Infinitesimal computations in arithmetic dynamics (Adam Epstein)

is a pleasure and honor we speak we here at campus so many so many of the books that were cited here yesterday are with texts that were fundamental and not that l’├ęducation it much many of the many of the notions introduced discussed those two actually configuring here so okay so just to review the standing here so you studying rational deaths from love you want to be one projector space on the bench with parameters basic to smooth out by algebraic variety of dimension to t plus 1 and there’s a lot of work with some group of run like a in german it’s also known as both these transformations we’ll grab one to group with exxon back d by conjugation is a quotient space it’s a version space is itself a private health break or I even lose three dimensions that motion has dimension to t minus two okay the quotients fantastical enemies they’re not so bad I’m going to talk about way around the singularities when they talk about as dis lectures not about that okay i’m interested in various detestable questions concerning such spaces and maps between them this is just a sample so we have this quotient projection if you pretend that project spaces on that van folder if you look away from singularities or you say something about trends are thousands of orbits you can ask whether this quotient projection submersion meaning that the derivative is subjective you can ask an iteration to iteration takes you from a primary makes a lower degree maps and grab a base of higher degree maps evidently an algebraic or prism is when does this inclusive immersion and it’s a more technical questions involving the behavior of various dynamical variants of cycles or critical operations and have their cycles mr you’re low liner and that cut out to pose a condition I that will cut out of locusts in parameter space so in our wallet size D generously index with a good correlations you get these low side do you do several at a time where they smooth like the intersect transversely okay so so I’ve already forgotten there’s a quotient projection one is one is all rational have one is recommend other countries will our little he said when does question about it’s more weird where and when these newbies these are about your questions when also means where I’m at for which all you can further no matter how much you like the question will be further of what got anything elastic so it just just please wait okay so after what circumstances can one and we’re circumstances means whatever it is but I’m so in order to do this in order to do this we need to address certain basic issues you start running all this down if you handle various cases and coordinates but starts to get very very messy and bonding so to actually make any progress we need to describe the tangent cotangent spaces of our parameters based in intrinsic way and i want to emphasize this is absolutely not merely a matter of chemistry or aesthetics we’re interested in results valid over fields ok the talk says arithmetic okay so when we are made for example also include what characteristic thank you so even though this is a whole banana starfish and even though of course I’m secretly drink these oversee because that’s sort of what I really care about I want to do things algebraically because actually what I’m going to prove over city or something I’d like to prove over the upgrade rational numbers without reference to see and limit imagine that much of that argumentation about all that a lot of formalism might actually make sense / / characteristic p and it’s interesting to see how far one can go without having to commit oneself to a given characteristic so to do this we need to step that instead of working in the category of public sentiment folds

we need to build an outrage geography so I feel sort of give a quick review of some floral is owned this is just a gap is up here if you are not interested in characteristic paid or if you find this generality absurd just ignore it think seagull the piping you’ll notice i will never make reference to anything happened about see i would I can make reference to real and imaginary parts okay so basic talk members of up to isomorphism there’s a unique small town trip because field given characteristic rather than characteristic zero and is the build-up and numbers or is the closure of finest regardless cake and it turns out that in for various purposes including all of our purposes these are the only skills you need to consider okay many properties of such fields and wall first order properties which is just some type of food vision nothing all while saying it if you fix the degree essentially everything is saying thurs first back and sort of codified so many people heard of the sort of Elections principle basically said there’s just sort of one algebra geometry negative characteristic in particular the results that you get Percy of the jump your receivers sort of Jabbar are going to be the same statements whether it up whether or not you improve them the same way and enlighten me it might even be that’s a proof a result for over Q Bar the only way a moment is to in focus principle move over to see and do some complex analysis for example the proof of thirst and rigidity which of course is my secret motivation in all of us I would like to get around that a little bit to you right now I can’t but not happening in this talk but are setting up all the formalism there so that if someday somebody figures out how to do it you just plug it into it okay so just some words this is doesn’t this is just so in case anyone is bothered by what if I we need to consider a tangent vector to somehow great variety the flag will work the closure of the finite field there’s a formalism for this you can get out of an algebra this is what what is set up in basic algebra geometry classes so i’m going to discuss things in the cabin context of algebraic varieties which is probably not optimal if i were really our professional rounded faking it i would do schemes but there’s nothing i’m going to do a get that requires schemes and it would annoy people so just varieties and these ones being smooth varieties you can forget all this in a minute and just go back to your preconception about travelers base of rational maps except when i say would be legitimate / q bar or people so for example I will be I will be talking about can smoothness actors derivatives and so on and so forth but it’s all okay hook it up at a commutative algebra and this last point here it is just this is more or less this is one thing that distinguishes Brian schemes and it’s anything about transversality