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>> Thanks for the invitation please stop me for questions if something is not clear. To me if you understand where I want to say. So I'll talk about the said locally the callable codes. And some connection for Editor comment very explicit where it might be progressions and random savings this is joint work with the operate from C.W.I. OK so.

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OK So let me the plan and maybe I'll start with the demand result the main taking cold tool 1st which doesn't make any sense light self but I will show you why how such a result would be useful in. Several of these obligations OK So let me go for the main result of the me me notion is that if a caution with.

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So suppose there is a subset P. of. The K. in say gosh and that of the said D. e's supreme and so the expected value of. Your sample random question. Can demonstrate one. Or T. and D. in a product of unity going to have been absolute. So that's the definition of aggression with it's going to deal with what it's saying is OK let's say this is my say D..

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Then what he's saying is bigger random direction. Of descent in this random direction and you and you average it for all the time that's called the mean weight and they are close related so the garden with the D. roughly square K. times the mean weight of the. In a random direction like what is the width of the set in the right.

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OK. So I want to tell you thank you so. So for example maybe it's good to look at an example towards the garden with the. The L. 2 ball. So I think be a little ball it has the same with in every direction and the width is like a constant so it's like roughly where so this is.

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The disc or. If I or does a gosh in width of. L one ball. So if you just go through various things so what is it is the expected value of a random direction the Supreme. Or D. in the long ball. Of G. and T. So this is a deigned for rent so you should put all your weight in the largest needle Do you feel this is just an expected or D..

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Maximum or I do. And this grows like square block. So yeah so that is the reason it's much smaller as you know like there are some places where you know that has very high grade but on an average it's much smaller this ball the ball is much smaller than the ball 3 should understand so it's like some kind of air is notional with.

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So. So before I tell you the main result let me tell you some of the patients. So real mostly talk about uniform hyper graphs So let's say edge we use a D. uniform hypergraphia. So let's say the uniform. You say and then they dig their degree is the maximum.

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Member of this incident on a murder next it's the man in the max degree of the griper graph and let's define B. at. X. So suppose I give you X. and so let's say this really is and Mark this is and I give you X. in 01 to the and he gets all facts he's the some nation or all ages and.

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So so be it off like some indicator function office said is just a member of the new status of it and yes. OK So let me state the main result. Is so simple that. It's one it's going. To be the hyper graphs of low degree it's a think of the as constant cations it is on and where does this and let it die from 01 and 2 are to the K B U and by the Sub be it.

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OK so the coordinates of this map by just be a type it's just counting the number of induced. Then the gushing width of the image of the snap. Is. The sum constant depending on the. Square root of a. Like this. It's not about the looks very scary so what it's roughly saying is falling is so what N.T. is just an organization factory so if the number of maximum number of edges on the New York this graph says.

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So this is just the maximum of a do so if you divide by that then or and D. are like some numbers between 0 and one. Right. In fact or so OK let me just write it in a different way this is the expected value or G. supreme or all X. in 01 and if you write it out this is what it looks like to one of K. D. B..

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Of things. Right so this is a way of saying so let's just divide by this and here so these and then various something between the wrong one. Number and you said this fractional number of them. You saying that this is that most Square or times of this memory here.

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Yeah decoded to the graph. So it's giving you a log and squared log in this factor completely disappears and you get to work 8 times were done so why is this good so what is so weird adding up. Numbers with some random random questions you can also think of cautions you can also think of random science roughly the same so instead of random go independent cautions you can choose an independent science plus one minus one thing the same month or.

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So is saying is you're adding up these numbers with random sayings if there is no supreme om you should just get square. Right you're adding up independent cautions with some corporations at most one the magnitude of this will be at most working but now we have a supremum or several such corporations right there like don't demand such things but we are saying that it should not increase the thing by 2 months there is this 1st big bond on how much use this is going to create you are.

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OK so. Called. Norm and no even that one ball it goes click a things right it's like a diamond from object. So this is a subset this is a subset of R. Thank you. Yes thank you. OK So yes so let me give you some of this by itself looks kind of you know looks just take a nickel I'll give you a very simple application and turns out more or less equal and or you can say what is the operational meaning of such a bound.

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So for this let me. So let's say H. one. Is before that so let me define this notion of approximation for hyper graphs. So so let's. Do uniform hyper graph. Let's define draw a sub edge of a set S. is the number of induced edges So S. is a subset of the word this is.

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The number of used of in S. and the normalize it by the total number and this and so the fraction of edges of it would lie completely inside my set this. And I say that the hypergraphia is approximates another approximates another it's prime if for every subset of our there on the same word back said let's say on and where does this every subset for every subset that density is this Roy it off as is approximate absolute approximate to Roy It's.

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Sort of voyage prime office. So if you look at you know the number of windows read there's any subset the fractional number it is the same approximately the same in both the hyper graph so this is a notion of approximation. Yeah but yeah this person or mine isn't.

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So OK so now it's for the main Param can be interpreted as follows so suppose. One K.. Are bounded degree. D. uniform I progress on. The same on an arc business. And let support and let. To be the union of these I progress so here I am assuming that if some I just repeated in it so it's just disjoint union kind of thing like there can be paddle edges in it.

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Then if I sample. Yeah exactly so just you can think of more like a distribution and it does that it's just multiplicity. If we sample. Least many so if your sample and to the one minus one or 2 times logon. Be called to this sample these many high progress from.

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And it's it's I one. Uniformly from this it's one point. Then it is epsilon approximated by. You saying no matter how many initially hyper graphs I have in it like it can be a union of several of them like and the 100 or whatever. I can always aspires to fight I can just use a really small number of them uniformly from here then it will be really good approximation to it in terms of all Cottrell you for all this usage.

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And. This statement is more or less weak on you the way you put it to you can reduce it to. Any Question So for example for graphs it's really saying something strongly defended equal to or saying no matter how many initial like it's a graph which is a union of bounded hyper graphs wander graphs then I just need to choose a log enough that proves prox made it about by cut and since torque for graphs approximating all the cut values you say my spectral things insult approximate spectrally and.

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So. So it's a good question so the trivial bound is just like you know and or and logon is like trivial because you know we want to approximate So the notion of approximation involves 2 but then tests if you do these many samples for every copy you can get it right with probable the exponential in and then you can Union bound or could demand things so this is kind of to.