Okay. Yeah. So thank you very much for the introduction and invitation. It's really a tremendous honor for me to speak here at this conference. I first had the opportunity to meet Luke maybe, I think maybe it was about three or four years ago. And since then I've had a lot of exchanges with him and I really appreciate how kind and generous and encouraging. He's always been one of my last trips before the pandemic.  And Luke's mathematics has certainly had a great impact on me both from exchanges with him and from from studying his work directly. So yeah, so thank you, Luke and happy birthday. So what I want to talk about, I want to talk about some results, I guess, in the area of logarithmic algebraic geometry. So this is a subject of which Luke is of course, one of the principal architects. And in particular, I want to talk about some versions of log-syntomic chronology and some comparisons with paedic nearby cycles. And start by talking about logarithmic homology theories in general. So what I'm going to talk about is joint work in progress with Bargov-Baud and Dustin Klosson. So I want to talk about log-syntomic homology and sort of an integral version of log-syntomic homology. But to get there, I want to start by first just talking about logarithmic paedic homology theories in general. So I think somehow the starting point for this is the following example. So if x is a smooth variety over a field k and d sitting inside x is a normal crossings divisor and j is the open inclusion from u meaning x minus d sitting inside x, then one constructs the logarithmic Diron complex, sorry k is a field, constructs the logarithmic Diron complex and make up our star of x log d. And so this is, I guess this is a quasi-coherent chief of differential-created algebra on x and basically it's generated, well, so this is, sorry, so I guess I should say that it's a sub-sheaf of the Diron complex of u. So it's a sub-sheaf of the Diron complex of u and it's a sub-sheaf which is generated by, well, differential forms on x as well as sort of dxi over xi where these come from local equations of the divisor d. Right, and so I guess it's a, so this is a definition of the logarithmic Diron complex in this context and in characteristic zero it gives a, it computes the Diron cohomology of the open complement which is one reason this is a good definition. Right, so this definition I guess it generalizes to the theory of log structures. So, right, so maybe I should say this computes cohomology u in characteristic zero. Okay, so Cato and Fontaine Lucie define more generally the notion of a log structure on a scheme, so on a scheme x. So it's given by a sheaf of monoids which I'll write as m sub-x on the etal side of x and a map of sheaves of monoids. So I'll call this alpha which goes from m sub-x to to the monoid which is just the structure sheaf of x but with the monoid structure given by multiplication instead of addition with the property that alpha inverse of the units is isomorphic to the units via alpha and so then this structure is called a log scheme. So for example, if we're in the setting as above so if d sitting inside x is the normal crossings divisor then we have a log structure given by m sub-x is going to be those functions which are invertible on on u. Okay, so I want to fix some notation here so if we have a log structure like this some notation for further on so if we have a log structure this is also going to be we also obtain a sort of a well so we also define or we notate m sub-x bar to be the quotient of m sub-x by o-x o-x cross so this is sort of measuring the the non-triviality of the log structure and I guess usually we're going to be in a setting where o-x cross is acting freely on m-x so this is again some sort of sheaf of monoids and now m sub-x bar is naturally going to come with a map so so so from the structure map of the log structure m sub-x comes with a map from m sub-x to the following following sheaf of monoids which is the sheaf of monoids of sorry line bundles with a section so this is I guess the this is represented by a1 mod gm and effective effective cartier divisor right so this will be useful later on okay so so in general I guess it's a sort of general phenomenon that a lot of homology theories that one can define in the world of world of schemes naturally can be extended to the to the category of log schemes and so the example is right so so maybe the first example is that if you're given a log scheme over a field then one can define its log diram complex so I guess we had called this omega upper star of x comma m sub-x and this is well this is defined sort of similarly as the diram complex is defined similarly via generators and relations so and you start with the diram complex well I guess I'll say this sort of informally it's and then you add you sort of formally add objects d log of m for every dm divided by m for every element m of this this sheaf of monoids so this is supposed to be d log of m and then you sort of enforce the you sort of impose the the appropriate relations for for the diram complex and for what d log of m is supposed to satisfy so this generalizes the example of the logarithmic diram complex so this is the first example but there are many other piatic homology theories for which there are also also log log versions so for example there's log crystalline homology developed by kato so let's say we have a log scheme over fp then can define the log crystalline homology and this is defined right so sorry so maybe what I should say is that this is going to be some sort of lift of well in in nice situations of the log diram co-comology to zp from fp now and this is defined and this log crystalline homology so I guess this is absolute log crystalline homology is defined using well I mean crystalline homology is defined using the crystalline site and one sort of defines a log version of the crystalline site to to define log crystalline homology so adapt definition of a crystalline site um so I guess there's the example there's also the examples two other examples that I want to keep in mind but maybe I'm going to come back to sort of later our symptomic homology uh so this is going to be some sort of filtered for danious eigenspace of uh of log diram homology but in in mixed characteristic uh and piatic nearby cycle homology so this is for schemes over zp and this is going to be the homology the etel homology of the rigid generic fiber so for formal schemes over over zp so these are also two examples you want to keep in mind and we'll come back to more later but I guess the most powerful of these piatic theories is given by prismatic homology and so a logarithmic version of prismatic homology has been developed by by koshikawa so so given a prism a comma i well then prismatic homology is some homology theory for formal schemes over amod i that lives in modules over a and one can one can generalize this to log schemes so can take the prismatic homology of log formal schemes um and actually so koshikawa considers a say more general situation where where a comma i is is also allowed to have some sort of log structure um and uh and this this homology theory is is uh i mean there are a lot of you know structure theorems in prismatic homology like the hud shape comparison and so forth and these these have adaptations to the log setting where sort of differential forms you have log differential forms um so there's a there's a log prismatic homology and this was also preceded by um work of uh sasnovicius and koshikawa in the case of semi-stable schemes uh over oc and where the prism is is a prism a comma oc um right so um so this is an important special case where i was done earlier um right so so in general why you know why does one want what is the motivation for having these log um log theories well at least part of the motivation is that uh i mean these various types of phoetic homology theories um are are related to each other in various ways uh which it's governed by i guess part of the subject of of uh phoetic hodge theory um and having so a lot of these sort of comparisons can be extended to the log setting um and so so so there are relations that you can you can obtain between between phoetic homology theories and in the log setting uh as a result so so let me just explain a corollary and I guess this is a corollary of the theory of log prismatic homology um so let's say um right so so let's see the uh complete algebraically closed non-archimedean field ring of integers oc and residue field k and let x be a smooth proper scheme over oc uh and let d be a normal crossings divisor and then the conclusion is that if you take the the dimension of the etalchromology groups of x minus d excuse me is it a relative normal crossing divisor sorry is it a relative normal crossing divisor yes thank you thank you yes relative normal crossings divisor thanks uh so then the the dimension of the etalchromology uh the dimension of the etalchromology of the of the complement is going to be bounded by the dimension of the diromcomology uh of of the special fiber with log poles uh along the reduction of the special fiber of d so dimension over k so um right so this is the type of comparison theorem that that follows from the theory of prismatic homology actually the the a of homology so this is is proved by batmorrow and schultz uh when d is empty and and sort of a log consequence of the log theory is sort of being able to make some more examples like this and i guess also we talked about torsion sorry torsion also uh yeah yes i think yes you have bounds for the yeah the amount of the amount of torsion in terms of the homology so yeah so the first thing i want to do in this talk is i want to explain how one can build these homology theories so as i mentioned um so for example um log i guess log crystalline homology and log prismatic homology are defined uh using logarithmic uh sort of logarithmic versions of the original site theoretic constructions um but what i want to what i want to explain is that there's also a way that one can sort of reconstruct these logarithmic theories uh from the non logarithmic versions um and so this this will become useful when i uh when i want to talk about this log symptomic homology later on um and that's and the strategy is to use this uh use this construction uh use a use a construction in in logarithmic geometry called the called the infinite root stack so the first goal is to construct or to give an alternate construction of these theories um from the non logarithmic versions using uh using a general construction called the infinite root stack um okay so this is a construction which is due to talpa and vestoli uh and it takes those inputs so so let x comma m sub x be a fine and saturated log scheme um and so then what they do is they construct a stack that they call uh so then there is a stack uh which uh so let's we'll denote it um the infinite root stack of this log structure um let's denote it like this uh and there's a stack this the stack comes with a map to x um which is the isomorphism where the log structure is trivial and somehow the stack is encoding uh is sort of encoding the geometry of of the log structure um um so let me give the basic example the basic example is that if you have a one uh and the log structure is given by the origin so it's given by the divisor at zero um so then the infinite root stack of a one comma zero it's going to be the inverse limit uh it's going to be the inverse limit over n of the stack a one mod mu n uh inverse limit of n under divisibility uh and right so I guess what's going on is that the transition maps are they're raising to the appropriate power on a one and they're raising to the appropriate power on mu n so raising to powers um so I guess right so I guess we can think of this as it's it's like spec of k brackets x to the one over n modulo mu n at at each stage how does this compare to the koumari tall site uh yeah thanks for the for that question I will I will come back to that