everybody for this very happy meeting, mathematical meeting. So we are very happy and lucky to have Professor Carolina Rao from IMPA, lucky also despite the effort of Lufthansa that was trying to steal her, to keep her immunic for all the yesterday and today, but in the middle of the night she met then. So we are particularly thankful for her to for this complicated trip that brought her here. So Carolina is a good friend of ICTP, I mean, but before that, I mean, she's, as I already said, she's a professor at IMPA. She's one of the very prominent figure in algebraic geometry all around the world. And she's an affiliate member of the Brazilian Academy of Science. In terms of our ICTP friendship, I mean Carolina is one of our most distinguished associates of this part of the Simons Associates Schema since 2015. And she has been, she has been actually giving a number of seminars. Some of them are actually on our YouTube channels, our basic notion seminar, for example, and but number also research seminar. And she has been lecturing in a number of schools, I would say actually all around the world in Chile and Mexico and in Trieste a few times in our activities here. So we are very, very happy of this connection. And as just a highlight of, of course, of her standing in the mathematical community, I just remind the fact she's been an invited speaker in the International Congress of Mathematicians, which in our communities, of course, is a very high recognition. And she's also the vice president of the Committee for Women in Mathematics of the International Mathematical Union. I think for in all these capacities, for all these reasons, we are really very happy to have her here. And she's giving this, the talk of today on Algebraic Geometry, History, Application and Current Trends. So thanks a lot. Thank you, Claudia, for the introduction and for the invitation. It's always a great pleasure for me to be here at ICTP. So I, so today I'm going to tell you a little bit about Algebraic Geometry, which is my area of research. It is one of the oldest fields in mathematics and still very active, because it has connections with many other areas within mathematics and other sciences. So when I send the title and the abstract, I was planning to talk about Algebraic Geometry, History, Applications and Current Trends. And then when I started preparing my talk, I realized that in fact, I probably don't have time to tell you about all this. I will tell you something a little about history and applications, but I will mainly focus on current trends. So let me start by telling you what is Algebraic Geometry. So Algebraic Geometry is the field of mathematics that studies solutions of polynomial equations in several variables. So this is where the algebra comes from, because polynomials, they have a very rich algebraic structure. And so to manipulate them, there we have many tools from Algebra. But this set of solutions, they actually have a geometric manifestation. And this is where the geometry comes from. And so what we will see today is that Algebra and Geometry, they actually play a dance together to one complimenting the other. And from this point of view, I like to quote this quote by Michael Atia.   So we really need all our senses to perceive the world. And this is how Algebra and Geometry play here. Okay, so what are the objects of studies of Algebraic Geometry? So as I said, these are, we are interested in solutions of... Number six, Algebraic. Okay. Okay, so we are interested in studying solutions of algebraic systems of algebraic equations. So here I get some very simple examples. So this is the equation of a line. So this is a line. So maybe I should just for completeness, so probably we like it. So this is a linear equation. The first one, the other one is just, you think of the equation of the circle. This one here has this equation. So these are polynomial equations. So here the degree is one, the degree here of this polynomial equation is three and so on, and they give us this picture. So the point here is that what we want to see, so this is the algebraic part. So these are the polynomials that define those set of points. But when you look at the picture, you really have some other filling and some other grasp of these sets. So they do have, so for instance, this point here in this third example is what we call the singularity. So here the curve is not smooth at that point. But also these are simple examples because these are solutions of one polynomial equation in one variable, but of course you can put more variables, more equations, and you have much more interesting varieties. So now polynomials, they actually, they are used to model many real-world phenomena. So they really appear in many parts of mathematics. So let me just mention one application here that could be interesting, for instance, in terms of computer vision and so on or computer graphics. So imagine that you want to plot a certain algebraic equation. So how would the computer, so computers only deal with polynomials. And so how would you say you would plot one surface like this? So one idea is that if you have a surface which has a certain equation, and let's suppose that you can parameterize this equation. So you find polynomials, say X, Y and Z polynomials in terms of other variables. Here I put U and V. And so this will help you to parameterize this variety. So a computer could plot this surface and you could actually visualize it in 3D or in 2D, in a projection through 2D. So for applications, it's actually very important to be able to parameterize a variety. So now let me just give you this question and we are going to come back to that later. So a very natural question which is important to application is to know which algebraic varieties admit algebraic parametrizations. When I say algebraic parametrizations, I mean that, so let me just give you, so I was working out this right before the talk, so I hope I got this right. So for instance, for this curve here, this curve can be parametrized. So let's say with a parameter T to T square minus one, T cubed minus T. So hopefully I got this one right, but this is what I mean by parametrization. In other, in more dimensions, you would have more variables. And here I also allow not only polynomials, but ratios of polynomials. So this is what I mean by an algebraic parametrization. So this is, sorry? So you mean what you usually call rational? Right, yes. Right, so yes, I mean rational parametrization set. This is what I mean, the quotients of polynomials. Okay, so now, so we are going to come back to this problem. So you can think of this problem, this is a very, I think of this as a very algebraic problem. You want to determine whether you can find a parametrization. And the point that I want to make now is that now we are going to move to geometry. And we are going to see that a simple algebraic question like this has a counter part, a counter part, a geometric counter part that is very interesting. So now let's move to geometry. So let me just start by talking about curvature. So maybe I think you all know what a curvature is, but let me just remind you very quickly. So a curvature, by a curvature, I mean it's a measure of how much a curve, so this is a curvature for curves, deviates from being a straight line. So how do you associate a number to a point on a curve that makes such measure? So what we do is, so suppose you want to measure the curvature of this curve at the point P. So I take the circle that best approximates the curve at the point P, it will have a certain radius, and the curvature is one over this radius. So you see, if you have a straight line, you can think that it is approximated by a circle of radius infinity, and this gives you the curvature is zero at that point. And so at different points, I can have a different curvature. So this is how we define curvature. Now, let me just mention that the curvature is not something intrinsic to the curve. It depends on how this curve is embedded in the space. So for instance, let me just give you an example. I have this, I can make this into a straight line or I can make it curve if I change the way that it's immersed. And so this is going to be very important. So this, the curvature for curves depends on the embed. So now let's move to higher dimension, and now I want to explain what it is the curvature, one way of measuring the curvature of a surface. So let's start with a surface, and I want to measure, I want to associate to it some number that, sorry, that measures how this surface is curved at this point. So this is, I will do the following construction. I have, at this point, I have a tangent plane and I have a normal vector. Now I take any plane that contains the norm, passing through the point containing the normal vector. So that is going to cut my surface along a curve. And then I can compute the curvature of this curve. And now I can vary my plane and compute the curvature for all the curves that I obtained in this way. So if I do this, there is going to be a maximum value of curvature and a minimum value of curvature. Here it is important, one point is important, is that when I do this, I have to assign also a sign to the curvature. So we can say that if something curves upwards, then I will tell, I will say that it has a positive curvature while if it curves to the other direction, then it will have a not negative curvature. So for instance, for this surface here, I have here the two planes that cut the curve, cut the surface in the curves with minimal and maximal curvature. And one of them is going to be positive and the other one negative. So if I call this two extremal curvatures K1 and K2, I can define the, let's say, the Gaussian curvature to be the product of this, to the minimum and the maximum. So this point that you see here, what we call a saddle point, is a point of negative, negative curvature. While if I had a sphere, this would be a point of zero curvature. And one important, sorry, positive curvature, and one, positive curvature. And one, if I have a piece of paper like this, this has zero curvature. Now one important feature of the Gaussian curvature is that it is intrinsic. So it does not depend on the way I embed my surface in the space. So for instance, if I curve this sheet of paper a little bit, you see that most directions it will curve downwards, but I still always have one direction here where it's flat. So in the end, my, in this case, the Gaussian curvature is always zero no matter how I move this, I move this piece of paper. So I will never be able to make it shape like a saddle unless I destroy the, or fold the paper. So this is the definition of curvature that we will need for a surface. And from here, we have this great trichotomy. So these are the three different fundamental different behaviors of curvatures in surface. So we have negative curvature. So this is a saddle point, a zero curvature and positive curvature. So this trichotomy is going to carry on when we start to talk about classifications of algebraic varieties. Okay, so now let's move a little bit to algebraic. So this is, so this, these are concepts from differential geometry. And now I want to apply them to algebraic geometry. So to do so, let me talk a little bit about Riemann surface. So if I have