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The last talk.
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Of the morning.
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Is by Nicholas cockers at the pvp fl and he's going to talk about observable persistence modules are gamma shoes and take it away.
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Nicolas Berkouk: Okay, thank you very much Shannon Thank you thanks to thanks to the organizer for the really nice opportunity that's pretty rare in those moments to speak so today i'm going to talk about so observable persistence merger so here.
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Nicolas Berkouk: let's say the main features of here will be first to ask Okay, so there is a well defined a theory for persistence with the entire existence, but there is.
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This question.
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Nicolas Berkouk: Of what can we measure with the intervening distance and what I propose to explain in the stoke, which is a joint work with Francois pity is that there is a very nice structure on the let's say.
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Nicolas Berkouk: category that we can observe with the entire existence of persons Madrid which, precisely, is a certain type of sheets, which are called gamma sheets.
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Nicolas Berkouk: So all of the talk is an extract from a paper that just got published in algebra and geometry topology and you can also found find the print on archive, and here I will just present some results of the paper, but there are more things within the paper.
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Nicolas Berkouk: So let me start with a quick recap on one dimensional persistence so persistence modular one parameter pistons module is just a founder from the Posted category.
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Nicolas Berkouk: or to the category of Kay Victor spaces, so I just fix it feel the key and in this section I will just consider two persons Madrid M and N.
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Nicolas Berkouk: As point wise finally dimensional, which means that the victim species at each point of our is funny dimensional.
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Nicolas Berkouk: So there is this very foundational theorem for persistence, which justify the existence of code in the.
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Nicolas Berkouk: Continuous case which is the theorem of quality movie which was proved in 2012.
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Nicolas Berkouk: which basically states that whenever you have one dimensional persistence over all it decomposes as a direction of interval modules here, so it means really persistence module.
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Nicolas Berkouk: That are completely determined by their support, which is an interval of all and if I just recalled the collection of the interval.
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Nicolas Berkouk: appearing me this decomposition I get the bucket of em and they really, really do the strength of the theorem is that this bucket complicated that our minds em up to either Marxism, so this is why we can work with precedence diagram etc.
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Nicolas Berkouk: So then, so this is a very every slide but I had to introduce intelligent distance in one slide So there we go so.
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Nicolas Berkouk: Now that we have this algebraic object of persistence merger one important question that was answered by cesarean quarter in 2009 is out, we measure distance between persistence merger in.
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Nicolas Berkouk: let's say meaningful meaningful way.
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Nicolas Berkouk: When we have in mind.
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Nicolas Berkouk: Application for machine learning, and so the answer of shuttle encoder is to define the entire living distance between two persons Madrid.
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Nicolas Berkouk: And so the entire existence is based on the notion of epsilon intelligence, so I start with two persistence Madrid in an n Okay, and I will say that F that M and N or epsilon interleaved if there exists to Marxism F and G from em to n shifted and from n to me shifted by it's such that.
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Nicolas Berkouk: They basically the composition commutes up to the transition Marxism Okay, so it means that when I go from n to n insulin by F.
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Nicolas Berkouk: And then I go from any discipline to me too excellent by G shifted with epsilon I get the natural transition Twitter and Marxism and the same thing should be to hear.
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Nicolas Berkouk: So I think now the definition of intelligence distance I could have sort of interweaving I can define the notion of interweaving distance between two persons Madrid.
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Nicolas Berkouk: which will just be the infamous of the values, such that my two persons who are intuitive Okay, and so this quantity can be infinite okay if.
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Nicolas Berkouk: My 2% mature or not intuitive for me excellent so First, there is a nice observation that this quantity satisfies the triangle and liquidity OK.
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Nicolas Berkouk: And then, or so, this is the main let's say.
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Nicolas Berkouk: reason why we like interviewing distance is that it is stable with respect to the operation of taking several events, a persistence Okay, so if I have to function on the topological space X.
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Nicolas Berkouk: values in a valued in all I can look at their ice sub level set persistence Madrid okay so it's just the ice emoji of the reveal set.
