Хорошая удовольствие, чтобы участвовать в целибрации. И, да, мой talk is about wild coverings of Berkwich Cures. Я не надеюсь хорошей familiarity with Berkwich Spaces, так что я попробую, как-то, дать базу, без давления слишком деталей о Berkwich Spaces. Ок, теперь... Ок, теперь просто... А? А, я... Спасибо. Так что целибрации, я думаю, мы всегда работаем в геометричных ситуациях, так что мы фиксируем альжебракт, в полной реальной целибреции. И мы будем участвовать в Berkwich Cures, только в таком целибреции. А теперь фанетморфизм, геоналитичный Berkwich Cures, может быть... В принципе, вы можете тоже думать об эквивалентной языке, с этой языком формы модели, или рюкзитной. В конце концов, мы все изучаем, более-менее, то же самое. Объект от чуть-чуть различных тенголов. Так что это может быть довольно сложным. Я могу дать какие-то фотографии в сервисе, когда я попробую convincing you, но, на самом деле, не очень просто, хотя в данном данном случае. И они могут иметь очень... что-то я называю, топологичную ромфикацию локусу. Все эти нотения будут представлены позже. Итак, мы хотим дать какой-то комбинаторийной описания. Для обычных курсов стабильная редакция дает более-менее лучшую комбинаторию в детстве. Для морфизм вы увидите, что есть определенная стабильная редакция. Она не описывает морфизм в данном случае. Так что, мы хотим какую-то хорошую информацию, когда стабильная редакция дает. Минуты будут следоваться в варианте. Первая функция это, что я называю, различная функция. Просто, в любом случае, она комплюет в различных, в том, что в том, что в том, что в том, что в том, что есть файна-заморфизм, который просто поднялся экстенцию и обладает более-менее, в реакции романтистерий. Так что, это будет Eating Brook و selfish function, currently, I cannot explain what it is towards the end of the talk. We will introduce and I'll formally some results. Ok, this is just the goals and sources. Это к альтервертой? Нет. В данном случае интересный кейс. В принципе, это можно. Если рейс, то характеристика 0, то это не интересная, потому что все это так. Но если это кейс, то это both mixed and equal characteristic, это equally interesting and we will not distinguish them at all. Вы не поняли, но эти маномиальные объекты, они маномиальные, в которых, в чем? В коэффициенте? Да, в коэффициенте 0,1. Просто у вас есть функция 0,1-0,1, в которой есть маномиальные объекты с некоторыми коэффициентами. Вообще, коэффициенты будут от абсолютной цели кейса, но я не оформлюрую так. Окей. Теперь, другое функция, она училась в социальной работе моего пшд-студия, и форма порта, эта пара пейперов была о профильной функции. И также есть небольшая веревка 2 пейперов, которые мы сейчас обрежем, и, ну, эти пейперы тоже будут на IHS сайте, а также я поставил на моих сайтах. Окей. Теперь план. Ну, за половиной пейперов я will talk about more or less basic results, what is known to experts on Berkowitz geometry, but again, I would like to recall various things. So, first of all, we will discuss Berkowitz curves, then we will discuss morphisms of curves on what is more or less well known. I cannot say classical because Berkowitz introduced all this stuff in 90s. I used to say classical, but it's sort of classical for Berkowitz geometry. And then I will talk about different function and profile function in the last two quarters of lecture. Окей. Let's start with conventions. Probably, I will try to put some conventions here on board, because the only problem with slides is that you know, I can put very few information simultaneous on the board, so I will try to keep something. Here, ok, so evaluation always is taken real maybe I will put in such real evaluation, if it's not obviously, otherwise also we'll use notation k-circle, this is the ring of integers it's customary for rigid geometry and Berkowitz geometry, and the residue field will be denoted by tilde. Окей, so with notation with respect to valued fields now also we fix k, which is an algebraically closed ground field, complete ground field. And evaluation is not trivial? Evaluation can be trivial, when all results will be trivial, so it depends on you, if you prefer to allow or not, but in principle it's Окей. k-analytic curve X will be called nice, if it is smooth proper connected, in principle one can instead of smooth, only say, rick smooth and this boundary for simplicity I ignore this. In the papers one considers slightly broader notion of nice, so for us just smooth proper connected, and then it's just a notification of some algebraic curve, which is also nice, it is smooth proper connected. Окей, and f will denote a morphism we want to study so it is finite of nice curves. Окей, Good, now Go back one slide. If X equals X, is that what you need to say? If What is if, in particular? In particular it's a property of such a thing. If it's an it's not always an aleutification, it's a very curve, is it? Under these assumptions it is always it's written in particular what is the aleutification. Окей Now About points of I don't want to give general definition but maybe I assume a little bit with you often literally a little bit with region geometry. So Mercury spaces are defined similarly one takes affinoid algebras define some spectrum, which is richer than in region geometry and then glues them, gluing is a little bit subtle, like in region geometry so I want to skip the details but it's important to mention that points correspond to same evaluations to real same evaluations on affinoid algebras and to any point one can associate a complete residue field, a good invariant point unlike algebraic geometry where points are just evaluated and fields here are evaluated in complete fields, so a really a good invariant of point is completed residue field so for a point x in x, we are given H of x, completed residue field I think Valkovich consider more general affinoid algebras than than intact, so you draw the streak That's correct, but I decreased generality here from beginning, so we can assume everything is strict and I'm not going to use this to any detail, so affinoid algebras will not appear in this talk okay for any k-variety, so in principle since in our generality everything is algebraisable we can only think about an elitification of some stuff and it's maybe a slightly simpler way so for any algebraic k-variety Berkevich-Fontorel defines an elitification and a map from an elitification x to a variety curly x a fiber of a point z consists just of all real valuations on the residue field of z and for any such variation we can complete k of z with respect to this relation and we get completed residue field of a point so in particular for any closed point since k is algebraically closed by assumptions residue field is just k so we can put only one valuation compatible with valuation of k valuation of k so the fiber is just a single point and this is a classical point appearing in rigid geometry so we'll call it a rigid point usually now in particular if curly x is an algebraic integral k-curve when said theoretically a curve elitification of curly x consists of classical rigid points and everything else adjust valuations on k of curly x on field of rational functions okay now points of k-analytic curves okay so in principle points are divided to four classes this is more or less necessary five but okay let's try to do some other with this so type one adjust usual rigid points which are sort of classical ones everything else should correspond to non-trivial extension h of x of k is a non-trivial extension so when we are given a non-trivial extension of valued fields we can ask what extends if residue field extends or if group of values extends type two is the case when residue field extends so h of x tilde is strictly larger than k tilde and it turns out it requires an argument but it's not complicated in this case group of values is just the same as group of values of k I am using that k is algebraically closed in this claim and the residue field complete residue field of x actually is function field of a k-curve so it's translated as degree one this curve c of x so x type two and c of x is the residue curve curve of h of x tilde of k tilde okay now type three is the case when the group of values extends in such case one can show that residue field is trivial and the factor of groups of values is just z and fourth case is when we have a non-trivial extension but nothing extends neither group of values nor residue field this is the worst case, it's called type four now a small remark fortunately for us type four points will not be essential at all so I'll mainly ignore them throughout this talk which tries to prove stable reduction or resolution in high dimensions stable reduction in high dimensions this is really your enemy but once we have stable reduction for curves we are not real trouble okay now a fine line so let's consider example how these points look like for a fine line so take x is a fine line of k and fix a coordinate t then it turns out that any point is a very simple description just the norm corresponding to this x same evaluation when we shift coordinate by a is just a sort of monomial valuation is defined as maximum on monomials so it's a sort of generalization of classical Gauss valuation when r equal to 1 and a equal to 0 is Gauss valuation okay in particular if x is of type 1 this happens if and only if r is 0 so this is just classical point equal to something and otherwise x is the maximal point which satisfies the inequality absolute value of t minus a is less equal r because obviously any point which satisfies this should also satisfy less or equal in wet equality so x is the maximal point of subset given by absolute value of t minus a equal r so it's just a disk so any such x is maximal point of a disk so points of types 1, 2, 3 are just classical points and maximal points of disk what do you mean maximal point? the same evaluation it defines on polynomial in t is larger than larger equal than same evaluation of any other point contained in the disk okay now we can parameterize point by a so the question is what is the redundancy of such parameterization into sound redundancy is very simple the disks must coincide for points to coincide so non-archimedian disks coincide if and only if radius is the same r equals s and a and b are close enough any point is center of disk so this is and because of this we can draw a picture of x very easily for example for any point a we have a line of points p a r for any other point b we have a line of points p b r