the to the to the to the two to the end to the deal okay so this this two to the N to the D this is 2 to the N log n okay when D grows this is very very far uh of course in dimension three not I we just said that every simplicial sphere is polytopal but in general most spheres in high dimension are not polytopal okay and this is something funny well funny like we know that if you take any random sphere it will not be polytopal just the numbers don't like the polytopal are we know very very few spheres that we can prove that they are not polytopal okay it's very hard to decide whether a sphere comes from a polytop or not so actually we have very few examples that are handcrafted that show that this sphere is not polytopal even though we know if we take one at random it will never be polytop uh and let me tell you how you prove this and we you use this theorem by Miler and Tom that's bounding the number of sign patterns of pols so you take some pols okay and now every point will evaluate so real polinomial will evaluate either positive negative or zero on that polinomial so that's what I Tred to describe with this small picture okay and this serem bounce the number of different sign patterns that will appear okay so this point for example is negative on the first actually this is wrong I think this is positive so sorry for the mistake so this point should be positive for the for the first polinomial negative for the second and positive for one so you have number of sign patterns and this theorem bounces number of sign patterns if I remember correctly the way of proving this is you bound the sum of the bety numbers this bounds the number of connected components and uh so actually first you do it for those that have no no zeros this bounce the number of connected components and this uh bounce the number of sign patterns without zeros uh and how we connect these to polytopes and and I I think forget maybe there's a lot of information in this slide let's just look at this picture and what I want to tell you with this picture is that if you know the orientation of every Simplex spanned by your points so here I have a triangles and I want to know if they are positively oriented or negatively oriented with respect to the order what whatever order that you have chosen what I want to say is that if you know this information you can tell whether some think is in the boundary or not and this is the picture I want to you know like if I have this Edge which is in the boundary now if I do look at the orientation of this triangle with respect to this order is the same as the orientation of this triangle with respect to this order and the same as the orientation of this triangle with respect to this order so all these orientation match and this tells me that this Edge is on the boundary whereas here if I do this one this one and this one if I look at the orientation it's opposite from this one this one and this one and this tells me that this Edge is not on the boundary because it has points at both sides of course orientations this is just looking at the sign of a determinant so now here are the polinomial systems I was looking at the previous slide uh so this I can make a polinomial system given by these determinants and the sign patterns is uh again very coar upper Bound for the number of polytopes okay and if you plug in the numbers you get an upper bound of n factorial to the d^2 plus uh o uh plus something which is two to the end of nothing uh and this is like the the end of gr bounds uh introduction almost uh it says that well now Research into commun St of comx poop has growth in at an as rate and nowadays I would dare to say it's an established field and with a very active Community then uh so this doesn't end introduction then he says what he will teach you in in his book but I didn't copy that uh so this ends the introduction and let's start with the talk H so I want to tell you about something that happened after grun bom and I was involved with respect to this problem and is uh so I just sketch an upper Bound for the number of combinatorial types of polytopes uh and I want to tell you my work on the lower bounds on the number of Comm poops and this is the story of the problem almost so uh the first B there was a first big bound of order two to the n loog n so these in some s this this this solves the problem because the upper B was also two to the n log n okay by shemmer and something which is amazing is that there's the word neighborly here so you know mskin conector there's only one neighborly polytop which is the cyclic polytop and it turns out that the largest family of polytop that we know they are all neighborly okay so this was wrong by by far okay there's two to the end loog and many neighborly polyps and then this was improved for simplical poop so we lose this property of being neighborly to n factorial to the D4 by alone so see that once you've settled this uh uh asymptotic uh dependence what you're looking here is essentially what's the constant in front of n log n this is a constant that depends on D and if you are looking at this game now shemmer proved a one half and alone proed the D and this uh topic was uh let's say dead for like 30 years until I did my phc thesis uh where I slightly improved alone bound to theal I want to highlight that I mean it's very tiny Improvement but it's a lot of work because there are very large numbers okay so you really need to produce many many many politops and also I wonder what neighborly here again um like I have to confess this is what not what I was looking for when I was working on this actually like I somebody gave me a problem where they needed to have a neighborly polytop with certain properties and I tried to construct it so I couldn't and so I worked very hard on making a construction that was very flexible to construct new neighborly polytop I never managed to find the one that has night property so I had to go to my backup plan was just count how many constructed and it turned out they were more than nobody ever constructed before later I proved that a polop with that property didn't exist so um but anyway but this is not the recent work uh I wanted to tell you about it's something which is more recent just last year with Francisco Santos and Eva Philip who is my PhD student until next month when she's defending uh where we managed to further improve this this bound to n factorial to D minus 2 uh without neighborly uh polytopes okay and what we do is we take the polytopes that we had here and we prove that all of them have many triangulations so we go back to this Catal and combinatorics so when I tell you about the triangulation in higher Dimensions I mean it's subdivision into simplees uh special family of triangulations are those that are regular which is you take your polytop and you lift it into one dimension higher you lift it as you want and you look from below and you'll see some subdivision and this is a triangulation this like if the heights are are arbitrary enough then this is a triangulation for example all triangulations of polygons arise this way but not all triangulations arise this way actually many most triangulations are not regular example one nice exercise is to prove that this cannot be done this way and I still have like two minutes so let me still uh give a hint of what I did uh so uh the relation between regular triangulations and polytopes is that if you have something lifted like this and you add a point at Infinity then you have the combinatorics of a polytop in one dimension higher and um we go back to the proof okay of the Catalan numbers but I want to think about it in a slightly different way so what happens here is imagine you have two vertices which are very close then every triangulation you can think of it as a triangulation of the polytop you get when you merge the two Pol the two the two points and you have to decide how to divide the the triangles that you have here some will go to the vertex W and some will go to the vertex V okay and you have to be able to decide which go to which side and this uh works also in higher Dimensions if you have two vertices that are close enough and this is like Clos and off has a very precise meaning then every triangulation like this one you can understanding understand it by taking a triangulation of the two Pol the two vertices that are merged so this is a triangulation of this so see the two vertices there are merged here and now in my triangulation there are some that will go with the with the red red vertex and some that will go with the with the green vertex I I don't want to give more details especially because I'm soon over uh over time essentially so what we do is we are actually prove we prove that they if everything goes to the red vertex is a regular triangulation if everything goes to the green vertex is a regular triangulation and you can continuously go from one to the other being always regular and you know you have as many as like small flips that happen in this path from one to the other and since my poops were neighborly and there were a lot a lot of faces you have to flip through a lot a lot of of faces and this gives you the the viability H and this is the proof and something very interesting on what happens here with respect to previous constructions is that previous constructions we had control I I I can give you the F here I know that there are many if I go through this continuous path but I don't know who this continuous path is I don't know anything so I don't know I know I've constructed many poops well we have constructed many poops but I don't know who they are and uh let me finish here thanks you for for your [Music]