in in just a in just a moment actually so um yeah it's closely related um okay so but yeah maybe let me just let me just do two more examples um so right so this example can sort of be generalized so uh so suppose the log structure uh comes from a from an effective Cartier um sorry comes from a divisor uh from so from a single divisor so let's say a smooth divisor um so uh right so log structure on x so then that effective Cartier divisor is classified by a map to a one mod gm uh and then uh so this infinite root stack of x comma m sub x or we could also call it x comma d uh is given by uh so what you do is you form the fiber product of x with so you have a one mod gm mapping by multiplication by n to a one mod gm and you you take that fiber product so you're taking an nth root of the of the divisor of the blind bundle with section given um well given by this uh the structure and you form the inverse limit of n along to visibility um right so so these are two basic examples um and in general the phenomenon that happens that you sort of see here is that when you take this infinite root stack of x comma m sub x uh mapping to x then uh the fibers at a point well x and x uh is given by the classifying stack of uh so if you have a if you have a point x in in in the base scheme x then uh uh then there's a rank r which corresponds to the so then if you let r to be the rank of the monoid m sub x bar the sort of residual monoid um then the fiber of this this this map from the infinite root stack to x is given by the classifying stack of the tape module of uh the group scheme of the group scheme of all roots of unity so take the group scheme of all roots of unity form its tape module and uh take its classifying stack to the to the power of r so so that's what the fiber looks like at at each point um right so I think this is uh this is uh analogous to the construction and partially motivated by the construction of the kato nakayama space for for a complex log structure where this would be replaced by a circle um okay so so let me just give the definition let me just explain that the definition in general so uh so if you're given a map from t to x where x is the log scheme if you want to lift it to the infinite root stack of x comma m sub x then what you need to do is well so the map from t to x is is giving you so f yields uh a map from f upper star of mx bar uh to the to the groupoid of of line bundles with section so I guess this is like a symmetric monoidal functor and what you need to do is you need to you need to make this map infinitely divisible so in order to lift it to give a lift need to make this map so let's call this map phi need to make phi divisible i.e uh so let me move this down need to extend it over the rationalization of the monoid f upper star of mx bar um and so that's exactly what we saw sort of going on earlier which is that you you know you had like a log structure given by a single divisor and so that's that's like a effective cartier divisor that's like a line bundle with a section and you want it you want to take the roots of that structure okay um but so so when one formulates it in this way that's on top of in the Stoley's paper um it's it's a definition that that's not sort of it's not really in terms of the the given divisors really in terms of the log structure okay so um right so the general sort of idea or sort of principle is that uh a homology theory uh with uh for a log scheme so uh is given by its value on the stack with and the stack with no log structure and i guess this is this should say with profinite coefficients um so i guess this is sort of a general sort of idea in this in this business um so yeah so you want to go back to um so so right so this this idea has at least a couple of antecedents in the literature um so one of them so to go back to to Arthur's question uh is that this is related to um this is related to kato's comor flat site not quite the comor atoll site because we're going to be taking my p power roots sorry so the antecedents are given by kato's comor flat site so i believe the the comor flat site is closely related to like the maybe the fppf or finite flat site of this infinite root stack i think this is discussed in the top of the Stoley paper so i believe it should be possible to phrase this construction in terms in terms of the comor flat site instead of the stack as well um and another antecedent is uh is Olson's work on the stack of log structures and in Olson's setup as uh so instead of instead of uh instead of forming this infinite root stack there's there's a map from x to the stack of log structures and one considers some sort of relative theory um and so i believe that's closely related to the setup because this the sort of infinite root stack and i think it's it's i think you can you can build it by um by sort of well sort of by making by starting with the the map to the stack of log structures and sort of making that infinitely divisible uh or sort of perfect and that will you know so so that will that will sort of make some comparison so haven't worked out you know details here but i i believe that this should be closely related to the setup of Olson uh where you where you sort of use the fact that making something perfect is going to get rid of sort of differential forms um okay but so um but basically the example that um wanted to um sort of an illustration of this principle is the following calculation um so let's let's consider the case where x is smooth and d sitting inside x as a divisor uh is a normal crossings divisor right so i guess it's a it's a relative normal crossings divisor if we're over some base some given base um and then there's the following right so there's a following calculation so if you consider the infinite root stack uh of x comma d um so this is right so i should say this is not quite it's not quite it's not quite an algebraic stack because you get these i mean you get these like big uh stabilizers but it's not so far from one um and in particular so we can do something like take the differential forms on it and so a calculation is that if we take um global sections of uh differential forms on this this object then this is naturally the same as differential forms uh on x with log pulse along d and this is with profite net coefficients so excuse me so when you say stack of course you have to specify the topology so for example you can use since you're with all the finances you can say f p q c topology but then you want to do a differential form so this not so i'm not sure what what the foundation so for for a usual i think stack of course we use that it's given by a scheme model who uh smooth something i mean smooth group way but here of course you have to work in some with bigger things maybe formally smooth so i mean so so i think one way of yeah sorry sorry i think one way of saying this oh so is is a stack in the flat topology for example sorry you mean f p q c uh sure or you could say some sort of in the centomic topology yeah f p q c topology or some sort of in centomic topology and right so yeah so thanks for that question so i i should yeah i should probably clarify what i mean here um so one way of saying what i mean by differential forms is that uh so um i guess it's a theorem of uh bar gov that uh if you if you form the wedge powers of the co-attention complex they are sheaves in the flat topology so it's a theorem of thought that um wedge powers of the co-attention complex say on rings are sheaves uh in the flat uh in the yeah f p q c topology and so it's going to make sense to evaluate these on any um on any stack like um like this um but in fact it's a little so i said i did say differential forms here and the reason is that uh i guess you can sort of regarded as a stack i think in the i think i want to say in the in smooth topology so so you can really just say differential forms instead of the co-attention complex um um sorry i didn't particularly catch what is that there's a quiet topology so it's a joke a quiet uh okay so i uh thanks yeah i don't i don't um not completely sure i think that what i understand is that that that the kumar flat topology is closely related to this construction but i didn't know about fontans um quite topology a lot thanks i'll have to yeah i'll show look that up um yeah but this is kind of a fun sort of yeah sorry so just to go back to also to open this question i mean this is kind of a fun sort of calculation to do like if you have i mean if you have k brackets x and you have the log structure given by x and so so in fact for what i'm saying is you don't actually have to go to the infinite root stack you can do the p typical version where you take extract p power roots and then what you're considering is you're considering something like spec of k brackets x to the one over p to the infinity and you're quotienting by uh by the group scheme which is uh uh the tape module of mu p to the infinity um and so when you when you go up to mu p to the infinity or when you go up to k brackets x to the one over p to the infinity differential forms go away there but the this group scheme is going to have a cotangent complex and so when you form the and when you form the limit um when you form the stacky cotangent complex sort of from the limit of these things you end up recovering this log um uh cotangent complex um yeah so um so um yeah so i just want to go back to this um this principle so in particular it's going to follow that one can uh can define sort of log prismatic homology or log crystalline homology or the others uh of say so if we're in the setup of a prism a comma i of an a mod i formal log scheme uh in terms of the infinite root stack as um but there's a bit of an asterisk here which is that in general the construction of the infinite root stack um if you if you really sort of follow it it it it's going to involve some sort of tensor products and those tensor products might not be derived um and so i think that doesn't isn't going to cause a problem for some sort of etal theory but for prismatic homology that's going to be an issue so this might not this might not work in general um so if you want to define log prismatic homology i think this is going to this kind of thing is going to work very well this is going to work well in say a very nice situation if you have a formally smooth a mod i scheme with log structure given by a relative normal crossings divisor um but it might not quite give you the right thing in in general if you do this because of these um these higher tours um but what you can you can what you can do then is um is is you can this is sort of the fundamental case uh and and then you can sort of animate in general so so you can define log prismatic homology or log crystalline homology in in this special case using the infinite root stack and then you can animate in general so here animate means that um um um you sort of simplistically resolve a log ring over a mod i um and so i yeah i want to i want to emphasize this this this idea of taking a homology theory um so so simplistically resolve a log ring over a mod i over sort of their by sort of very special cases and the special cases could be something like you know a mod i brackets x1 through xn and y1 through ym where these are giving these variables are giving you the log structure um and then and then you sort of animate in general to to define it for for a general um log um log ring over a mod i or log scheme um and this yeah so you want to emphasize that this this construction um of of uh of animation in this way uh for for piatic homology theories um goes back to uh goes back to luke in in in the context of derived dirom homology so this is as in eluce's construction of derived dirom homology so right so as a result one can recover this construction of of log for example this construction of log prismatic homology and the sort of various types of comparison results uh using the infinite root stack um in this manner um so i but and it is yeah also for like crystalline