is somehow a statement along these lines so there’s only trivial Club elements in other words and others reduced version on reviews is the difference between looking at the zero locus of the equation of x equals zero vs x squared equals zero the difference you can attach a difference because of where the second one is a multiplicity I want multiple head about multiplicities alright this if i’ll get you more systems more fizzles what do you think they have to think they are but that you say everything you can’t just discuss these in terms of apology these objects come equipped so far structures use officially tell you what counts is a full working function even here there is a definition of catching vectors and turns of various it well did you do cocaine in space to get derivatives so you can talk about all that little works the Stanley the chain rule all that good stuff no quite you’re not needed but there are ways around that that are more elaborate just economy okay yep this is just this is the most arrogance of the most acknowledge I have your through good continues actually need to take this seriously and not and are doing the characters to key stuff for real then there is further complexity that comes about your because of is a notion of what gratification pains in fun ways so

for example you can have a up just easily write that maps now I have wonder maybe I’ve done oh they were you want to take these egos disease at the what happens you take the derivative let’s he goes to see in the Dean we all got to take derivatives the outbreak derivatives going to work late you’re saying is so the derivative is going to be P times seeks to P minus 1 okay there’s worse if weird characteristic p that comes out to 0 which means that for the further player vo taking derivatives that map behaviors it’s the constant compliments report is not okay so practically inseparable and they’re a pain in the neck from where it is just simply something which is also in the way and not quite as in length as inseparable but still some of the ladies wild travel occasion so in general there’s a there’s a sort of a you know unholy conspiracy that’s happened from the characteristic of your field and for example what we use for global degrees and you basically you here you have to think much more carefully and there are funny things that have mystic happens to divide any of these characteristics zero it’s not an issue so again if you’re only interested in see your Q Bar forget about this but just to show that check things certain statements by making are actually accounting for exceptional behavior here okay and the point is that if you restrict attention to the separable case and in other words where you don’t have to protect hero then a version of green apartment supplies things to hate more or less as you’d expect particularly if you count critical points properly which does not necessarily mean by the local pretty this is the business of wild gratification if you calculated according to revocation the application divisor of the crib three on the other hand gives a wild place here and you just take derivatives now your characteristic p the derivative of that comes out as one which means at least at least not here if you’re a half line space you have a critical point so the only places that you can have amplification is the boy okay that’s a wild thing because what we’re seeing you need to be rattled by our leads to a distinct ones see if all the ramifications sitting over that so this company’s the formalism someone I will lock them talk around wife releases alright she’s are your friends I know there is this allergy and irritation is over there’s a learning curve achievement honestly it might sound scaring the beginning that saved work much later on cold sheets and cold ology are simplifying devices to avoid unnecessary excursions into coordinate attendant asian bridge against 11-0 anyway Seether said look right at the end but he never do those occupations ok so the formalism of shields was discussed in panels talk I’m just giving some examples i’m interested in such types of objects I’m interesting typical horizon sections of all homered flied bubbles this might be a whole different functions or what the vectors neons will work differentials and then for every homomorphic version is a meromorphic version because you can talk about mariners exceptions what it means for a section when otherwise of such a bundle that important somewhere what the borders oh they see the modify these you can take for example we’ll get a look at vector fields upon dilution to that it’s a mnemonic lie and say well actually I really want to be requiring zeroes at certain points in other words being homomorphic but not zero might sort of countless meromorphic rights of the formalism fanaticism okay and then every once in awhile o’clock veteran quotient sheaves these are called skyscraper sheaves they may have a little bit essentially this is this came up an elastic question it’s long stalk it’s just part of the formalism that conform exact sequences of sheaves and get potions and you just have to compute them from a formalism oh yes you will you really I’m not even regular but but the work will in here so that people will feel comfortable halt the secret oh very ok ok so i prevail translation ok so i should say here’s the secret I’m saying I’m saying I said I said mirror or homework of years so that those dad because some of them say mera more beginning with his regular mirror it’s sure that you hold more thing evokes the analytic data of it when is ecology arguably this

statement defines comal in fact this is Grove leaks definition or funk torial characterization of homology homologies would ever got this okay and this property characterizes call balaji up to uniquely up to independent organisms so I don’t really it justifies us to know that it can be computed if I ever do need to compute it I’m going to compute it and we will show you the occupations but a fairly down is a classified vehicle partitions ordell’s where you essentially essentially getting your hyper apology by looking at at singular parts of for example polar parts of your of global Central’s and up to some equivalents one scholar anyway I can spell out a deal just lettin actually illustrate the computation but this is not been the most it’s not my preferred about it’s a cute definition this is this bro team statement this came up in Southie a stock that he used this formalism so if the tangent was the tangent space tangent space to to a rat d this is parameter space now that my boy space though it’s cannot be isomorphic Nativity to the space