Singularities are points in a real analytic geometry where a function has a value of zero. Hiranaka's theorem is used to resolve these singularities by blowing up smooth centers to make the function normal crossing or monomial in local coordinates. Sigma is a real analytic isomorphism used to transform a function into a monomial. The process of resolving singularities is an important concept in algebraic geometry and includes the G-Huber scheme, which can resolve a poison singularity without an iterative blowout procedure.

A singularity is a point in a real analytic geometry where a function f has a value of zero. Resolving singularities using Hiranaka's theorem involves blowing up smooth centers to make f normal crossing or monomial in some local coordinates. This process is difficult and requires careful selection of where to blow up. If an infinite number of real analytic functions are studied, it is not possible to reduce them to a finite number and have the same set. The default colour of the boards is blue, a compromise between white and black.

Sigma is a real analytic isomorphism and birational map constructed using an algorithm to transform a function f into a monomial in a local coordinate system. The coordinates k1, kd, j1, and jd are all non-negative, and h is non-zero. Sigma is an isomorphism almost everywhere, except for the exceptional locus where it is not an isomorphism and the determinant of the map vanishes. This locus has a nice form, being the intersection of coordinate planes, and the zero locus of a function f when pulled back also has this nice structure. A single blow up is necessary to resolve the function f(x,y) over the real numbers.

Function f has a zero point which is blown up to the complete projective line, e, using x and y coordinates. Hironaki's theorem is used to factorise functions in local coordinates, such as x and y on one patch, and x and y/y on the other. Resolution of singularities is an important concept in algebraic geometry, and is typically done by iteratively blowing up a singularity at a point or hyperplane. Famous examples include the G-Huber scheme, which resolves a poison singularity without the need for an iterative blowout procedure.

A singularity is a point in a real analytic geometry where a function f has a value of zero. The speaker discusses how this can be resolved, and how the algebraic and electric worlds have different ways of resolving singularities. He also mentions how the default colour of the boards is blue, which was a compromise between white and black.

A real analytic set can be defined by a finite number of functions by adding their square sum. If an infinite number of real analytic functions are studied, it is not possible to reduce them to a finite number and have the same set. A singularity is defined as something that does not look like a manifold. Resolving singularities using Hiranaka's theorem involves resolving it to a simple normal crossing.

Hironaka's theorem states that any analytic function f can be made 'nicer' by a process called resolution of singularities. This involves blowing up smooth centers to make f normal crossing or monomial in some local coordinates. This process is difficult to execute as it requires careful selection of where to blow up.

A map, referred to as sigma, is constructed such that it is a real analytic isomorphism and birational. This map is used to pull back a function, f, such that both f and the determinant of sigma are locally expressible as monomials in the same local coordinate system.

Sigma is a birational map that can be constructed using an algorithm, transforming a function f into a monomial in a local coordinate system. The coordinates k1, kd, j1, and jd are all non-negative, and h is non-zero. Sigma is an isomorphism almost everywhere, except for the exceptional locus where it is not an isomorphism. The determinant of Sigma is h.

A map Sigma is not an isomorphism when the determinant of the map vanishes, which is known as the exceptional locus. This locus has a nice form, being the intersection of coordinate planes, and the zero locus of a function f when pulled back also has this nice structure. As an example, a single blow up is necessary to resolve the function f(x,y) over the real numbers, as it cannot be factorised over the complex numbers.

A function f has a zero point and a map which blows up this zero to the complete projective line, e. Algebraically, this is done by replacing x,y coordinates with x and y on one patch, and x,y/y,x on the other. Substituting these new variables into the function, the determinants of sigma indicate that the equation defining e is x=0 on one patch and y=0 on the other. The function then becomes some copies of e.

Hironaki's theorem is used to factorise functions in local coordinates. An example is given where a function is split into two patches, and x and y are replaced with x and y on x, respectively. The x is factored out, leaving y on x squared plus y on x cubed. On the other patch, y is factored out, leaving x on y to the fourth power minus x on y squared. This factorisation is done to determine the pullback of the function on both patches.