or dirom homology and yeah i just want to emphasize that the the essential feature that to do this is that you have a homology theory on the one hand you need some sort of flat descent so as as a no first question um so that you can um you can you can evaluate it on a stack in a good way and the second is that it behaves well with respect to animation so anything that's sort of built out of the cotangent complex um is going to behave well um with respect to animation so yeah so when you animate the the you are seeing about some geometric object but within in the like the cotangent complex where we don't have the results by uh some free polynomial thing but geometrically it changes the spec here you work with something maybe you want to to complete it along the original locus or do something so or do you do you take this and so do you have some simplification resolution and do you do you complete it along the kernel to the original thing or do something to localize the original thing or it doesn't matter no i don't think i don't think you need to i mean so i think for example in this in this case i think so it's a i guess an insight so i mean this is sort of i guess comparable to we're using like the fact that if you um so this is an insight of uh of bot that if you take derived dirameclomology in the piatic setting uh then your it gives you the right theory say for more generally for smooth objects by the by the cartier you know by the cartier at least and more awesome and so yeah so yeah so maybe i should also add uh i should also say that um so here yeah so you don't need to do any sort of further completion i mean you need to do the iatic p comma iatic completion as usual but um so here we use i mean we essentially use the hodge tape they're into cut here for log the dog's moving to the cutty type right yes yes um yeah so i mean in general it's going to be i mean so you're going to get a theory that is you know controlled so i mean this like this this type of theory you're going to get if you have if you have a theory that's controlled by the cotangent complex then this this sort of procedure is going to produce an analogous theory that's controlled by uh well gabber's uh log cotangent complex so use the hodge tape filtration and it's i guess it's a bot that log piatic derived dirameclomology works so it gives you the right answer for smooth schemes not just for polynomial for smooth algebra so another natural thing i just to recall from some discussions i don't remember the explanation but when you do derived dirameclomology as far as i remember you have some big complex and either you can take the total with direct sum or direct product it gives you different things or the same thing so what is the answer so just about derived dirameclomology it's not so you have your your so either you do it in characteristic p or when it's important to maybe over said i'm not sure so you have some nice implicit resolution to take the complex of each and then you put it out of course it is not in the first quadrat it is another one so the total complex is there are two ways at least to define the total complex so they give you different theories yes so sorry so i think i think what you're saying is the question of like the hodge filtration but i think it's right so i think it was the insight of bar gov that if you that in the in this patic setting it's it's not going to matter at least at least if i understand your question correctly i mean if you form derived so i think i think that the so i think the distinction is equivalently between derived dirameclomology where you i think you you don't where you don't complete with respect to the hodge filtration and when you do and the answer is that first move things that doesn't make a difference which is not true i mean it's not true in characteristic zero but it is true in characteristic p by the by the hodge by the by the by the Cartier isomorphism so you mean you start from any ring any fd algebra in characteristic b and then the two patic the the two derived dirameclomologies are the same this is what you're saying using a simplification right this is in this all this blueprint of valga one on wait sorry i think there was some confusion uh i think akhil was i mean what akhil was saying only applies in the smooth case and then for general rings you just use the direct sum version that's what animation does for you so because it's a conjugation which is better so somehow right i mean you that's what the you don't think about the direct product version after uh proving this theorem in the smooth case okay so you prove in the smooth case what you said but it is not true in general probably probably it is not true you say yeah it is not true in general because one is going to be complete for the hodge filtration and the other is not in some singular setting yeah sorry that okay so you take the direct sum okay thank you okay yeah thanks bar go for car fine thank you i'm sorry for interrupting no no thanks uh yeah okay all right sorry so i guess i have maybe a little more than 15 minutes so so now what i want to do is explain the um let's talk about symptomic homology um so this yeah so this is sort of a general procedure of um defining logarithmic versions of these pietic homology theories and i want to sort of apply that to um to this this context of symptomic homology um and right so this is another one of these pietic homology theories and so let me start with the following definition which is due to uh frontend messing and kato so let uh r be a um formally smooth uh zp algebra um so uh right so first let me define it for i at least p minus one we're going to define zp of i of r to be the following uh to be the following object so this is going to live in the the p complete derived category of zp it's the fiber of phi minus uh p to the i which uh which goes from the sorry uh the i stage of the hodge filtration of so you take the dirham complex the p complete dirham complex of r and form the i stage of the hodge filtration and map that to uh uh the dirham complex of r so here we use we use a p complete dirham complex of r uh together with uh it's uh with the two pieces of structure given by the crystalline forbenius which is an endomorphism of the dirham complex so crystalline forbenius sorry uh and the hodge filtration uh so this this naturally produces an object uh living in the p complete derived category um of zp um and uh right so let's sorry let me call the zp of i uh fontan um i guess fontan messing in kato of of r so um right so this this definition is uh one can do a little bit better when um uh when i is small so if i is less than or equal to p minus two then one can define in addition to the forbenius one can define a divided forbenius phi divided by p to the i which goes from the i stage of the hodge filtration to the dirham complex itself so this is this is something that's going to be natural at least in the derived infinity category uh and then so let me define zp of i of r to be the fiber of um phi divided by p to the i minus the inclusion from omega r greater than or equal to i into omega r of dirham complex okay so this is i guess this is what one might call symptomic homology which are not no in the first one i is bigger than i can throw less than i is larger yeah yeah where is the hypothesis at number two on i used uh so i guess it's being used to define this phi divided by p to the i so well i guess yeah uh right so i mean i i guess you want to make this sort of natural so that okay yeah yeah i i think one can say it so for example one can you know one can define this sort of locally in the i mean one can define this using like a chris and then i think uh i think you will have an issue with the forbenius not being divisible enough that yeah thanks um okay okay so so right so this is some general construction and okay so we can define it let's i mean we can define this for all p complete algebras by animation and then we can also define it well so using the well for example using the procedure that i just explained or in this case you can just do it directly define on a log p attic algebras um right so using this infinite root stack so yeah so i guess one of the reasons these come up is that there are comparisons to there are comparisons from symptomic cohomology uh to uh to p attic nearby cycles so there are various comparisons i'm sorry you could do it in the law case you could either use the infinite root stack or suppose you just use log with differentials instead what's the how do they compare yeah i mean it would give you the same it would be yeah it would give you the same thing yeah because because because the logarithmic differentials are also the differentials on the infinite root stack um but i guess the infinite root stack would be helpful for when i want to do the integral version of these things um a little bit um yeah thanks okay so there are various comparisons in the subject uh so various comparisons between symptomic cohomology and uh p attic nearby cycles uh after truncation um so i think right so there there are many instances there are many uh examples of this in the literature so um for example uh so kato shows that if r is uh formally smooth over oc so c is algebraically closed non-arcomedian field you can i think you can weaken this whole bit so c only has to be perfectoid um then uh z p of i of r is the truncation of so you can consider this as a i mean z p of i i mean you can think of this as some sort of i mean as you let r vary it becomes some sort of sheaf on the pro etel site of r and uh well it's the truncation uh truncation tau less than or equal to i in the pro etel topology of the the p attic nearby cycle so of r j lower star of the usual z p of i where j going from spec of r join one over p into spec of r is the inclusion so if you if you take the generic fiber you have the the usual z p of i the take twists and the etel or pro etel topology you push that forward to z p of i and uh well i guess maybe you i should say spuff of r maybe i should say spuff of r because uh we're really restricting to the special fiber and uh and the statement is that it's the truncation in the pro etel topology of spuff of r of this of this p attic nearby cycles so this is for i less than or equal to p minus two you should say um so this is this is provided kato um for you thank you with dvr in kato what did the original of course it was less sophisticated than what you said but did they start from a dvr it's perfect the residue field and just considered this case i mean c oh yes sorry yes uh kato in and and if you but this technique you said is through using using found things i mean what you say is true that for any yes sorry what i'm saying is true but i think i think yeah so you're right i think kato is probably assuming that c is a completed algebraic closure um of like a p attic field but yeah thanks for that um and then uh there's work of kuri hara uh which uh which proves i guess again this the same statement and i won't keep repeating it but in the case where r is now is now formally smooth over dvr but now you give it the log structure that comes from the special fiber um so if you don't assume the law if you don't put in the log structure from the special fiber you'll get some contribution coming from logarithmic diram that forms from the special fiber which i think is the setup that kuri hara considers uh and then uh there's work of suji uh which considers uh are maybe formally smoother or semi-stable so let's say yeah r is semi-stable over oc and i think you're also allowed to have a divisor divisor at infinity here uh and in all ways so no uh bound on i uh but up to isogenic so uh right so somehow the zp of i's are for i uh greater than p minus two are not really the right thing they're only sort of the right thing up to isogenic uh but there's a comparison with piata herbicycles in this case um and uh uh and more recently worked by komez and nizio uh where r is semi-stable over dvr and again