 All right, so everyone, welcome back to the graduation seminar, and this is going to be the last seminar this semester, and it's going to be the third annual Markov lecture. Okay, with that being said, let me just randomly choose our next person to introduce our speaker, and he's going to be Clyde. Such a random decision. Because of the British accent, right? Yeah, right. Okay, oh wow, great turnout. Okay, welcome to the third annual Markov talk. This talk, Jacob Davis, who will be presenting today, will be talking about pseudo-stochastic time in Pundit. I like a Frisei polygraph. Okay, Jacob. Goal speed. Thank you. Could we have a short interlude in which the possessor at the Slapper talk tells me how to do this? You click. Click. Okay. Right, so this is going to be a talk about pseudo-stochastic time in Punditianianianianianianianianianianianianianians of the random model of Frisei polygraph, which is a fascinating variation of the usual stochastic field, so there are lots of interesting behaviors in the pseudo-stochastic field that don't occur in the stochastic case. So off we go. You click. You click the other way. Try right there. Okay, let us begin with some definitions and some motivations. So the ground of Markov-Fray's polygraph or IMFP is a natural object for testing truth values in a mathematical context. So let's immediately know the universe of sets as of course as usual. Then we have this function. Note this is not a function into the set zero one. This is a function into the real interval zero one. So it's not giving you a binary assertion of truth. It's giving you a sort of continuous sense of how probable is the set X to say the truth. Now perhaps a little clarification of how set say things is in order. This is reflecting you can take of course any mathematical object and encode it as a set and in particular you can take assertions of information, so for example proofs and so forth and encode them as sets and then they say stuff. And of course some sets are more honest than others. So the existence of this class object is equit consistent with the existence of a pink unicorn which might lose some imagination that clearly sists thus meaning the RMFP itself exists. The process of constructing the RMFP from a pink unicorn is quite complicated and frankly quite gruesome. So as you may be aware there's the principle of eternity in mathematics which asserts that every game played on the real numbers where the players will tend to be playing natural numbers to form a real number is going to be determined. So one player or the other has a winning strategy. This is the abbreviated AD. A variation of this AD is the principle of pseudo stochastic determinacy where you play your games not just in the natural numbers but through points on the RMFP. So you're playing points on the graph instead of natural numbers. And during the game do the points say what they are saying which may or may not be true? Well that's quite interesting. So actually we have a process so the players are alternatively playing statements until at the end you've got a collection of sets, you've got a collection of statements which is called a quorum and then the quorum votes on whether they want player one to win or whether they want player two to win. And of course the players want to try and establish a quorum that will vote in their favor but the ways quorum votes can be very complicated so it's not just a sort of monotonic thing so the strategy can be very complicated. Yes. So the existence, it says it's equiconsistence with the existence of a pink unicorn. Is this a particular pink unicorn or does any pink unicorn? Any pink unicorn so the definition of pink unicorn which is you're looking for is a unicorn that is pink. Yes. Why does it have two dots for the AD? Well that's an unnamed. How's it pronounced? AD. I should warn you if you ever attend any talks given by a speaker whose native language is not English it becomes very difficult to distinguish AD from AD. Okay that is why of course yes because you see the dots were transferred through what's known as the dot transference function. Under this axiom the theory and structure of pi and homogenous automorphisms of the RMFP is trivial as all the randomness disappears because you have ways of well one player or the other has some way of sort of always ensuring that they get the opinion they want to be the result so it's sort of there's a canonical opinion and then the other player really doesn't have much to say anymore. So we're going to construct a random Markov Frisei polygraph so the model for random choice depends upon a lemma equivalent to the axiom of choice omitting the footnotes sadly I cannot omit the lemma. I see the following lemma which uses some results from outside of mainstream that matter which ensures that one can construct upper bounds in certain circumstances so of course again we will omit the proof because the limitation is very cumbersome and the delegation is annoying and the interpretive dance is several hours long. We're going to denote synapsic stochastic determinancy which of course is also known as AD sometimes we find that too cumbersome is an notation so we have an alternative variation n backslash omega phi the sum of a t of x z to the pi i to the p y j lambda alpha beta that should be adults instead of a camera sorry once again I say I must in the