and they meet precisely at the radius absolute value of a minus absolute value of b so the whole structure of type, points of type 1, 2, 3 is just a sort of tree which probably even if you have not studied York Visionary, maybe the picture of tree you saw because it's very easy to draw reviews you just see such a tree now a small remark not important for us, type 4 points correspond to embedded sequences of disks without intersection classical intersection so it may happen that we are given some intersection of disks without anything and when it is type 4 point I put it in LeBrakis this is not important ok now skeletons so let's say with a subgraph gamma of x by a subgraph we mean the following it's a connected subgraph whose vertices are only of types 1 and 2 maybe on this picture I should also explain what are the types so all these points are type 1 and type 2, 3 so the intersection points they are type 2 the interesting points disks of rational radios these are points where we can specialize to different points, to different directions and something like this is a type 3 so we want vertices of the graph to be only type 1 and type 2 and such a graph is called a skeleton if the complement of the set of vertices is a disjoint union of open disks and open annuli or maybe semi annuli by semi annulus I mean disk open disk with a punch, so disk minus a point because my vertices can be at type 1 points I also allow such a creation so a typical example take something here make a cut what you have between 2 cuts will be an open annulus can you also join 0, infinity and p1 through all the count no, ok I am trying to keep a little you want at least one type 2 point you are absolutely correct ok, I try to you want one type 2 point any skeleton gives a good combinatorial approximation of x because of a felling fact first of all x without whole gamma so for any annulus we have a central chord like here which connects the two infinite directions of the annulus so if we remove the central chord from the annulus we get disjoint union of disks and fact is that maybe I just I did not spell out we want the vertices to be as I explained and we want edges to be central chords of the semi annuli so if we remove the whole graph the whole skeleton from x we just get disjoint union of open disks so curve becomes extremely simple without the skeleton indicates the skeleton knows almost everything about the curve at least any combinatorial information any larger subgraph any larger guy which satisfies these conditions also is a skeleton only for curses just low dimensional phenomenon and one can compute for example genus of x as the sum of genus of all points plus the first betting number of x or one can only sum over vertices of the graph and add the first betting number of the graph it means that the graph contains all points of x with nonzero genus the graph contains all loops of the curve so graph knows a lot about the curves it follows immediately from this observation that x without gamma is disjoint union of open disks now by a genus of a point of a curve I mean the following if point is of type different from 2 when genus is 0 we cannot assign anything interesting but if it is associated to x a curve cx of a residue field and we can take genus of this curve what do you mean by h1 of x which kind of comorrhage is it h1 you can take usual with integral in which space as usual as usual topological space in fact I don't need it but okay as usual topological space okay now semi-stable reduction theorem asserts that any nice curve possesses a skeleton okay probably this is not a formulation you are used to but it's equivalent and this is so called skeletal formulation of semi-stable reduction theorem we do not really need its relation to formal covers to formal models I will just briefly indicate it so classical formulation is equivalent to skeletal one and it claims that x possesses a semi-stable formal model of a formal spectrum of k0 or since we are even in algebraic situation even of a spec of k0 we can take algebraic model and relation between two formulations is as follows we always have a reductional specialization map of closed fiber of formal model and then preimage of generic points of closed fiber of any generic point is just a single point of type 2 preimage of a node is an open analyst and preimage of a smooth point is an open disk so actually if you are given a semi-stable reduction and a skeleton and moreover you can naturally embed the impedance graph of the closed fiber of semi-stable model into Bjorkovich space and this set of vertices will be precisely parameterized by irreducible components of the closed fiber now local structure of nice curves actually it can be obtained from stable reduction but one can also go in other direction one can prove by direct analytic techniques local description and when it's not so difficult to glue it and to prove stable reduction by purely analytic method so first of all nice curve is a huge graph it's something like this but we should think about it as a sort of graph By the way I did not discuss the question what is the topology again it's not so important for intuition I think it's enough there is some fancy topology but let's ignore it okay any type one or type 4 point lies in an open disk because on the we can only get type 1 at vertices we can have only type 2 and 3 any type 3 lies in an open analysis and description of type 2 points is a little bit more fancy so it turns out that any point of type 2 has many directions and these are parameterized by points of the residue curve so in the picture I placed here every ramification looks like p1 over residue field because all genuses here are 0 but in general points can be of type 2 can be more complicated and they do not have to be locally embeddable in 2p1 so there might be many points which contain some genus and okay well now one more basic fact about curve so any curve possesses a canonical minimal metric so that logarithms of absolute values of functions are piecewise linear with integral slopes so under this condition there is a single minimal metric its existence can be proved by few but in essence you can use stable reduction and you can avoid using it so its not as difficult as stable reduction now I will prefer to work with exponential metric or radius metric so radius of an interval is actually exponential of the metric of the length of the interval with respect to metric then absolute values of functions will be piecewise monomial as we should think about them so there is some dominant on any subintervals there are dominant monomials so in essence we avoid playing with logarithms going back and forth with logarithms if we use radius exponential metric but how can this be let us say for the curve there are some cases where the semi-stable reduction is not unique for p1 if you want to I never said anything about me but let's see p1 so here is the metric the distance between points with the same center of radius s and t is just s over t whereas depending on which one is larger than one so it does not depend on any choice given a skeleton gamma inside y we denote by r gamma the inverse exponential distance from gamma this will be important in the last part of the talk so maybe I'll put it here so if we are given gamma inside y a skeleton when r gamma is the distance from gamma so actually it means equivalently we can remove gamma from y when everything breaks to disks we can normalize them to be unit disk and when the distance inverse exponential distance from point y to be skeleton is just the radius of y inside the disk normalize to be of so the same normalization I think was used by Francesco in his work on solutions of differential equations good now so far this is the picture of course now let's discuss morphisms so first of all I would like to introduce a multiplicity function of morphism which is a morphism so maybe I'll put it here and f from y to natural numbers so for type 1 points multiplicity is just usual ramification index for other points the maximal ideal is trivial but the field is not complete residue field is not directly closed it can extend so the degree of this extension is multiplicity of map at y this is a very simple fact where f is a local isomorphism at y if and only if a multiplicity is 1 so actually it's an interesting question for example to describe where f is not a local is not a local isomorphism I'll call such set ok let me first say about this I'll call such set topological ramification locus so we consider multiplicity loci of f to be all points where multiplicity is at least d this will be denoted nf а? no no no it depends at some stage it will be but a priori you don't have to so all y such that nf of y is largely equal to d ok so one of our answers to describe this also we'll say that f is topologically tame if multiplicity is invertible in residue field ok now fact for any interval inside y if we consider it parameterized with radius with radii parameterization when first of all the image of i under f is a graph so and this graph we have natural exponential metric with respect to this metric the restriction of f is a piecewise monomial map and the degrees of this map of slopes strictly speaking one has to say degree but I'll say slope in this example these are just multiplicities so if we know function nf if we know everything about metric structure of the map we know completely what are the slopes so we completely describe the metric aspects of ok now simultaneous semi-stable reduction so skeleton of f is a pair of skeletons on y and on x so such skeleton on y the vertices of skeleton on y contain this set contains the ramification locals so in fact the only reason why I insist to allow vertices of type 1 is that I want ramification points to be inside the skeleton ok now theorem any finite morphism between nice curves possesses a skeleton why? because we know that once you can find one skeleton you can find as many as you want just in legend you still get skeleton so you can easily play with x and y and find a pair which is compatible so the theorem is not essentially strong it can be deduced relatively easily the price will pay is that it does not give such a good description of morphisms as simultaneous reduction provides for curves ok so this is this argument how to prove deduce with theorem from stable reduction and one can also formulate it in language of formal models will not need it so just a size remark it's more or less equivalent to existence of a finite