semi-stable allows some uh log structure um at infinity um and again rationally in all weights or rationally or up to isogenic in all weights um so it was only sort of more relatively recently that there were integral versions of these types of results uh because the because symptomic symptomic homology defined this way is not uh somehow it's not the right construction and integrally uh so only recently uh there are integral and the reason is that one has to replace these zp of i of fontan messing in kato with something a little bit more refined that comes from prismatic homology instead of diram homology um and so there's a definition which was made by batmorrow and schultz so they define object zp of i uh for all i of r so r is again this periodically complete ring and these arise as the associated graded's associated graded's of a filtration on uh on top on the I guess this object that arose in algebraic topology on topological cyclic homology of r with dp coefficients um so these can be also defined in terms of absolute prismatic homology these are defined as so zp of i of r is uh so this is discussed in the paper of batmorrow and schultz it's also a fiber of a divided forbenius minus the inclusion so minus the inclusion which goes from absolute prismatic homology of r except that now you've uh you've neigard completed completed with respect to the neigard filtration and you take the ith piece of the neigard filtration and you have a brachis and twist mapping to again neigard completed prismatic homology of r with this brachis and twist so this comes out sort of very naturally or this expression comes out very naturally from the expression in terms of topological cyclic homology and the sort of apparatus of uh of tc so so this is this is sort of an integral theory that works works in all weights um and um right so um so so for example in in work with antio uh moro and nikolas we showed that these zp of i is defined by batmorrow and schultz agree with the frontend messing and kato definition in low weights or sort of up to isogeny integrally um but these these zp of i is defined by batmorrow schultz work better integrally because uh so um sorry right so they they also show that uh they show that the zp of i's of r are truncated piatic nearby cycles uh in all weights uh if r is formally smooth over oc uh and this was extended to the semi-stable case uh r over oc by colmez uh dospenescu museo so r over oc semi-stable um right okay so yeah so what i wanted to explain is that right uh one can formulate a version of uh an integral version of these types of comparison results want to add one more item to those lists which occurs uh in uh in the case where you're over a dvr so uh and and where you do have to take into account the log structure coming from the special fiber as well as some sort of semi-stable log structure um so first is you can define the zp of i's in the log setting for example using the infinite root stack for example um and i yeah i guess i want to end with this theorem um with uh botan clausen which is going to state that basically the the the theorem of colmez and museo that you have this comparison between this rational comparison between uh truncated piatic nearby cycles and uh symptomic homology works integrally if you use these zp of i's so if r is semi-stable over oc and you can generalize this a little bit like you can allow these changes of semi-stable things as in colmez and museo um and you can allow a divisor infinity then the zp of i of r with log structure with the natural log structure are tau greater than or equal to i of piatic nearby cycles um so i just want to say a little bit uh i just want to say what goes into this i mean so somehow it would be really nice to have a purely sort of prismatic uh and the piatic proof of this um but what we actually do is uh is to is to use a connection between topological cyclic homology so these zp of i's are related to uh as in bms2 they're related to topological cyclic homology uh which turns out to be related to the piatic k theory of the generic fiber and then what we do is we uh we use we use the the block hadoe or bellinson licton bomb conjectures which give you a filtration on the k theory of the generic fiber and then you sort of try to match that up with the the filtration on on tc uh of log tc uh and and the piatic k theory of the generic fiber and the piatic k theory of the generic fiber is controlled by like these piatic vanishing cycles thanks to the block hadoe conjecture and so i think i believe this this type of idea and using k theory in this way is is due to museo um but uh in this context it would it would also be very nice to have a a purely prismatic approach so okay so i will stop here so thank you very much but there is no conjecture involved here oh sorry yeah i should i suppose you you don't need any conjecture sorry sorry sorry sorry i guess i'm yeah i should have said that so you're using like the block hadoe conjecture uh but yeah what i should say is this is a theorem of of uh weavadsky and roast yeah thanks okay uh but somehow this is like an elatic statement and it would be nice to sort of stay purely in the piatic world um but yeah so in the previous result so in this result you have the log structure uh uh used and in the the kind of semi-stable the number six or before no no before number six the the one where you had the uh uh okay you have the smooth case uh but then the you had also the uh the case of semi-stable which uh like the semi-stable case i will scroll by the spin this didn't use the log structure it uses the log structure sorry i maybe in a smooth case over oc you don't need the log structure because a log structure is sort of uniquely p-divisible so it you can put the log structure but it doesn't affect things and in the semi-stable case of an oc uh then i think you need the log structure yes um yeah i i have a question i don't know whether you can hear me hi yes i can hear you yeah okay so um i have a question so this um uh this uh way of of presenting things using infinite rule specs does it allow you to describe hiodokato theory uh i don't know i i would love to yeah i would be very interested in understanding that better but yeah i i don't know but thanks thank you for the question i was curious about that too so okay so it's a general question so in the arson's approach uh defining log structure on x is sending x to some some special stack which is locally like towing stack for example a 1 p.m and in the other approach instead of sending x to some stack you send a stack to to x stack yes so it's a sort of dual approach like uh i think in the terms of sanity in the Takeshi's expose you have two two things so even uh either you will send x to a curve or you send something to x yes so it's sort of a dual dual approach so in arson's approach uh it's a property of morphisms of uh or even uh objects log structure it could be read on the uh on the on the map to to this log log thing log stack and so i wonder where there are uh similarly you can uh so there are two questions first of all which type of stack would give you a log structure you said that uh somehow you can cut the log structure from something algebraic just a stack it's something like a root stack so what are the properties which would characterize the root stack and then and give you some some log structure and also uh what properties of log structure you could you read off on the this uh root stack like uh log smoothness not flatness et cetera um yeah thanks for those questions i think yeah i i believe that so in talpo and vestelius paper they explore sometimes so in particular i think they show that you can read off things like log flatness and log et al my kumar et alness in terms of what happens on the infinite root stack um i think i think they also discuss the relation um i think they also show that you can recover like a fine and saturated log scheme from the infinite root stack uh but it's not fully faithful but you can still recover it up to isomorphism and i think they also probably discuss which stacks are of this form i mean they for example they and they always have this particular form that like they they live over a scheme and the fibers are these like i i guess sort of profinyte tori um like classifying spaces is sort of internal characterization of these root stacks uh i'm not sure but it might be in their paper but i'm not sure off the top of my head but thanks so so the infinite root stack like if you have a if you just take a maybe i uh you said something about recovering the fine and saturated log scheme not not as an equivalence but only after isomorphism can we remember what it was or i believe i believe what they prove is that if you have if the if the infinite root stack if two infinite root stacks are isomorphic then the log schemes are isomorphic is my understanding um but on the other hand it i think they also show that it's not fully faithful that functor from log schemes to to stacks in this manner but for isomorphisms it is that is the isomorphisms correspond but not the maps is it the idea or that okay uh what i think the statement is just that if they're isomorphic as stacks then the log schemes are isomorphic i think it's not necessarily yeah not that the delta isomorphism on the set of isomorph okay okay yeah that's my that's my understanding yeah so here the in the last thing you said that you you interpret the zpi internal topological cyclic homology theory and paper of the general fiber so do use results relating to the k theory of the general fiber to some material to something involving a chemical what did you expect to see i mean which i i think that i think it's okay now but do you use such results yeah yeah i mean i guess yeah so we're using like the k theory the generic fiber is related to etiological homology and by the like blockado and valence and lectin bomb conjectures and so we're we're using that um and so yeah so basically the point is to match up like uh i mean you have you have something like tc of r i mean maybe with log structure or yeah so uh which is which is you know which is filtered by these zp of i's and then you have something like the k theory of r join one over p i mean let's just assume that r is like strictly Henselion so this is close to k of r to begin with uh and then then you're using the fact that this has a filtration by blockado valence and lectin bomb and this has a filtration by botmore and schultz and then the claim is that those can be made to match up and to do that one has to sort of take left arrived or animate the the right hand side and actually use these results of block and kato and followed by others of uh on controlling peatic vanishing cycles or peatic nearby cycles in terms of differential forms but you also need the k theory results which which compute the the k theory of our one over p yes we need the k theory results yeah okay you just explained to me that several years ago okay so uh is there any other question yeah i have a question so uh the tc of r would you would you wrote here is this just in a philosophical sense or did you really define it in the case where there is a log structure i mean so right so i i think you can define tc you can define log tc sorry i think maybe this also came from the chat so you can also define uh i mean you can define like log hoaxial tomology log tc and so forth using this this procedure because tc satisfies flat descent and behaves well with respect to animation and i think this will agree with you know i so i think has to hold nuts in uh for example consider version of log tc and i think it will be equivalent um so okay so you're using the if it looks that's the different tc etc of logarithmic things but does it also give you a definition of k theory of log log log schemes or log rings right oh yeah so one thing i should say is that it's not it's not the it's not the tc or the k theory of of sheaves on the log on the infinite root stack um but it's this thing defined by