a, I start the simplest Riemann surface I want to talk about compact Riemann surface. So if I start with a plane, but it's not just an, I don't view it as an R2, I view it as a complex plane. So I have the complex plane. So the Riemann constructed what is called the Riemann sphere which is a complex, it's a complex variety that is obtained by attaching a point at infinity. So one important thing here is that, so if we change coordinates at this point of infinity and write w for one over z, so we see that once we add the point of infinity we can make this change of coordinates and see that it behaves as any other point of the complex plane. So we can, we can, we can multiply and we can divide points in the, in the plane. And then we can, we make this change of coordinates to look at the point at infinity as a point in the complex plane. So this is a Riemann's, this is a Riemann surface, the Riemann sphere, and of course this has positive curvature as we can see. And now let's look at more interesting Riemann surfaces. So now what I will do is, I will do the following. I will take, I will take a polynomial f of an x and y and here I just have to be a little bit careful so that the associated curve does not have a singularity at any point. So that means that the two partial derivatives of f cannot vanish simultaneously. So if I choose such f, but now, differently than I did in the beginning, I'm not interested in real solutions but I will be interested in complex solutions. So now x and y can be complex numbers. So if we do this, we actually see that before we looked at this as a curve but now we can view it as a surface because each variable can take any complex value. And now, okay I start with this and now just like the previous slide I had the complex plane and I could attach a point at infinity and make it compact. So I can do the same thing here. So let me just explain quickly the construction of the complex projective plane. I will draw a picture for a real projective plane just so that we can see. So the idea is that we want to, we want this to be our plane, c2, c cross c. Union, I want to add one point for each tangent direction. So one point tangent directions of the plane. So if you, you can think of this in order to visualize, let's think of this as R2. So we have the plane and we want to attach one point for each direction. So the way that I can do is the following. I can look at, so this is actually defined. I can look at this, all the space of all the lines. So if I take all, so if I look now at c3, so we coordinates x, y and z. I can look at all the lines that pass through the origin. If I look at the space of all the lines that pass through the origin, notice that if I take a general line, it is going to cut, so this is, let's say that this is the plane z equals to one. So if I take a general line through the origin, it is going to cut this plane at exactly one point. And the lines that do not cut this plane are precisely the lines that are contained in the x, y plane. And there is one of those for each direction. So the way that we can define this then is to define this as a c3 minus the origin when I, so I remove the origin and then I just quotient by the equivalence relation that identifies two points in the line through the origin. So it identifies a vector v with lambda v for any lambda different than zero. So this is the projective, this is what we call the complex projective plane. So our, and it contains as an open subset here, our plane c2, viewed for instance, are the lines that cut this plane. So now if I do this, my s lives inside c2 and I can take the closure. So this is a compact, complex, a compact variety, compact complex variety, and I can take the closure of this s inside here. So usually what I will do is I will write the equation, what we call the affine equation, but I think of it, I will always be thinking of this as the compactification obtained by taking the closure in the natural projective space. So if I do this, I will then my, what is before it was a curve, we saw it as a curve, it is actually a surface and it will be compact, meaning that you can think of this as being bounded, but without any boundary. So here's for instance an example and all Riemann surfaces, so all the surfaces that we can construct like this, they will have this shape, but I may have another different number of holes. So this is what we call in topology the genus of the surface. So this again, this is an intrinsic invariant of the surface, you may try to deform, if you look at this surface that I have there, you can try to deform it and you see that no matter how much you deform it, you will not get rid of these two holes and you're not gonna make another role. So this is what we call topological invariant. But now let's go back to the question of curvature. So suppose that I want to know what is the average of the curvature of this Riemann surface. And there is the famous Gauss-Bonnet formula that tells us that the average of the curvature is well, two pi times two minus two g. So it depends on this topological invariant. So you can change the Riemann surface, try to deform it, but you're gonna make it more curved at some points and less curved at others, but in the end you are not going to be able to change the average of the curvature. It only depends on this topological invariant. The g zero, right, correct. So for the Riemann sphere, the g zero, and see if you put g equals to zero, you're gonna get four pi, which is the average of the curvature in the sphere. And so see, for genus, so we have genus zero, the curvature is going to be positive. Genus one, the average curvature is going to be zero. And positive genus will have average negative curvature. So this is the trichotomy that we have in mind. But now let's remember that this surface was defined by an algebraic equation, a polynomial f in two variables. And the beautiful thing is that both the genus, which is this topological invariant, and the curvature, which can be computed from the genus, they are actually only depend on this, on very simple algebraic properties of the polynomial. So what is the simplest algebraic invariant of a polynomial one can think of? So this could be, so this could be the degree of the polynomial, right? Even if you change variables, the degree is not going to, if you make a linear change of coordinates, the degree is not going to change. And so as it turns out, for Riemann surface, which from now on I will call a complex curve, because it is really, the complex dimension is zero. Sorry, it's one. So it has real dimension two, but complex dimension zero. And so if, oh, sorry. It has real dimension two and complex dimension one. So I call it a complex curve. And we want to compute the average of the curvature, depends on the genus, but as it turns out, the genus only depends on the degree of the polynomial. So it can be computed with, so this is what we call the degree genus formula. So this is one of the first things we learned in the first course of algebraic geometry. We obtained this. So we see that, so what we see is that, that trichotomy for complex curves, say plain complex curves defined by a polynomial like this. So now I put the degree D here. So D here is the degree of the polynomial F. So the curvature of the Riemann surface that we obtained this way, we'll have positive curvature if the degree is one or two, zero curvature if the degree is three, and negative curvature if the degree is greater equal than four. So we see that in a sense the simplest curves, so the lines and conics, they have positive curvature. So positive curvature, maybe what I want to think of in the algebraic geometry, varieties having positive curvature, they tend to be very simple. And now let me go back to a question that I posed before. So it seems that this has nothing to do with the question whether which curves admit algebraic parametrization. So this seems not to have anything to do with curvature, but in terms it does. So the answer is the curves that admit algebraic parametrizations are precisely the ones having positive curvature or genus zero. So from one side you have the geometry that is measuring the curvature, from the other side you have this question of parametrization and we get, but they are actually, this is a niff and only if. Now, maybe if you will remember, I think I gave you the parametrization of a cubic curve, this one right in the beginning, I showed you that this curve has a parametrization, it has degree three. But this is what I said, I have to pick up my polynomial carefully so that it does not yield singularities. So when you have a singularity, then there is an adjustment to make. So this is still a curve of genus zero if you define genus appropriately for a singular curve. So I don't want to get into that, but what I want to say is that, so here in this discussion I'm considering a smooth variety, so I'm ignoring those that contain singularities. Of course these are very important and they are very much studied, but they will bring new ingredients to the discussion. Okay, so now what I want to do is to move to higher dimensions. So this, so in complex algebraic geometry, one of the central teams and the most important directions of research is towards a classification of algebraic variety. So what do we mean by a classification? So we have our algebraic variety, so this is a zoo of algebraic varieties, and we would like to classify them. So we would like to divide them into certain boxes according to their shapes. And what we want to do, so for curves, this is a accelerated trichotomy that I told you before, that depends only on the genus, and we have these three classes of varieties depending whether they have positive, zero, or negative curvature. We want to do a similar thing in higher dimensions. Now of course this is much more complicated, so I'm, but this is possible to do, and this has been some of the most important developments in the last decades in algebraic geometry. So let me just briefly tell you how it goes. So we start with a complex projective variety. So again, I always consider something that is closed in the projectives, in the projective space. And I will define the canonical class of X. So the canonical class of X is the first-turned class of the tangent bundle of X. So for those that know, so for those that know what I'm talking about, let me just tell you that our canonical class lives in the second cohomology group of X.  