formal model ok with both curly y and curly x semi-stable now to which extent simultaneous skeleton trivializes morphism ok the answer is written here on the complement of skeleton our morphism is just disjoint union of finite italic covers of disks by disks but such covers can be complicated that's all the stories so they can be really complicated here you exclude the characteristic pre-radical maps because you want ramification locus to be content yeah yes in this theorem in the end in the realisation theorem we will allow I agree with you ok let's ignore this point, it's minor you can always split to purely-radical description of tamed morphism so any topologically tamed that is morphism which is topologically tamed at any point italic cover of disks by disks actually is trivial so and also for annuli any topologically tamed italic cover of annulus by annulus is kumer so it's given by a simple formula and it follows that if f is topologically tamed then the morphism splits outside of the skeleton any action can be only on the skeleton and it's constant along any edge on any edge we have just the same power e, the same kumer map because any edge is just a skeleton annulus so this gives very good description of combinatorial description of tamed italic covers so in particular actually what we get we get a map of graphs with multiplicities and it satisfies natural conditions I'll run through first two because they're really obvious you have some local constancy of multiplicities so let me not comment on these two but just sort of a little bit more subtle condition is that we also have a local Riemann Hurwitz condition for this map of graphs so we are given a map of graphs probably maybe I'll start here map of graph gamma y to gamma x I know something like this and for a vertex here its image we can relate with genus of v and genus of u using the ramification along the edges by usual local Riemann Hurwitz and not surprisingly this thing follows from Riemann Hurwitz formula for reduction curves so it's not stronger than usual Riemann Hurwitz and in fact these three conditions where the only numerical conditions the map of graphs should satisfy there are lifting results that if you give a morphism of graph which satisfies this, that and that then you can lift it to be a skeleton of a morphism of burkwish curves ok now problems with the wild case ok, as I said the tal cover of this by this can be complicated and the extension of residue fields can be purely inseparable if it's purely inseparable we can say nothing interesting about the map of reduction curves so Riemann Hurwitz just completely breaks down third, even if it is separable the local terms of E is now larger than E and E-1 and it involves different in general so in case of wild ramification even if extension is separable we have trouble ok and two more examples the non-splitting set can be huge for example and here I would like to draw a couple of pictures for example let's consider a map T goes to T to the P over let's say C L it's C P over there but let me take here just general L and then maps looks something like this and it turns out that there is a metric neighborhood of the chord zero infinity the map is not split and it splits outside so this size is absolute value let me write it just immediately now it turns out that this radius R equal to 1 if L is not P if we are in the same case but it is number which is equal to absolute value of P 2 1 over P-1 if L equal to P both facts can be explained very easily you should consider radius of convergence of P-th root of 1 plus T this is 1 if L different from P and this is this is absolute value of P to P over P-1 if it is if L equals to P so just you try to solve this equation in H of X and you cannot solve in this radius but you can solve when you are outside so we see that in general the non-splitting set in non-time case it can be a huge chunk of a graph ok and the situation becomes even more complicated if if we consider for example C2 mixed characteristic 0 2 and we consider double cover of P-1 F lambda from E to P-1 of K in such case we have 4 interesting points infinity lambda 0 1 and it turns out that if absolute value of lambda is large and absolute value of 1 over 16 the number which is written here becomes absolute value of 2 squared so if this thing of exponential radius 1 4 does not touch this locus then the situation is relatively simple we know that we have 2 disjoint here we can extract root of T-lambda here we cannot extract root of T-1 everything else is split and ok and above this interval which connects the 2 non-splitting regions we have 2 pre-images so what we get here we have something like what I draw so what we get is a bad reduction a bad reduction curve and one can even compute the length of the loop in terms of absolute value of lambda so this happens if absolute value of lambda is large when 1 over absolute value of 16 but if ok I don't have time to explain what happens if we are on the border but it's easy to see if these 2 guys touch if just they meet at a point when instead of this you'll just get one point which is an ordinary reduction point now but the interesting thing is when absolute value of lambda is less than 1 over 16 this is equivalent to the fact that absolute value of j is less than 1 because there is scaling factor between j and lambda and this is equivalent to j tilde equals 0 which is the curve reduction curve is super singular super singular reduction and in this case just by hands one can compute the following situation lambda infinity 0,1 equals 0, infinity the same locus 0,1 but at the touch point there is first of all there is some direction I think something like square root of lambda in direction of square root of lambda maybe not completely certain where you'll get a this distance or distance from this point to the two axis is smaller than absolute value of 4 so this point is located much closer than you would expect you have meeting of two things and suddenly they cancel one another ok so this picture is definitely a little bit strange when you see it for the first time now I'll try to explain where it comes from and give some logic to ok so the different ok just motivation these last two remarks actually especially the last one says that different might be an interesting invariant to look at and especially because different measures the wildness of extensions if residue field extension is inseparable we have wild extension so it's natural to check how wild it is ok so we'll indeed look at different now the definition which I'll use in this talk is very different of separable extension is absolute value of annihilator of omega of integers of L over integers of k and before author says that this is incorrect definition let me make two remarks so first of all we use multiplicative language small remark so usual different will be minus logarith of what is written here like exponential distance and state of distance so we use multiplicative language and important remark is that this definition is the right one only because in our situation omega L0 over k0 is of almost rank 1 it's a subquotient of L0 so in general you should use a smarter definition which involves fitting ideals or something like that so it can be it's doable but I prefer a simple definition because one just proves by hands since we are sort of one dimensional situation we are only studying curves of algebraically closed field in our case one can show that this model is a subquotient of L0 this definition works also one can consider log different it's defined similarly but using logarithmic differentials if k is discreetly valued when these two are related by uniformizers and in general there are no uniformizers and they are just equal so classical Riemann Hurwitz formula for morphism of algebraic curves is written using difference so I just put it here but it's in Robin's book it's very classical ok and now the different function so now naturally I will denote by hyperbolic the set of points of types 2, 3 and if you wish 4 but again 4 is not important for us and to generically tile f from y to x we assign different function that is to any point y we just assign different of hy h of f of y now small remark, side remark we work with not discreet valuations unless we consider trivial valued case which is not interesting otherwise valuations are not discreetly valued and then between different and logarithmic different in fact what I define behaves better if you interpret it as a logarithmic function but we do not have to distinguish because just they coincide but maybe ok now this guy measures wireless of f in particular is just constantly equal to 1 in topologically tamed case and it easily explains all phenomena or so far I want first of all to give you intuition why this is helpful after that I will formulate and stuff like that ok so first of all here it turns out that different in mixed characteristic the difference is always larger equal than absolute value of p for extension of degree p so there is a minimum shift on the on the very bad locus on the curve which connects with two ramification points it's very natural and the different we measure different on y so this is y2x so here and also so this is absolute value of p and it decreases it increases in all directions with constant slope p-1 so on the distance absolute value of p to 1 over p-1 the difference becomes 1 and this is precisely the boundary of the wireless ramification locus when different reaches 1 wireless ramification stops so it completely explains what goes on here and it turns out that here also one can describe the picture a little bit more funny way again it's absolute value of p here and here so just slope 0 it goes with slope 1 in all directions outside of here so it goes here and here with slope 1 but from here you have slope 3 and this leads to the super singular point where you have more ramification everywhere else you have just slope 1 so from this picture you can completely describe the wireless ramification locus this will not be a metric neighborhood now because the difference is changing but it will be sort of conic neighborhood ok now general facts so, first of all it's piecewise monomial on intervals probably this is due to little bit bomber we proved also for type 4 we improved a little bit the idea of proving this is very simple on intervals divide interval to pieces which are embeddable in annuli between annuli you just write down a series and you compute it extends to a piecewise