Resolution of singularities is an important concept in algebraic geometry, as well as many other fields such as statistical learning theory and physics. It is typically done by iteratively blowing up a singularity at a point or hyperplane. Famous examples of resolution of singularities include the G-Huber scheme, which resolves a poison singularity without the need for an iterative blowout procedure.

Resolutions can be used to analyse physical processes that are governed by potentials or Hamiltonians related to polynomials. Vector fields can be written as the gradient of a smooth function and where the vector field is zero, the particles will flow from or to. Singularities can be shifted by a constant and the internal structure of the singularity affects the dynamics. Differential topology and singularity analysis are used to understand this.

Mathematical physicist Arnold was the first to classify elementary singularities and their physical relevance. This has been further explored in modern string theory, where the dynamics of strings can be described by resolving singularities. In the real world, singularities are often seen in caustics, where light waves intersect with an uneven surface and form a singularity. The orb cam can be used to check what it is looking at, and it will go to the closest of the boards if it is misplaced.

now you've got 28 minutes to prove resolution of singularities jesus sorry guys i just went back from another meeting yeah so i'm gonna so so you can knock people over now with this new advertised yeah that's a bug that's been around for a while i don't know why that is did you just knock someone over they just sort of fall flat on the ground yeah like they won't see that happening but from your point of view it looks pretty amusing oh really so i'm the only one who says this i don't know why that happens but yeah okay so um yeah so i'm gonna let's let's let's say some stuff about resolution of singularities i'm going to be super informal here uh let me find my pen i'm not going to be too uh what's the word pedantic about a lot of things and interrupt me if you've got any questions um oh yeah actually i have a question why is it that every time i go into the boards i default as blue like the color is that because is that for you guys as well or just me it's set up that way that was from when the boards were sometimes whiteboards and sometimes blackboard so neither black nor white was a good default uh but we like so for example in the other world deprecation they're kind of whitish boards oh yeah they're white yeah for a default that is never the right one [Laughter] there's a power of compromise right yes i want to talk about resolution of singularities and this is not going to be super deep um it's going to be as i said uh uh informal and so you know the setting we're looking at right is uh we're going to be looking at i guess um you know real analytic right geometry now i know absolutely nothing about real analytic geometry besides what i've learned from from you guys so um this is going to be this is going to be um this is going to be uh uh uh probably some of it might be wrong right so you got to tell me if i if you catch me doing anything doing anything uh that's not quite legit but uh you know i feel like actually if you're used to analytic geometry and you go on the algebraic world then you're more likely to to to to just be cavalier with things right going back the other way you know a lot of things that you know you you you know you can do a lot more things in in the electric world that you can't do in the algae break world right so i'm gonna define um a singularity right in a most pedestrian way right so um i'm just going to you know consider right i'm going to consider function f right in a right i'm just going to these are just analytic functions

and i mean i i might actually want to say you know them as being analytic on some domain u right where u is some open in our d right and so um you know to a function like this you can associate uh you can associate a set of zeros right so uh you can also uh you know given f right you can define right x which is the set of zeros right which is x in i guess i'll say it's in u right and f x equals zero now i'm not gonna you know talk about more than one function right we've kind of talked about this some time ago i don't know whether it's part of this seminar series or some conversation we had before it definitely in in watanabe right if you have um a real analytic set right defined by a finite number of uh randomly functions you can just combine them by just adding the square sum right and then that will define the same analytic set right at the as a set of functions so if even if we're interested in uh zero sets right we can we we can still think of you know we can still study functions and try and resolve their singularities and so that's the that's kind of the the context uh which i'll present some of these results uh so there's a question maybe i i i didn't know i couldn't find the answer to maybe dan you know this right like so if you have an infinite number of um real analytic functions and you look at the common zero set can you always you know reduce this to a finite number and have the same set like does that work you're asking if the ring of analytic functions is an ethereum and the answer is no okay yeah so that that's it right so so i was trying to i was trying to say is that true well i couldn't i couldn't find a proof so it's probably not so okay so so you know we're starting off with finite numbers of defining functions right we're going to you know cut this back down to one you know like this this x right so we say that uh so we say that x in x is singular i'm going to just have a very simple uh definition if if x is not a manifold now i'm not super happy with with saying it like this actually because um you know when we actually go about resolving singularities using hiranaka at least in the theorem as presented in watanabe we're not really doing this right we're actually resolving to a simple normal crossing and that's a slightly more involved definition and i guess i'll just i'll just wrap that into the the uh statement of the theorem uh but in any case i feel like i should say what is the singularity and what is not right so you know basically it doesn't look like a

