Thursday, May 26, 2022

PDEs vs. Geometry: analytic characterizations of geometric properties of sets – Svitlana Mayboroda

Well first of all huge thanks for inviting me again big thanks for both peter and camille camilo for its own visit um this is actually an expansion of my uh mass conversation so from last week because yes as point without that books recently .

Um it's uh i'll try to keep it colloquium style since i talked about localization last semester which normally would have been a better topic to be honest but since i talked about optimization last semester i chose more geometric um geometric topic but i'll try to keep it colloquium style nonetheless .

Um so because this is a colloquium let me start with a few naive comments we are going to talk about the director problem for the laplacian about the harmonic functions on uh general domains to start with as per usual you can think about this um in terms of many applications it has heat conduct you know the .

Temperature surface of the room you want to know the temperature inside um psychics you are trying to look for electrostatic potential or something which might be closer to people in this room than minimization problems for instance shape of the soap field on a given frame on a given line .

What do we know about the solutions in which situations can we solve explicitly and by now i haven't reduced f on the right hand side so we're speaking of the general solution to laplacian equals f not necessarily functions well explicit solutions are here in all cases .

Are basically reduced to iron or half space or a ball which is morally of course the same thing and in each case the solution can be written as an integral of theta against the green function in um around the green function is of course one over x minus y to the power minus .

Two times the constant in half space you can still hook something up using the um reflections plus the integral of the position against the boundary data and on the ball you can go all of the same thing directly or use the half space and complex analysis .

To pass from one to another change in the way this is basically um you know 19th century mathematics and this is basically all you know as far as explicit solutions if your uh boundary is smooth and you are still on a regular manifold what happens is let's say for example you have a graph boundary so this is .

Your few facts and the domain is over here and he's smooth basically uh well you don't have explicit solutions anymore but life is still good uh for instance you can't think of it as uh the same problem after the change of variables which would lead you into half .

Space with a piece of good news that now you're on the half space and there's a piece of bad news that now you have an operator different from the location you have device with a which potentially carries elements of the gradient of your graph but again if life is smooth since it's .

Still good and you can write green functions representations you don't have explicit formulas anymore but you have the same estimates as you would think you should have and the solutions will have the same dependence on the data as roughly speaking and the top space case all this has been sorted out and there are 50 .

Cents how far can you push this well at the very least uh from the point of view of this change of variables but honestly from the point of view of median of other sense you need gradle the fee to be defined almost everywhere solutions domains are fine the remains with corners the demands in which .

Gradient of the graph is defined almost everyone anything beyond it is uh possibly not a good idea at least among the graph domains and already this um while it really sounds like just pushing the smoothness to the edge of fred already did this to create the .

Developments of the harmonic analysis of the ages 90s um some elements of geometric measures there is certainly singular integral techniques control sigma city sense like this so it sounds similar but it's realistic and very difficult already to extend this to the electrostatics and this is pretty much the limitation .

On smoothness you can learn though worse than that of course there is this word knowledge in mathematics and uh while in terms of grand demands your worst situation is richards you could in principle envision uh the boundaries which are much more .

Complicated than that which are you know crazy unions of liberty's graphs of all sorts so the structure the topology of the south would be much more difficult really push your knight away from the even of the pieces ellipses and could be really portioned at every front election fast forward .

30 40 years i mean this is really uh i would say about 90s the beginning of 90s when the question started appearing seriously enough in harmonic analysis right now we have finally basically in the beginning of 2020 there has been a series of results in 9 2018 19 20 21 and at least at this point finally we are at .

The stage where we can say that we know the necessary and sufficient condition and this is the main gist of my toe so there is one message that you bring out of here today is that rectifiability is necessary and sufficient i will define what rectifiable means i will define everything else i will .

Define define necessary institution for good but basically in the world of the estimates that i'm talking about and the solubility of boundary problems brain function nastiness demonic major estimates of all of those the necessary and sufficient condition on the side is finally known and .