So an element here in the second cohomology group, what does it does? It associates to a compact surface inside my variety, an integer. So here I take, now when I say a real surface, so a real surface, as I said, this is a complex curve. So it associates to a complex curve, always a number, an integer. And this number, if I take minus K, this is exactly the multiple of the average of the richer curvature of X along C. So what I wanted to know is that we have this gadget here and this is intrinsic, this is canonical, that's why it's called canonical class. So it's the first-turned class or minus the first-turned class of the tangent bundle. And in a sense, it measures the curvature of X. Now, and so what we would like to do is to divide of us, classify our algebraic varieties into three classes, according to positive, zero, or negative curvature. Well, this is unfortunately not well, maybe fortunately, otherwise things would be too easy, but this is not possible because in general, so this is, you can think of this as a function that associates to every curve a number. And in general, the sign of this function is not well-defined. You could have some curves for which this number is positive and other curves for which it is negative and other curves for which it is zero. So in general, this is not well-defined. When the sign of the canonical class is well-defined, so these are very special varieties, then we have these three cases that for simplicity, I will call them, so those for which the canonical class is positive are called of general type, and they're called of general type because most varieties with well-definite sign of the canonical class will lie in this class here. We have those K equals zero, which I'm calling calabiol. So these are the, for instance, the calabiol varieties that appear in connections with physics. And the ones with negative K, which are positive curvature, are called funnel varieties. So these are very special varieties, exactly because the sign is well-defined, and the tools to investigate, to use to understand these varieties are very different. They have very different behaviors and very different, many different properties. So let me just illustrate for the simplest case. Now again, suppose that my variety is given by a single algebraic equation. So I'm looking at complex vectors that satisfy a certain polynomial of degree D. And again, I choose this polynomial carefully so that I don't get any singular points. And if I do this, I can easily compute what is the canonical class in those cases, and I get exactly this trichotomy. So canonical class negative are those with small degree. So you can think of these as the simplest, simplest algebraic varieties, for instance, planes and connet, quadrics, and low-degree hypersurfaces all lie in this case. Here, again, I still have something finite. And then here it becomes wild. So as soon as the degree is large enough, everything that I, every hypersurface will have positive canonical class. So I have been studying funnel varieties for many years, so I'm not going to tell you, I'm not going to get into details, but let me just tell you that there is, we can define what is called the index of a funnel variety, which measures how positive the anti-canonical class is. So see, this is something that can be made precise. For instance, there are many ways to define this index, but you can see, for instance, what is the biggest integer dividing the minus k inside this cohomology group. So the biggest, this integer, the more curved variety. So this is measuring positivity of the anti-canonical. And so for instance, funnel varieties with high index are extremely special and classified, so high-rational homogeneous spaces all appear in this class. And I think one of the biggest achievements in complex algebraic geometry in the last decades is what is called the minimal model program. So what does the minimal model program say? It says that these very special varieties, the ones with a definite sign of the canonical class, they are the building blocks to all algebraic varieties. So this, I assume that I'm saying this in a very vague way, but so the jargon in the area is that all complex projective varieties can be built up from these three classes. So how they can be built up from this, then maybe this is a bit mysterious and it would take more time to explain. But I heard that Mori was here last year, right? So Mori was the one who set up this program and who completed in dimension three and he probably spent more time explaining the details here. But here, let's just assume that this is what we get. Okay, so now what I want to do now is to, okay, maybe go back to a question that we had before. Oh, okay, so maybe a little bit about the minimal model program, some history. So in dimension two, it was so for, so first in dimension one, Riemann surfaces, this was just done by Riemann, so this is 19th century. And then dimension two, this was done by this Italian school in the early 20th century. And from dimension two to dimension three, some very important new ideas had to come up. And so this was proved by Mori in 1988, you see here a huge cap and that won him the Fields Medal in 1990. So what Mori realized is that, well, among other things is that if you look at dimension three and higher, there is no way that you can avoid singularities. So for curves and surface, you can do the classification scheme without dividing into these three classes and making the constructions to get there by avoiding singularities, starting in dimension three, this is impossible. And this has led to an important theory which is a theory of singularities, of pairs of the minimum model program, which is very important, also in algebraic geometry. And dimension four, this was proved more recently by Birkart, Cacini, Hayk and McKernan. And again, this was a major result. There are still some things missing in this theory here, but for purpose of the classification, their result is enough. And let me just go back to the question that I posed before about which algebraic varieties admit algebraic or maybe rational parametrization. So these are called rational varieties exactly because they admit this rational parametrization or uni-rational depends on your definition of parametrization. And so for instance, assume that we are in this in one, we consider varieties in these three classes. So which of those can admit a parametrization? And the answer is that again, the only ones that may admit a parametrization are the ones having positive curvature, the funnel varieties. Now here, this is not a niff and only if, and this is one of the most active also problems, also in algebraic geometry currently is to decide which varieties are rational. So there has a lot of great deal of results in this direction and connections to recent techniques from Hodge theory have been applied into this question. And so this is very active to decide, given a funnel variety, whether it is rational or not. And one important, maybe future directions that I heard recently is that not only, so is the question whether is there, can there be an algorithm that given a variety, there is the size whether it is rational or not. So this for instance is a problem. This is a question that is not known. So which is again worth investigating. Okay, so this is what I wanted to tell you about the classification of algebraic varieties. And what maybe what I want you to bring home from here is that algebraic geometry study objects defined algebraically but which has a geometric manifestation and geometry and algebra they interplay in this game. And that one of the important problems in algebraic geometry, the classification problem is to distinguish these three classes of varieties and understand how every variety can be built up from these three. And now in the last minutes maybe I want to discuss different problem which have been studied a lot like since that 10 years ago or so, which is the problem of foliation. So this, so let me start with something very simple which is algebraic differential equations. So these are differential equations again that only involve multiplication, quotients and polynomials and ratios of polynomials. So for instance these are two examples of algebraic differential equations. So these are very, these algebraic equations are very simple to solve and these are the solutions. So we see, so this is, so this is a, I'm sure that all of you have seen this but what I want to point out here is that we have two algebraic equations but the solutions may not be algebraic. So this is clear from the picture. So a fundamental problem, a fundamental question is if you see an algebraic differential equation, so when can you guarantee that the solutions are algebraic? And again I'm going to answer this question but so this seems to be sort of an algebraic question but we're gonna make, put some geometry into this picture. So before I do this let me just reinterpret this. So how can we reinterpret a differential equation? So this differential equation you can see it as it is given at every point of the plane, this is given a tangent direction to the plane. So this is, this ratio here gives a tangent direction at every plane, at every point and the solutions are just curves that are everywhere tangent to this tangent directions. So this is what we call a, a foliation. So let me tell you what in general, so the foliation is something that generalizes this algebraic equations. So a foliation, so I have here say an algebraic variety and a foliation is a partition of the variety into what I call leaves. So these are sub varieties here. As we saw in the example, we start with an algebraic variety but when I take these leaves they may not be algebraic. They may be transcendental but the algebraicity condition that I put. So if you go back to this, what this is, this equation is telling you is that this tangent direction to the curves, to the solutions, they vary in an algebraic way. So this is given just by this polynomial. So I do the same thing here. So I consider these foliations but only those for which the, so here I call the tangent bundle of the foliation. So this is at each point, I consider the tangent space to the leaf at that point. So this gives, and this is a subspace of the tangent space at the point. So what I get is a sub bundle of X. And so here what I mean here is this is an algebraic sub bundle, meaning that the tangent directions, they vary algebraically, even though the leaves may be transcendental. And the question is, when are the leaves algebraic? So this is the analogous to the previous questions from this point of view. And in order to answer this question, given a foliation, when are the leaves algebraic, we are going to again introduce the canonical class and measure the curvature of this foliation. So this is something that we, together with my collaborators, the Fandrouelle from Grenoble, we started to look at this some 10 years ago. And so we've been studying this and I'm just going to give you just a hint of what kind of result we can prove. So again, you start with a foliation. So the tangent shift of the foliation is going to be some sub bundle of X. While here, just to be precise, there are some points here where this foliation fails to be locally a product. And these are called the singularities of the foliation, but that's okay. I do admit some singularities. And again, I can define the anti-canonical class of the foliation as the first churn class of the tangent bundle of the foliation. So this is, again, this is sort of a measure of the curvature along the leaves of the foliation. And so this is the canonical class of X. And then again, we can study foliations for which, so in general, the sign again is not well-defined. There will be curves that will intersect it positively and curves that will intersect it negatively, but you can look at those very special foliations for which the sign is well-defined. And for those foliations, so this, and so this is, we started 10 years ago to, we defined the fun of foliation to be foliations for which the anti-canonical