monomial function that is on any interval it behaves like a piecewise monomial function also to type 1 points and one can describe the limit behavior at type 1 points so it turns out that it becomes constant if you have tamer ramification for example like here in positive characteristic you may have wild ramification type 1 points and different vanishes and you can describe the slope and the main property you have a balancing condition at any type 2 points and this is the most important because in order to understand this situation we would like to understand how it happens where there are points where we have slopes 0, 0 and everything else is 1 and so on what is the combinatorics of this picture the answer is here it's just a sort of Riemann-Hurwitz form or an analog of Riemann-Hurwitz form so this is precisely the genus part of Riemann-Hurwitz and local contributions are also sort of something very similar and in particular almost all guys here are 0 so almost all slopes equal to in separability degree of extension of raise to fuels minus 1 so almost all local terms vanish ok now what about the proof I will really really give a very brief indication of the proof so proof of balancing maybe I have to make one remark it seems that this formula is a close relative from my discussion with Ahmet very probably it's close relative of formula for vanishing cycles of Cato and probably this formula was described by Rino for covers of curves over DVRs they seem to be independent we work here not over DVR and so on so formally probably there are no implications but it must have relative but again I do not have any formal claim here but our proof is very simple it takes about 2-3 pages idea is that delta f is a family of difference so just if by definition of different so we consider a lattice in omega x we consider a sort of lattice of integral differential just minimal OX0 model which contains differential of OX0 and then consider omega y integral over public of omega x integral this is a torsion shift of K circle models whose stocks almost cyclic and precisely measures with different and then ok, then choose some element of K0 of absolute value equal to different and compare reductions of omega y integral and public of omega x integral but shifted by this number and these two reductions produce non-zero meromorphic map between over reduction between omega c sorry, it should be public of omega c y c x to omega c y and the balancing condition just boils down to computing degrees of this shift y poles and zeros the degree of this shift is precisely the left hand side of balancing condition and local terms will give you ok so I just should say here that if extension is inseparable in general the natural map from here to here is zero so there is no interesting map but if such a picture appears as reduction of map between analytic curves then we can produce an interesting meromorphic map and it is responsible for slopes of difference ok, now minimal skeletons so we say that a branch at the point of type 2 is trivial with respect to different if the slope is the expected one if you remember almost all slopes are equal to some number and when they do not contribute to local terms to balancing condition so in such case we say that this direction is trivial from point of type 2 if we are given a skeleton of x and we want to enlarge it and put inside a skeleton so we fix something on x to avoid situation like p1 where we can choose different skeleton so we just fix some skeleton of x and then we ask what can be the minimal skeleton of x and y simultaneously which contains given skeleton of x and the answer is given by a theorem if we have skeleton and we take its preimage gamma y, a priori gamma y does not have to be a skeleton but it turns out that we do get a skeleton of f if and only if ramification locus of f is contained in the vertices of gamma y and for any point y all branches are pointing outside of gamma at delta f trivial so this thing actually produces you at least in theory and algorithm we need to give a cover how to find stable model of what you have above for example here you start with obvious for candidates for bad points, ramification points so we should include them we should take the convex hull but we see that there is a strange point on the skeleton where the different behaves not normally so we must include this direction when we include all this stuff non-trivial different we get skeleton of both ok ok and also different controls we set of if a degree is p when the different as I explained here for example and on that example different completely controls the non-splitian locus but if a degree for example is p2 or pq when it may happen that there is a huge locus where a degree is p and where a degree is pq and so on cannot be controlled by a single invariant so the question about all non-splitian locus is still open different cannot control it in general and this leads to the last part of my talk so let's say that a closed subset of curve is gamma radial with respect to a skeleton gamma if there exists a function on gamma such that s consists of all points whose exponential distance from gamma is bounded by r ok, maybe it's better to draw a picture to explain it just by single picture so let's assume this is some edge inside gamma and there is some function r on gamma from e to r and the set s is required to contain all points whose distance to gamma is controlled by this function on gamma so it's sort of something like radial set so in all our examples the locus was such a radial set with respect to different ok so theorem, there exists a skeleton of f such that its white part radializes all sets all multiplicity sets all together moreover if we found one such skeleton we'll do it's job and moreover in three cases you can choose any skeleton whichever you want these are cases when degree is p when f is tame and f is a Galois or even you can take normal curve which is composition of Radical and Galois example ok, if degree is p when actually the different trivial decreases gives you the radius of the set mfp ok how to prove it it's really very simple I call it splitting method it's used a lot in theory of valued fields so if you're given a problem about valued field you often want to solve it as follows for tame extensions for degree p extensions and do it by hands when extend to compositions and you get Galois extension because any Galois can be split to wild and tame and wild can be split to degree p and finally use some descent to get with no normal case and the realization theorem is proved just by this method but sort of globalized a little bit globalized one uses that category of italic covers of a germ orph at the point actually is the same as category of italic covers of spec of h of x so you can split locally you can split morphism like extensions of valued fields ok and finally ok, maybe I'll just since I'm starting to be out of time in a minute I think I'll just the last thing very quickly without giving details, spelling out details the last question is once we have a realization theorem we would like to know what the radii are we solve it for degree p this is different so is it a natural invariant of extension of completed residue fields and the answer is as follows just consider this radii r1, r2 and so on it's a better invariant because it's not compatible with compositions I give you composition of two functions it's difficult to express radii for composition through radii but you can rearrange it in a clever way ok, it will take two minutes to explain so just believe me you can rearrange it just as a sort of function from zero one that I call profile function and it's just equivalently original with radii and then the theorem says that if y2x is generic when for any point y of type 2 the profile function which precisely measures all this radii where splitting where we have locus of degree p to the n and then p to the n minus 1 and so on it's precisely the Herbrandt function of extension so last two comments even to formulate this theorem one has to extend Herbrandt's theorem to non-discrete setting actually it was not existing for non-discrete you just take proof of ser from from local fields and write something about almost monogenous extensions it's doable and when you can formulate the theorem and the proof is again straight forward by use of splitting method and for simple extension of degree p you just compare the two theories because both are controlled by different and the last is that this radii a priori defined only type 2 points but obviously they form some piecewise monomial family so there is a result about this ok thank you for your attention thank you very much questions just so you mentioned the fact that this is the difference so along the edges it is piecewise but is it so now what about it cannot be continuous in Birkwich topology for very simple reason the locus I describe you metric neighborhood it's not closed and not open in Birkwich topology it's closed in metric topology but metric topology is much stronger Birkwich topology your curve is locally compact so it's not ok at Birkwich conference I would stress this point that it is semi continuous for the Birkwich topology by easy it's always less than or equal to what you think yeah yeah yeah yeah semi continuity you have but that's all and it's continuous for metric topology it follows from this radialization theorem and it is definitely not continuous in some cases ok like what you said how does your high ramification behave in towers airbron function in this case again because you are almost monogenous you are sort of rank 1 composition of airbron when you have tower airbron function is a composition but for profile function which again it has very clear geometric interpretation profile function you just consider an arbitrary interval here from gamma to type 1 point you take its image this is 0,1 ok 0,1 and you have a restriction of f gives you some map this is the profile function if you are radial of the choice of this interval so for this definition it's obviously compatible with compositions for airbron function you should prove but it's like in classical theory of high ramification