Achieved viability of the cell so basically that's the gist i promise to talk about the green function so let me give you just one slide of you know tiny bit very incomplete introduction to the story now i'm talking about the necessary and efficient conditions before it was on .

The sufficient one so now we are asking ourselves not only the direction from smoothness to the properties of pds but the direction the opposite point so what's known well one example and the most obvious thing to study of course is um going from classical smoothness spaces so if the boundary is lipschitz .

Then you have green function in cellophane and vice versa green functions and some builder class and boundaries order as well with the loss of smoothness the laws of smoothness is important the converse is not true meaning that you cannot possibly have a criterion in these terms the necessary .

And sufficient one but actually um to me and to a certain class of you know applications this is not even the bigger problem the biggest problem is that this is not sufficiently quantitative it's all about isolated singularities it doesn't give you the global picture of .

Sins and as any atomic physicist will tell you you know one atom will always be degenerate you know defective it should not matter for the big picture of the system so this is sort of the late motif of this talk we are trying to understand sense from the view of big physical .

Systems we are not trying to understand isolated singularities it does not matter i mean it does not matter for this sort of questions most of what has been um attacked before in this terms is really the behavior of green functions of solutions near small singularities this is the opposite point .

Of view this is really there the entire game must understand what matters at big scales sort of what matters when you look at the system at large or locally you know locally and globally simultaneously and scaling barrier terms ideally with minimal geometric .

Assumptions we are not speaking of products or anything like this they are not restricting dimension or anything like that and with the minimal assumptions and operator that is very speaking of manifolds at large so the main result for today is um .

The theorem from basically last year in any open and connected domain omega i will define what i mean with a dimensional boundary there are no restrictions indeed there are no restrictions on n yet again i will define about what pdas i'm talking .

The green function is almost defined if and only if the boundary is uniform at x5 so that's the end of it this is the minimal possible assumptions on the set you could introduce to possibly talk about this sense this is unnecessary and sufficient .

Conditions there are essentially no restrictions behind it this is both this despite the looks of it i mean basically the rest of my talk is reducing every single word on the slide so i promise i will give you all definitions .

But let me tell you that there are no dimensional restrictions or you know any other sort of restrictions as you can see from this theorem this is a criterion this is different only goes both ways this is both local and global and as a matter of fact scale environment .

So it really classifies the situation both at small scales and at large scales at the same time and despite the looks of whether she's quantifiable there are norms coming with each side so there is a very careful normal control at both local and global scales in both directions .

And basically the rest of the talk will consist of all the definitions and some historical contacts which will hopefully show you how we came to this result because it's really the end point of a huge line of developments which i will not even be mentioned today in every answer there is a question hidden i mean you could say it's natural .

Or it's not natural i hope to prove you that in some sense this is a very natural important result okay but in order to talk about this since let's uh go back a little bit and look at the object which is my opinion actually more complicated .

But which hasn't started much more because of well basically because of the history of the problem because geometric measures which is harmonic so harmonic measure of the part of the boundary let's say e is a very bad color uh let's say e is a part of the boundary some set on the boundary harmonic .

Measures the probability that the brownian traveler started from point x and working randomly will answer the boundary through e as opposed to the complement uh it's the simplest building block of the harmonic functions any harmonic function would be written as an integral of data against harmonic .

Much of the problem is in order to talk about this you need to know something about it and um i mean the simplest question is of course to which extent is a absolutely continuous or proportional to the sigma to that the measure of what you actually see so the question um in other terms what .

Is whether the whether omega is absolutely continuous with respect to the bag of hosts of natural on the side um what is the dimension of structure of harmonic measure how the brownian travelers see the cells on the boundary now uh .

You would say that you know looking at this set of course the answer is yes the bigger is the set e the bigger the chances of brownian travel to exit through the side so more or less the boundaries seen proportionally to them um with that size what can happen well this is a picture of actual real molecular scaffolding of .