class is, so the canonical class is negative. And we started to study what are the special properties of these foliations. And it turns out that they have very, they have some very special properties and I'm going to just tell you one of them. But before, let me tell you another current trend and an important future direction about foliation is, so this is something that I'm also getting involved with, which is to find a minimal model program for foliations. So is it true that as in the case of varieties, all foliations on algebraic varieties can be built up from these three classes? So what we have so far is for foliations on surfaces, this is okay, this was the work of Brunella in the 90s, early 2000s. And for dimension three, this, we hope this is true. There has been a preprint in the archive in the end of last year that claims that this is, that the minimal model program for foliations on three folds is T-O-K. This is certainly a very interesting result. But in dimension greater than four, this is completely open. So this seems to be, so the techniques used for varieties don't all apply in this case and some new ideas and new techniques have to be developed. And okay, so now let me, let us go back to the true, the differential algebraic differential equations that we started with. And we have these ones with algebraic leaves or algebraic solutions. And this one had transcendental solutions. When you compute, so this is a set, this is a foliation and we can compute the canonical class. So if you compute the canonical class of this one, this is negative. So this is what we call funnel foliation. And this one has zero canonical class. And so one may wonder whether this positivity of the anti-canonical or positivity of curvature has anything to do with algebraicity. And this is what we have been studying for some years. And let me just give you a hint of the type of result that we can prove. And the answer is yes. In this case, this foliation here has algebraic solutions. This differential equation has algebraic solutions because of this positivity of the curvature. And so what we prove is the following. So this is just one example. So we start with the funnel foliation and a funnel foliation has negative canonical class or positive anti-canonical class. And again, we can define several invariants that measure how positive this foliation is. So I don't want to get into the details, but for instance, if you think of this, again the anti-canonical is something that lies in the second cohomology group. And you can ask what is the biggest integer that divides this class there. So this is a measure, this is an index that measures the positivity is a very rough one. So we have more refined ways of measuring positivity. But for sake of this talk, we can think of this as a measure of positivity. And what we prove is that the algebraic rank of the polliation, so that means this is again a measure of how algebraic the leaves are. The algebraic rank is bounded from below by the index of this foliation. So, foliations which have very positive curvature tend to be very algebraic leaps. So this is roughly, this is some of the results that we obtained, but we investigate this in more details and make some classifications of foliations with high, funnel foliations with high index. But I hope that for the sake of this talk, this is all that I want to do. So thank you for your attention. Thank you. Comments or anything? Usually on P2, we have this result that generically, foliations don't have algebraic leaves. But if we have more than a certain number, then suddenly all leaves become algebraic on P2. There exists like a similar result on any kind of surface or foliation. So what is, if you have too many algebraic curves, then they have infinitely many. So, maybe I should make it clear that if a foliation is not funneled, it still may have algebraic leaves. But if you look in the space of foliations, these are going to be very small dimensional spaces inside the space of foliation. So, I don't know exactly what kind of a similar result you were thinking of, but. We have more than a. Ah, so you mean in general, for other surfaces. So I don't know, but I believe that there are some results for surface. And I would hope that in higher dimensions, one can, maybe the invariant that one has to measure is a little bit different, but I would expect that it's possible to have the same results in higher dimensions also. I need that. Can you give us any feeling for what you mean by saying that all dryness can be built after these three times? Yes, yes, okay. Very roughly, it's like first with these eight geometries. Yes, I will, I can tell you something. So, okay, let me give you a crush course on the minimal model program. Okay, so, if you start with a smooth protective variety X and you would like to ask whether the canonical class is positive, okay, let me just put it this way and then I will explain better. So, one thing that you ask is, is the canonical class NIF, NIF means greater, it's greater equal than zero. If the answer is yes, you stop for the moment. So, you make this question and if the answer is yes, then you stop. And there is a way to produce a map that is going to separate the zero part from the positive part. So, here you can define a map from X to Y such that this has, again, greater equal, K greater equal than zero and the fibers have K equals zero. So, you separate and here, well, okay, maybe you do this, you do it many times and then in the end you get a vibration where the target has positive K and the fibers have K equals zero. Okay, so, in this case you are happy, you obtain your variety. So, if the answer is no, then that means that there is a curve for which K dot C is negative. And now, what you want to do, you want to find a minimal curve in the sense that can be made precise but there is a minimal or many minimal curves with the property, this property and then what you try to do is you try then to contract it. So, if no, then you just try to contract. And when I mean contract C, I mean I contract all the curves that are in the same cohomology class or proportional to the, have a cohomological class proportional to C. You contract all of them. Now, if then what you would like to do is to, so you contract that means that you have a morphism from X to Y where you contract this curve where C is negative. And what you would like to do is to come back here and keep running the program. However, this contraction always exists. So, this was proved by Moriv since very early that this contraction exists. However, this variety here in general is very singular. So, you cannot go back because I mean some of this, some of these tools, so in principle we started with a smooth variety. And now you get to a singular variety, you can, some of the tools that we applied here don't apply anymore. And now there are a few cases. So, some of the singularities that appear are, we can handle and we can make all this machinery work for singular varieties. So, here there are two cases. So, singularities of Y, let's say Tame, and singularities of Y, Wild. So, one of the important developments in the area is how to actually quantify and distinguish these two. And what is the smallest class of singularities that you have to admit in order to have this? And this has been achieved, so for this Tame once again, you can go back and keep running the program and all the theorems and the existence results that we obtained in the middle, they still hold for Tame once. Now, when you have wild singularities, there is no hope to be able to run this program here. And then here what we do is what is called a flip, a flip instead of contracting a bad curve. Well, I'm calling the bad curve, but it's not a bad curve. It's like this negative curve. You contract this curve, instead of contracting this curve, you make a surgery. You make a surgery, we remove it, and then you introduce another curve in a different direction. And after this surgery, which is called a flip, so instead of doing the contraction, you do the flip. And this flip here, so you change something, a curve, which was negative, by another curve that is positive. So here, you remove this C that is negative, and you replace it by another curve on X prime that has positive intersection with this canonical. And again, this variety here, the contraction is bad, but this variety here has tame singularities, and we can make this, and we can just go back and run. So one important, so two very difficult results. So this part here is the singularities are tame, then you're okay, this is very easy to handle. The two difficult theorems are the existence of the flip, the existence of this algebraic surgery, and the termination of flips, to say that this program terminates. As what I said that in dimension three, this is all known, in dimension greater than four, termination is not known in general. But what they prove is that you can run a very special, instead of choosing here an arbitrary extremal curve, minimal curve, if you choose this curve very carefully, then this is called a minimal model program with scaling. If you choose it very carefully, you can prove termination of the flips. There are other questions? I had, oh, yes. It seems that there is a notion of irrationality between foliation, can you explain if there is? Yeah, so the notion, so you have a notion of a birationality of varieties. So two foliations are on different varieties are birational, if you have a birational map, that on the open set where you have an isomorphism, take one where there is a big open subset where it takes the foliation from one to the other. And then again, you want to do the same game here, but now you want to contract always curves that are tangent to the foliations. But in principle, this scheme for foliations we hope should be the same. But now you are just contracting curves that we want to contract curves that are tangent to the foliation. Yeah, so what this is showing is that every variety is going to be birational, too, either. And here, of course, maybe I did not say how there is a third possibility here, sorry, that when you contract this curve, yeah, so this is an important point. So it could happen that when you contract this curve, you obtain a vibration. So this would mean that the canonical is negative on the fibers. And then again, you have a, the canonical is going to be negative on the fibers and then you run the MMP for this one, and then so on. But yeah, so every, maybe one way to say is that every variety is going to be birational to, let's say, it's birational to a vibration where you have canonical positive, canonical of the fibers, zero, canonical of the fibers, negative. So you separate these three parts. Well, I have a couple of curiosities, but they can't wait for later. So I think it's also a little bit hot. So I would like to thank, to thank Carolina again. And actually I would like to invite everybody for a small refreshment on the terrace, just outside here, this one. Okay, so I hope this was, yeah, this was. Wait, you're always playing the key one. I hope you get the current trends. And then you know, if you're able to jump to the end of the slide with this one, I don't have you there. Oh, thank you. It's an amazing lecture. Okay, thank you very much.