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Nicolas Berkouk: of F and G and then that I know by this web form that they are interested in distance will be bounded by the soup no between those two functions so basically if I start with to function that are let's say.
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Nicolas Berkouk: F and noisy version of F, then I know that the entire distance between the opposite ends with you will not be further apart, then they are the soup known so in some sense the size of the noise of my measurement.
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Nicolas Berkouk: So okay this distance has really nice properties, but now, one question is so at the algebraic level what can we measure with this notion of distance okay so, for instance, here, there is a notion of a phenomenon of ephemerality.
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Nicolas Berkouk: Which is.
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Nicolas Berkouk: In some sense, something that we.
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Nicolas Berkouk: cannot measure using gentlemen distance so a very simple example is let's start with just one.
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Nicolas Berkouk: One point in URL or and I can define this persistence merger Okay, which is just K, valued at the point X and zero elsewhere okay.
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Nicolas Berkouk: Then it's super easy to show that the entire living distance from disparate systems Madrid 2020 percent Madrid Israel, but this persistence Madrid is not zero okay.
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Nicolas Berkouk: And so, this phenomenon also olds if I take arbitrary their exams of modules like this okay so, for instance, I could have a persistence module that is supported over all but which is a distance zero from zero Okay, so there are some things that we cannot measure with the entire existence.
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Nicolas Berkouk: So now, I will explain the foundational work of shell quality movie and discover.
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Nicolas Berkouk: on defining a kind of category which mud out properly.
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Nicolas Berkouk: The fema all your systems mature.
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Nicolas Berkouk: So in this work, social selling quota defend your category of fema on persons major to be just persistence matures edge that all the internal map are worth zero Okay, and so there is.
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Nicolas Berkouk: kind of Nice algebraic property of this category, which is that it is, if I have an exact sequence here of persistence mudroom.
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Nicolas Berkouk: It is the same thing to ask that m is a female or m m prime prime prime is a female okay.
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Nicolas Berkouk: And so, when we have this property for subcategory we say that it is said subcategory of the category of prisons, with you and so now, the point of interest of having this property for a female person smudging is that there is a well defined notion of categorical cushioned of.
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Nicolas Berkouk: Persons major by if a man versus machine Okay, so this is.
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Nicolas Berkouk: Just a very general statement of category theory right, and so we can define.
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Nicolas Berkouk: The observable category, so this is done by your by your Sutherland quarters, so the observable.
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Nicolas Berkouk: category of persistence material as the category of questions mutual merge it out by the fema at once Okay, and so this is really just some abstract getting really machinery right, and so the get the now the construction of.
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Nicolas Berkouk: have discussions is unique only up to equivalent of category and it comes with.
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Nicolas Berkouk: A projection factors okay so really just think of like, for instance, cushion of group, if you moved out to group by a normal subgroup, then you have a projection map from this group to the to the questions okay so here it's really the same thing.
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Nicolas Berkouk: Right and so once or think of Marxism in this category as family inverting.
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Nicolas Berkouk: The the Marxism of persistence merger was countered and cookie on in our html Okay, so we decided that Marxism was counted in cook county is.
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Nicolas Berkouk: fema fema to be either Marxism in this category.
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Nicolas Berkouk: So.
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Nicolas Berkouk: Now, if you want to do today, and you are interested in two more let's say you apply the questions, it would be like Okay, but.
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Nicolas Berkouk: When those category happens in real life okay I wouldn't put question, and so this is why, also in this work just selling quarters give an explicit description of this category Okay, and so now in some sense.
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Nicolas Berkouk: So, to define this story to to introduce this.
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Nicolas Berkouk: This explicit description they they rely on the interval decomposition through him so you must first do the assumption that you are walking with point twice for any dimension or persistence major OK, and then basically we can express it.
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Nicolas Berkouk: One more detail for the observable category of funny dimension of point twice a new dimension or persons Madrid as dress.