An organic semiconductor from one of the collaborators which are friends to show pre-heat from the cambridge and as you can see the real geometries are not as fun as that and in particular it's really not clear how the brownian travelers how the electrons are going to arrange themselves near the boundary and .

Uh if you look in terms of the harmonic riser you know you start from the interior point you put the brownian traveler it's really not completely clear what are the chances of reaching all parts of the boundary whether it would be able to go through the antenna in this picture .

Whether the end points would have bigger chances to be hit in this picture or the granular travelers would actually go inside whether the other side is accessible and all of that and secretly i'll tell you that the picture is even more complicated it's actually a three-dimensional one this is the second 3d .

But we'll talk about the separator it's already not falling into here so be able to talk all right so what's the dimension of the set that the brownian travelers can see and is it through that face in the of that kind they can actually enter all the way to the set .

I don't know if i'll be able to draw the fractal by my hand i don't want to spend too much time but you know i mean you have something like this which definitely doesn't look like a fractal point is your assigned and brownian traveler from here cannot go all the way into the fjords can it go all the way to .

The crack well um in 2d the answer is known there are a few results i'm mentioning here to give proper tribute but actually the main theorem is due to macaroon 1985 and the answer is the dimension of harmonic measure on the plane is always fun so no the brownian traveler can not ever reach a fractal set .

And having faced the fractal it will always choose a subset of dimension bond where it lands probably looking you know something like that closer to the surface so this is uh the situation on the plane given the situation on the plane you might contracture that in .

The dimension of omega is n minus 1 and this is not the case so first of all the only positive result known towards this is due to per game 1987 the only result known as that the dimension of omega is less than none this is not a typo and i'm not crazy all we know is that it's really strictly .

Lies and then dimension there is literally nothing else which happens since we're again in the positive direction of the subject and what's worse we know due to wolf and 1991 that in dimension three there exists a snowflake which gives a dimension of harmonic .

Matches strictly bigger than two so this is not true and already in three dimension there are more than two hard and two dimensional sets which give the dimension of the harmonic matters if you think that this is a mathematical .

Artifact just for the front of you i have put out what i promised in my conversations a snapshot of a real wool acoustic barrier built actually by my collaborators um which is a fractal acoustic wall on a real one on the hardware on paris basically coming from the gold snowflake .

It works it holds that it still holds i mean it's been about 15 years it still holds the world record of performance when it was built it was 17 times more powerful than any other noise abatement well known out there i don't know what are the numbers now .

But it's a it's a very real scene and really mathematically we still don't know what's going on it's a snow plane what is it it's a snowflake yeah i mean i have a close close-up but it's basically yeah so this is literally there you know the wolf snowflake .

And this is the it's generation six i believe but it looks pretty well so mathematically speaking somewhere between dimension somewhere between 2 and 3 there is a number given the dimension of harmonic measure we still don't know the number all right how about the structure .

What do i mean by this let's say you are in dimension n minus one is it true that brownian travelers would necessarily reach the entire set of which is an n minus one dimensional set can every set of dimension n minus one holds the harmonic measure answer is again no and in order to talk about that i'm .

Finally starting to introduce proper geometric notions so first of all i will be talking about the so-called gear force regular sets this is just this is just the concept of the dimension it's not really regularity in the sense that all i'm saying is that in every bowl there is about our quality of myself so this means it's uniformly .

Dimensional so this is the only dimensional um concept and then of course rectifiability rectifiability is the property that basically yourself consists of countable union afflictions graphs modal or set of dimensions zero we will be more occupied with the .

Quantitative notions so in particular we will look at the stats which are uniformly rectifiable and what it means is that for every point on the bond the reverse of all i mean it can be you know crazily disconnected but for every point of the boundary for every scale for every ball centered at this point .