So, here's question number one. Okay, number two. Number three. Read about this one. Alright. Number four. Okay. And then, last one. Number six. Okay. Uh, what's that? Uh, just my treat. Okay. There's no rush. Take your time. Back to one and two. No, you're late. No time. Strav, this might be our strategy to figure out how to get pie. I'm not even trying. Yeah, you are. Strav's too cool to try. That's right. Is that gonna be able to say good evening, Strav? Yeah, good evening, Strav. I'm a nice guy, so we'll take it back. You got that, Strav? One of you, Strav. It's okay. We're a family here, right? Alright, we'll go over these at the end. So, we did. One of you, Strav. You're gonna get that book from me, right? Sure. Shimmering black areas in the vision of both eyes, they last about 30 seconds each time. Um, in terms of his eye history, he's had some almyopia. He wears glasses. He's recently diagnosed with ocular hypertension, and so he's on the TANAPROS. No real significant medical history. Sometimes it takes artificial tears. So, what do you want to know about this guy? What questions would you ask in my clinic? How long has this been going on, and how frequently does it happen? Yeah, so it's been going on for a few months now. And it happens probably every day, maybe a couple of times a day, but not necessarily. How do you associate it sometimes? You know, Disney has no vertigo, no pain, or anything like that. It's really black, or is the shimmering color white? It's black. He thinks it's black. What he notices is that it's got like a shine to it. How do you treat it? He used to get a headache like once a year. Well, he gets a headache like once a year now. He used to have headaches when he was a teenager, but no longer gets frequent headaches. Do you ever get a warrant for headaches? He used to get a warrant for headaches. Can I draw what?  I know, he says that once, like if you ask him specifics, he'll say that once he was weakening, and he then like moved his head into a weird position, and he saw like these splotches of black. He's not sure if it was in one or both eyes, but at that point they lasted for a little bit longer, and then it went away like a few seconds, and then it went away. What is he doing when these come about? Well, that was one time he was weakening, but it's not necessarily associated with that. Okay, so he could be sitting down? He could be sitting down. Well, we'll get to an exam in a second. But that's okay. Okay, so he's 2020 in both eyes. He has some APD in the right eye, but his pupils are normally sized. His pressure is 18 in both eyes, and his ankles are open. I'm going to ask if they know that they should know evidence of BAS. No DIDs. So let's dial in. Can you ask him an exam now? Oh, yes, yes, you can. He's going to have a field later. This is what his optic nerves look like. How do people describe his nerves? It's like double mask. Very red. It looks normal, but maybe anomalous vessels in the disk. He does seem to have... He's a healthy guy. He does have a bunch of vessels, and that left eye doesn't look kind of like a little anomalous in the configuration of the vessels. He's got a little bit of a sheen. He's young. I don't know if this is right, but I was thinking about this. It looks like there's blurred margins, and so you always think, okay, blurred margins. Oh, is this papillodema? And so thinking of things that are pseudo-papillodema.  Oh, yeah. Like... Oh, here. Anything over here? Not as much. So, he had an auditory license, and he definitely did have trusen on an auditory license. So yes, very good. And let's see, I think... Oh yeah, he had also told someone else that he'd had visual field defects to leave out an MRI, because the MRI was pretty darn normal. This was a case where the guy had had transient vision loss initially, and then started having visual field defects later on. But just to talk about optic disk trusen for a second in terms of how it can cause transient vision loss, disk trusen is thought to be due to impaired axonal transport, and then the cells kind of die and extrude their contents, and they get stuck and calcify. I think that's a theory, but I think that's a well-established theory. And the trusen are mostly bilateral, but in this case, they were asymmetric, so he did have an APD really well with regard to that. And it's most often in white section of rare and non-white people. So you can get an ultrasound to see if they're calcified trusen. Autofluorescence was what worked for him. You could also get an OCT EDI to see if they're buried trusen. He didn't have that. And then, as you said, we want to distinguish this that disc leakage on fluorescent angiography that a true optic discidema would have, there's none of that in optic trusen. And I didn't know until recently that optic trusen is associated with a few other ocular conditions. Does anyone know a couple of what those might be? Not that they're always found in these cases just so they can be associated with it. So I didn't... Yes, it is, because in rare cases they can go on and it is kind of like a risk factor for developing an ION. I wasn't thinking of that one, but yes, you're absolutely right. So actually two other ones are like retinitis pigmentosa and suizanthoma elastica are both associated with disc trusen. So in his case, to answer your question about the field strap, he did have a pretty large peripheral archaeo-defect in his right eye based on the the trusen that was there. Any questions about that case? Back to that photo again. Which one? So I agree, his are not immediately obviously do look like his nerves are kind of weird and they look like normal, but you can kind of see that on the edge, if you correlate it with where the trusen is on autofluorescence along this edge here, then you can kind of see that that's where a little bit of the margin is blurred and it does look a little bit more nodular and bumpy there. I think it's a little harder to see on disc photos too, rather than undersolding. Looking at it, probably too. Okay, case two. So this is a 60-year-old woman. She calls the page response line intermittent blurry vision in her right eye. She says that she sees this like wax paper over her right eye and it's associated with this achiness. It's happened every night for the past four nights in the evening time and it lasts for about one or two hours at a time. So she sees a glaucoma doctor for asymmetric cupping. She's on Latina breast for that. Her pressures were recently checked like a week ago as 22 in the right eye and 14 in the left eye. She has a history of hypertension and hypothyroidism. What are two questions that you'd want to ask her? No diabetes. Is she faking? She is. Has been on other glaucoma drops in the past and they've all caused this horrible irritation to her eyes and so, yeah, her eyes are not at, your coordinates are not in pristine condition. So right now she's kind of just on Latina breast and only been able to tolerate that one. Good question. What's the achiness like? It feels like it's on her brow. Like she has this like brow ache. She sometimes does rarely. Not that she can tell. No, not terry. She has noticed that her pupils sometimes dilated when this happens though. We call her to come in for an exam. She's 20, she's still having an episode of wax paper over her right eye when she comes in. She's 20, 80 in that eye. Her pressure is 40. She has no ABD, but she's got microcystic edema in the central cornea of her right eye with one plus cell. Her angles are open on the right eye and she has moderately kept nerves more on the right than on the left. So this is kind of an interesting case. It's not the classic ankle closure that can cause intermittent transient vision loss, but in her case the glaucoma team thinks she has this weird inflammatory glaucoma and when she has an episode of inflammation or pressure spikes and she was off everything, but the Latina process at that time and so her pressure just kept on going up. So eventually she, this is what she had this RNF also just a little bit of thinning and then kind of an unreliable visual field on that right eye but they thought maybe that she has some losses that could be associated with glaucoma but her pressure was uncontrolled off of drops and she couldn't tolerate the drops so she just got an Ahmed valve recently. Did she have an episode of wax? She didn't now. I don't know if they have a reason for why she has this so it's kind of an unusual one, but ankle drops can do this and inflammation can do this. So the other one is 81 year old woman comes to clinic because she's been having black out vision on both eyes she notices this when she gets up in the night to go to the bathroom and she turns on the light and she says I just can't see and she doesn't think it happens without turning on a light or going from a dark room to a light room. She had cataract surgery 17 years ago, she wears reading glasses, she has a history of heart failure and hypertension and her vision is 2060 2150. Any question that you want to ask her? She turns the light on and she's like I just can't see anything. Probably lasts five minutes or so and I think that's better. No eye pain. One question. No, no jumping. Did she feel like her vision has changed otherwise? Her visual acuity is not very good. She knows that her vision isn't really good but that hasn't changed in a long time. I did her cataract surgery. 