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Nicolas Berkouk: In some sense choosing an orientation or a choice for the endpoint of each interval okay so here, I will define this.
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Nicolas Berkouk: overland as just persistence mature that are direct some of interval of this shape Okay, so I asked for the Left endpoint to be closed and for the right one to be open, right and now one one of the results of social selling quarter is that so this category is actually.
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Nicolas Berkouk: To it satisfies the universal property of the sacrament of the observable category of questions, Madrid and we can explicitly the projection factor.
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Nicolas Berkouk: dressed as basically if I start with a persistence, which will m here, I will in some sense to compute its value you add a point, as I will just take the limits over all points that are strictly.
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Nicolas Berkouk: Greater than so okay.
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Nicolas Berkouk: So this is kind of.
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Nicolas Berkouk: hybridization.
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Nicolas Berkouk: And now a really nice results or so by.
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Nicolas Berkouk: By cesarean quarters in this paper is that, so the projects, the production from tour here preserves doing turning distance.
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Nicolas Berkouk: So all of this work has been done in the one parameter case Okay, and now the natural question would be okay, how could we generalize those definitions and.
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Nicolas Berkouk: How could we also give an explicit description for the observable category of persistence merger, knowing that for municipality or persistence merger they exist, no.
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Nicolas Berkouk: backup decomposition in general okay so here the answer will be.
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Nicolas Berkouk: So one answer will be by using a shift theory.
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Nicolas Berkouk: So let me do a very quick recap on shifts in sort of the goal of this section will be to explain our we can actually see persistence mature as shifts Okay, and just use a shift framework.
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Nicolas Berkouk: For business Madrid so so sheets are just really an object so counters that models data over topological species Okay, so if you start with a topological space X.
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Nicolas Berkouk: You can look at the category of its open subset Okay, so it will be the category with object open set of X and Marxism increments of open subsets.
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Nicolas Berkouk: And here we will be interested in to the opposite of these categories, which means that we have more freedom from an open subset to another one if we have.
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The.
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Nicolas Berkouk: Last, one that is included into the first one Okay, so we just reverse yellow.
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Nicolas Berkouk: And so we define oppression of Kay Victor spaces, just as a filter from this category, so do the opposite category of footprints upsets to vector spaces Okay, so it means that the pressure is just a machine that takes open sets to.
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Nicolas Berkouk: Victor species and.
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Nicolas Berkouk: Restriction of so when you have a big.
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Nicolas Berkouk: Open subsets into a smaller one you have an arrow from the to be vector space to this model vector space.
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Nicolas Berkouk: Okay, and now chief sheaves will be just some kind of continuous spreadsheets Okay, so you want that, basically, you do not have.
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Nicolas Berkouk: Any random ways of.
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Nicolas Berkouk: associating Victor spaces to open subsets you want to do it in a kind of locally consistent way Okay, and so what does it mean in a more rigorous way is that so you have.
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Nicolas Berkouk: A property that is that is called the gluing yet excellent so appreciate, if we do chief if for each weapon subsets in each covering have a fan of subsets.
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Nicolas Berkouk: it satisfies the green yes excellent and So here we just explained shortly what it means so imagine that here, I have a big open subsets, that is, the reunion of three.
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Nicolas Berkouk: Open subsets with also the intersections Okay, and the brewing action will just tell us that I can recover uniquely the vector space.
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Nicolas Berkouk: associated by May shift to the big to the Union of open sets just by knowing my shift on open sets an intersection and coming dissection of the these open seats Okay, and so this is formalized just by this limit expression.
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Nicolas Berkouk: So she's really just are ways to associate.
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Nicolas Berkouk: To local data, the entrepreneurial space spaces in.
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Nicolas Berkouk: in a coherent manner.
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Nicolas Berkouk: Okay, and now, so I will introduce.
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Nicolas Berkouk: The topology on RN Okay, so I equip our end with the the project order okay switch the usual order when we do multi parameter persistence, so I will say that X one X is less and we call them why one way and if, and only if it is true component twice okay.