There exists electricity image so that your cell coincides with the eclipse of so much to one percent or half a percent or whatever it is that you want so there is a literal image somewhere here which changes from scale to scale it changes from point to point so that one .

Percent of your set lies on description this could seem like a very you know the intersection in this case is somewhere here this could seem like very little information it's only information about one percent of the set but somehow the fact that it's true at all scales for all points for all r is immensely .

Powerful it turns out to be um very interestingly exactly the correct condition else and another way to think about this which is actually probably more relevant to the subject is that uh the stats such that have so-called tangent lines tension planes .

And for every scale for every point there exists a hyperplane so that again one percent of the set is close to the hyperplane and this slightly improves the scales in a certain and technical way i'm not going to talk about this notice that even the concept of attention to .

Sort of uh efficiency here in the sense that it's more like approximate lines than tangent lines generally speaking but basically that's the idea there is a control of one percent of the sun what's not rectifiable well whether one percent is a fixed proportion or it's .

Yes so i mean of course our uh there is a trade-off secretly in the library's constant so the literature's constants of graphs are uniform too if you take half a percent ellipses constantly so what cells are not rectified well the most emblematic one is the um garnet four corner counter set so you .

Take a square you eat up one fourth on each one one half of each side the middle one half uh instead you put it into corners so your next generation is actually four corners .

And you keep doing this you eat up again one half you put it into corners your next generation is four corners and so on and so forth the set you have now we use as a counter said there is no on it and any line mean i don't even know what to draw .

Because there is you know whatever line you put it's a bad line and it's not going to improve its scales for itself similar so basically there is no approximation line there is no approximation okay what definability well rectifiability was identified as an appropriate notion .

Already in the beginning of previous century 1916 whenever memories prove that in a plane for every rectifiable side um the harmonic measure is absolutely continuous with respect to each one the quantifiable result is due to lagrangia and then .

Many many results finally brought it to the heart dimensional sentence this is not even the full set of it but the punchline of the story is that in any dimension uniform rectifiability plus some connectivity tiny bit of connectivity is actually sufficient for um absolute continuity and sometimes i .

Write the infinity i mean the harmonic measure is an infinity weight just take it as absolute continuity if uh if it's not friendly for you if it is immigrants that are not talking about that i'm talking about the congress right i would .

Like to understand if the congress is correct i would like to understand what the brownian triangular see so i would like to understand the opposite whether i can understand the set which hosts the harmonic method this turned out to be much more complicated but you know i found them .

Reasons 1916 and you're saying that in 100 years we should be able to do and in 100 years we did 2016. this is actually a combination of two separate papers but in the end we publish this together this is the first converse staff and memories and it has absolutely no .

Restrictions so for a new open set um n minus one dimensional just in the sense that h and minus is not real if harmonic measure is absolutely continuous with respect to sigma the certain multiplier actually i'm right and the supporters rectified with them mutually absolutely .

Continuous the sentence rectifiable i'm not getting into that level of details here so this is that as far as the converse goes in this particular setting um this covers the so it all there are no restrictions coming with it i'm formulating qualitatively there is a quantitative analog again with all the .

Norms in in place but one thing that i want to mention and um sort of your intuition to this towards the subject should be in terms of those tension plans it's a question of what harmonic functions can possibly see and the idea is that .

They they need to see some sort of philanthropy at all scales it's sort of like flying an airplane and uh um i apologize for the same example as in mass conversations but since half of you were not there it's a good visual um you need to see where you're going at .

All scales it could be faster you could have mountains at one scale and you know you would have to change to buildings at the other scale or trees or whatever it is but at every single scale you need to roughly speak and see the direction you need to roughly speak and see your landing strip you need to roughly speak .

And see where the brownian travelers are going and this turns out to be the right notion because the goggles of the harmonic functions are not all they're good they don't see the details when you say they're omega associated to each rectifier is it isn't like exactly something .