metaphor locally we say singular um you can state this in terms of the tangent space jacobian criteria and whatnot but essentially that's that's what it is so what is a resolution of singularities right so so the resolution of singularities all right so what we're trying to do is we're we're trying to make f nicer so we're going to make f nicer and that's like making x nicer right because f defines x uh nicer i'll define it at x right uh and you know make what does make mean so this this is really like you want to pull back and nicer means uh normal crossing or in the language of a kind of whatnot based uh presentation of him results monomial right you can always express it as uh a monomial in some local coordinates which is just like u1 to some power u2 to some power the ud is some pow and and what happened there yeah you know this is because i blame apple for this sorry about that i was trying to scroll your board which you can't what did i write i just i just introduced the palm rejection thing and you managed to hit the exact limitation of that little piece of code with that little maneuver so well done a pump so you mean if i just put my hand down it won't you know make it all messed up but i used two fingers too close together and that's not that's exactly what that's exactly what i did to test the limit of it and you managed to do it oh you've got a big hand in me uh singular x if wait what did i write there if we say that x is singular at off is a singularity of x if x is not right right it's not a matter for that all right thanks [Laughter] okay so all right super vague what i've written um but i'll stay here in arkansas uh and then i'll do some examples where we actually you know use the main tool in all of this is blowing up um and i think in the statement of hironaki's theorem uh he doesn't talk about blowing up in the statement of the theorem but when you go through the examples that's that's what you always do and in fact the the i don't actually know the full proof of perinaka's result but um i think you're able to construct these resolutions of singularities by just blowing up smooth centers of course which which smooth center would you blow up is really the heart of the algorithm you know to say that it's kind of a trite way of you know saying the theorem right it's actually quite difficult to select where you blow up so uh the the statement of theorem looks like this so this is a theorem hironaka right so uh we let f be analytic right then

there is a neighborhood of zero so i guess um neglected to say that f is zero at zero maybe i'll i'll say that uh and uh assume f of zero is zero f is not constant right so it's not the zero function otherwise that would be that wouldn't work right then there is a neighborhood a neighborhood w of zero a birational so i'm gonna use the word by rational because i think it makes the statement a little bit simpler and birational is not super crazy uh essentially um it's just a map which is defined i guess this map will be defined everywhere on the on the on the source um and it'll be almost everywhere uh and isomorphism and almost everywhere um a real analytic isomorphism actually so uh so that's you know what you can think about and in fact once we do the examples with blowing up right all of the maps that we construct using the blow blow-up will be by rational by construction right so i'm going to just pack some of the technicalities in this world by rational right but it's real analytic morphism uh and i'm gonna draw it here they call it sigma my notation is slightly different from uh what uh from what the knight base just because uh i guess i guess i guess this is how i learned it so it was hard to uh how to follow his notation i got confused all right let's walk over to the next board and finish the finish the statement okay okay so you have this uh you have this sigma right right we haven't said anything about how to construct this or we know that it's biracial uh such that so such that right what we said earlier about making f nice i'm going to pull back f along this side you're a bit closer to the third board so the orb is looking over there at the moment back this way yeah cool thanks let me just do that again right so what are we doing right we're we're going to pull back this function right along uh along sigma such that sigma star f and and the determinant of sigma so that both um are both locally expressible where sigma star f is the fancy pants way of saying if composed with sigma sorry i i'm trying not to i'm not going to have a name for the function without the variable right so yeah yes or i could have i could have written it like this right yeah okay both locally expressible uh when i say um expressible man my noises are terrible all right look as uh f h well i guess uh i'm gonna be slightly precised i'll write down what i mean here as monomials in the same local local coordinate system so so basically more precisely oh this is a terrible way right