17 years ago, that was impressive. So you dialied her and she looks like this. So she's got a lot of Druzen and then she's got a whole bunch of geographic atrophy in the left eye in particular but also in the right eye. So what do you want to do to try to explain her episodes of vision loss? Yeah, sounds like a good idea. How do you do a photo stress test? Shine a light in her eye and measure visual acuity afterwards. Yeah, what would you be expecting to be normal? So so you shine the light in their eye for I believe it's 10 seconds and then they should a normal response is gaining two lines at least two lines above their best corrected visual acuity in less than 90 seconds. I think it's 30. I think like over 90s for sure abnormal, right? Yeah. I think BCSE says 28. But I think you're right. I think they have to kind of have a visual acuity of like at least 20 or so for this to work because if it's too high then it's kind of a little bit harder to measure. But yeah, within like definitely normal less than 30 seconds they get down to it then a line or two of what their visual acuity was before and if it's like 90 seconds to two minutes or something like that you're definitely in the abnormal range of that's more of a macular problem because if it were a nerve problem she could have a normal macular stress test. So the other thing I absolutely agree with you Mike, if she has having transient vision loss in both eyes and she has 80 and she has any GCA symptoms I would definitely get labs on her but if she doesn't then I feel like you can kind of explain this as with a positive macular stress test. Also maybe you want to think about like ocular scheming syndrome vessel imaging. Not that I think this looks like that but it could be a little tricky because you'd be having lots of hemorrhages but. Would you expect ocular scheming syndrome to have a abnormal or normal? Abnormal. Abnormal. Could write on the issue. Yeah, exactly. Okay, last case. 46 year old man comes to the ED because his left eye has been blurry since this morning. He had a kind of episodic bilateral blurry vision when he was exercising a few days ago. These are all exercising vision loss, sorry guys. But at this time his vision is not returned to normal. He occasionally wears reading glasses. We have MS that was diagnosed seven months ago. He's on copaxone but he was not sure that the copaxone was doing anything and so he's doing this experimental intranasal stem cell injections up his nose where they tried to get to his brain and the last one was four days ago. This is a true story. His case. His vision was 2020 on the right eye, 2030 in the left eye. Colour vision was down in the left. He has a left APD. This was before I knew how to measure APDs so I don't know what it was logarithmically. He has a painkillability but his pain and abduction of his left eye and his normal DFE. So you're seeing him in the ED on consults July. What do you want to do? His episode of MS was when the only one that he's had so far was when he had numbness and weakness of his hand and he was found to have cervical spinal cord lesion and some demyelination was in his brain at that time. That's the only time he's had an episode you said? That's the only time he's had an episode you said? I've already gone for these intranetist stem cell injections. Is that two of them? The stem cell things? No. What's the uterus phenomenon? It's an MS when they get overheated they can have various neurological deficits and weakness or even transit vision loss episodes. Yeah. So actually that's a good question. So when he had his first MS flare seven months ago he did say that he was very heat intolerant his weakness would get worse when he was a little bit, you know, warmer or trying to exercise and he did have some blurry vision occasionally like episodically during his previous episode of MS in both eyes. After that flare went away it went back. This time it always returned until right now. The left eye hasn't gone back. About eight hours or so this morning he found some hang time. Only when he moves his eye abducts his left eye but yes he does no double vision. He says my left eye has always been my better eye and now it's definitely my worst eye. The stem cells aren't there or what would you guys say this is? Yeah. So he had an MRI there wasn't a lot of enhancement of the orbital portion of the optic nerve but you can see on the canolecular and pre-chiasmal part that his left optic nerve was rightly lit up compared to his right. So he was admitted before salumedral and the stem cells were just why he came in because he thought that was what caused it. But he was then he had to work up for NMO which the aquaport for antibody was negative and then he was supposed to follow up an MS clinic and he never did. So I don't know what happened to him. In the coronal section on the left side there seems like a dark spot like underneath the vasculature is that just how it happens to look? Below that, inferior and medial and inferior and inferior and medial Yeah that's like this whole circle like inferior to rear. Do you know what I'm saying? Like you going down? Oh I see what you're talking about. I don't know how to interpret that. They don't know what that is. It's just normal. Can you point out the optic nerve? Oh sorry. So his right optic nerve is right here and his left optic nerve is right there. So just some brief things about optic neuritis, Utah's phenomenon staff already described that one. They think that it's because of these like weird funky sodium channels that are affected and kind of abnormal after demyelination. I don't know if that's proven though either. The other thing that optic neuritis patients can get are basically flashes and photisms where they hear noise like the doorbell rings and then they see a light in response to that. So this kind of strange wiring that can happen after an episode of optic neuritis. And it does happen in about three quarters of MS patients at some point in their disease course and sometimes as the presenting symptom in like a quarter to a third of a quarter to a fifth of the patients or something like that. I think that's all I have. Okay so let's start off with a case. I'll just read it out loud. So patient presented to his optometrist following an episode of transient vision loss in his left eye. He describes a slow blurring and darkening of the vision of the left eye with a similarly gradual return to normal. The whole episode lasts 10 minutes. He described similar episodes every two to three months for the previous three years with no associated migranous aura or headache and exercise was not a trigger at this point. And so this is not meant to trick you at all so this is actually a really cool case report I found. So this is his fundus photo of the left eye before the episode and then they actually caught one during the episode here so you can see kind of diffuse retinal whitening along the inferior and superior arcades. And then this is 10 minutes after the episode. And so we're going to come back to this case so just kind of like keep that in mind. But to start off I'm kind of going to go over transient monocular vision loss secondary to more anatomical reasons and so I wanted to start off with just like the anatomy of just basic anatomy of the vasculature of the eye. So we have the internal carotid artery and of course that branches off to the ophthalmic artery and that further branches off to the posterior ciliary arteries and then the central retinal artery. The next slide I have a kind of a better picture here where we can see the central retinal artery coming from the ophthalmic artery and then the short posterior ciliary arteries coming to kind of feed the coroid. And then just remember that it's the central retinal artery that kind of gives off these collateral branches that feed the optic nerve and this becomes important in a couple of different disease states that we'll be talking about. So central retinal artery and posterior ciliary arteries are N arteries and reduced perfusion and either of these will affect the retina. So the territory is actually overlap which is a safety zone for the optic nerve head but it also creates a watershed zone in the retina. Does anybody know what layer in the retina this watershed zone is? So Rob I know you know inter nuclear. Yeah we talked about that like my intern here I think you taught it to me now I'm teaching it to you. So it's the inter nuclear layer that you have this watershed zone. So when you're evaluating somebody with transient monocular vision loss I think we can gauge a lot through history which I don't know about you guys but that's not necessarily why I went into ophthalmology but in neuro ophthalmology we do talk to the patients and it has a lot to do like how the vision loss occurs. And so the pattern of vision loss may give a clue to the diagnosis. So when you're talking about altitudinal vision loss so when they describe this kind of veil shutter shade or dimming this was actually a study that I read sorry I didn't reference it but these are the four most common symptoms or ways to describe people's symptoms when it is actually this altitudinal vision loss and that is strongly suggestive of vascular etiology so this is opposed to like a constrictive vision loss so when you have this altitudinal vision loss very very strongly suggestive of vascular etiology and then we talk about positive or negative symptoms so positive being scintillations or photopsias and then negative symptoms being black or gray this gray is kind of up for argument there's once again like this other paper that was all about gray being a positive versus negative symptom is very riveting but I think they kind of concluded that gray was maybe a negative symptom but nonetheless these negative symptoms are you know this isn't a hard and fast rule black and white but negative symptoms do often lead one to believe that it's more of a vascular etiology rather than you know a cerebral migraine kind of etiology for migraines what do people have do they have positive or negative symptoms generally positive exactly yeah okay so this is kind of a really cool drawing of patient that experienced transient monocular vision loss so he describes you know his whole vision going gray or let's just say that's black for the lack of we don't have you know black as a negative symptom and then this is kind of how his vision came back kind of like in these white blobs what what artery do you think was occluded to make I'm telling you that this is in fact caused by like an arterial occlusion what do you guys think actually caused this exactly yeah so that's yeah exactly what this is is because you got to remember the posterior artery kind of branches and has all of these tiny other branches that feed the coroid and so this is basically the coroid reperfusing and so I thought it was a really cool this was actually drawn by a patient when he was describing his visual loss and so you know people can also describe these like lobules blackout vision with these lobules of vision reappearing and then of course the valuation so looking at the patient's eyes this isn't as valuable in a lot of these transient monocular vision losses because unlike the photo yeah and especially like the reperfusion of it too it's going to be kind of like you know like whitening coming out not so much like these blobs appearing back in the vision yeah so they actually published a whole article about that I mean it's really hard because we don't catch these people in the act a lot of the times of these like transient monocular vision losses but at least with this with the posterior ciliary arteries I don't know about the central retinal artery if it's been like evidence proven or not well yeah that's that's the point of this is this is not like this is what patients say and how it can help like us further evaluate what the patient has you know like they're