Thank you so much for coming today. The title of my talk is misdirection thoughts on the future of programmable money is just the Generic title I give for every talk I give and then I change it just before I get up on stage because this is all improv folks I'm going to be talking about Universal basic finance. That's the topic of my talk today Where at the moment at a crossroads where we're having a great debate About the future of cryptocurrency about the future of finance and a lot of people are expressing some very strong opinions Myself included the backdrop to all of this is All of the people who aren't in this room who will never be in a room like this in In 2013 I visited for the first time the beautiful country of Argentina and I did a talk in Buenos Aires and It completely changed the direction and trajectory of my involvement in Bitcoin and open blockchains For the first time I didn't need to explain why Why Bitcoin why open blockchains? Why freedom of finance? Everybody there was interested in one and only one question. How? How do I do it right now? Because the why is obvious It's obvious to people from Argentina Someone came to me at the conference and said you've inspired me today. I was terrified to come to this conference I was afraid of what might happen if our government changes again My grandparents were kidnapped By the previous regime. We didn't see them for months. We thought they'd thrown them out of an airplane I Think about that for a second We talk about financial inclusion. We talk about the banked and the unbanked and we have no clue In our position of privilege what this means It shook me to the core and I've had conversations like that every year since I've started working in this space People who come to me and say I hear you Because that's my story too What does it mean to be unbanked? The World Bank defines unbanked as the two and a half billion people Who have absolutely no access to financial services and live in cash-based societies? Conveniently they only count heads of household Meaning the spouses and children are unimportant in this calculation Only the primary earners Matter and yet we know that in the vast majority of the world finances controlled by women They're not counted as the head of household To be unbanked is not to simply have only access to a cash-based society to be unbanked is to lack connectivity to the world it's to lack the ability to participate in Trade and commerce to be able to get a job that connects you with other people who want your services around the world To build a future for your children It is to be condemned to poverty and when we look at that. What do we think? They don't have money. That's why they're unbanked wrong Absolutely wrong They don't have access They don't have documentation They don't have the necessary literacy to fill in an application form and Sometimes they don't have the clothes or the shoes or the appearance To even be able to enter a bank without getting kicked out by the security guard That's what it means to be unbanked And what does that do it creates enormous poverty around the world just so we can persist in This petty bourgeois idea that as long as we have every participant in every transaction Prove their identity and we can track and control Every transaction through this totalitarian surveillance system. We will end crime and In order to believe in that false idea of safety through totalitarian control We condemn billions of people to poverty Billions of people not two and a half billion many many more We talk about Financial inclusion from the perspective of the rich and privileged world. We live in the the world. I live in as An American citizen. I have access not only to open a bank account But also to trade in multiple currencies without restrictions Access to unlimited investment opportunities all around the world access to capital liquidity and credit The stability of a currency that doesn't destroy itself overnight taking my entire savings with it and Hopefully most of the time access to institutions that are not actively stealing my money Or Governments that are not actively destroying the currency in order to pay off their debt through shadow inflation hyperinflation and then finally financial collapse How many people have that Everyone in this room probably But if you count all of the people around the world who have access to that kind of financial service It's maybe a billion and a half And that's why after 2013 I started saying strongly and Loudly This is about the other six billion That's what financial inclusion means Our regulatory system is actively excluding people from access to finance We have finally reached a point Where access to the most basic financial services has become a privilege a Privilege that the average person has to exercise this dance of Proving themselves worthy in front of a banker a Filling in reams of paperwork and application forms in order to be granted the privilege of financial services and We even turn around and condemn cash the ultimate peer-to-peer Anonymous fungible mechanism that for millennia has provided financial services to everyone basic financial services and Cash has one fate of law and it's not that it's anonymous that it's the greatest feature and it's not that it's used by Criminals criminals get a banking license and defraud people by the millions its greatest feature is That it is available to everyone without vetting its greatest feature is that it is an open transparent neutral verifiable peer-to-peer transactional system and It's greatest weakness is that it is constrained by geography and locality. It doesn't have range and scale and Now we have a new form of digital cash Which is also open and it's also neutral and verifiable Unforgible transportable but this one is borderless It's censorship resistant It can be used even when your government doesn't want you to use it It can be used without privilege without identity and It can be used everywhere in the world by whoever wants to use it by simply downloading the software and Running it on whatever computing device they can afford That is the real revolution here and while we're having a little Privilege discussions about should we regulate crypto? How much should we regulate crypto? Who should regulate crypto?  They will build a future for their children We are paralyzed by the fear of a few bad actors Blinded by the fact that the worst actors are the ones who act with state privilege and endorsements With the intelligence agencies hand-in-hand Inside the surveillance capitalism mechanism to fund dictators and drug lords all around the world With our tax money to the tune of trillions The real terrorism the real drug financing doesn't happen in cash it happens in millions of barrels of oil and billions of pallets of dollars Transmitted through wire transfers by the banks who get called again and again and again and again and they pay a fine That is a fraction Not only of the criminal behavior that they profited from But of the tens of thousands of deaths they've directly contributed to and not a single person goes to jail and at the same time Some people have the audacity to say that we need to end cash in order to stop crime hum Magnificently self-righteous in the United States 18% of the population does not have access to banking services. That's 60 million people. I recited this fact at a banking conference and one lady in the back raised her hand and said Why should we give? Illegals bank accounts That's a chilling statement to make let me translate it to you in words that will make more of an impact those people Don't deserve the privilege of financial inclusion those people When your neighbor says those people don't belong in our neighborhood It freaks you out because you suddenly realize you're living next to a bigot But when a banking regulator says why should we give? Illegals a bank account. I calmly responded You shouldn't We will at the last conference. I attended After I gave a speech to crying the lack of security and financial inclusion a young man came to me and said This really spoke to my personal story Young man in his 20s has been living in Western and developed countries for the past 15 years Has not been able to open a bank account in 15 years Every time I write my name and place of birth in the application. The process has already ended. I Was born in Iran. I didn't choose that. I didn't do anything. I Pay my taxes I have a job. All I want is to deposit my paycheck In a bank account so I can buy groceries For 15 years. I've been unable to do that That is the face of the unbanked There are people walking the streets around you in this city of privilege in the independent district of the city of London bought and paid for by banking corporations in this little enclave That is the Vatican of capitalism a city within a city People walking around you invisible people your janitors your service professionals the people making you a sandwich who don't have a bank account who take their paycheck in cash and they go to Various places in order to ask for payday loans Maybe they take a check or all the form of payment they can deposit and they get charged 10 or 15 percent Or 30 percent to send money home to their loved ones so they can support a basic life of subsistence That's the unbanked they're right here in this city. Don't imagine the great masses of Africa, of course, they're unbanked or Southeast Asia or South America right here your neighbors and They have been made to live in that position Just so we can continue this illusion that safety comes from totalitarian surveillance This very conference is being sponsored by a company that is spreading around millions of dollars in marketing That does surveillance that is actually promoting and advancing the state of the art You all have their name on your badge look down and check it out a Company that builds state-of-the-art surveillance and then shares and sells that data to companies That further progress that data to intelligence agencies and this lovely form of trickle-down surveillance eventually reaches some regime That likes to cut up journalists with hacksaws