of global form of exceptions of a certain life male model of order cheap associate ocean liner left f with the five-pointed star for tangent well now what is that this is well called a pullback bubble the trouble is the word pull back it’s used in two slightly different ways in this discussion and the usual right we need pull back we pull back differential forms remember Tucson is really what’s given in the second one and the point is that in the separable case there’s a canonical isomorphism that let you go from the first presentation which is always which is always available the second presentation which is somehow more congenial for what I want to do here and we do it all the time okay so this this this step happened inside the airlock I’m going to do it too which means that once I start actually spelling out formulism detailed I’m going to begin the inseparable case again of them and probably the same base and of course secretly over people are because I really accomplished okay and this is close to a proof of why I talking you gave some details about this too it’s just what how do you how do you figure out what a tangent vector isn’t anything you take a curve in that space okay and you just somehow figure out from the definition general definition of educated vector should be what that is and here you can just follow individual points individual points if you can’t infect us but they are not happy they are not attached to the bookstore with her attention the image or so if you just take the derivative if you might be landing here you don’t actually get a vector field well kitchen tenders you getting filled up Bevers try to transfer them back using the inverse of the derivatives because I will take the derivative will take effect attacks and Kenya vectors and that’s of X so so long as you prefer you can bring it back where you want them when will this go wrong well when it says critical points you’ll acquire some poles who’s border is a key for the death is having pie and the other by the critical he knows what the pay it there is a sheet of Homer dependent fields under you another gap that funny things of all day together yeah lists of that critical divisor event okay so let’s just criticize all the pretence of Finance kills gratification all info nice elbow but if you’re working over see here are gamma irrational map has to be minus two critical points Calgary multiplicity if there’s no multiplicity it’s just too d minus two points full stops you can say gamma the sub app does that sin but more generally some of these points which was really that device remotely sin okay so the other earlier settlement JV babe you looking so theta sin gamma said that means so each year of that is the space of merit or vector fields which are a whole morphic away from the critical points and such that the poles for example at the critical points are all simple well you have several points versus of the water polar borders is less than equal to be the order of the course so this is data twisted by the divisor ok what’s the library and the side is right I mean when I was

relationship so you are lying you are allowing it’s quite a nice their review sort of mandating zeros of it this is this is a warm up what can you do in this formalism I mean it’s like a two line computation if you actually want to do it right but composition I mean what you have maps right so we’re gravity bradley is a space of a nap so if I propose that should give me some math between travelers basis right and you’re right that look of your cases by hand using matin or something so he certainly isn’t map do a regular pap and your sustain about the derivative there is a corresponding formula of course this is derived from women from a formula in the pullback bubble presentation we’re going to separate building sumption but you want to transfer things over to these twisted performance with bubbles here than you need separate bility for that and this is just the chain this is just what but notice of simpler formula okay I can apply this now Cora Liberty first corollary if I fix a map and then imagine progr get hit by Mobius transformation in fact I’m actually really really concerned with doing all this in a small neighborhood of the identity because remember I said it’s going to talk my way out of water more business this is how but I performed my quotients I’m really only interested in pushing account by a lead small my motivation or congregation near the identity and this allows me to pretend if I ever need to that my coaching spaces are actually manifold so if I go write something that has got a cotangent space or a tangent space you think I’m lying that’s what’s happened but you as an immediate correlated one or two left computation previous we get this derivative the work of the identity is given by Sophie what is the H year of theta another name for each year of theta is little sl2 of those are does the globally globally regular vector fields as three dimensions work with them right you can write the bound you know the quadratic polynomial x t by DZ that’s all same vector field we ought to pull back vector fields that’s what’s giving the previous formulism I coordinates you compose with the people as map / derivative ok says that operations please pull back here by Danny lines all that and very good formula another foreign militant this was used King subbies but wasn’t quite written this way but a lot like that so the derivative iteration again the use of the separable a hypothesis is just because a lot of things in a form of theta sub things rather than pull backs so it’s just someone who acts like the inner it’s up to one less intimate so V goes to be plus have pulled equity plus down the bot plus F to be n minus 1 mole that ok this is called the theorem here this is actually no great shakes it’s just that one distinction food horrors this is an easy thing if I were neglecting all of this average managed nonsense i can write a statement the proof lots but I want to just illustrate to you in one case what it means to pay the expectation to find a characteristic in a case where I actually knew how to do it okay so basic so audible are there going to be pulled out invariant global meromorphic vector fields and the answer is basically not let’s just say first of all if you’re working in characteristic 0 let’s say over see if you are same thing the answer is going to be now what you can actually probably just figure that out directly the arguable next page so well known facts stronger versions is that those are used for example in to say that the characterization of pullback invariant quadratic differentials right this is weaker than that so if you prove that if you prove that there’s effective essentially no pull back in the period what I’ve differentials of the purpose ill then apply this that Rachel vines we when will this attention square okay so I’m just so expelled it out here I’m really interested in showing that that is not one whenever I can show when and we are ever in whatever okay so this is these are the facts