precisely and that was a little bit too more precisely right we can write sigma star f s well now i'm going to be i'm going to have a u which is a local coordinate system right u equals i'll write that in sigma u is equal to plus or minus u 1 uh k1 u d k d and the determinant right this is going to be some function h uh and u one why is it j one u d j d right so the same coordinate system um and uh the kind of assumptions on these k's and j's is that they're all non-negative so k one k d j one j d right they're all non-negative right and the h it's not zero right uh i guess h is not zero on uh yeah some neighborhood right of uh uh well not not well no sorry nonzero at u right at that at that point where you've chosen local coordinates you want to uh continue d so i'm a bit confused about the relationship between u w w tilde and rd is your rd actually your w or what uh what's rd well looking back on the first board you had sigma from w tilde to w but what's w oh all right so w is a neighborhood of zero i guess i could have just said that was you right but um yeah so i guess w is an open subset of you right so okay yeah also yeah let's i'm guessing determinant of uh sigma means the jacobian yeah yeah yeah okay that's right so any any other questions oh so so where these are normal crossing that's how we're going to define normal crosstalk and so um so yeah so basically the theorem says that you get this function f right it could be a mess um at a certain point right you can but there exists um this birational map right that you can construct using some complicated algorithm that when you pull back f right it has this very nice form right and locally it's just a monomial and uh and um and so the geometric way of of kind of looking at this right um is that uh so what is this what is it saying right so the second um the second uh statement here the the jacobian determinant of this uh of this map maybe i should have said sigma this is not quite what i want to write is debt sigma right yeah mr mr sigma yeah so that's basically saying that the uh so i didn't define this earlier but uh when you have this map from uh w tilde w right it's an isomorphism almost everywhere um but there are they're gonna be there's gonna be a locus where it's not an isomorphism and we call it the exceptional locus so um so uh let me maybe i'll just write this in word so we can kind of refer to it so so the map sigma is almost everywhere and analytic isom right so so there's going to be a locus

where it's not right so uh i don't know if this is the best way to uh to define this so maybe we'll do it like this right so um so there's uh there's gonna be like some i'm just trying to see whether i can just say this right yeah i think i can right um basically i can define it on the source and say that's where the uh the determinant vanishes right so um yeah so so the uh the exceptional locus right i guess it's it's the vanishing locust of right right so that's where it's not an isomorphism and so what this is saying is that the the this this exceptional locus where the map sigma is not an isomorphism also has a really nice form it's just normal crossing devices if you look at kind of the local structure of the zero set of of such a function it's just in the section of coordinate planes right you're just you know looking at u equals zero u one equals zero or u two equals zero or u three equals zero right not counting multiplicities it may not include all of the coordinate planes because uh some of these exponents can be zero but that's kind of what it's saying right like at any point right the local structure of this exceptional exceptional occurs is just um you know intersection of sum of the coordinate planes and moreover if you look at uh the function f when you pull it back the zero locus also has this nice structure where um it's just you know uh the intersection of some coordinate planes it could be just one right it could just be one you know just just like a hyper surface or the hyper plane right or in more complicated situations you could have like hyperplanes intersecting at the point but you know only only well maybe not at one point but like intersecting you know somewhere right okay so so that's all i want to say about uh resolutions uh and the theorem uh what i'll do in the last uh few minutes is to you know compute some examples and i think i talked for too long it's already eight minutes to the end so uh let me just do this quickly uh these are all from watanabe's book by the way so we can i'm gonna put the numbers there uh so let f x y be this is the most basic example you can you can take right so over the the real numbers um you have to you know you you know you can resolve this over the complex numbers you don't this is already normal crossing right because you can factorize this over the complexes you can't do this over the rails so you have to do some blowing blowings up or on a single blow up i guess uh to to get a nice form for this guy so um so what