Well as i said if it's if only guys absolutely continues with respect to h n minus one then omega0 is rectifiable but if they are mutually absolutely continuous then the set is rectifiable right so what does it mean that omega is likely support i mean like you just it's it's the part of the it's the part of e which .

Is supposed to follow basically excuse me i don't understand what little omega restricted to e is rectifiable means little omega is a measure i think you want to say that it's capital omega restricted to e that's rectifiable not really but let's not get into this let's .

Let's go as follows so if omega is mutual absolutely continuous with respect to issue minus one in both directions mutually absolutely continuous then e is rectifiable okay so let's not get into that so when i'm saying that the measure of one .

With respect to another one this is a one-sided condition and this one-sided condition implies something on the restriction point even instead you say that their mutual absolutely continues with respect to each other which is a two-sided condition then the set is rectifiable in the sense .

That i defined so far so let's get there everything is honest i promise i'm not cheating but there are there is more i mean like it's actually a richer result i mean there are all partial pieces into this which is why the formulation is like this .

In any case the mutual acts of continuity and classifiability in the world for the set in the way that i have actually defined so let's not get them to use here but back to the green functions on the story so first of all the result itself did not appear in the middle of nowhere .

There have been many predecessors and there have been um results by now results after that i mean this is since 2016. the clues were on the moon the predecessors is you know kenyaktor when they were trying to uh characterize right-hander flat cells in terms of oscillations of the porcelain kernel .

I would also mention pokemon martel that goes closer in terms of techniques afterwards finally the necessary and sufficient topological conditions were identified in 2018 so now we know that topologically the appropriate condition is the so-called big local drone condition which has to accompany rectifiability .

I won't go there we also know that this is true for much more general operators than just the atlassian dashes on manifolds or otherwise for divi grad operators and i will define shortly which operators are allowable which are but back to the story of today uh first .

Of all this is not the green function result and that you could say the question of paste which results you actually want but more to the point um and i will tell you what exactly i mean by the problem in lower dimensional sets for any case when the dimension of the .

Boundary is lower than n minus one this result shatters just dramatically i mean however you define your operators however in a sort of meaningful way however you define your brownian traveler's cover you define your approach to lower dimensional boundaries this is just plain not true in any circumstances .

So whenever you work with for example with a complement of one dimensional set you cannot do that you have to go back to green functions we claim and another problem is that all the reactions have restrictions on the coefficients that is it's only semi true sometimes through on the manifolds to add to the confusion .

Let me pinpoint that um so we just discussed that the status erection fabric and only is absolutely continuous with respect to sigma which means also the opposite that on the counter side of the purely unrectifiable set this is not the case and singular with respect to sigma .

But now now if you are on a manifold or um you know as a pda person if you work with a given operator with a certain a it turns out you can build an a so that your harmonic measure is absolutely continuous with respect to sigma all the same kind of set .

This is bits and pieces of the construction but the point is that the counter set is the same so there exists an a scalar a again no triton you know real plain one such that omega is again absolutely continues with .

Respect to sigma so you can have an encounter set at hand you can sort of build the manifold which would guide your brownian travelers in such a way that they would arrive honestly in uniformly so the situation is really much more um faster than it seems to be and i believe .

This is the first counter this is actually the first example a few months ago but i think that's the first example of absolute continuity on the character set it's just a complex analytic construction you know you basically do it by hand is this a generic in any sense .

What do you mean by generic amounts of by hand's construction in the sense that i take a counter seven i mean these are literally the love allowance of the green function so i'm building them by him it's unique once once i give you the the counter set you get just one or do you get a bunch of conditions we didn't try .

I mean that's sort of you know i mean we just did it by hands so what is the picture there you're just having some plugs green is so these are level lines of green function so green are the we are building it you know it's a two-dimensional thing .

So we start on the counter set we are building i mean we basically draw the picture first and then we are proving that what's behind the picture exists in some sense that this is really you know this is a real collision and we can really introduce solve the inverse problem .