spelled out here and I’ve got a couple of examples illustrating some of the funny business characteristic illustrators for example for example so in characteristic 0 you’re safe right because you’re forgiven if you have them that T goes to C squared then you can pull that care about it here if you take this ichigo’s disease of the day and this vectors bill isn’t varying up the scale but the scale is one of a big which isn’t one except the characteristic is fine delightful one right so startup are comin over now doesn’t that was my breakfast ready you see the square root that would have half-integer water but you’re to think the personal anything would be powers and DC bar if you take a left as example the quadratic differential is not coming from the square root of n that’s why the question that’s why the quadratic differential discussion contains the vector field discussion it’s sort of like you know uh not necessarily oriented vs oregon anyway truth is because the proof the proof comes immediately everybody helps his level up but probably improve but knows uses the word trulia center what else but fortunately you know you do it i remembered work things out like so many many years ago it’s probably show so principal so fortunately of them to reconstruct the years of proof of completely valid proof and the Apple disease of the minor characters all right well let’s face it i doing well what does usually I think I i agree but that’s why searching for big all right fine yeah well which sexually mean fundamental theorem of algebra covering it that’s all the same wednesday i do not want them unhealthy or improve that’s that’s all like your top electrical proof is no doubt the same as the algebraic proof but anyway here is and that the two that two of four sure to an earlier 78 said I should even show this slide don’t breathe this like this like this letter is just in to show that I have to check something I mean it’s essentially the arguing that you’re using various places to classify the pullback the eigen quadratic differential under pull that except it’s a little bit more involved because since CC they usually okay that everything function that what’s on top there so the invariance part it’s just that equality and now what’s the inequality for if you’re working over characteristic zero that’s going to be in a quality you just work it out honey what is one selects the border of a zero or hold a fullback event there’s a formula and that’s the formula it’s just that in a case of wild ramification right they weigh the ramifications of I Zuri’s not what you read off naively trouble why is that become y’all have example what about because if you have like Z to the if you had see to the p+ either something else okay you’ll see the P is your local degree what we want to compute the rampaging divisor you’re really looking at the vanishing the an order of the derivative and that’s really about the higher power that you see after the term that goes away okay so that’s why the inequalities here’s you have to fight against that quality if you actually want to prove this so there were two extra lines of discussion but morally morally it’s because you’re gonna pull sets up the right assumptions are both for our both going to be backward invariant because of this poles always and zeros and the tame days and the finite obviously and flat rate variance as a nice new pleasant only two points and it’s not proof and then you just work through this and see what needs up and see what what closer mine is whatever d might work out to be there is characteristics ok so just coral just just have immediate corollaries of the the formal statements i had that about what these derivatives are and that they’re addictive filled our position is this so using the answer wait wait very rustic went with does wedding where or what characteristics so it means a statement like that I’m not saying that this is sharp I just wanted to sort of spell out reasonably digestible statements that for example only mention came this earlier so Samuel is similarly under under the under under such assumptions here and with the correct interpretation of you know the tangent space so to speak of 50

less space this is when you can expect the torsional rigidity submersion which by the way is always in characteristic zero right there’s nothing that you did you doesn’t even come up why is this because you see the reasoning is you can come up in characteristic zero I guess I guess the statement be locally that means again according single wing away from singularities moduli space you have it’s a local product in this in this in middle school category right the moduli space and the flavor there’s no so that’s interested testicle that’s because this Deborah vanishes because you can’t have this but except you could have this in certain 20 cases in characteristic that the assemblers a similar discussion for immersions it works out the fine point the flight great works out to be slightly differently different but again there’s a basic statement that’s true over every 60 and then the way you have to modify it because you know plus or lights DVD and like workout one because the characteristic anyway as I court earlier ball is now here’s the snake I mean that I mean the reverse of this oversee using Julia sets for example this is visited state with a mathematically arises an iterative of fire the other men fighting in the other maps okay and its various groups of this oversee and the one I seem to remember those there are no developers have you ever seen an algebraic proof so I’m giving a healthy brain now of course if you’re interested in characteristic p so the above standing into some flagrant of excluded your item i have not thought about this long enough to know to do to wonder if there’s actual if actual violations of this i don’t think there are they’ve got but in any event if you’re just taking over see this is how it is a way to improve planning in that lower degree and iterate up to a given again okay now here’s where it struck once you start to approach the interesting questions about local dynamical variants and what happens to damage your group travelers baseball with critical orbit relations it starts to