we're going to do is you know there's this point where f is zero right it's just zero um and then you know we have this map where this zero blows up to be this e which is um complete the projective line and so how do we do this so typically what we do is we we just blow up right and what does this mean i'll show you how this goes algebraically right and basically you have um you replace the coordinates x y with x and y on x on one patch and then you uh do the other um the other thing on the other patch all right x y y right and so then you pull the function back right so this these are coordinated patches for the thing on top right so this is a sigma right and so sigma star f what is this on this patch you just substitute in these new variables and then you factorize it out and then you do stuff right so so here y becomes y on x over a y or x times x so you can pull out the x squared write 1 plus y on x or squared right and over here you do the analogous thing where you just take out the y instead of the x and then you have the x on the y i think i think the um orb might be looking at the wrong board at the moment oh did i move sorry um i walked back is is it fine though um i think [Music] so my screen you're standing next to the previous i went to the third board without going sorry guys where's dan telling me off me see sorry i had to go yeah so i moved through that no problem you know i was doing much better when i was by myself doing the euclid when you went here i'm messing it up coming back to this right so i guess i drew all of these um basically right blowing up this very simple function we're just gonna do this algebraic trick um i'm gonna just do this algebraically you replace x uh x y with x and y on x on the other patch y x on y y and then you just substitute in new variables and and this is what you get and now you can see that um the uh the the functions right look almost right right they're almost normal crossing you've got this really kind of strange term here that that we can get rid of in the analytic category but let me just write down right the determinant of sigma i guess here that's x squared right and the determinant of sigma right that's y squared so what this is telling us is that the x on this patch on the left right the uh this the this you know the uh the equation defining e is x equal zero on the other path the equation defining e is y equals zero and um and the function just becomes you know some some numbers of copies of e and

some other stuff that doesn't touch e and so there we go now this is still not quite what hironaki's theorem has quoted in whatever says because um because uh you have this you know he had plus or minus multiplied by a monomial here we have a monomial multiplied by some positive function but positive analytic functions you can always just wrap this up square root whatever invert it and wrap it into local coordinates so you're okay so the only thing we're going to be changing here is um i'm not going to do it i'm just going to say it is that we can change change coordinates and wrap this term up uh into i guess x and then what this will do is change the defining equation uh well it doesn't change the defining equation for e but it changes that sigma in these coordinates right so um so that's kind of just a little bit of of detail there that uh you might need if you were to do this um and follow and get exactly the the result as quoted in what's not big so um the only uh probably one more example that i'll write down if i have time is to do this one more time with a different function and you see it's really the same procedure right and the function i had this i'm unable to see my noise properly somehow i can't zoom in i'll zoom out but the function is uh so this is example 3.14 right so the function here is wow my phone is really locked up i cannot sorry all right got it so function is um so if i don't finish this you can you can try this out like literally it's just the same kind of procedure you replace um the coordinates with the ones that i've written before right you change x um well x to x and y to y on x and then you factorize and you do the same thing on the other patch and you see what happens to uh to these functions on on both what so the pullback of f f on both of these patches and then there you go right so i'll just do this quickly right i'm going to factorize the x out this becomes x squared minus this is again x squared is gone and it's going to be y on x uh i guess times x and then plus y cubed is going to be y on x cubed i guess there's an x so really i could have pulled out an extra x right x cubed x minus y on x plus y on x cubed right so this is already normal crosstic right you can verify this i guess this is on one of the patches i call it u b on the other one you take out the y right you take out y and i think you can take out y cubed here so you can take a y cubed and you get x on y 4 i guess with a y minus x on y squared and then you get a 1 here

which means this is this is not going to touch the exceptional so you're good so i guess i'll leave this i check these are both normal crossing okay so i think i'll end it there um there are of course like much more complicated complicated schemes of blowing up you may have to do this multiple times but you know essentially once you've got these new patches right you just examine them and you can choose some blow up to to try to simplify this it's not always going to be blowing up at a point but for the most part all the examples i've done have been blowing up at a hyperplane right and and so that just means that instead of replacing um y with y on x right you just replace some subset of the uh of the coordinates with you know whatever the code whatever subset you choose over x right over one your choice of one of them and you can do this you know multiple times and see uh if at the end you get something that is normal crossing and the iterations of these procedures for simple examples are usually not super hard and of course doing this in general is really is really much more difficult but in a lot of the examples that you see in what nabe or you see in any of the places you find online right that's really all you have to do um yeah so i think i'll end it there uh maybe you guys can fire off some questions at me that i can't answer okay let's do it thanks thanks kenneth that was good um yeah are there any questions feel free to ask broader questions uh those will be especially entertaining to watch kenneth answer but technical questions are also welcome i guess it seems like a fairly technical procedure but resolution of singularities is extremely famous in algebraic geometry and plays a very important role in many areas outside of algebraic geometry right statistical learning theory is just one but it's also you know as a as a role in physics and all sorts of other places yeah but i think some of the more famous examples of um of resolutions of singularities is not not constructed by iterative blowouts right like despite what i said that that's all you have to do is really yeah like you know these g hubert schemes is probably the most famous example that i know of that's the coolest example i know of where you know you just you just do this you know g huber's given oh there you go it resolves this poison singularity and it's like magic right but there's no like they don't do this iterative blob procedure like i do this is just for learning and for baby examples