flat descent and so since k theory doesn't define doesn't satisfy flat descent i wouldn't uh see how to define yeah okay yeah thank you yeah but yeah i mean it would be nice if there was really a category here that one could just sort of apply it to the category and but here it's really being defined by flat descent uh well i think at least yeah i think at least you like for a log scheme you will you can do what yeah but i i don't know for yeah i mean so yeah i hope that for example some of the techniques uh that uh you know that the bar gov was explaining for working with these uh you know with absolute prismatic comology where um i mean so right so i mean this is you know this the statement about the zp of i is i mean these types of statements are i mean their statements equivalently about absolute prismatic comology um or some sort of filtered nigh and i guard eigenspace of prismatic comology and um so i mean these statements were proved for example by batman or schultz and comministus minuscunizio over oc uh using i mean using the fact that you have like nice expressions uh for like for a nth comology like these cutarum complexes and given by laida in particular and so forth um and i yeah i mean it would be i certainly hope it no will be something like that there will be some sort of something like an laida explanation for these types of statements so for okay and then maybe one could get hope for a purely pietic approach to these things um but um no okay so i don't know if i can just remember another technical point so when when you looked at the where it was going to like when you try to do the um omega i on some well maybe put it in this talk or in another maybe the lighting no you you mentioned the result the the the the wedge i of the cotangent comology is a shift yes but this was uh yes uh but this i think means that well of course this is in some derived sense but this means that you do uh just check uh cover or or hyper cover i think it was pulled for just check covers yes comology usually is defined in general as a limit of hyper covers which you cannot avoid i mean if you do unbalanced things so so so it uh so here this could be well i'm not working with this but this doesn't cause a problem here yeah so thanks for yeah right i mean so i think that is uh i mean as far as i know that's an open question of whether the cotangent complex and its wedge powers are hyper sheaves for the flat topology i think this was raised maybe in batmorrow and schultz uh uh it's paper second paper um but so i think in the setting in this particular setting like where you're taking at least when you're taking uh these infinite root stacks of sort of pretty nice things and then from what you're animating um there's no issue because everything for example is going to be like in the symptomic and then that's going to force everything to be uh um sort of bounded and formally bounded um and so so these issues of hyper completeness are are not going to not going to show up because you're not going to have uh stuff in comological degree going off to comological degree minus infinity um so nice thing yes so nice things like you start from a smooth with normal crossing divide okay then everything will be okay i understand then the convergence questions okay so the other and the other questions okay which case uh in the computer okay so let us uh speak it again for the illuminating but technical thank you thank you very much


What's going on guys it's your boy DPJ here today with another destiny video and today I want to talk about the recent secret code announced to exist on the crucible map last exit but before we get into the video if you guys do enjoy at the end leaving a like is very much appreciated let's aim for 4000 likes that'd be absolutely amazing okay so last night during bungee stream they mentioned that of a code hidden within the crucible map last exit yeah take a look where we will leave you that's where we like you best have they broken games have they broken the code in last exit yet uh what what nothing never I've said too much anyway is this really a thing what of course there's a code in last exit sure I have not been told about this they don't tell me anything okay so they mentioned that a code does exist within the last exit PvP map and ask if anyone solved it yet so what do we know so far about this code well it's really early days and at the moment there are so many theories about what this code could be where and when and how it takes place and what it actually does unlock it's absolutely crazy I personally have explored the heck as to last exit and have come across a few real strange things for sure many could be map mechanics but knowing bungee and the way they implemented secrets into the rise of iron you just never know okay so within the map last exit the primary design is the fact it's basically an old train station as such a station called off world transit something we've seen on other planets before as last exit is based on venus now on the side of some of these characters we do see states m line which obviously stands for mars line and on some we see v line which means venus line but yeah off world transit this all makes sense now like I said earlier this could be map mechanics but approaching these carriages we see lights come on where it states regional locations as regio in latin is the word for a region now this does happen when approaching both carriages but yeah for sure it could be a map mechanic and not much else another real strange thing but interesting is the reflection shown in some of these carriages now some show the correct reflections but others show a completely different location some reflections show the m line station so I did go and explore mars and where we find carriages within the off world transit station I didn't find anything of real interest so back to last exit it was now back to these reflections in these carriages some of them show what looks to be a capture point near where a would be so I tried loading into a private game of control within the last exit map capturing all points and seeing if anything changed and it didn't seem to be the case now this massive screen within the station does show a couple of images which do flicker which you can actually make out the first image is a little more scrambled but it clearly shows venus the second image shows a cable car which could be an indication of a connection with the first mission within the rise of iron the king of the mountain mission where you have to ride a car to foul winter but something else of interest is on the map frontier we see the same hammer logo which is shown in the flickering screen and last exit but we also see cable carts but that for me is where the trail kind of ended so again back to last exit to find some more clues as to what this code could be so a reflection on another train shows what looks to be a silhouette of five or six gardens placing something now where this is at the moment I have no idea but this could be an indication of some sorts another thing which could be of interest is the image mantis tweeted out yesterday it basically states how the stream was fun but he also tweeted with it an image of something green to me it looks like a green screen used when editing and it could be maybe just something to lead us on who knows but I went back to last exit to check all things green the location indicated by green within the carriages is of the regio interesting on the small billboards that of the regio billboard is flickering if you watch it closely it does flicker within a certain pattern but it could be nothing at all of that when searched on google does indicate that it is a place on venus as are all places the carriages state it just opens up many more views that one can come to and yeah now there are many more things to try and explore and figure out if you have anything of interest let me know another thing I will add though is regio is the Italian word for royal today's mission is the queen's ransom is to retire there I have no idea that mission though is on venus so exploring that daily mission is one thing I will do as there could be something very well hidden within this daily mission which leads us to many different locations who knows now reading more theories unread and such people are trying all kinds of things to work this out it's really exciting to see where this leads and hopefully it is for another new exotic weapon I am seeing many people saying this could indeed be the quest for that new exhaustive we've seen so many images of but have no idea of actually how it's earned this weapon is called the dubious volley a weapon a lot of people own two mines or whether or not it's actually in the game now its perks can be found within the destined database and you can see them on screen now so by the vast me included this weapon is believed to be in the game whether or not it's tied into this new hidden code which is found within the last exit pvp map is something we have no idea on at the moment now it is really early days but over the next coming days expect much more info to come out on this and if I find anything else new you guys will find it here on my channel so subscribe for daily destiny if you're new here turn on the notification so you don't miss a thing and I'll catch you guys on that next one


Hi good morning and welcome to today's products in focus as you be able to see there the US 30s continuing its move up to 16598 but its pause there today potential support remains at 16401 and with Twitter disappointing and eBay moving down there's a little bit of a dulcet tone and some of the US markets there today Twitter actually dropped 10% on a deteriorating picture a lot less users log on times eBay actually beat estimates but they gave a kind of a revised outlook for 2014-2015 saying that competition was going to be strong we don't have another bit of US earnings until the 1st of May we've got Exxon mobile reporting so the interesting thing is go UK100 however is going great guns this morning and re-challenging potential resistance at 6774 touched it yesterday touching again today a breaking close above that opens up the road towards next potential resistance at 6874 so things that certainly look interesting there and we do have I think we've got Lloyds on the 1st of May and a raft of other foots to 50 companies reporting so be interesting to see how things progress today tomorrow and the next day so UK100 looking very good this morning looking at Japan to do five it's come down slightly as recent economic data there has actually disappointed things industrial production came out slightly less than expected 0.3 versus 0.5 dollar yen is slowly creeping up which normally a week or yen would be helping to pan to do five but they're the economic data is kind of weakening this in the short term so as you can see this level's been quite strategically important for some time 14209 still in play it's a level at Japanese index traders will want to look at looking at dollar yen dollar yen actually showing a bit of volatility this morning you can see the long-legged candle was indicative this was much lower and it's now trading a little bit bouncing around that 21 and 55 period SMA potential remit resistance remains out one or two spot 87 but we really need a break and close above this to rechallenge 103 spot 77 a little bit away from there right now could all West Texas continues to come down really ugly candle there yesterday you can see a hit potential resistance at 103 pushed all the way back down it's lower again today as oil production increases in Libya though from what you can see in the news the Ukraine crisis seems to continue to deteriorate with pro-Russian supporters storming other government building across eastern Ukraine but crude oil is not really having much of it this morning the back of the fundamentals and it is a crude oil Wednesday of course you do have the oil inventories due later on today at 330 UK time gold as ever is drifting a little bit lower a lot of volatility around these particularly important 1295 we're trading blow again today the potential next potential support remains at 1267 today's close will be very interesting and will help pave the way directionally for gold for the next session looking at your dollar to finish things up a little bit of weakness there yesterday will actually a lot of weakness as German data disappointed yesterday and you can see we're bouncing around that 21 period and 55 period SMA next potential support 137 72 cable however still trying to be a bit more confident above 68 20 hit one of the highest levels since 2009 on Monday and has been unable to really show that that that next stage at next leg up to help it challenge 170 spots well sorry one spot 7042 we need a break and close above here I really come from breaking clothes before we're going to see the next leg up otherwise expect a drift down our potential drift down I should say I come at data wise there's tons so we've already had the Japanese data later on today you've got German employment data you've got a consumer price index flash data from Eurozone you've got ADP private payrolls in the US at 115 you got US GDP at 130 crude oil inventories at 2 sorry at 330 you've got Chicago business survey at 245 and then you've got the federal funds rate at 7 p.m. UK time so today there is so much macro data is bound to have some sort of impact in the market so just bear that in mind keep your eye on the chart for loads of cool stuff going on there from our global analyst team and make sure you make insights part of your layout going forward and join me again tomorrow to find out what happened next