get a little bit hairy and bobby zamora formalism work right photos that won’t report it so here I just want to give some ideas give an idea about the sorts of things when we get out of that formalism with more or less minimal effort ok so this statement here this is actually not particularly hard to prove this is this is an exercise on the formulism a simple dominion capital a recipe for you I am not using Thurston’s theorem in here i am not using contraction i’m not using anything simple form a population notice i say fix points much much more elementary than if I said periodic was notice I also do not say longer Kelvin ok so there’s a proof of this using basically playing around with things formerly miss Fleming here versions of this lab la not these particular came up in saviers thought precisely in this formalism and what’s interesting here is wicked I found that I found when I sat down to do this while ago I said that I was really thinking of multipliers and I’m sure let’s just ignore the case very parabolics was just look at maps to have d plus one simple fixed points and report their multipliers and let’s show that those multipliers are important one activity not quite we made a very important is because you know there’s at least one relation the homework index for young many of these clots which can certainly Jack’s discussion of the whole working in Victoria probably led to half might work and of course so if you take a lower for communiqu├ęs formula it’s a heavily relation you just how great way to differentiate into few a formal relation gives you one in one constraint like one group is other constraints now if you step back and just remember your complex dynamics it’s very easy to get yourself there’s no other constraints you can just exhibit maps with people of arms but the spirit here is to do things into the testicle in general setting so the fact the intestinal over critics from override this was like this is a very very funny funny fact if you just consider formal expression they get your actual FF and you look at the formal expression f i know im using coordinates but I sort of happiness f pregnancy / the difference z minus F of Z squaring they have normally you wouldn’t spirit but you’re taking the derivative so we can

square with nominator if you have five on top and just it’s supposed to be a quadratic differential you just trust me slap on an easy squared and miraculously enough despite how it looks but that expression discourse corner covariant in the sense that in the sense that if you conjugated yet by mobis transformation and then wrote follow the recipe for the new map so of course that conjugation Reds have it with the subtraction and typically you think that this would miss you welcome you only get up to some correction but actually works on the militants it’s a neat fact ok so the faculty that the fact that that thing right now have these properties is the instant decimal over the index formula if your constraint and then some dimension account with residue theorem from its properly organized if you’re this statement here that Brooke that subject to that relation there are no others so now about the indices right so far work with multipliers it was somehow much messier so the correct move is to go to multiply to the indices and this automatically recovered of parabolic basis okay so of course i’m going to have fewer than eight plus 16 points if I have multiple fixed points but however however these six points I have one less than that is the number of independent local ojala analytic ordinance we get from this and of course ones what are going to be when local because we in verse theorem in the algebraic and it needs a recipe pilot actually working late oh all about that but okay it’s because i did not prepare a slide on how to compute age one line repetitions so this is a repetition but they’re here this is if you can see I’ve been looking at the set of all fixed points of effort for each to fix point I look at that expression polar part of that expression so that thing with the bracket means take the polar part at the fixed point starting to simple folks later on the bullet proteins in it slightly differently here and storing it simple I do that overall the fix point I Adam up okay and then an object object of such a such as simple to pull aparts automatically gives you an element or can be interpreted by Lee you know the Adele repetition presentation of ecology that gets you that gives you an element of each one in and that’s must achieve that’s consistent this is quadratic differentials requiring zeros imposing zeros appropriately along critical replication visor they have against global ones right because they need poles first of all right but you will have personal knowledge okay and this gives you so an expression of this kind of would give you always an element of that first of all logical thing that this would always gives Europe that’s the old organisms this one always interested in room the perfect what’s the proof the proof is exactly here’s the proof the proof is that the Royal restrictions are the same local roofers remove the brackets there is a global object there right and by the definition of h 15 repetitions it follows that it’s serum that’s the proof this is h1 my son adele’s then look at the Opera bells can I suggest please turn conscious I just wanted to translate from is that if you think of ecology in terms of coverings by open sets and things to find no differences of coverings if you have a bunch of objects to find on the individual element sets of a covering and you can take the dip differences and their values on every pair of intersections and that automatically is an element in each one and so you go from so this DJ needs to the boundary of chains and that’s this map page 1 which is which is what this is just a bad data path doesn’t necessarily explain what I mean by that notation I mean my notation really delicious express what I’m coming so laboriously major Sarah may just say one work so I didn’t say where the dalit look you take you consider you allow yourself at every point and every point if you want you allow yourself to write down a polar part okay except except you only be only allow yourself to write