Don't fool yourself There is no ability to separate your position within a company like that from the moral implications of your actions and I say to you if you're hiring people and you see names like that on the resume Just like if I was hiring people and I saw journal dynamics Ray Theon Lockheed Martin Or any of the companies in our space that are promoting and actively pursuing surveillance technology to sell it to the highest bidder They have declared moral bankruptcy If you declare moral bankruptcy you should have seven years of Trying to find a job and being interrogated about your ethics Just like one of the poor people whose life has been stuck in a box of poverty We'll have to spend seven years trying to prove to a banker that they have the level of privilege to get banking We have to make some deep moral choices at the moment We stand at a crossroads because right now the world governments are trying to abolish cash They're trying to abolish the very last lifeline that remains for the billions of people who have a life of poverty Not because they don't work. They work harder than all of us not because they don't deserve But simply because of where they were born or what documentation they have or what level of literacy They have or even how they look In Saudi Arabia women are not allowed to own property and bank accounts And this is the case in half a dozen countries around the world and we look at that and say Shame on them. That's immoral. That's disgusting And then we turn around and we do the same thing to immigrants in this country and my country I'm half British half American. I have to apologize to a hundred and ninety four countries around the world the moment I arrive and I know In my heart that every time I pay taxes. I'm killing thousands of people by proxy. I still pay Because I'm also helping people who are in welfare People who need help most importantly when we make these moral choices. We have to understand Surveillance never stopped crime Surveillance is the license given To the people who are on the top of that To control our lives They will commit crimes. They will commit the worst of crimes What I call mega crimes and I know Britain doesn't use the metric system So mega is the prefix we use for millions a mega crime is one where for example You foreclose on a million homeowners and don't go to jail. That's a mega crime We're doing surveillance and analytics To catch a petty drug dealer who's selling pot for Bitcoin Who's doing surveillance and analytics on Lockheed Martin? Who's doing surveillance and analytics on the money laundering banks? Nobody. Do you know why? None of them ever go to jail. The regulators are completely captured and the very system of controlling finance from above By having levers of power over the lives of millions of people billions of people of having the audacity to cut off entire Countries and say well, they're under sanctions They're not privileged enough.  What's happened in Russia? What's happened in Venezuela? What happens every day to billions of people around the world? Let me end on a positive note because you're probably a bit freaked out by all of this You should be this is serious stuff Open borderless public transparent neutral censorship resistant Strictly private Cryptocurrencies exist They will not be regulated They cannot be regulated Not by committees because they're regulated by mathematics. They're regulated by algorithms They provide certainty of transaction. They provide programmable customer protection. They provide reputation management They provide access without identity They give billions of people eventually not just a bank Account in their pocket, but a bank in their pocket They democratize the function of banking and turn it into an app that everyone can access without vetting Because they've already Prevetted They have agreed to download the software that follows the rules of consensus and that is the only vetting required in these systems but we shouldn't Allow that We did But we can't have people make anonymous transactions They will But we must Regulate this you can't and you won't Because six billion people need this and you have neither the moral authority Nor more importantly the practical capability to stand in their way or Even to stand in the way of what is going to be the greatest revolution in financial services in three centuries Universal access to basic finance. Thank you Thank you


10. My fellow solutions. It is with humility and gratitude that I address you on this occasion of the 45th anniversary of our country's independence. My government is committed to assisting young solutions and all those who have ideas for businesses to become successful entrepreneurs and to be the new standard bearers of our future economic development sees the opportunities that exist and let us unleash our creativity for the benefit of our country. Happy independence and Lucia and may God bless you. I thank you. 9. Have you independent ILO? 8. The Saint Lucia social development fund via the basic needs trust fund has again helped transform the learning landscape of staff and students this time of the Laquamengo preschool with the retrofitting of a pre-K classroom. Provide them with facilities and spaces that would stimulate their growth across dimensions. Having a properly equipped classroom with all the tools and the conditions that are conducive for learning will certainly set off our students, our children on the right footing. 7. The Pan-American Health Organization donated defibrillators to the Ministry of Health, Wellness and Early Affairs. The health organization continues its commitment to the improvement of the health sector in Saint Lucia and around the region. The Pan-American Health Organization is pleased to hand over these 10 defibrillators to the Ministry of Health where they can be used to aid in resuscitation of persons who may have suffered cardiac arrest or arrhythmia or irregular heartbeat. 6. Agriculture Minister Nobel Alfred Prosper together with agricultural development officers from the Taiwan ICDF on Tuesday engaged with farmers at Union Okik Garden where they participated in a subsidized sale of agricultural inputs. Today I'm hoping that our farmers can take advantage of this opportunity. I'm certainly sure it will help boost the agricultural sector to reduce their cost of production because of the fact that they are getting their inputs on other subsidized products. 5. The dispersal of a total of 34,000 bags of fertilizer across the island's eight agricultural regions has begun with farmers set to receive four bags of 25 kilograms of fertilizer per acre. This is for me a very good level of support to our farmers under the program and I'm hoping that that will help us boost our food security. 4. The Royal Saint Lucia Police Force wishes to inform the general public that the immigration department's passport office currently located at the Lamar Building on Bridge Street Castries will be relocated to the second floor of the Cox Building located on Cadiz Street Castries. 3. Saint Lucia played host for the first time to Relay for Life, an annual event hosted by the American Cancer Society. So the event really is bringing together of teams, sponsors, especially survivors so that we do what cancer requires which is to educate, create their awareness, get everybody advocating for what needs to be done for cancer. 2. The Saint Lucia Marketing Board recently celebrated a significant milestone with the official opening ceremony of a newly constructed compost house. This compost house will be used for the Lucia Marketing Board to turn wasted fruit and vegetable into fertilizer. 1. Today we are going to witness the official turning on of the light at the Parkes Blade Field and this is a project which embodies progress, unity and opportunity for all our young people. It is my sincere hope that these lights will serve as a catalyst for growth, empowerment and comrades among our youth instilling in them the values of teamwork, perseverance and sportsmanship. With that said, I am free to officially declare the past line on.


You have to take 100% ownership for your life. Nobody owes you anything. Nobody's going to give you anything.  The minute you start playing the victim card and blaming somebody else, you might as well sit down, turn into a ball, and cry yourself to sleep.  Nobody cares. The only person that cares about yourself should be you. The only person that's going to make you a better version will be you. You're the one that makes the choices every day. You need to take ownership for your choices. I could have chosen to. Nobody will ever love me. I'll never have women in my life. I'll never make any money. I'll be ugly, broke, whatever, da, da, da. I can't think of a time in my life when I've been happier than what I am today. I stand before you today at 43. I'll be 44 next month. Divorced, got an eight-year-old daughter.  Multiple streams of income. Women come knocking on my door if I want it. It's not hard. You can work on whatever you think is holding you back. Pain plus reflection equals growth.   Sit back. Take a deep breath. Breathe and reflect. There's a lesson there. Don't lose in life. Learn. Popular video on my channel. Sorry about the resolution. You're not broke and I spent a lot of time putting this one together. You can go look it up on the channel. You don't want to play it, but you aren't damaged or broken. Where you sit right now is where you're supposed to be.  That's your starting point. Okay, you can go up from there. As an adult, you get out of bed unless you've lost your limbs serving your country and you've got to roll into a wheelchair. Everybody in this room gets out of bed the same way. You roll out of it, get up out of it, stand up out of it. You start your day the exact same way.   You don't get to play the victim card. You're in charge of your life. I don't have any sympathy.  You're the one that's gonna get the results that you want if you make the changes in your life to get what you want. You got to do the work. Nothing's gonna come to you. Lesson number three that I've learned in my life is listening to your intuition.