So this is the picture of the green function greener the um level lines red are the gradient lines orthogonal ones and then we are basically solving the inverse problem showing that you know like everybody who would do the inverse problem will tell you you know that .

Showing how to build the conductance matrix corresponding to this first i mean the conductance matrix is kind of freezing but then you need to make it interest color and that's actually possible let's really find by hand so the domain is yes .

And the harmonic measure is actually proportional to sigma but we start it's super cute does the electricity have to be very far from one no no it's like between one fourth and four numbers coming from the center what you're saying a quarter are you .

Using are we using what are you using some pyramid complex analysis between the number of quarters input are you just saying one fourth no no no it's no we are not using problem yeah i answered no .

No where i mean this is a this is an explicit you know we don't need code because it's a fractal so we know how each scale depends on the previous one adjust the conversions of the series all right so back to the setup which i promised to state .

Uh let's now define the let's now go back to the green function in which case i claim there is no funky topology involved there is a much bigger set of i mean we can work with all electric operators at least in one dimension at least in one direction and we can work with all dimensions so let's see how that goes .

Let me define since first of all we already discussed d dimensional air force regular workforce day with regular boundary is just the dimensional i sometimes say unfortunately regular sometimes i'll force regular because betavid is one of the collaborators on this term so he doesn't let me .

Stay in his serum with his name but um otherwise it's just the dimensionality um so what is quantifiably often unconnected well first of all if the underlying dimension is n minus one it doesn't really matter we don't have any restrictions it's avoid saying you have enough access there are .

No issues in the dimension of the boundary to start with this n minus one what we need and i won't be drawing this is a little bit of access to basically talk about brownian travelers to talk about the sins uh these are just standard mouth topological assumptions you need like .

Through points and so-called her not chains but poland is you basically need enough access to the time uniform the concepts here now this one hurts i know but i promise this is the only slide which will hurt what is almost defined this is sort of the gist of the story this is the right way to think about .

That sort of the boot properties of the green function in it so let's start from the karma sometimes you have a set again i promise this is the only technical slice for bear with me you have a set which is a subset of the boundary across .

Real lines so cross the scales we say that she is uh satisfies the carcass and pikan condition if there is less and less of the set as you go towards the boundary so you want the characteristic function to be integrable against one over t basically the point is that you have t and .

Denominator so you have a tiny little bit to make the same conversion the characteristic function so if you have yourself and you have a this is your boundary this is your scale uh what you want is that there is sort of less and less of yourself as you approach the boundary and even less than .

The distance to the boundary again justin the way to make the interval conduction that's right this is the bad set the good set is the complement of it so the one we are talking about is basically everything else and if this is satisfied if there is more and more and more of your status .

You are approaching the boundary then you call it the currency prevalence so all i'm saying is that there is more and more and more further now in which sounds is the green function close to the defined functions well it literally can be approximated by distances to planes on the carlos and turnaround .

So what i'm saying is that not only there is a defined function there is a plane and there is a distance to the plane well which is sort of the same thing okay but of course distance to the plane is a fine which approximates green function in the scarless and prevalent sense so other .

Than the small jump of which there is less and less and less as you approach the boundary then function is close literally the distance to the plane in this completely crazy domain if you think about this um so first of all i mean defining the scarless and prevalent .

Condition is sort of one of the achievements of this paper to start with because again that um the trick alone was to understand what matters and what doesn't i keep insisting this and this is the point that small things do not matter from .

The point of your physical processes and the important issue was really to understand what does it mean that small sense do not matter how do you measure what's small what's big what matters and what doesn't is the importance of one over t as opposed to other powers it's the exact uh i mean .

It's the exact one which fails i mean c is one dimension or just scales so what i'm saying is that there is tiny bits you need tiny bit better since you are you know close characteristic function is not just uniform and see okay um .