So you know how you're always asking about the Federal Reserve? No? Oh. No questions at all? Okay. Okay. Let's start there. That's Ben Bernanke. He's the Chairman of the Federal Reserve, or the Fed. The Fed is, basically, America's national bank. This means that it makes loans to the U.S. Treasury, processes government checks and bonds, collects all payments, like taxes, and holds accounts the government can use to make deposits. These accounts are important. The Fed is required by law to hold some form of collateral for every dollar in circulation. Some of it is in gold. Most of it is in bonds or other government securities, which means, basically, the dollar is a promise to pay backed by another promise to pay. This is important to remember when you think about monetary policy, which may be one of the most important things the Fed does. Monetary policy is how the government attempts to control inflation and unemployment. The Fed's mission, by law, is to assure stable prices and low unemployment. The target they aim for is around 2% inflation and 5% unemployment. Now before we go any further, let's stop for a second and discuss what inflation is exactly. Inflation is money losing its purchasing power over time. One sign of this is when prices rise sharply. This can happen because everyone loses faith in the value of their money, or for other reasons. For example, if the government printed a trillion dollars tomorrow, people wouldn't sell as much milk to you at the same price as before. Because there would be more dollars than before, but the same amount of milk, if their price stayed the same, they would come out behind when they sold to you. At the end of the day, money is only as valuable as people's belief in it, which is why promises to pay can be backed with other promises to pay, or IOUs. I have tried that in my own life though, and it never seems to fly. The Fed's chief method to control inflation is by setting interest rates for the transfers of money between banks. Banks move a lot of money back and forth between each other, and this basically controls how much interest you get for borrowing or loaning money, at least indirectly. Changing the interest rates keeps banks from doing too much loaning, and it affects how much money there is overall. It's tricky because a little inflation is actually good for the economy, while no inflation at all, or too much inflation, can interfere with economic growth. The Fed can also affect inflation by selling government debt, or by setting the rules for how much money a bank has to keep in its vaults at all times. More money in the vault means less that can be loaned out. The Fed can also loan money directly to banks or other financial institutions. This helps money move around, especially when the banks won't lend directly to each other, like during a financial crisis. One of the Fed's most important jobs is to loan out money when no other banks will, so the economy doesn't freeze up. The Fed also controls the electronic systems that banks use to transfer money between each other. Before there was an electronic system, the government had to print bills in large denominations, so large sums could be loaned back and forth, which is why you read about huge bills in the past, like the $10,000 bill. The Fed distributes the cash to the U.S. Mint prints, but only 8% of the nation's money is estimated to be in actual physical cash. Aside from monitoring the stability of the economy and the flow of money, the Fed also regulates banks and financial institutions, along with the regular federal government agencies that do it, like the SEC, the FDIC, the CFTC, and the Treasury. The Fed's part of this is to ensure that banks are stable, which involves measuring what assets and loans the bank holds, how risky they are, how the bank's procedures work, how easy it is to convert the bank's assets to cash, and other similar factors. The Fed is also required by law to review banks in low-income areas to make sure that they give out the same percentage of loans to poor customers as they take in from them in deposits. This law was passed in the 1990s because Congress believed that the banks were taking poor customers' money, but then refusing to give them any loans. So that is, in a nutshell, what the Fed does. And there you have it.


Welcome to Coast to Coast. My name is Lily Weinberg. I'm here joined by my colleague, Lillian Corral. How are you, Lillian? I'm good, Lillian, how are you? Doing well. What's new in LA? Nothing much, there's lots new, but nothing new that we haven't already talked about, but excited for today's show. I am particularly interested just in sort of hearing a little bit more about this topic of cultural bridging, and one thing that is new in LA is just a conversation around how not just white Americans, but all Americans and then immigrants, in particular, really think about being a part, an ally and a supporter of the work that's happening, especially being led by our black communities in the U.S. to address racism and to bring it to light and to talk about it and make sure we're really thinking about transformational change in our states. So that's a lot of what we've been hearing here in LA, and so I'm excited for today's show. Yeah, me too. I'm really excited. I'm happy that you are the one who called attention to this subject, what you're seeing in LA. We're also seeing this in Miami, and Miami's an incredibly diverse community, but we also have quite a bit of anti-blackness, and so we will dive into that. I think what's also been really interesting that I do wanna highlight, we talked about this in the opening of the last episode about the transformation that's happening at a municipality level in cities, and the actions that are being taken, but we're also seeing that, and I'm seeing this, and I'm sure you're seeing this in the tech world, organizations are also, they've made their statements, but now the real work is happening, and hopefully the real work and the actions, and I think that's two problems. One is looking at leadership, but then second is really doing a deep dive on where the money is going, and an audit of that with respect to vendors and all sorts of things. And I do wanna just call out that I sent this to you earlier today, Jay Pitter who's gonna be joining us shortly in our conversation, she had a letter, a call to courage, which really outlines very specific practical steps and actions that urbanists can be taking in their work. So really, really love that, and we can link to that in our chat box. But can you tell us a little bit about what the purpose of Coast to Coast is and why we've been doing this? Yeah, so for our guests, Coast to Coast has become sort of this weekly sort of topical show around the issues that are really driving discussion in our communities, and we're really focused at night and thinking about how both the information that's circulating in our communities, but also the way that people are engaging really helped to build the vibrancy of the place that we live in. So a lot of the topics that we've talked about have focused around placemaking, have focused around public spaces, especially in the early phases of the COVID response, and then obviously now we're dealing a lot with issues of race and equity, and are these spaces in these cities, the spaces that really are inclusive to all of us, or just some of us, and we think about it really from a perspective of like the future of the city and how are we making sure to your point that the tactical approach of that Jay or others really help us drive new strategies in our communities so that we're building those future cities with everyone in mind, so. Yeah, that's great framing. So I will move forward with talking about episode seven. Lily and you did a great job of setting it up, but I'll just say a few more words. So for episode seven, we will be looking at how we build connections across race and ethnicities. We'll do a dive into how addressing anti-blackness across diverse communities and forms our conversations and action, really looking at that action piece. We'll examine what it means practically. We'll give some historical context and we'll look at how we can move the needle forward in our communities. So I'm really excited to do that. And I think we have the right guests today to join us today. So we have Jay Pitter and Anna Sophia Pillies. Please join us. Hey guys, how are you? Great, how are you? Thank you. Doing well. Well, I'm thrilled to have you join us today. I'm gonna just do a brief introduction for who you two are. You have amazing bios and then we'll just dive right into the conversation, okay? So Jay Pitter is an award-winning placemaker whose practice mitigates growing divides in cities across North America. She spearheads institutional city building projects and specializing in public space design and policy forgotten densities, mobile equity, gender responsive design and inclusive public engagement and healing. Jay is also just a friend of Knight Foundation, both of you are friends of Knight Foundation. So Jay, thank you so much for joining us. And Anna, co-founder and executive director of the Miami Freedom Project, which aims to transform Miami political culture through educational outreach, communication strategies and movement building to bring about a more just and equitable city. Anna is also an award-winning food and culture writer who has organized Latino Get Out the Vote efforts since 2016. Anna, thank you so much for joining us. Thank you. And of course, you're based in Miami and Jay, you're in Toronto, right? But work all over, so fantastic. Thanks for being here. And so I just wanna take a couple of seconds to talk about how we're gonna structure this. We're gonna take 15 minutes to have a conversation between the three of us. We're, I'll ask a few questions and then we're gonna open it up to the audience. And that's really a big part of our conversation. So audience members, please put in your questions into the Q and A. We also have this live streamed on Facebook. You also can put in questions in Facebook, hashtag Night Live, and let's get started. Does that sound good? Okay, let's do it. So question one, and Anna, I'm gonna have you set some framing for us. So what underlying tensions and conflicts has the current moment