sense that if I wanted to give you a fully formal proof then of course in addition to the interpretive dance you need one of these four alternative lemmas yes those I'm going to omit into my lack of experience in the field of dance mathematics so now we apply our lemma and this is this is left is to exercise if you do want to do it at home please ensure there's no one else around and when I use we here of course I'm not claiming credit for the result it would be slightly presumptuous to expect to be awarded the prestigious mrs. fields prize we mean that we expect the authors of the paper to be awarded the medal for that I mean really the combination of their profound mathematical skills they're really quite striking dance moves and they're very disturbing I believe we were talking about the mfd of the random mark of fries a polygraph but of course this is quite a similar object to the brand name in mountain police which you can see a sketch of that this is an example of a polygraph in which truthful statements have been established and the royal canadian mountain fleet police is in fact an entity which where the strategies establishing the weather is true false will always be the truth because the police in Canada are not committed to the north west mountain police in 1873 an extraordinary fact as denoted by the exclamation mark then in 1920 it was realized that actually there were parts of Canada other than the northwest so the organization was expanded condensed train in Virginia Saskatchewan which sounds quite cold the principal elements on the RCMP regimental badge of the bison head making a scroll and a crown as I'm afraid you can't really see that because the picture is blurry but if you zoom in you can see these these objects so these four objects are the main things that are used in the construction of the strategy so the fact you have all four of these available makes it relatively easy to have to create strategies that allow you to prove the truth of statements if you if you ever find a police officer who lacks one of these things that indicates the truthfulness cannot really be provided so any questions so far okay so in the next slide we'll introduce the RCMP problem one of the most important open questions in in the pseudo stochastic field I'm sorry yes why do you expand upon the connection between the RCMP and the RFMP um yes so the RCMP yes I'm sorry I didn't put that on the slide the RCMP is a special client of RMFP uh or uh yes in which the the truth of the statements always comes out as being positive remember in the normal one you're simply establishing the truth of statements which so they could be true or false yes I'm a little rusty on this but I think there's some sort of obligation that this should be done in French too this is this is a bit of a problem actually uh because as you know most mathematics is published in English but in order for a proof to work uh concerning RCMPs then you have to write both French and English copies otherwise the arguments just sort of fall halfway through so this is quite frustrating for mathematicians who don't speak French against what's for say French? Sure so uh recall from the previous side the RCMP was founded in 1873 though of course it was just the northwest country at the time 1873 as I draw your attention to the importance of the exclamation mark earlier this is a very large number remember of course this thing is built on the universe of sex so you would expect this was founded in the future this was founded in the future that's that's a very subtle point that I was planning to state over yes it was founded in the future and this is often confusing to beginners in the subject the thing is because these are objects in which we can fully establish all the things to be true then in a sense we don't need them to exist yet because we can always already see everything that they will ever do so we're sort of current conducting an analysis presently of uh the RCMP uh keep order in a future Canada many years from now yes is 1873 having units associated with it or is it a uh no what years yes one thing that is a little unclear is how the RCMP survived the heat death of the um of course uh the RCMP are famed for their links with Google that's equivalent to I'm trying out a new presentation technique one word at a time it's quite exciting yes I'm not quite sure why the brackets there like that okay I mean you might think to yourself uh that this is just some very basic arithmetic and it does not really merit a complicated if and only if definition that add appeal to the fundamental theorem that says the activity but in fact uh Google is rather is a rather tricky mathematical object its value is rather unpredictable so in some cases it's 10 to the 100 and at other times it takes other values so you have to be very careful uh so uh this uh so as I mentioned here uh we're writing a proof about the RCMP so part of it has to be in English and part of it has to be in English so uh by we can get that stuff uh a Google factor uh I should point out Google factor has no relation to Google in regular script that's a common point what's the correct pronunciation of that Google exactly oh good question so we know that 1873 is prime moving swiftly on and that in 1873 Charles Zenit proved that he is transcendental uh actually uh Charles Zenit as you might have guessed uh was one