So the catalyst condition itself in the strong sense is of course all that goes back at least with his thesis actually in 1971 um but it was used in a slightly different context regardless and measures are those characterizing harmonic functions with vmware data so the concept itself is natural our .

Version is bigger just the weak analog of the harness and condition much less wrinkle b is an analogy but the point is that the concept itself is different i mean you are not looking at the harmonic function with some data you are looking at the proximity of the green function to the distance to the .

Boundary so the distance to a plane i'm sorry not to the boundary the distance to the plane and what i'm trying to say is that it's sort of from using because the result is simultaneously more powerful and less powerful compared to what you would probably want based on this experience and the subject .

On one hand uh it's um you know it doesn't tell you what happens on those small inclusions on those small sets on the other hand on the good side on the big set it tells you not just that the green function is um you know it's better than lichens i mean in particular the green function is .

Richards it's much better than that it's basically almost defined and you would say you know weight weight weight weight water when near the corner the green function is continuous i mean you can't do it explicitly it's actually you know fractional power of distance to the .

Boundary so how can it possibly delicious how can it possibly be defined well there cannot possibly be that many corners that's the exact point that bad things do not happen often the rabbit sounds happening to you but bad cells do not happen often that's almost the chairperson of .

The day at least let's just control them somewhere um what about the catfish ones so what about the what do i mean by practitioners again depends on your favorite saying i mean i'm thinking about diversions from operators so minus divide right but you can think .

About many poles so what what if you go on the manifold in the free boundary direction from almost defined to regularity to 35 billion there are absolutely no restrictions on the coefficients now it's just an electric for any electric operator the same result is true to the best of my knowledge this is the .

First free boundary result in the subject in the you know whether we are talking about the green function or not it's literally the first three boundary results which does not carry restrictions on the coefficients so it's true for all electric operators .

In the opposite direction there is a restriction which is again in terms of discoveries and prevalence sets um i will not talk about this but you know long story short this is this is sharp there are appropriate counter examples so in in the opposite in the direct direction this is what it should be .

And finally how about dimensionality so i have something like seven minutes to talk about the dimension i'll try to run through this super quickly in the case when dimension is n minus one i basically wrapped up the story at this point so this is this is what it is the beauty of the result is that .

Contrary to everything you can possibly know about harmonic measure it also applies to set of dimension less than n minus 1. and by this i mean far less than n minus one i mean the interest in one series of much lower dimensional sets so what what you actually should be .

Imagining well you should be imagining since like a curve in r3 so it's a one dimensional certain australia or a dna chain of which i'm drawing a bunch here and again not going into details they do actually tend to be together rectifiable when they need to attract molecules so there is a certain i mean this is you .

Know whatever is coming has real applications in the context of dnase and finally you know the big data i mean whenever you have quite often you have the data coming in lower dimensional cells you need to understand that the harmonic function so your your .

Domain is a complement of a lower dimensional set the harmonic functions do not see them at all just not do not acknowledge their existence the probability to hit the set of law of dimension is exactly zero so what do we do well you need to reward the branian travelers for .

Approaching the boundary and that's by the way what the dna changes sometimes though in some electrolytes so what you need to do is instead of that i mean either in terms of coefficients or i'm centering of this in terms of the energy you need to .

Introduce some rewards and mechanism and you need to introduce the coefficients or energy which all you know which gets higher and higher in this case goes to infinity as you know the the same approaches as distance to the boundary goes to zero so it's the inverse powers of distance to the boundary .

The exact power is you know whatever is dictated by your homogeneity you can do something much more general thirdly you have to do something much more general so when you are starting to you know really get into this you understand that you need more general operators you need many folds right away even if you are trying to study the .

Simplest one you need the means of the mix rather than just local dimension because you are going to have to shield pieces of your boundary from brownian travelers and if children in dimension one gives you still you know the domain of how dimension one .