down not reveal all or parts at finally many points and then you lot out you might help by what would happen if you had it only pull away only trivial roller cords everywhere or the system of polar parts for any global object okay and because if you remove the brackets here it’s the same formula that the global object and therefore as a repetition is trivial and references you know I just gotta be I thought that’s what it was called I thought that’s what they call of interest in surveying Sarah moment and then it’s essentially Adele’s but it does involve affection an extra completion which is gratuitous in this case so it’s a ritzy go for late to restricted product motor okay alright so now I mean this point that’s where to start waving my hands because the commutator sometimes are suppressing some of the computations but actually you no way I’m not as many of the computations that one like to think one might go about doing happen automatically by virtual form the form of symbology I mean for example doesn’t mean the fact that you embed a discussion hit to call knowledge ebay realm should may well already take care of checking that certain things you worry about are well defined and as a bonus phrasing there between terms of ecology plugs win it’s like putting up to the internet the plugins the larger world of other subjects that are all serving the language of global live algebraic topology class field theory etc but right so not let have any connections and to make you but maybe someone else by a formalism because they’re somehow only one color ology from these Americans Asian okay anyway so here is going to go a little bit that this is where for the experts but this is all sup housing is here so if i look at this is what is this upper row near me with the exact sequences here i thought i can look at the sheaves of exact evens of shoes you want to think of this sort of point wiser will really see how what I have a better if you look at that home every vector fields right we be talking about homework vector fields are requiring with glue required 30 skirt various orders and various places or ever look at the quotient of vector fields over Fitbit reveals by that right I can do it along who said be for a long ass day and you’re reasonably arbitrary but if I want to have a diagram between them for the diagram is something that I don’t have to write down it just brings down from the theory then you need these condition condition here the fact of the cardinalities per page rate is for something on the next page right by me these conditions came up for example in Sarah’s god they’re coming up for the same reason here but here we first see them in terms of what does it take to get this tie but does it take to be able to generally write this diagram down ok essentially I mean what these the existence of these vertical arrows is all due to the fact that when we know how many borders event your fields are supposed to behave when we pull them back when we might acquire what happens to be stronger this year oh my you might require a moment so and so forth and so if I really want to take this at face value without worrying ejecta that everything that I’m going to want to assume team ramification I think there’s probably a version of this diagram of the next diagram for water ramification but it may not be the version of now because because remember that they would begin equality in here so I’m not going to go there since his tail ramification or its characteristic zero or its see okay but there will be no analysis okay so what is called all of you Cole modalities the unique thing up to unique item or Commission which turns short exact sequences in for long exact sequence is full spot so that’s nothing to improve its really definition all right and then okay because because the things I’m talking about really our best phrase in terms of cotangent vectors I’m going to be utilizing things the way it’s called spirituality there’s a relation between it turns out this essentially if you’re talking about sections of sheeps socially the line models and 80 google up that an 801 each one sort of abstract page 1 of 8 from i buckle is God needs will do h0 in something related to that line going to take the goal line buckle and you cancer with the cotangent that’s there too well that’s what did this for sections of wine bottles and behaves a little differently with the skyscrapers use all discussed that half the for a bright line diagram so you take this diagram you dola as you think a little bit about what’s supposed to happen in at the co Colonel at the Scotch greater again hit this diagram now this diagram if I forget regret this

work at this point in the insecurity of Jack fest spirit my previous chat fest on right i mean this diagram appeared in that this is one of the diagrams appears in my transversality discussion although this is not really this is this is the one that relates this is the one that’s there so to allow you to transfer transversality computations done in the defamation space where somehow things are work fans parent to the results that you probably really want which are things happening in parameter space or moduli space but it’s so and which is just a walk through this diagram here so this diagram is presents itself when you followed make those moves the thing I labeled up there with T star bullet of death that that’s a very tough in theory ketheria Joseph’s coat for earlier and that is a kernel here but we’ve already identified those well we already identified this one in previous slide and basically that means in my thesis right discussion with the different space are credited of effect with the cotangent space the defamation space is is by definition or provide let’s morphism be h0 so pull more four quadrants of marital for quadratic differentials allowing simple poles in de Milo the image bye bye Danny mindset push forward acting on a quadratic differential allowing simple schools and then and you know open era the experts are the experts who were asking about my test examples of persons this is where that would happen so go look at the diagram turn everything on this side without agreement particularly this column over here is where are those times where that discussion of fixed point multiplier was happiness it was quite easy whereas over here when people are interested in in for example computing variational photos for multipliers our site almost warble Norfolk indices or discussions of peril Ellen DeGeneres guessing saviers clock and there’s they’re probably writing think they may be writing things down in terms of them probably differentials because of informations whether it’s realized or not those computations are having an addictive information space where can you and that that is where we know how to give criteria and arguments for transversality involving them for dreamers healthy love games like critical or populations and higher period multipliers and those things all rely on there’s no rigidity or certifications of that sum of any statement