of the founding cadets of the RCMP and uh this was his uh submission uh the graduation as a member of the RCMP uh which was later proven to be the least possible transcendental uh okay yes those uh it's quite an interesting story actually so by later we mean I think only in the 1960s there was a very long proof there were many attempts by many very distinguished math petitions to find transcendental smaller than e and bring many techniques uh in uh numerical analysis and in number theory of pride but none of them were successful and then ultimately it turned out that he is in fact the least possible transcendental uh so of course they were wasting their time uh with an irrationality measure of two uh which means it's it's really pretty irrational so if you say something has an irrationality measure the obvious next question is what is an irrationality measure um uh yeah so we're going to appeal to some of the work of Charles and Lee Cooper uh who of course is a theoretical physicist whose work has found some surprising applications in uh in number theory and in super simplistic uh theory uh so we'll consider a measure different from a rationality measure namely Wiener measure and uh then we'll consider the Wiener sausage uh and then we're going to do some work with this measure in order to construct the irrationality measure any questions yes yes yes uh I forgot to mention actually uh that this is also a necessity for talks so you you have narrowly avoided uh a mathematical problem emerging on the next slide so that's that's extremely helpful thank you please remind me to throw in random comments in french uh yeah that is uh a sub-portrait uh by distinguished french artists so of course the rcmp uh famed for their their study of mathematics i'm familiar with this kind of thing uh so we're going to introduce the rcmp problem uh as we mentioned one of the central problems in rcmp theory rcmp as you recall is a variant of the uh of fray's a brownie problem uh okay um um it also helps it's not essential uh but when you're doing work with the rcmp in addition to sometimes speaking in french if you have an occasional slide in a needlessly fancy font so uh uh there the the uh wiener sausage uh w delt from t a radius delt from left t is the set value to random variable on brownie and pass b defined by the delt's of bar for the brownie and path b uh r from time t uh so put another way it's a tasty random variable taking value in the space of nowhere differentiable uh poor things uh the nowhere differentiable there is quite important the notions you can't really formalize it if it starts to come to french so the following fascinating and unexpected theorem is of central importance in probabilistic animal training uh which is uh one of the uh well really the main famous application of pseudo stochastic theory uh i mean various attempts were made to train animals using just stochastic theory but these were all not successful but pseudo stochastic theory was much and more effective uh so we have the sausage representation theorem by ys 19 uh that's actually 1890 maybe a little bit sleeping the way it's written so uh g2 and g3 are 2d and 3d independent immunosossages respectively uh and then they form a martingale we call a lure point equipped uh with an integral uh volt there i should mention is voltage uh and then of course you're integrating across g2 you're integrating d3 across g2 so 3d object across a 2d object which sounds almost plausible what what is gamma what is gamma uh gamma is is the gamma function uh you know as used in the rear map offices uh yes so you can see a sketch of the area of integration there and uh most of the interest uh most of the weight of the integral is going to lie in the central area where there's a picture of a dog and it's about the side uh i mean they're not necessarily major zero but they're generally kind of negligible by comparison uh so uh that's how to build a martingale color lure pair uh conversely if you have a martingale color lure pair with a standard pairing uh then we can construct uh sausages as above uh and in addition uh they're equivalent to the homotopic taste which is which is always useful uh okay so if you want to apply this there and then of course you have to perform an integral across uh sausages so you need to know at least approximately the volume of the sausage or have some kind of limit uh tending towards it uh so naturally uh if you wish to perform an integral obviously what you want to do is to count the integer points the sausages in the finished space okay so uh the sausage reference equation theorem uh is is very simple for convex lattice sausages that integral is very easy to perform uh so we're going to consider that example first uh and then look at the more difficult examples with so uh a convex lattice sausage uh is is a geometric object playing uh what's simultaneously is an important role uh i should stress it's not playing an important role actually it's quite unimportant in the combinatorial commuters uh a combinatorial commuter is a kind of function that transitions back and forth uh once a day these things occur in torrent varieties uh where they correspond in particular to polarized predictive torrent varieties i don't want to give you the full definition of torrent variety because that would be a little technical uh but i want to give you some