Then children and heart for dimension produces this stats of next dimension but anyway long story short uh we have by now the elliptic theory which says that you know for any boundary measure subject to a certain very simple condition this is no which is a boundary measure that's just m which is an interior measure so for any .

Too much of subject to some very mild control condition you can build the elliptics area um no matter what was the dimension of the boundary with regular you know the georgetown definition of harmonic definition of suggestions like this ironically this ends up being a certain extension of .

A certain generalization that you want of course investor extension operator for those of you who know what i'm talking about so in the case when you have a flat boundary and it's n minus one dimensional the fractional laplacian as a matter of fact is the delicate normal operator .

Of something in the spirit of what i'm talking about depending on them so this has you know this has been generalized by alice chandler for collaborators who it has to do with fractional finance operator on um even lower dimensional sets but also assured that it's part of a big .

Theory but all of this can be generalized and basically what we managed to do is to define differentiation and integration because these are powers of the laplacian this very very very general sense appropriately with appropriate estimates .

You still have for a certain very special distance which i don't have time to go to you cannot use euclidean distance from the setting that's going to fail you dramatically it's a really really bad idea to use your creation distance but you can't define a different distance to the set and in .

Terms of that different distance the final coefficients you will ultimately have the same properties as what you expect on every uniformly rectifiable set again no topological restrictions um the results and harmonic measures are softer continuous with respect to sigma .

Problem is the converse fails congress fails you know like it shatters it fails dramatically it gives you no absolutely no chance to actually survive basically there exists an alpha so that i mean in this newly defined distance which certainly i won't be able to define for you .

But there are existing choice of coefficients for which the distance gives you all the noise the green function is the polar infinity and it gives the i mean it gives the property that the harmonic measure is basically one in this case so i'm sort of having um you know .

Trouble explaining this and finite time but that's a little bit similar to that counter counter example your guidance sense also perfectly in some sense your guidance and so you have you have worked so hard to guide your brownian travelers so that they even reach the boundary that you guided them so that they're collapsing on the .

Boundary in some sense not really but there is going to be the reality in the uniform and that kills the idea of the converse at this point one should be depressed it means that everything was for nasa and probably it's their own notion to start with but there are no other options you .

Know you you're trying to get something at least from the points of your imagination that's the most natural one until you realize that you have been working with the wrong object all this time it's not the harmonic measurement matters it's the green function and for .

The green function for all of those operators yet again you have an effect on great result this is the appropriate way to think about this and with the very same definitions as what i gave you without any restrictions and any dimension yet again green function is almost defined in the very same sense .

If and only if the sample is rectifiable no matter what is the dimension of the set and the such as far as i know this is the first and the only to this day characterization of rectifiability and multi-dimensional cells if in terms of pda properties no matter you know in what world of pdf .

Properties this is clearly the first if and only characterization so this property really appears to be i mean we actually went into studying the green function from this point of view from the desperation with the harmonic measurement only then we realized that .

Everything else is new too already in the function international actually but i'll finish here i think thank you very much a question so what happens inside a way to you could use a higher order operators or p is going to bite you i mean in order to .

Give if and only if conditions your in order to give free boundary conditions you need something positive because you need something pushing you know if your solution oscillates you will really have a hard time formulating a free boundary result right because the free boundary result and the sense as a result about the levels of a positive .

Function about the zero level of positive function the problem with the bilateral as you know very close to my heart is that it doesn't satisfy the maximum principle even in the corner the harmonic functions oscillate infinitely so you cannot possibly create a free .

Boundary result based on the parallel passion i mean i'm slightly cheating that's actually where we started and we did something but it looked so beautiful after half a year of work that we never published and we're always ashamed to talk about it like i can tell you you know what exactly it says but i mean .

There are i mean i'm slightly cheated there are some people under the results by the question but they're really much weaker because of lack of maximum principle you need the positive face to be pushing you great well i'd like to thank the sneaker again for wonderful

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