Have you ever wondered why, when we look around us in the natural world, we see very few straight lines, but the straight line is ubiquitous in the systems we engineer. From buildings to circuit boards it appears to be the default position. One way to understand this is that the systems we engineer are based upon our scientific and mathematical understanding of the world, which has inevitably started by describing the simplest and most orderly systems, that is to say those that are composed of linear forms and relations. From Euclid to Newton and on, science has been focused upon the orderly systems of perfect squares, triangles and linear relations of cause and effect, that can be encoded in beautifully compact equations. Thus we describe the real world as a kind of approximation to these perfect linear forms. But let's take a look at some of the basic principles underlining the theory of linear systems. Linear is basically a fancy word for line, and a line is often understood as the shortest or most direct path from one point to another. To say there is a linear relationship between two things is to say that there is a direct relationship of cause and effect between them. Let's take an example of this. Say I'm playing baseball and I hit a ball with a bat. If we make a model of this system, we see that the energy inputted by my swinging of the bat will be directly proportional to the output of the ball's momentum as it travels off in the opposite direction. This is a simplified model, but it illustrates the direct or linear relationship between cause and effect, my swinging of the bat and the ball's motion in response. There are of course many examples of linear systems, particularly within physics, but we can capture the underlining logic of linear systems in general with a model that has only two simple rules called the superposition principles. Each states firstly that the output to the system will always be directly proportional to the input. So if there is a linear relationship between the amount of fuel I put in my car and how far it will go, well then if I put twice as much fuel in the car it will go twice as far. And secondly that if we add two or more systems together, then the output to this combined system will simply be the sum of the two outputs of the original systems. So say we have two tractor factories, each producing a million tractors a year. Well if we merge them then we will get a factory that will produce two million tractors a year. Linear systems are deterministic, meaning that if we know their present state we can fully determine their past and future states. This can be seen by plotting a linear system as a graph where it will always be depicted as a straight line. Although linear systems modelling has proven highly successful in many areas and is often a very good approximation, the reality is that we live in a world with ecosystems economies societies and physical systems that are not governed by the superposition principles and thus are what we call non-linear. An example of non-linearity might be listening to two of your favorite pieces of music at the same time. Because there is a relationship of interference between them, the result of the experience will not be a simple equation of adding the enjoyment from listening to each independently. This illustrates how non-linearity arises whenever there is some relationship between elements within the system that can be either synergistic making the output of the system greater than the sum of its parts or one of interference making the output less than the sum of its individual components. To illustrate this further let's take an example of four workers producing clothing. In isolation each seamstress can sew a given amount of clothes within a day. Now if we put them together we might get one of three results, firstly they might not interact with each other very much meaning the whole would simply be the sum of its individual parts. But equally likely they might form some cooperative relationship which lets them each specialise in a particular function making them more efficient as a whole and thus the output of the system would be greater than the sum of its parts due to these synergistic relations. Or inversely they might start all talking with each other, getting little work done and thus the total output would be less than the sum of the individual outputs due to these relations of interference. Non-linearity can also arise from feedback loops whereby the same process is iterated with the output fed back as the input to the next cycle. A classical example of this is compound interest where at the end of each period the balance plus interest is fed back into the formula to compute the next cycle of interest accumulation. Non-linear functions are an important concept within non-linear science and have been used to create a whole new type of geometry called fractal geometry whereby iterating a simple function generates irregular organic looking patterns that can model many of the geometric forms we see in nature from the structure of seashells to the rugged formation of mountains. In these non-linear systems superposition fails meaning one cannot break the system down into smaller sub-problems and then add their solutions. We must consider a non-linear problem in total. It is this need to approach non-linear systems as a whole that is giving rise to new, more holistic approaches to science that are developing under the canopy of the term complex system science. Non-linearity in all its shapes and forms is at the heart of many of the 21st century challenges to science as we try to extend the scientific framework beyond its dependency upon linear systems theory to finding new ways to embrace the complex world we live in on its own irregular and imperfect terms.


Welcome everyone. My name is Dr. Rashmi Karnajani. My pronouns are she and her. My presentation is called from Mrs. to Ms. to Doctor. Mothering work and talking back. I'm presenting at reimagining our worlds from below. In session 6, May 21st, 2022. In the interest of my bandwidth here and to continue with my presentation, I shall be turning off my camera shortly and I will rejoin after my presentation during my concluding comments. In 1980, Foucault said, the gaze. There is no need for arms, physical violence, material constraints. Just a gaze. Why this title? I situate myself as a global South woman. I was born in Mumbai, India. I'm a third world woman. I'm an immigrant woman. I'm a mother, an educator, a researcher. My trajectory from a dependent spouse to a married woman to being a single and South Asian mother is embedded in my talk today. I would like to invite your attention to examine this visual. This is The Small Hero by Dorothy Smith 1987 in the everyday world as problematic. In this figure, which I have also used in my doctoral dissertation, Dorothy Smith talks about the work of mothering in a complex of relations, sketching and institutional ethnography. Here, it shows a single mother as a small hero looking up into the governance machinery that is towering over her. Right above is the work organization of mothering and teaching. To the right is interlocked the political and bureaucratic organization of education. To the left is professional discourse. Right above her head is a single parent discourse. To the left is social class and hovering right above everything is the state. Dorothy Smith examined the single mother standpoint in her work and I have taken it up and developed my own understanding for my research which I shall share with you in this presentation. In my PhD, I looked at mothering work and parent engagement policy. Here, I looked at the Education Act which is right at the top, which is the boss text. And then I looked at institutionally coordinated professional discourse of teachers as experts, political and bureaucratic organization of education in Ontario and the parent engagement policy. Ontario has a parent engagement policy which makes its presence felt in all curriculum documents from kindergarten to grade 12. In Ontario, elementary education in publicly funded schools goes from kindergarten to grade 8 and then grade 9 to grade 12 are the high school years. In my research, I took the standpoint of mothers and the standpoint informant here and the researcher are standing together. You can see the interlocking factors of gender, race, ethnicity, social class, linguistic prowess, etc. The question marks are placed in the place where the questions arise with some may also see as the policy arena. This is where the education policies get taken up or the gaps begin to emerge. The blue arrows that are coming down from the Ontario government and the Education Act are the ruling relations that penetrate various aspects and coordinate and regulate the work of mothers. Returning the gaze. This phrase appears in my title. Banerjee's notion of returning the gaze is what I take up here. It is a way of looking back at our oppression. Institutional ethnography as an alternative sociology is an excellent method of inquiry through which I can do this. Institutional ruling relations and the institution are in view for an institutional ethnographer. I'm looking at the educational work as a South Asian mother and teacher. The institutional presence as a South Asian researcher and my social presence as an immigrant woman. That Roxana Eng says is a labour market category because some of us in our brown bodies as South Asian women known in the Western world will forever be seen as immigrant women. I would like to begin in a good way with a land acknowledgement. I remember every day that we in this land now known as Canada are all treaty people. It is my spiritual practice to greet the sunrise for guidance, grace and courage in my everyday work. I'm joining you from Markham in York region which is in the greater Toronto area of Ontario. The school board where I work is located on the lands of two treaties. These treaties have been signed with the Mississaugas of the Credit First Nation and the First Nations of the Williams Treaties who are the Mississaugas of Alderville, Gervlake, Hiawata, Scugog Island and the Chippewas of Beau Soleil, Rama and Georgina Island who is our closest neighbour and partner in education. Since my arrival here in 2002, I'm learning my responsibilities as I live and work on traditional territories as I notice my presence here as a racialized settler. I acknowledge my linguistic, academic and institutional privilege in this place. A silenced history, residential schools