examples of torrent varieties which will give you a very good sense of what these are so uh in particular torrent varieties from corns uh so we started the corn germany denoted by sigma sometimes by tau in a lattice uh and then uh we're constructing a complex affine variety uh so uh you let as sigma be the computer's seminal group is this complex variety of variety of sausages or a variety of what kind of variety uh are you or you mean an algebraic variety yeah and it's actually both that's the very nice thing uh so it's been possible to take algebraic varieties and generalize them to sausage varieties uh so there are slightly fewer constraints in sausage varieties but then they cover a wide passive examples yes is it possible to do something like compute the cohomology of a sausage link these techniques um i'm i'm afraid i'm not very familiar with the cohomology theory of sausages so i can't give you an answer to that one does anyone know anything about the cohomology of sausages okay i'm sorry about that uh oh oh yes so we have uh so we're forming uh c s signals so you take the complex numbers uh and you append to them uh this group uh so then uh you have uh what's known as a complex agenda uh yes complex agendas of course combinations of a real agenda with an imaginary agenda which has a tendency to greatly complicate anything you attempt to construct on top of them uh so uh as you as you'll know if you've started the area uh then you'll get an affine variety uh which will uh denote by u sigma and then uh this affine variety uh is as i mentioned so uh you you have uh these affine varieties which you can view as sausage varieties so after that uh somewhat involved discussion uh we'd like to look at some of the previous work on the material again just to get you a little more comfortable uh with working with sausages so uh given any agenda uh of the form we just generated uh then we can saturate it with some number generally denoted n of sigma of convex lines of sausages so you're very interested in the question of how many convex lattice sausages you need uh so record from earlier that the case of convex sausages is very easy to address uh because those integrals we were looking at a while ago uh are very easy to compute for convex sausages so very often uh you want to take uh sometimes not convex and saturate it with some things that are convex which will allow you then to approximate that integral uh in the non-convex case so uh if uh if sigma is is empty here uh here denoted slightly excitingly by ascise error uh then the problem is of course trivial um then uh we have worked by salami in 1992 uh that in general uh you have uh there's quite a large bound on it so this is obtained uh by some common tutorial theory uh in fact by some applications of some uh relatively simple brand theory which is why you see these very large bounds appearing uh then this one was generalized uh by chorizo who was uh i believe a student of salami so uh getting some lower bounds uh so uh the same uh citing misera cases trivial so then if you move to a sigma equals two case uh so uh five sausages uh do not suffice uh and this was proved uh by exhaustive brute force search and uh application of the seminar budget uh this is kind of a funny story actually uh because uh this uh salami's work was just sort of theoretical in age i mean he's just proving these very large bounds uh but then chorizo was actually doing more practical constructions and trying to get lower bounds uh and he actually i believe they're at the university of toronto and he almost bankrupted the university by excessive spending on sausages uh and he he sort of got his got his phd just moments before oh so it wasn't even computer simulation he was actually working hands-on with the sausages yes yes uh he was a uh he was a groundbreaking technique uh but again uh salami salami was kind of a more old-fashioned i mean that's kind of repetitive but i mean he preferred theoretical he wasn't really comfortable working with physical sausages uh chorizo has of course since moved on to better things so uh of course then you have you still have this gigantic gap between the two if you stick two into that equation you still get these numbers this is a very interesting question and in order to compute n of sigma one of the things that would be very useful when you're trying to improve your lower bounds uh is if you could have a polynomial time algorithm for partitioning the sausages because the problem that chorizo encountered uh is even with this really quite small number of sausages uh the process of partitioning them among uh the seminars uh was excessively time consuming i mean it's exponential one term so this was why he wasn't able to get better than sausage and also the fact the university okay uh so no one has been able to show that it's possible to define uh n of uh i'm now using delta instead of sigma just to mix things up a bit uh tonight and a delta sausage is among uh and a jamber um yes uh i should have said uh so so sorry yes thank you again uh so uh uh this is this is another interesting application of uh pseudo stochastic theory actually so you'll notice it doesn't say there exists a polynomial time algorithm but just that there almost certainly exists a polynomial time algorithm so this is one of the cases uh where attempts were made to construct a proof using