any statements the event for example that that that this operator he was in champion so I could for a scientific label turtle here and 50 in there that’s using it’s a desperate person virginity okay so people are also probably more used to the presentation of this in the analytic context for that for the tangent space to love this is uh it’s gratz expressed in terms of helped Ronnie differentials rather than first ecology and she effective use of zeros and again because there’s only one goal bubble tea right so there’s some people adult always a morphism which will which will enable you show that that writing things down beltronics will also computes ecology okay so if you like Meldrum before you to do you like analysis you can do it that way which is safe if you’re over sea but you don’t have to do you can phrase it go home illogically and make sense of it much of the time overall their fields and all the time over a few bar it’s just that this is 0 this is the place and the only place in my discussion were already used in see that works my secret agenda for the rest of my life is to trying to get rid of that which guy happened was they not what the talk is not before the objects that talk about the rolls down the statistical programming so I think I can do it I can do it I can do it right now oh please please please let me please let me do it right let a take the boat take a postcard you find in that but let a equal people to post greater sin okay okay to claim to claim that that map is an ice and water ism that’s Thurston Richard gets into this aberrant here say that this arrow here from that to that when I some more physically this would be h0 this would be quadratic differentials allowing simple poles along with most critical

said same space there to say that my dancing lines have pushed forward as my some organism is the instant test will step in for each of the NFC it is there’s no well that would like to bet that’s what I call in for the test motors your agility I don’t know if that comes to the diaphragm statement so will you know what else right no in terms of vicious question so okay oh I can say at this point at this point all the pretty diagrams like for example you notice isn’t raining here is Dave Hanson visible on the back that will be helpful era there’s a slightly pink shading there any of any of these colored buggin these are these were made by style yay and some of them are quite aggressive so you have that subject so this is the exact sequence which is expressing containing the relation between the infinitesimal deprivation space and the infinitesimal moduli space home whatever okay yep here’s another one actually this exact sequence i play was play with even earlier this is the one that’s used in like the two sure Baker so here now this is a little bit more so now I’m going to be a ladder okay so on the vector field side it’s mandating higher order zeros but once i go by quality it’s going to be allowing appropriate higher wormholes my room tour or so this is best expressed language visors think that there’s a multicenter so we already have am sets a and be satisfied with previous conditions beating things eppa betty and said critical values at ASAP three points a nap suppose that you take those sets and you just add into higher just adam heyer the complicity at some of those points as your life with that and deep everything fine just being up to someone implicitly say okay and then do the same thing for me or go to a compounded thing for me and then again without actually having to do anything the theory gives you again why because of the look so is this what is this upper relations its operations of look if I if jurors a vector fields requiring zeros of a high multiplicity and al injecting two terms of vector fields requiring all the zeroes of Oklahoma City one and then there’s a would be and you can for the a and the existence of these netword arrows follows 12 lectric Lee from the relations between be granted before okay and there was there a snake I think less about muster the previous they’re supposed to be all of the most important both reporters I think I’m this this slide is supposed to come in after this when this is funny so this is the snake of escapism we have this diagram here and when you have a diagram with this night you can applaud to stay clever but good job the curls on top so you have curls on live in here I’ve suppress these because I don’t care about okay I really is gonna be 0 goes the kernel of this goes to the colonel this goes to this which goes by the connective an organism to that coterminal which would then go to let’s go cordial and the next poker but this is really all i care of him so again the interesting part i mean this is all thought logical but the total of departments at least interesting is coming up from this and so in other words the transfer from the cotangent space the quotations base modulus to the contagion face of death nation’s faith comes from now by the way Wyatt why has this go to change space in the deprivation space yes well I mean at this point let’s say if I’m working oversee remember this is what is thirsty rigidity if a is equal to B rigidity be is and personality staton so in other words the sting that the deprivation space is as smooth right is reduced and the poor sense of the word I had in the earlier slide that is that is implicit in that notation there otherwise you have to be talking around that the way we’re talking around program I’m not sure I can because I have a nice example if there’s time to go again where that is spacious absolutely reduced their for obvious reasons are now this slide should come Africa and so here this presents it’s a different thought I ramp shade in a different way this is the relation now this is what I use in my proof of the cycles and turn to a number of most critical tales so this D of F a