yeah that's right i guess that's true that like many things in mathematics there's kind of existence statements and then there's okay the thing exists but what's the right way to think about it so if there may be there always is a resolution but there's often a way of constructing it that makes clear that what you've done by resolving is something deeply interesting [Music] yeah and also resolutions aren't unique right there are you know many paths to finding a resolution and like trying to understand the relationships between these different resolutions is actually uh quite an interesting topic in itself um i don't know whether this is true for the real analytic category but uh is it true that for surfaces is unique in the for analytic surfaces the resolutions are unique do you know that i don't i don't know yeah that i don't but that's the famous result um in algebraic geometry that uh for characteristic zero surfaces um i think even p uh but don't quote me on that that resolutions are unique uh and like strongly unique in you know in that you can always it always factors through the minimal resolution uh for surfaces but here i don't know what happens in the real category um you know as i said i know i know nothing about real geometry are there any other questions uh dan you mentioned that it comes up in physics i was just wondering about where does resolutions come up in physics oh uh for reasons that aren't too different in the final analysis from how they come up in statistical learning theory i guess uh but so there are many physical processes that are controlled by potentials or hamiltonians that are polynomials or can be related to polynomials and um if you think about a vector field right so you've got you could easily imagine a particle that's evolving in time according to a field of force given by a vector field uh generically if you just pick an open subset of whatever space that vector field lives in you can write your vector field as the gradient of some smooth function and then well where the vector field is zero tells you where the things will flow from or flow to right that's the basic point in analyzing odes for example okay so suddenly you care about functions smooth functions and where their gradient is zero well up to shifting by a constant that's a singularity and then you can show that the sort of internal structure so to speak of that singularity affects the dynamics very strongly that's a sort of fundamental point of differential topology and singularity

theory which grew out of physics for example arnold is a mathematical physicist and was the first to classify the most elementary kinds of singularities and then it's just a short step to see that the resolution of the singularity sort of has physical relevance it's i don't know a short way of explaining why but um but i guess maybe the strongest connection is in modern stream theory where you can make the argument that well if string theory were true then the fundamental degrees of freedom which are these strings propagating in space-time when they encounter a singularity of the potential that lives in their action um they they kind of probe the internal structure that is the dynamics of the strings can be described by doing some mathematics involving the resolution of singularity so that's pretty interesting um as far as we know i don't i don't since string theory doesn't maybe appear to be true i don't know of actually any convincing argument for why resolution of singularities in the hieronaka sense is is connected to something very deeply in physics that describes real things i mean often often the singularities you encounter in the real world are fairly simple ones um for example the cowsticks caustics that you see at the bottom of a pool right where the light hits the surface and the surface is uneven and therefore deforms the the light waves and at the bottom of the pool and you intersect a surface with those light waves you get singularities you can classify what singularities are possible and so thanks um i'll make a note kenneth that uh yeah sorry i wasn't around earlier first of all if you maybe it's worth checking just by clicking on the orb cam when you're connected as speaker to check what the old cam's looking at or you can see this black ring around the orb right that shows you what the orb's looking at if you move over here you'll see that just so that's a kind of quick way of checking that it's actually looking at what you think it's looking at yeah so like in front of each board there's a there's a fixed location for the orb right so if you if you go over a little bit to the other region it will just go to the next board it won't yeah that's right i think i think the the sort of misplaced here but it's too easy to get it on the wrong one here so i wouldn't say this is your fault but um it's natural to kind of stand where you did but yeah that's right yeah the almost go to the closest of the boards right they're kind of if you if they were all in a