stochastic methods and this was not successful so instead they went with a pseudo proof uh in pseudo stochastic theory which almost showed the design result is this almost certainly with respect to the irrationality measure yes excellent question so yes they're almost certainly uh or pseudo certainly isn't sometimes said it's a polynomial time algorithm uh it gives an answer uh so again you have further applications and now we don't have uh is correct we don't even have nearly correct we just have it's likely to be nearly correct uh with high probability uh if uh between four and four plus epsilon of the following conditions hold so yes you do have to bring in some external assumptions about the conditions yes i'm a bit confused at the moment there are three things that seem very similar on the board we have a sassage a sossage and an osage could you illustrate the difference between the three for me um yes okay so yeah the notation here again is not ideal so uh because this is uh so the sossage uh as is pronounced by french people uh that's the necessity i mean that's really if we were doing if we were on an english side of the sassage so that's the sassage object uh then uh you have the osage object which has nothing to do uh with sausages at all uh yes it's uh it's just uh i mean there it's it's just being uh or maybe it's in the sad ass if i'm just being used as a dummy variable uh so technically you don't use anything but the use of osage is traditional oh well that's an empty sum right um oh wow this is where i should i should draw attention to the use of almost certainly likely and high probability so uh we haven't written it there because it's a little tricky to write in but really this is just less than zero almost certainly nearly likely with high probability and so it could actually be really large okay so that's why we have to risk something and does the log have anything to do with the sausages uh the log is is the regular log well that's an iterated logarithm uh yes so that's the osage and then over here um that's s multiplied so this is another of the conditions we assume we need a little bit of uh vector calculus and again you'll know we're working not with real vector calculus because you know we're working with super vector calculus so the thing is that the grunge in it's just kind of has highly grunge unity which is a much easier thing to do okay uh and then we want the dimension of the specter here of the force saskian anti-bond season of harmonic fibers of manifold uh with pennant the time of the time of the specter here of the force saskian anti-plurist sub harmonic fibers of manifold with bergman kernel that should say should be non-trivial i'm sorry i mean yeah so and there's there's the obvious other conditions that you're all down as familiar in these contexts the total list has um about 961 elements of which you need between four and four okay so um you can read my paper for them it's it's quite a long okay so um you remember uh the aim of all this uh was to provide uh tighter numerical bounds on the number of sausages needed the number of convex sausages needed uh to cover general sausages uh so uh numerical methods have recently been applied there uh and uh you'll remember the upper bounds we have are highly exponential so this is trying for a polynomial bound uh so you would use a a highly polynomial f and an analytic extension of the previous algorithm um the process of analytically extending algorithms by the way uh is is is quite new and quite exciting uh so the case of f square brackets n of delta sausages uh so uh ah yes so um this is useful in cases uh where we're not able even able to get near Lagrangianity or Lagrangianity kind of almost kind of Lagrangianity uh in cases where the Lagrangianity is simply awful then we can still we can still get things if we bring in this of this polynomial so the main lesson to generalization uh is that uh of course uh as this comment is super stochastic word you have to introduce the word kind of uh in your results so here the convergence any kind of holds are only offered in s deformation uh so uh if you want to know the the definition of kind of in this sense please see the seminal work by uh professors s reference and s clause in uh 21 34 forward to it uh and of course s deformations uh are often known as squinting uh the term s deformation is is preferred these days because it sounds more there is interesting about bounds of course but uh the field is generalized beyond just trying to get about bounds uh where to be honest we still haven't been that successful uh but considering uh fractional and unsuba agendas so uh this is uh as uh as you probably noticed in your mathematic career sobriety and sausages are rarely found uh can rarely be obtained from the same uh the mathematical entity so uh there's uh a functional theorem uh which is often attributed uh to hubert dingle and herbert berry uh why it's attributed to give me a spoiler no one really knows interesting in no one's ever been able to track down professors hubert dingle and herbert berry uh but we're working on it uh so this is uh for the sober case uh you'll remember the sober case uh is when uh relatively few sausages occur and uh as you'll recall we're interested in cases where we can cover sausages with comeback sausages so the smaller number of them we can do with the better uh and