brought to the surface both within and between racialized communities? You know, I think it's, if I can speak specifically to the Latinx context, in South Florida, for example, we have a majority minority city where part of our origin story, what we pride ourselves on is being a city of immigrants and very much a city that we want people to look to as an example of what immigrants are able to accomplish. But the immigrant story is very complicated for everyone and the Latinx community in this moment, particularly because in and of ourselves we have racial, we bring to this country racial hierarchies, colorism, inequities between us, a very rich culture that represents all of these communities that we're very proud of. But then incoming to this country and has part of that assimilation process we're asked to make different choices. And some of the work that we would have done within our community to unpack these racial questions that's almost disrupted in the face of assimilating to a new system. And that new system requires you to make a choice in a lot of ways. And in assimilating it, you almost are asked to become a model for the system as being representative of what that system makes possible. And in exchange, you're upholding that system and you're celebrating that and you're very much part of what's keeping that system in place. So to the extent that that's not there, the inequities in that system, they very much become a part of your story. They've become part of what you're defending and seen as representing. In the Cuban American example where a Cuban American led organization with this moment brought to us was how can we, we're very proud of our Cuban American success story in the United States. But then now that that's being challenged, how can we speak to this moment as a community? And what we found was that we had to redefine who that we was. We are a very diverse community in and of ourselves and we wanted to make sure that in speaking to the Black Lives Matter movement in a way that shows allyship and listening and a proximity to, we recognize that there are people in our community that are very much represented by the Black Lives Matter movement. And there is no distance between the movement, what it's asking for, what it's trying to correct and address then to what they're seeking in their own community. So we needed to find a way of representing, having a Cuban American platform that could speak to this that represented the diversity and the richness of who is in that community. So that was the first challenge and that was the first question we had to ask ourselves. And it was a long time coming. I think when you're in the immigrant experience you can kind of put those kind of soul searching questions on hold and that there's no more time for that we really need to speak to this moment. And you feel like it is happening now within the Cuban community in Miami. There's a movement to have this conversation. I think it's happening in our community. I think it's happening in other Latinx communities. It's happening across the country. But as you said in the introduction it's the challenge is gonna be how it can continue in the healthiest, most productive way. So that it keeps moving forward. It goes beyond the statement to real action. Yep, yep. And we'll get into that more. Thank you for the framing there. Jay, same question. And if you can reflect a bit on what Anna said too. I think this conversation is so overdue and urgent. One of the reasons we haven't been able to have a complex and rich conversation like this one is because generally speaking the story about America is a black and white story. And what happens in that framing is that a lot of people who are not black or white get erased from that narrative. And also what happens is that people who identify as black or white have different dimensions of their identities erased as well. And so I think the first thing that I just want to underscore is that we need to think about these stories and these spaces and these cities where we're aside each other. We need to think about expanding this narrative beyond a black and white story and the spectrum that is incredibly polarized with nothing in the middle. So that's the first thing. The other thing that I would say that really resonated with what Anna said which I thought was very courageous to say because we don't often say this out loud. Black people tend to make the distinction between sort of being people of color and black people because it is true that black people have a very distinct experience of racial discrimination just based on our histories of displacement and how we came to be in the US and how we came to be in Canada where I live. And it is also true that we are often put in a terrible position where we don't want to call it anti-blackness in communities who are also facing discrimination and are also equity seeking groups. So we see anti-blackness not only in Anna's community but in among the feminist movement, right? Among women we see it with disabled peoples but we're hesitant to call it because we also are very aligned with those equity seeking groups and we'd like to get justice for all. And the last thing I'll just say to this question is that as a way of framing this conversation the two things that I would say is that it's very important to remember that in 1968 when the Civil Rights Act was advanced, people see the civil rights movement as being a movement to advance black people. Very few people recognize that the civil rights movement actually supported the feminist movement. It supported movements against homophobia and even in that act it speaks to immigration. So a lot of racialized communities living in North America enjoying privileges and having model minority status, actually that was made possible because of bad non-model minority black people. So our civil rights movement paved the way for immigration and I just wanna underscore that so much because if you think about the civil rights movement it was very focused. The people on the front line were the people who were descended of enslaved African Americans. So certainly immigration was not their problem. If you think about like what their fight was but they made it their problem because when we start to talk about equity if we are not intersectional and truly talking about everybody and if we're not fighting for everybody then it actually isn't equitable. And I think that I'm so proud of my community for modeling that and in this moment I think we have the moral authority to call not only on white people but to call on other racialized and equity seeking groups to mirror our dedication, our values and our commitment to wide scale wholesale social equity. So we've been failed not just by white people but also by either racialized groups. And so this conversation is very urgent and important. Thanks Jay. And so we dropped in a few links earlier discussed the call to courage, your amazing letter but then also some historical links that you provided Jay. And one thing that I just wanna before pivoting to the next question one thing that you're really great at doing Jay is looking at the history and especially with your place-making work and you just discussed the history of the Civil Rights Act and all that. Why is that so important to highlight and elevate when we're talking about this subject matter? I just think that in my place-making practice overall we're all given this dominant story and very incomplete. And so we end up doing harm inadvertently. I don't believe that people are bad necessarily. I think that people are incredibly misinformed. And so because I'm both a place-maker working on public sites but I'm also an author and so story is so central in the work that I do. So I'm always thinking about who are the whose stories are we not hearing? What voices are not coming to the table and how do those stories help to inform the ways in which we perceive not only our cities but our relationships with one another. And I just wanna make another quick note about relationship. In this conversation we often talk about allies. I have very mixed feelings about that because that is war language. And while I understand that we've been combating anti-black racism for centuries I do not want my identity or my experience to be framed as war. There's so much more to my black experience than that. And in my experience in doing this work across many, many cities is that the allies I'm not looking for allies. I'm looking for friends. I'm looking for authentic relationships. And so just as we even go on with this conversation it would be good if we could center friendship, relationship, not just allyship because what are we saying when we use that word and are we not just replicating this thinking that keeps us in battled social locations with each other? I'm just not interested in that. That's a great point, language matters. And it also matters who's around the table. So I'm gonna move on to our next question. And so Anna, so what are some of the examples of cross-cultural bridging that could be the basis for reconciliation and solidarity between racialized communities? And Anna if you can kind of, if you could also build in what you think will happen next to in this answer. I mean, I think it's a really incredible opportunity for us to look at how, I think we think of ourselves as a very multicultural country. I think it's something we pride ourselves in. I think in the Latinx community I think it is a very much a point of identification. We see ourselves as a multicultural community. We celebrate Afro-Cuban, if I can use the Cuban example, Afro-Cuban writers, poets, politicians, voices, as much as we celebrate people who we think of as European descent, it's very much, and as we look back on our culture, we very much identify with and celebrate a culture that's very diverse in and of itself. It's a point of identification, a point of pride that it is so. But we also can't let that be a substitute for the real work that has to happen. So when you think about Miami, for example, as a city that's very diverse, multilingual, but you still see these inequities. So if we're comfortable with each other's music and we're comfortable with each other's food and we're comfortable with each other's languages, that's wonderful. And is that gonna be the foundation for the real work that has to happen? To the example that Jay gave us of the civil rights movement, are we looking at the issues that are impacting Black communities as being also issues that are impacting our larger community? So if we have, get to the point where culture has brought us together in a way that we see each other's humanity, we identify with each other's interests, loves, joys, can we get to the point where we then also want to support each other in these actions? So Miami, for example, we have climate change. Are you going to, that's going to be inadvertently affect Black communities? Are you only gonna advocate for climate change or are you gonna advocate with it the intention that it is having a disproportionate effect on members of this community who are of African descent? So it's also doing that cultural work and then working with intention to advocate to bring about the equity that should be the kind of natural outgrowth of that positive experience of just enjoying being with each other and seeing each other and understanding each other. That's really fantastic. And that climate change example is so relevant within our community in Miami. How would you, how would you, how do you rank us in Miami for this type of work? I mean, how are we doing with this type of work? I think what Miami has going for is that it loves to talk. And I think that's the beginning of any initiative, any time you need to, you really need to get people in communication and being forthright and honest about where they're at and how they can see a best way going forward. Something we do at Miami Freedom Project, and I know that Jay has, I've