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Nicolas Berkouk: And given one point in RN I will define its downsides to be just the subset of elements of our end which are just lower than X.
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Nicolas Berkouk: And now I defined the Alexandra have to put a jian around on our sensory as the topology which as open subsets the subsets of our brain, which are stable by the operation of going downwards Okay, so it means that if I have whenever I have a point in my.
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Nicolas Berkouk: In my subset X it's done said is most automatically be included.
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Nicolas Berkouk: Within the subset.
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Nicolas Berkouk: And now I will use, so the notation this notation to denote the topological space already equipped with the Alexandra have to purge, so this is kind of we outsource, for instance it's definitely not our staff or or tissue or anything.
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Nicolas Berkouk: And to see, for instance, one we are the ever have this topology just observed that, for instance, each point for this topology as a man urban neighborhood which, namely is its downsides okay so, for instance it's a property that is definitely not satisfied by the UK gentle Polish.
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Nicolas Berkouk: So now there is.
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Nicolas Berkouk: A technical.
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Nicolas Berkouk: Research that I will maybe just explain very briefly, but not in a very detailed way that basically tells you that.
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Nicolas Berkouk: persistence major over RN are nothing but she's on the Arctic tundra of approach Okay, so this is.
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Nicolas Berkouk: A wizard that was first proved by just intravenous stages and then also by case you haven't checked out in 1018 so it really tells you that you can consider persistence, would you as sheaves but on the special to publish OK.
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Nicolas Berkouk: So now we have all the ingredients.
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Nicolas Berkouk: To define.
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Nicolas Berkouk: The almost an ingredient ingredient there is less to project that I need to introduce.
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Nicolas Berkouk: yeah so now so yeah sorry there's a lot of energy of different coverages in history, so now introduce again a new topology which is called the gamma topology.
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Nicolas Berkouk: And so to do so, so gamma here stands for a Cone okay so it's it can be defined for any cross convicts Cone but he I will just keep it simple with the usual color coding in Iran.
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Nicolas Berkouk: Okay, so I introduced to the coordinate gamma here is just the Queen of.
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Nicolas Berkouk: elements which of all.
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Nicolas Berkouk: nuggets negative coordinates okay.
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Nicolas Berkouk: And I will declare that.
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Nicolas Berkouk: An open subsets so a subset you in RN will begin open if it's if it is open for the usual topology so let's say the UK gentle Fujian RN and, in addition, I asked that it is stable by addition with elements of the coin okay so, for instance here if.
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Nicolas Berkouk: Our does.
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Nicolas Berkouk: Typically, typically again open set looks like well, it looks like something if gamma is this going here Okay, then, for instance, again open set will be something up.
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Nicolas Berkouk: like this okay.
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Nicolas Berkouk: So I will note RN gamma delta vertical space aren't equipped with the gamma topology.
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Nicolas Berkouk: So now something to be noticed is that the gamma topology is actually an intermediate topology between direct some draft apology and the you can gentle approach okay so here it is.
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Nicolas Berkouk: So it's not nothing to intermediate but again metal project is both a sub topology of the Alexandra topology.
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Nicolas Berkouk: And you can gentle prodigy so it's a way to in some sense mix that to to interpolate between a very discreet, you will be revealed in terms of topology case and the real of up gentle paradiso easy to purge right and so to see this, we can just hear link those by the identity maps okay.
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Nicolas Berkouk: In in this direction, I will denoted by Dita so from the Alexandra have to prove it to the gamma topology and from the UK gentle project to the gamma to Peru gee I will denoted figure.
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Nicolas Berkouk: Okay, and so now the the identity map from the 600 apology to the gamma topology which is continuous.
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Nicolas Berkouk: indices a derivative shifts perfunctory that I will you know to it's the push forward filter by data.
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Nicolas Berkouk: That, I will not buy a vista be tested desktop Okay, and it acts it act super easy on objects, because, as the gamma topology is a symptom ology of Derek sondra have to prove it, I can just restrict.