of course we had these very bad bounds in the general case but so we're we're looking more specific circumstances where smaller number of sausages was advice uh so uh any family of sober sausages uh is uh systematically almost destroyed this is a strengthening of almost just going to means you can do it in a systematic way a lot of almost just joining is really quite slapdash uh so an immediate corollary of uh what one what is often called the dingle berry theorem though uh more properly it should just be a folklore uh is that uh any sad family or a systematic is either sober or as the term is uh flaccid uh flaccid indicates uh i mean the definition is technical basically means highly non-convex it's possible to find many points within it for which you draw lines between the parts outside of your uh so uh let's see so we were considering uh systematically almost destroying system so we've shown uh we can get systematic almost destroying this uh and now we're considering uh a slight strengthening of this not a strong strengthening an almost strong strengthening to almost strong systematically almost destroyed families uh or as some doves as uh germany learn so we're looking at specification for such things uh by introducing the notion of strength any questions by the way uh so uh a family of sets is strong uh for x uh where x is another set uh if uh you can find a member of that family uh whose difference from f is countable and another member uh which is just joined from x that definition actually makes sense and then the thing is strong if it is strong uh for every x contained in the union of all those members uh so then of course it's almost strong uh if this family is strong for any set x uh which is uh weak uh where weakness follows uh the usual definition as you can see in reference one okay uh so uh a family of almost a strong systematically just general consensus called an asset family can't do the work of set theories abdul assad in 1960s abdul assad quite a controversial figure uh also had a political career but uh a very very noted set list yes i recall some uh well i don't recall personally but uh he used to host some very exciting conferences okay okay so this is this is the work that really made him famous uh well he did some other stuff so he never actually provided a proof of the following zero uh but he hosted a conference and after a seminar which was described by the participants as profoundly persuasive all the attendees signed a document deserting that they very much were convinced by this theorem so didn't actually need to provide a proof uh but we'll give one graphically on the board given the lack of such techniques uh today so uh let's first define some stuff so uh if we've got two uh sad families f and initial segment of f prime uh so initial segment in in the universe of sets of course so you can add larger things on top of f you can't survive small elements to f uh which have the same union uh and uh yep and then f is contained in a prime so yeah so you can you can sort of add larger sets but you can't add sort of small elements to the thing uh then and of course you also need this natural filter on u f uh such that for any weak set in in the ultra filter um um for which f is strong then f prime becomes strong so this is basically saying uh you can take your set f uh and then you build some more stuff on top so you extend it here to your set f prime uh and for any weak set uh over here uh for which uh you have strength uh then you can extend this up to here and you get strength here as well uh so uh now we have uh we have uh the theorem itself uh which as you recall was demonstrated at the conference in Damascus uh so uh given any symmetrically almost a strong family then the thing is maximizable in this ordering so essentially uh given any such set then you can perform uh an iterative process up to uh some f star up here where this thing is maximized uh yes so of course as I've offered uh to give uh an in-depth uh demonstration of the seiza to the conference participants uh and they they're shorting the details would not be necessary yes um that's the sadism follows from Pondslemmer from which Pondslemmer no uh because the one that's not in fact implied the other uh however uh the lemma does follow essentially from the theorem you have to you have to tweak things a little bit actually easy uh to get the theorem and uh this is another thing uh as I said uh asserted and again the conference participation participants were quick to assure him that demonstrations would not be necessary is that a duo of this theorem in an obvious sense what obvious sense can you speak of I don't know the normal so is every happy family minimizable uh happy families haven't really been much studied to be honest with you uh it might be an interesting research project actually um I'm not I'm not aware of any results of that kind on happy families but it doesn't sell any plausible reasons well there is an old folklore theorem that happy families are unique up to isomorphism that's a very good point uh so this I believe uh is attributed to uh Tolstoy perhaps so indeed they unique up to isomorphism uh so yes that would suggest actually the minimizer in would be a relatively trivial process because you're just going to get isomorphism between each of the steps as you minimize down so yes I think this is one of the