discussed with Jay how she brings us into her work as well, is about bringing small groups of people together around a dinner table to discuss a topic and have that be the direction and not have a sense of where the conversation needs to go, but really have a sense of what the questions are and that everything is open. And it's that power of representing that community as you think about what you wanna see happen in your community. So it's about bringing people into your home, it's about bringing people, having an experience through your culture, through the most fundamental way that we express that, through food, through conversation, through relationship building. So I think that's something that, I think people are very open to in Miami and they're very receptive to. So it's really about continuing that and seeing that just keep that growing and make that a part of this conversation. So if I hear you correctly, and I agree with you, that's the most basic form is breaking bread together, right? But if I hear you correctly, the conversation is there, people are talking, there's momentum, but we still need to get to the execution part, right? And that's where we're gonna have to work that muscle probably. Jay, same question for you. I'm gonna repeat the question for the audience. What are some of the examples of cross-cultural bridging that could be the basis for reconciliation and solidarity between racialized communities? Okay, I'm going to quickly speak to what Anna said about the limitations of celebration. In my book, Sub-Divided, I talk about the fact that, all of those diversity strategies from the 1960s, what they did was they encouraged us to celebrate each other's cultures and individual accomplishments. So people did a bunch of potlucks at work and they learned about, we got holidays where we got off, we were able to take our holidays off. What the stats show is that while we increased diversity and celebration, what also increased at the same time was social and urban inequity. So what that shows us is that celebration is not enough. In fact, celebration is a smokescreen. So we're celebrating people and then it gives us the false sense of actually taking action to create systemic equity, right? And so I just want to just very much agree with what Anna said about the limitations of celebration. Now, in terms of what it looks like to do some cultural bridging, Lily, I'll just mention an example from a project that you're super close to and was really instrumental in supporting and that was reimagining cheap side. So I work on a lot of projects where cultural bridging is essential because I work on Confederate monument sites and other sites with prop histories and a lot of pain. So most people would think that with a project like that, I would be engaging, again, black Americans and white Americans and especially because that's how the narrative sort of works. But the truth is that in healing the past and reimagining that site, I also needed to be very aware of other people in Lexington from other racial backgrounds and a large group represented was actually Latinx folks, a whole generation of dreamers. And so during that process, they really educated me about also feeling not welcome at that site with the monuments, not feeling comfortable or safe in public spaces. What we don't talk about is that black and Latinx people are often criminalized in urban communities. They often have more barriers within the educational system. Like there are a lot of overlaps in some of the challenges. And so in that particular project, I engaged Latinx folks from Lexington and they got to have a voice in reshaping that space as well. And so in terms of the cultural bridging, a major strategy would be to be able to hold two things at once. Let's in every single project hold the history and of the most impacted groups. So in that situation, African-Americans make sure that there's enough space for them to have a voice in the process and for them to shape the process. But I always say that there's enough justice for everyone. So I was able to do that while also holding space for people from other backgrounds who might not feel comfortable in that space for multiple reasons. So dreamers, single mothers didn't feel comfortable in that space. Poor people didn't feel comfortable. People who didn't have access to good transit didn't feel like they could even access the space even after it was reimagined. And so as someone who works really holistically, I'm always looking in every single project for the opportunity as we envision a future city to create space for everyone to have input so that we can create a shared reality together. And as you said before, who are we missing around the table? Who are we missing and looking really deeply at that? I'm gonna call in Lillian. There's a couple of questions. So thanks. Yeah, no, great discussion. And it's, you know, time is so short on the show. We can't always dig in deeper. But one big question that came up, so maybe both of you can address it, is it's a question around the use of the word minority. And so do you think if we were to, as it frame the identities of people and drop the word minority from our lexicon, like would that help us like make some forward progress in the movement to address issues of diversity and equity? Like any thoughts on the pros and cons of using that word? Okay, so fast. I don't identify as a minority. I think the fact that, you know, Miami's considered a minority majority city speaks to a lack of willingness to just acknowledge the people who are in a particular place. That framing is actually incorrect and it's an oxymoron that can't happen at once. If you are the greatest population in a place, you are not the minority. You're in fact the majority, but that term comes out of fear of having racialized bodies and black and brown bodies in a space. And so this effort to keep like black and brown people in this, in their place, in a minoritized space is where those kinds of terms come from. So I totally reject the term. I did just use the term. So I can say I totally reject it. I think we love the shockness, the shock value of it, which is why I think we do use it in Miami. I think as it applies specifically to Miami, its majority is defined by mostly more foreign-born residents than people born in the United States. So it speaks to the immigration experience. But I would never necessarily, to your point, identify myself as a minority. And I do think it forces you to lead with a less than. So I think that's problematic. I really think it's about kind of focusing kind of like where the numbers are adding up. And I think there's a power to that term because it shows that there's a certain kind of, an imbalance that people need to be aware of and they need to kind of speak to it in a really kind of forceful way. So there's a way of kind of using it in a powerful way. But yes, I definitely think it's something that we can evolve past at this point. If it's, I think that that's where we're heading in. Yeah, great. The other thing, besides numbers and demographics, in terms of creating urban equity, we need to think in terms of power. So for example, if we take a city like Detroit, which is majority black, minority white, but white people in Detroit shape Detroit, right? So even it can be misleading. So you can have a large number of people who cannot access systemic and social power to shape the cities where they live. And so I feel like we should be more focused on power than sort of demographic counts. And I think that demographic counts can again reinforce divisions across race. If we're living in a city, we are a city of diverse human beings occupying a city and then we can look at power imbalances so that we can address those. It's not about addressing or identifying the demographic as much as it is, addressing points of power where people have power or lack power and working with those people, regardless of the package they cover. Regardless of their racial identities to be able to create cities for everyone. Yeah, now definitely what's coming across from both of you is that the power question really is almost obfuscated by this notion of numbers, right? And the way that we tend to think about it demographically. So I'll bring Lillian, we are at half past the hour to wrap us up. But I think the last thing maybe I'll just leave you with we had a couple of comments that really were and more comments, less questions really around how you made sort of that immigrant experience. So maybe Anna, I don't know if in the closing remarks you wanna speak to this, but how coming into this country, I think there's a question there around sort of like the assimilation aspects of being an immigrant and coming into this country and sort of quote unquote imposing your own way and or assimilating to the way that things are. How do we address those kinds of issues as well? And then for me, the question is, how do we do it in a way that we're not perpetuating the kinds of racism and bias that's already built in here as immigrants? So I'll leave you all to wrap that up, but great discussion. Thank you, Lillian. And so Anna and Jay, I would like to invite you to have just any closing comments with this discussion. Anna, can I call on you first? I'm sure. I think that's a great comment. And I do think that's something that, like the minority question, I think we can evolve past. I think there's a very difficult choice that we're asking minorities to, minorities are immigrants to make when they come to this country. And now that everything's being challenged, we can, as people who have that immigrant experience, we can do the work to make our system more equitable so that when immigrants are coming into it, it's a, they're coming into a system that has, you know, it's a more honest system. It's something that they're not being asked to be lesser than, or to be, to make a choice to somehow, to, you know, feel, make a choice between being, you know, black and white, or trying to kind of check one box or the other in order to advance. It's a society where there is equity. It's a system that is allowing everyone to advance together. So immigrants are not being given this false, this false choice to make, that then they then, you know, exacerbate or they propagate it and they push it forward. Powerful. I think this is an incredible moment for every single human being to think about their personal accountability and not to hide behind, maybe not being up a particular race or having a particular type of privilege. All of us have implicit biases. All of us have inherited a terrible history and all of us are responsible for showing up fully and unpacking that history together. Black people, white people and other racialized groups as well. Thank you. And the opportunity is now. So thank you so much. Thank you, Anna. Thank you, Jay, for joining us on this important conversation. Lillian, do you want to talk about our next episode? Yeah. So we're going to be off next week, but, you know, for our audience expect to hear from us the following week on July 7th. But we'll start to look a little bit more at how do we build this future city? And some of the questions I think we'll be exploring in the month of July is just to give folks a sense of things like we talk about these smart cities and so how do we really balance, quote, having more information about our city, but also being really conscious of the things that are happening in communities like surveillance and then how that surveillance could be targeted against certain groups versus others. So kind of balancing that tension and then also things around issues of movement and mobility and how our mobility systems are being transformed a lot by technology. But in this moment how we're also seeing it really not rise to the challenge and relester everybody well. So stay tuned for those episodes and enjoy the July 4th weekend, which we'll be around for. See you July 7th at 1 p.m. Thanks. Take care. Bye.