we'll talk about that yeah yeah and then evaluating the anterior segment and the posterior segment can be really helpful too I spoke about this just last week because I don't want to go too much into depth about ocular ischemic syndrome but that's one cause of transient monocular vision loss that can actually have physical exam findings so we can have iris neovascularization we could have anterior chamber inflammation and can someone describe the anterior chamber inflammation seen in ocular ischemic syndrome in terms of like the cell and flare so it's flare at a proportion to cell definitely an ocular ischemic syndrome and then the posterior segment evaluation so let's see Marshall what do you see here in this fun this photograph the peripheral hemorrhages yeah exactly so that's kind of like the classic physical exam finding for ocular ischemic finding ocular ischemic syndrome you could also have cotton wool spots you could also have flame hemorrhages as well but these kind of mid peripheral Dopplot hemorrhages are kind of classic for ocular ischemic syndrome and then I don't want to sound like a broken record but you know with ocular ischemic syndrome as I talked about in FA last week you have dilated but not tortuous veins because the tortuous veins are found in which type of etiology CRBO exactly yeah so you have dilated non-tortuous veins and then kind of attenuated arteries and then trigger factors is a big key that patients with transient monocular vision loss can kind of lead you to the diagnosis so gaze of voc transient monocular vision loss is a tumor in the orbit until proven otherwise and so in this article Pezzel et al. described that optic nerve sheath meningioma are actually the most common cause of gaze to voc transient monocular vision loss and then bright lights leading to vision loss think of ipsilateral internal crowded artery occlusion and I wanted to put this in here because I misspoke last week and my explanation for bright lights leading to vision loss in ocular ischemic syndrome was that it requires more blood flow to kind of feed those photo receptors which then cause this is ischemia which then causes these dimm outs but in this article the physiological mechanism underlying this observation is not entirely clear because in diabetes it has been shown that dark and not light causes hypoxia in the retina leading to the suggestion that sleeping with lights on may have a neuro-protective effect on eyes with diabetic retinopathy so can you comment on that Dr. C? I have not heard of that yeah it's crazy I mean I've never heard of any retina doctor recommending a patient to sleep with their lights on but just something to think about what's that yeah exactly so diagnostic workup so most patients are not observed during the attack the rare situation with the case that I talked about in the beginning this patient happened to be so you really have to dig deep talk about the onset talk about the way their vision went away the way it came back aggravating factors of course EKG is really important ESR CRP super important especially in individuals over the age of 60 remember that ESR is normal 30% of individuals with GCA and then of course in DFE you know looking for visible emboli disc edema disc pallor attenuated retinal arteries cotton wool spots and attenuated veins this is kind of an old-fashioned technique just something I think we should know about so when you're applying digital pressure and the transient monocular vision loss is due to hypoprefusion you expect that the arteries are going to collapse whereas if they normally wouldn't collapse but there's one other etiology that actually has this same physical exam finding that is not hypoprefusion does anyone know what that is when you apply digital pressure seeing the arteries collapse so like high IOP so like well but it's just tricky because if the patient has already high IOP and you apply digital pressure and you see the arteries collapse you can like think of that to be like hypoprefusion from another source as opposed to from glaucoma like high pressure so just be aware of that if they have high pressures and you apply digital pressure it might collapse so ESR is normal in 5 to 30% people with GCA test like how do you rule out GCA well that's never how you rule it out no that's just a screen test if you're suspicious you start on steroids and schedule them for a temporal artery biopsy sometimes I'm asked to just like order ESR like labs for GCA where we're not really suspicious at all just because they're old well it's it's kind of like the sensitivity and specificity right so it's got a high sensitivity but low specificity so if you're actually that concerned that a patient has GCA then you like you should but it sounds like it's low sensitivity also I mean so he's right though it's all about pre-test probability so depending on your level of suspicion you know be smart about ordering this test because you're right you can open up a can of worms if you have a really low suspicion someone doesn't you know have GCA if you order this then when you do you're stuck like you back yourself into a corner so I'm not sure if you're saying like I feel like this test is not super useful I guess you know yeah no systemic symptoms and you get a normal ESR CRP that's really reassuring but if for some reason that comes back elevated you can totally have it all without any systemic symptoms so if this comes back elevated then that would probably push you to do a temporary biopsy in the absence of systemic symptoms or an ultrasound to hide your biopsy okay shoot I forgot to pull this up I think this is this thing it's really cool because once again you're kind of caught so this is actually you can see ret like emboli in the retina which is like super cool it look like so look down here and up here that's where I saw them the most yeah you see that one down there they go really fast yeah you see that one it's like shooting stars whoa like we never see that but be pretty cool whoa yeah it's really cool video to watch when you guys have free time yeah I watched it a lot yesterday okay so back to our case so what's on the differential for that guy that I talked about so yeah what do you guys think how old was he I thought you no he was young I took that out on purpose actually it was in the case but he was young he was like 22 years old yeah I would have just given it away what's on the differential yeah perfect okay awesome what else definitely sure well yeah I mean yes that's why I took out the age because I wanted us to kind of expand the differential likely not in a 22 year old but definitely sometime yeah yeah exactly so embolism retinal vein occlusion gca not in a 22 year old hyper perfusion migraine aura without headache or retinal vasospasm and I want to spend just like the last two minutes talking about retinal vasospasm because I feel like for me this is the one out of all of these that I don't know a lot about and it's I think it's oh yeah this was some other test here he did get this like whole work up so nlp during the episode were covered to 2030 no visual field abnormality no hematological abnormality and crowded cardiac investigations were normal so retinal migraine so the term was introduced in 1970 maybe by doctor degree who knows but so the definition the international headache society stipulated in 1987 that in retinal migraine a headache follows visual symptoms with a free interval of less than 60 minutes but may precede them so essentially headache can go before or after the visual symptoms and then obviously the diagnostic criteria for retinal migraine you must rule out any other etiology you can't just call something a retinal migraine because it could actually be something that can kill them very quickly and I thought this was really cool so systematic review they said in 142 patients that were diagnosed with retinal migraine and they concluded the diagnosis was probably wrong in the majority with convincing evidence in only 16 cases importantly all of these suffered from headaches and the absence of a personal or family history of headaches suggested of migraine caution should be exercised before making a diagnosis of retinal migraine and the reason why I put this up there is for the main reason is because you have to understand that the pathophysiology behind a retinal migraine is actually vasospasm of the retinal arteries but the incidence of migraines and retinal vasospasm is highly, highly correlated they're not really sure why but those are very much correlated so without that history you need to rethink your diagnosis so facts about retinal migraine monocular vision loss for 10 to 20 minutes it's a diffuser unilateral headache exercises generally precipitates the attack so that's something different about our case and it's caused by vasospasm of the retinal circulation or ophthalmic artery treatment is propranol and nephetapine of rapamil patients are having this all the time you could prophylactically treat them with aspirin and nephetapine to prevent these exercise induced attacks and then I had a couple other things about GCA but I'm not going to we're running out of time so just one thing I wanted to mention about GCA is that although most of them present with arteritic anterior schemic optic neuropathy remember that this can also include posterior schemic optic neuropathy, CRAO, BRAO, choroidal infarction, CNS stroke or diplopia the pathophysiology is exactly the same in all of these so vasculitis within the elastin of this small or medium to large arteries but they can present like this normal fundus finding but then you look here and this patient had a choroidal infarction secondary to GCA so just because the fundus looks okay it doesn't completely rule out GCA so in this sorry but in this one you're saying that this would have been like a posterior choroidal artery problem? Yes. So how come in the case that you're talking about at first like that guy had retinal whitening during his first I mean it would have to have been more than just the posterior choroidal artery? Yeah definitely it was probably like a branch of the central retinal or maybe it wasn't even the CRAO they didn't talk about which artery it was but yeah no that's a good point they say that most of the time like GCA doesn't only affect one vessel right like if you have symptoms of GCA you're having a lot of medium to large size arteries affected so most of the time we do see not most but you know something on exam some dyskidema anything