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Nicolas Berkouk: Any Alexandra she's to get to do gamma topology and I would get a game okay so basically here in a more explicit manner.
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Nicolas Berkouk: desktop of em evaluated on the on the set on the game open subsets will just be m evaluated to the inverse image by data of you, but data is just the identity map okay so it's just a few.
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Nicolas Berkouk: and
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Nicolas Berkouk: So there is a first result that I forgot to mention on the slide, which is that the desktop is a free faithful case.
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Nicolas Berkouk: here.
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Nicolas Berkouk: is pretty fast food.
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Nicolas Berkouk: or service, so we introduced so it's like super easy and abuse to define an internal insistence on gamma shifts and with Francois we showed that this filter data is isometric with respect to each of the entire living distance so here.
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Nicolas Berkouk: We call that this is, you must think of this as persistence, Madrid, because this is a these categories equivalent to persons, Madrid and.
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Nicolas Berkouk: So we proved that, so this is isometric point or even distance.
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Nicolas Berkouk: Okay, so now we have the ingredients to introduce the generalization of the definition of ephemerality okay so first one for observed that, in the setting of.
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Nicolas Berkouk: Fish as island quarters, we can actually rephrase the definition of a formality so here media will I will I will record it interested in the city of Jerusalem Kotor m is fema even underneath all of the Internet, I rose or zero okay.
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Nicolas Berkouk: And this property is actually equivalent to the fact of being a distance zero of the person's Maduro zero Okay, so now we will think of a female justin's Madrid, as the one that our distance rules you.
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Nicolas Berkouk: Now I introduce a subcategory of persistence Madrid Okay, who have Alexandra of shifts, which is the kernel of my mentor desktop okay so, which means it's the, it is the full subcategory of.
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Nicolas Berkouk: Alexandra fields, such that the image by star is a small fee to zero.
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Nicolas Berkouk: And we proved with Francois now that so in this setting, which is the multivitamin sitting this kernel as the same property as the other person's manager of shell and coders okay so.
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Nicolas Berkouk: A person's major easing the canon of beta style if, and only if it is at Intel even distance zero from the europa's transmitted.
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Nicolas Berkouk: So this is what lead lead us to define to introduce.
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Nicolas Berkouk: me female justin's module in the Multi parameter, setting as this subcategory OK.
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Nicolas Berkouk: And now we can do exactly the same thing as shows that in quarter, which is that the account this.
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Nicolas Berkouk: Is yourself so getting a we have systems, which are based on dollar resulting in she theory, so we can introduce the the quotient here.
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Nicolas Berkouk: The separation of category, but this question comes from merging out desktop but it's counted Okay, so we get a filter from to do counted here to Alexandra if she's Okay, so we to gamma ships, and so what we proved with Francois is that this is.
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Nicolas Berkouk: An asymmetry for the associated intelligent assistance and it's an equivalence of category, so it really so it means that this exhibit gamma sheaves wisdom intelligent distance as the observable category of persons with your indemnity bambaataa city and this.
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Nicolas Berkouk: model this expensive model does not require an explicit decomposition through.
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Nicolas Berkouk: So now, just to prefer a few discussions so so we yeah so in this instance i've showed you our we can prove that so we can give a sheaf.
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Nicolas Berkouk: Exploitation of the observable category of persons major and why is this interesting well gamma ships are not just random shifts fantasy of algebraic to privileges, so there is really a very.
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Nicolas Berkouk: agitated theory behind it, and so it is a very well studied objects.
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Nicolas Berkouk: In particular, by case, you are and and check er, and so we can use very powerful techniques to study them.
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Nicolas Berkouk: And also, just a quick remark that I have presented order results in the talk in an IBM sitting, but everything can extend to like the direct categories, but I wanted to be like you more it's to keep it simple.
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Nicolas Berkouk: And I thank you for your attention.
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um let's thank Nicholas for a very nice talk.
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and
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let's go with some questions and then we'll start back up I guess at.