this is the kernel of the error going on top this is what I hold the space of invariant divergences d for diversions supported understanding that what is for a subset of it because I haven’t necessarily allow pulls up all arbitrary high order because of weapons find that but wait up to whatever word right here event so what is that so if the Virgin’s by definition is a the class of cooler parts but for the purpose of this discussion hold are really starting at what at polar water too so in other words it’s a collection of polar parts starting with double and worst pole simple goals count as as a non singular the purpose of that definition yeah yes I got to try out of hope you know so and you know it’s just because there’s a natural issue of whether you can be very it’s basically over as Colonel these can be classified they’re completely classified into local data in my paper on the archive for ages and in fact this connector homomorphism here right so if you could show that it can act a little more physical is injective work you can show on subspace that’s injected that I can tell you that the dimension of the subspace is less varied dimension of the target and we know the dimension of this at least if we’re saying in character 60 because thirst and rigidity tells us that the dimension with this is the number of points and B minus n no number tails we take a large enough and the space of diversions essentially callous in its muscle whether be the United mcculloch cycles so okay now of course to get this injective we actually proved that theorem you have to restrict attention to the man repeller cycles because that’s where you can prove injectivity and you shouldn’t expect injectivity anywhere because it’s very easy to get into dimensionals physical Aryan things ok so again this is where this is where the formalism the formalism takes you so far right it doesn’t trivialize anything else but it sort of highlights where the difficult analytic content is that you might want to get around with new mathematics on bureaucrats at helping so now this tour de force to wa is the part of the final consideration really comes out of putting these things together so we have one of these up at one of these burgers about what the BBB and why we have the the diving that came the transfer between moduli spaces deprivation space and on this side on the vertical wall we put in the diagram used to bound the number of that repelling cycles and then if you take a look at where the exact sequences these things came from it’s actually the sort of issue there’s there’s news up there’s versions of this you know with bold face and someone hood on other levels here and there’s versions the vertical one so the whole thing effective nice ok so for silicon settling in here so this is if we’re gonna hit a three by three bike through diagram that’s the core of it and there’s actually zeros everything is somehow coming about an exact sequence so they’re zeros coming in everything and zero is going average to be suppressed in fear is over slides about that zero is essential there okay so this is what the front face the front three by three thing looks like that it’s basically the same things that the AIDS of turned that be the third phase SW through the picture Shane is coming that is great other projectors of this on my computer it’s all here you know later friend last night and also what are you doing a tiger then where I know what you do you get it with snake lemma this is to stand near they are all right what snake is going from the very diversion TSA’s the cotangent space that’s by fujikura construction and the other one is going from the tangent space the moduli space information space which how you transfer results by the way again that’s not always a nice and morphism either that you can show that this is a nice morphism when something here has no colonel and that’s precisely we have an apartment when you have in mark any any cycles of multiplier one and this came out like Sade’s talk mention get work network but okay so look here’s the thing i’m just as are going to summarize i’ll tell you it improves at the very end at this point we’ve got a diagram we’re just playing with the diagram now in general and in general 3 x 3 x 2 a thing here you’re not going to be able to go from any relation between this lower corner and that one but notice there’s zeros up there the fact that these two are serious in other words the fact that the fact that these these are isomorphisms in fact there are that the problem right so those portions are trivial but

that composes that that puts serious constraints what’s going on in here because of course there are also zeros from this direction here so actually so because of that it follows in fact that yeah I’m finished because of that it follows that this irritable this error which which in general he only goes this way is actually invertible so you can actually go from the middle up to the other corner and you can go through at a lower corner from the same reason and you can go to the front face you can do it back page on a whole diagram things are exactly well that commutes the airport you can do it for the Colonel’s in front and for the co kernels in back so we’ve done either for the kernels in front take the colonel going in here then yeah kind of taught a little map down there well as a map up there and it looks like your first of all might be well now we’ve got to commuter square where you sit down it compute its that may be commutative it’s close to it but not and that’s somehow appoints okay big up because I’m really finish I just need to show the last slide i’m finishing here did you get the stuff that will fit well there’s no point having gone this far without so this is that but this is not work so next diagram not return but now this has been extracted the whole purpose the rest of the day at this point and this one will look at the part of the diagram we watch so this is the colonel going to the middle bit these are these two snakes the green arrows are what we get to judge me at zeroes up there and the lawn area and what painting this is a kanay to the diagram believe I think it is and secondly actually think about where it comes from there is a katana logical nap and rest driver driver from k down to here and actually zero and that plus the fact that all of these ruminations your imply business of the zeros we were to apply that the colonel is gladly direct sum of the two corner bits and the fact that the god meant from the colonel down to where you want to go to 0 basically tells you and some be a habit is in this plot our version of it where he basically said there’s a residue fear of computation where you want something for the cycles you trade it for a certain orientation for the critical points and a minus sign comes up and that’s where it is okay to gallop that’s that derivation of something might look at what this is for example you can use this what you have this here you can use you can interpret this a special case of this is the variation important look for multiplies so purely algebraic a variation that’s valid they probably valid for routine maps of any character is that in any characteristic throwing a public okay so this is a business a section of the end of my clock there’s one more slide who’s operated I just hit return facebook so happy birthday Jen the questions comments based conversate okay