reasons why sad families have been so much more static than happy families because you tend not to get very powerful interest in results about happy families uh as indeed uh was witnessed by Professor Tolstoy who devoted very little of his career so uh this is of course further work coming out of what is known as the asset school so assets proof of the sadism theorem was of course uh controversial at the time you may recall the UN issuing a statement condemning but of course the modern mathematicians being at the conference at the time are roundly rejected the UN's condom name uh so uh Elmomaso another leading mathematician of the time was upset uh by the use of uh the axiom of choice uh he made in fact many passionate speeches condemning assets reliance on the axiom of choice uh but they never really obtained much popular appeal uh so uh he said I mean solid parenthesial approach but as someone who didn't appreciate the axiom of choice he wanted to do without it so uh he was joined by many of Assas former students which of course made Masso and Assas very uh bitter elements I mean you know how things go sometimes in the mathematical community uh so uh then uh the members of the Masso school uh did a lot of work on discretionary pre-alters that were using these objects as a way of uh attempting to prove the theorem or at least uh weaker forms of the theorem without having to rely on the axiom of choice uh and they were also further disadvantaged by not controlling a country because that way they had to actually prove stuff so in conclusion what have we seen uh so we've seen finite and infinite versions of the SADISM theorem uh I hope you noticed the distinction between them at the time so we've seen how this theorem has applications in the field of combinatorics logic and analysis and financial maths uh this is actually a little out of date um uh due to new regulations by the U.S. government I'm afraid the applications to financial mathematics are no longer real but still a lot of good stuff in combinatorics logic and analysis so we've learned how to use uh the compromises around this this uh now widely accepted theorem uh and how these led to led to the uh the work in the case where you don't assume the axiom of choice and uh where you work uh in a slightly more peaceful environment uh thank you for your attention yes well earlier you said that he was the least transcendental but maybe he could justify some of the math here seems a little suspect um I mean yes I mean you would obviously think you could just divide it by two and in fact I mean David Hilbert actually published a paper in which he said let's just divide it by two but uh some simple work showed that in fact this this is not effective uh I forget exactly what my problem with the paper there's some very subtle problems that arise when you attempt to divide another by two well it becomes unreadable like you just can't read half the paper it's that I'm sure that was a problem yes especially given the printing presses at the time would honestly be today is is it a problem that we get a smaller number that is not necessarily less um yes uh the the distinction between less and smaller seems seems like a like yes as you would expect uh a good answer yes a lot of philosophers have in fact become interested in this problem and have extensively studied the definition between smaller and less but mathematicians of goblin to be honest yes um sort of in the spirit of the sadism theorem uh could you uh demonstrate to us how the margarina or the moonwalk really can relate to your work well this is another thing that wasn't exactly uh that that hasn't actually been seen because uh as that uh offered to demonstrate uh before getting into what you might call the the nitty-gritty of his demonstration method he offered to perform the margarina for the conference attendees uh uh but uh shortly after his started uh they hastily assured them that they'd seem quite enough uh and of course the necessary things to proceed so this is another thing where we don't actually have the sort of written up version yes is there a connection between sadism and the study of sausages well you would think so wouldn't you because they appear in the same talk and it would be fairly weird if they there wasn't you'll recall uh so the people who were interested in particular in the relation between sausages and sobriety uh and uh the fact alike of sausages and also the presence of sobriety tending towards allowing you to construct these sad families to which you then apply the sadism results so it's not a direct relation but from the sausages you've then moved to situations where you become interested in knowing about these sad families so you become interested in sadism yes so what's your favorite variety of sausage um i have always been partial to combat sausages but yes i mean they're relatively simple but that's my preference different people like different things as you can see by the things yes the common to study sausages in say Polish spaces yes uh the theory of sausages quite right is more tractable in Polish spaces uh so and as they call Polish sausages uh in Polish spaces uh because they're they're an example of things that are relatively easy to cover by combat sausages so you can get much better balance on the number of combat sausages required very good well thank you for your attention