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1pm Chicago time but questions, please.
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Kelly Maggs: hi Nicole lovely talk night was really nice I was just wondering if you could give an example of some of the techniques that.
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Kelly Maggs: Or maybe one of the techniques that people use for gamma shapes that you think would be useful for persistent homology in the future, something that you think might be able to transfer over well.
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Nicolas Berkouk: So the other thing is that, so I did not.
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Nicolas Berkouk: Talk about it today not to be too too technical, but basically so gamma sheaves can be thought of as a full subcategory of Euclidean shifts.
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Nicolas Berkouk: So, it means that.
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Nicolas Berkouk: Basically, one of the problem of persistence Madrid when you so some people in a in a person's Community so to study multi parameter persistence basically one idea is to do some dimensionality reduction over the person's manager so, for instance, to restrict on two lines okay and.
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Nicolas Berkouk: So this corresponds in some sense to look at inverse image in a lack of Britain, the category in Britain, the cooperation feminism.
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Nicolas Berkouk: And in the category of persistence mature or of gamma sheaves there is a very limited amount of function of continuous map, because you have to restrict the the.
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Nicolas Berkouk: Other structure which is very, very tight and by looking basically at those elements as so as can achieve and then, as you can choose, then you have access to a way more diverse.
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Nicolas Berkouk: type of operations that could lead to dimensionality reduction techniques for persons with you.
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Kelly Maggs: cool thanks thanks for the talk.
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i'm Derek.
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Darrick Lee: And Nicholas, thank you for the very nice Hello.
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Darrick Lee: um, I just wanted to ask about this gamma topology um is, to me it kind of seems like the gamut apologies is very similar to the Alexandra topology except you're taking upsets rather than down sets and you also require i'm.
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Darrick Lee: Open sets to be opening and normal Euclidean space is this an okay way to think about it.
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Nicolas Berkouk: So I mean upset so don't set the are really not like it's just a matter of Convention of Convention that's really not an issue and yeah in some sense, you can think of.
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Nicolas Berkouk: gamma topology as Alexandra.
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To the boundary.
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Nicolas Berkouk: In some sense like you moved out the boundary of the Cone.
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Nicolas Berkouk: So basically in the gamma topology you, you do not care about what happens at the boundary of the cold weather index, and after prodigy it matters.
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Darrick Lee: I see So if I got rid of this condition that needs to be open in the normal Euclidean topology then with the Colonel be trivial of be star a beta star.
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Nicolas Berkouk: Sorry sorry say again.
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Darrick Lee: If I remove this condition that.
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Darrick Lee: The gamma topology has to be open, like gamma open sets have to be open in the usual setting um.
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Nicolas Berkouk: Then I think you would just get dirty sort of supervision.
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Darrick Lee: Okay.
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Nicolas Berkouk: I suppose because intersecting do like.
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Nicolas Berkouk: As you have an entity structure that it's almost the same thing to be stable by the Cone or to be stable bye bye don't see it.
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Darrick Lee: I see okay.
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Nicolas Berkouk: So when would you would get back to the direction of to.
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Nicolas Berkouk: purge cool.
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Darrick Lee: Thank you.
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Jen.
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hi.
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Woojin Kim: Thank you for Nice talk so I have a live question so in the case of one primary person module we had.
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Woojin Kim: This nice characterization about if the more person will just write it every Internet map is German, so we have that property for much parameter, setting as well, or why is the relationship between.
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Nicolas Berkouk: The two so that would be exactly the same, the same.
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Nicolas Berkouk: I mean it's just that it's not.
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Nicolas Berkouk: very convenient way too many like from the algebraic perspective it's not very convenient to talk about the internal maps of a person's merger, so this is why we are familiar formulated this way but it's perfectly equivalent to to adding that to the older or doing today miss you miss.
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Nicolas Berkouk: It perfectly excellent.
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Woojin Kim: Thank you.
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um.
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Any more questions.
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Then let's.
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let's thank Nicholas again.