like that okay Teresa you're up. Really quick here diagnosis migraine. Okay good why tell me how do patients subscribe to these whenever they have them based on the pictures flashing lights in their peripherals so scintillation that's like where that comes from right the scintillation and then the scatoma which you can see kind of in the middle picture so what about this like I was wondering I was like what's this like fortification spectra if you guys read this like so that is like the picture on the right so kind of like there's some zigzagging lines there and it was thought to resemble like the wall outside of an old medieval city and the like battlements up on the top and so that's like a long historically where that name comes from so migraine this is probably the most common thing that's going to cause a transient binocular vision loss however most patients are probably going to see it as just unilateral even though it affects both eyes and so you can see these you know with the eyes open and with the eyes closed classically so usually linear geometric and then the classic migraine aura is like I said that fortification spectra it tends to be black, white, or monochromatic and it will enlarge over several minutes so Dr. Degree in her headache lecture last week talked to us about how do you tell the difference between a TIA and a migraine do you guys remember what she said with vision changes yeah absolutely and then there was one more thing particularly with timing do you guys remember what she said and then the onset so the onset sorry I should say onset so the onset with the TIA is sudden whereas the onset with a migraine is going to gradually kind of come on and build and build and build and build so that can be helpful in distinguishing TIA versus migraine in the history so they can also describe heat waves or tunnel vision and they can't have micropsia, sorry that's in micropsia so small and big vision kind of the Alice in Wonderland effect as well so they're always usually at an hour if they're more than an hour you should be suspicious that something else is going on it's rare that they're outside of that and then they associated symptoms headache, nausea, vomiting sound and light sensitivity okay what's your diagnosis here okay so positive or negative positive what about the characteristics of the picture there's a pinwheel right and then what do you notice about the there's like stars formed is it black, white or colorful colorful what does that make you think of brain stuff good thanks John Brad so occipital seizures so these are typically unformed positive visual phenomenon so there's a whole other section on hallucinations for patients where they can have formed visual hallucinations and a lot of different diseases and that's in a completely different chapter in the BCSE that we're not covering today but classically these are colored swirling lights, a white out of vision can also be described with bubbles or a flash bulb going off circles and spherical patterns so that was actually a patient that just wrote through a picture of what he saw when he was having his occipital seizure generally these are very rare motion tends to be a frequent characteristic with them and then they can't have visual distortion and classically they're going to be in the contralateral visual field to the side of the seizure and they can have negative visual symptoms so these are going to grow over seconds in contrast to migraine aura which grows over minutes and of course they're more vivid and you can think of it this way so a post-ictal headache is actually a common feature of occipital epilepsy so just because a patient has a very vivid migraine aura and has a headache afterward you need to be thinking about occipital seizure instead and then their insights usually always intact to this but then your associated symptoms are a little bit different than a migraine so deja vu, somatosensory phenomenon, head and eye deviation motor activity and impaired consciousness and so here you want to get an EEG and neuroimaging is indicated so adults with this usually harbor a structural lesion whereas children it's usually benign diagnosis patient come to describe the visual loss to me is it positive or negative so negative sometimes you always want to think about occipital ischemia so complete binocular transient vision loss may represent a TIA that involves the occipital lobes so I think this is kind of another thing that maybe requires a little bit more explanation so a lot of blood vessels and our visual cortex takes up like our visual system takes up so much of our brains that there's a lot of different blood vessels that can supply it and so obviously you can think about the PCA territory with the occipital lobes but also the basilar artery is also intimately involved in supplying the system so if a patient had vertebral basilar system disease you're going to see more complex deficits so what kind of symptoms do you see in a patient that has like a vertebral basilar symptom or vertebral basilar stroke instability so ataxia imbalance staggering what about other symptoms nystagmus so ocular motor disturbances so gaze palsies INOs, SKUs, cranial nerve palsies, nystagmus all those things within the system too they can also have gray out or white out of their vision that lasts seconds and can have positive flickering stars and then also hemiparesis hemiplegia and hematosensory deficits and then also the bulbar symptoms dysarthria and dysphagia so these will typically last minutes they're going to be an older patient with vascular risk factors then they have a sudden onset in contrast to migraine and then they can't even have a headache in the brow contralateral to their hemianopia so just because you have a headache afterwards doesn't mean all the automatically migraine okay and then really quick not a visual hallucination but this is actually a painting that's in the National Gallery of Art it's called The Faint so I was looking through and I was like I remember this painting from somewhere so really quick I just wanted to touch on hypoperfusion so anything associated with postural hypertension or basal-vego presyncope has anyone ever passed out before so I had passed out in the OR this year for like the first time and had the like which was really interesting and then Stokes-Adams attacks so that's where a patient has an underlying cardiac dysrhythmia and then doesn't like a heart block and doesn't perfuse their whole head and so most commonly these episodes are going to occur while standing if they're occurring while the patient's sitting then you can think about something else so with the basal-vego and postural hypertension okay and then last one thing that we kind of talked about was occipital mass lesions and so you can also have episodic headaches and bilateral transient vision loss with these so I think this is kind of another thing too where you would want to have a field to evaluate it but tumors and AVMs can also present in the back with transient vision loss so it's kind of the same pictures before all right Chris Bear let's just review our quiz real quick for the sake of time and then let's get to clinic so for the sake of time we'll just skip the rest of this stuff here we've talked about a lot of this history's important there's some questions to ask, exam's important test and we'll skip all of this this is a nice differential for transient vision loss I can send that to you if you're interested alright so our first question young woman some white dots for revision a little bit of discomfort when she has these what do you guys think the answer is the most appropriate next step so if two answers B and A exactly so I agree the most appropriate next step would be gonioscopy because we're concerned about intermittent angle closure and someone like this with these kind of presenting symptoms so gonioscopy is the next best step and then we can pursue other things as needed but gonioscopy is the appropriate next step number two so 6-5 year old guy hypertension, smoking transient vision loss most securities in brightly lit environments so most important test to perform so I agree magnet stress test wouldn't be a bad idea um maybe this is a poorly worded question but the most important test is vessel imaging because oops, that's wrong most of the test is vessel imaging because like Brad said we're concerned about this is light induced hemorosis we're concerned about vascular problems and specifically in the carotids so vascular imaging would be the most important thing most common causes of transient bilateral vision loss natural woman transient vision loss headaches worse with lying down as though she was nausea um and so she's been taking accutane and so what does she have first and then what would be the thing that's not consistent with her most likely diagnosis right so we think she has IH right and so which of these would not be consistent with the diagnosis of IH the exact as you can have papillodema, you can have MRI we have to sell and you can have a cranial or six palsy but cranial or four palsy would make us think about other things so we wouldn't expect that absolutely number five, a six year old woman hypertension, transient episode of vision loss she's at vertigo um and she's had these episodes when she's at her hair salon getting her hair done um she thinks it's monocular but she hasn't checked so what's the most likely source of her pathology? exactly exactly um so this is vertiborbasin insufficient since Rich was just talking about um and I put it there the thing about the hair salon because it's kind of in the literature you can hear as like beauty salon or beauty parlor thing because as you have extension of your neck while getting your hair done you can have episodes of hyperprofusion causing these symptoms so um ask about where they get their hair done um and then finally uh you have an 80 year old guy hypertension diabetes he's a new floater kind of black spot that comes and goes and he had this episode transient double vision a few weeks ago um he's got a PVD with multiple floaters so what's the most appropriate next step for this guy? D I agree so even though he's got vitreous floaters he's got what he describes as a spot um you know his symptoms are concerning for um potentially GCA we need to make sure we pull that out and so um getting me a salon CRP would be the most appropriate next step um out of all these options here any questions on any of these? okay great let's go to clinic