in Canada. For a period of more than 150 years, First Nations, Inuit and Metis Nation children were taken from their families and communities to attend schools which were often located far away from their homes. More than 150,000 children attended Indian residential schools. Many never returned. I encourage you to look up the information around residential schools in Canada because this is indeed a silenced history. Public education in Ontario, Canada. A little bit about that. Ontario as home to over one third of all Canadians is the most populous of all provinces. Canada's largest city Toronto is the capital of the province, Josie 2008. Ontario's public education system caters to students from kindergarten to grade 12. My experience in Ontario is kindergarten to grade 12 education. Yes, my lived experience as a mother because both my children went through elementary and secondary education in Ontario. My field experience as a teacher and a consultant in publicly funded schools and my academic experience, my 10 years in educational research. I also want to draw your attention to this figure, which is the hierarchy of work processes in Ontario's K to 12 education. The institutional circuit, accountability circuit and ideological circles are three terminologies that come from institutional ethnography. The Education Act directs the work. Directors direct the work of supervisory officers, administrators, teachers and parents. And this is how the work accountability flows from the Education Act through the directors of school boards down to supervisory officers to administrators to teachers who direct the work of parents through institutional texts. Some are formal texts such as report cards and individual education plans. Some are informal texts like notes in the agenda book or an email or a phone call or a mark on a test or a rubric that is sent home. Immigration and mothering work are important parts to consider here. Our experiences as mothers did not teach us about the workings of the school system or about how our mothering work contributed to and was organized by it. Say Griffith and Smith in 2005 in their co-written book, mothering for schooling, which was the starting point of my research during my Ahmed in 2011. In Ontario, mass education has been around since the 19th century when it was first set up in North America and has been portrayed as a path that helps students become informed citizens, capable of true participation in a democratic society and a means through which injustices and inequities can be rectified by standardized opportunities to learn and advance. Griffith and Smith 2005 page 9. In my research, I take up or I theorize what I have called the blended standpoint. As a mother and a teacher in Ontario, I foreground the standpoint of mothers of students in grade 4 to 6, classrooms in the Greater Toronto area, who are primary caregivers and due to their location outside the institutional power structure experience the regulatory mechanisms of schools in particular ways. My blended standpoint is a combination of what I see as a mother and as a teacher. And it has taught me to notice how my mothering work occurred for my children's schooling and how the mothering work of my research participants unfolds as they engage in their everyday life. Noticing the disjuncture, looking inward and looking outward. So I enter this research through my blended standpoint as a mother and an Ontario certified teacher. This is a methodological stance that brings into light the ways in which my work was organized as a mother, who came to Ontario in 2002 to become a teacher in 2004 and learned about the ways my work was organized extra locally from outside my home and translocally between my home and other locations such as my children's schools, their principal's office and further and further out into the Ministry of Education. I began to learn this when I became an institutional ethnographer and I have been doing this work since 2011. Ontario has an excellent education narrative. It's called From Great to Excellent. Ontario's educational narrative is one that portrays Ontario as an educational leader. Education is also one of Ontario's key magnets in the immigration pipeline. Ontario's Ministry of Education has many policies and various related texts such as curriculum documents, informational texts and policy instruments that enter into the lives of many people of whom teachers and mothers are my areas of interest. A little about my MAD research. I had titled one of my presentations about my MAD research as Ka Kya Hoga. What will happen to him? This was a question that mothers often asked me, mothers of my students when I was a teacher. And my MAD research I undertook between 2011 to 2013. South Asian mothers and transition to high school was my topic. In 2013 and 14, I began to write up my research amidst the swirling waters and the unraveling of various things in my personal life. In 2015, I had to make a decision whether I was presenting my research as a thesis or an MRP. That's a major research paper. And then after all that work was done came the year right, the dismissal of the research and the researcher. South Asian women's mothering work. My MAD research brought forth these thoughts. The research data brought interview the barriers that South Asian women faced when they worked to support their children's transition from grade 8 to grade 9. Teachers would often suggest courses based on their children's ESL profile. Teachers would not always answer mothers questions. The school would send home forms without much conversation with families. Mothers also reported that their children faced ridicule from their teachers and the school office if the mother visited once too often. Because your mom always has questions. What does she want now? So is it silent South Asian mothers or silenced voices? Some mothers reported that their children would say, my teacher has told me what to do. You just have to sign here for their course selection papers. Some mothers in my research reported, maybe because I'm not a real Canadian, I don't get answers. One mother said, when the principal said to me, I don't carry educational assistance in my pocket. I was very hurt, but I did not talk back. I have the words, but I did not talk back. After my MAD research, I faced many barriers to dissemination of it. I faced in school barriers of gatekeeping. Because although there was a ministry funded project that would have allowed me to do participatory action research with other teachers in conjunction with their school communities. Without administrator support, the data could not be entered into this project. I was also told that South Asian mothers are the problem. They did not understand because of their language barriers and their unfamiliarity with Ontario education. The common sense sexism and racism here is that it is automatically assumed that South Asian mothers are educated outside Ontario and they don't get it. In my school district, I experienced dismissal and erasure. Many people heard of the research came and took their recommendations with them, but there was no invitation to collaborate. In departmental meetings, I'm seen through the lens of my employment role. The research and the researcher are erased. South Asian mothers are once again the problem. I continue to hear that the language barriers and unfamiliarity with Ontario education are the issue. Sometimes this common sense sexism and racism also comes from other South Asian people, educators, support educators, community partners and so on. Although there are many initiatives done to engage South Asian families as a policy initiative, policy implementation is not followed by program evaluation. So any data that filters up into the educational governance machinery of the school board or the ministry's records is invariably that which is showcased and cherry-picked. In the province, there are neoliberal mindsets, of course, and the immigration trajectory leads to pockets of raised class. People think this is the place where South Asian people live and therefore there is a lot of discourse around how people drive, how they behave, what they expect from their children, and also that they are the cause of mental health and well-being issues that South Asian children face once again. Sometimes these comments are also internalized racism and they come from South Asian educators and community partners. Money is invested with limited accountability. Once again, policy implementation is not always followed by program evaluation. Navigating the problematic, I moved from my Ahmed to my PhD from South Asian mothers to just mothers and teachers. Through its vast range of texts, the ruling relations at the Ontario Ministry of Education textually enter into and organize the schooling of children from kindergarten to grade 12. The implications of ruling relations on my research interest is that mothering work and teachers' labor get represented and regulated in generalized and standardized ways with no consideration for the local conditions in which individuals do their work. For my doctoral work, I looked at mothering work and teachers' labor and as an institutional ethnographer, I embedded the parent engagement policy into the flow of mothering work to see how mothering work changes. My research questions were, how does Ontario's parental engagement policy enter into and organize the work of mothers and teachers in the grade 4 to 6 panel of greater Toronto areas public schools? How is mothers and teachers work regulated by the Education Act as an authoritative or boss text to direct a course of institutional action within Ontario's educational system? My research findings were, mothers who are teachers or administrators used their job knowledge to help them support their children's schooling and they were not dependent on teachers to give them materials or to tell them what to do. On the other hand, mothers who were not teachers did their mothering work to support schooling for their children and worked very hard, but they did not always get timely, clear or accurate information from their children's teachers. For newcomers, prior to their emigration from their previous homes, although they did all their homework based on government websites, etc., actual experiences did not match the textual portrayal of Ontario and it's great to excellent educational narrative. Teachers also reported power struggles and workplace bullying from colleagues. As well, they reported apprehension and fear to advocate for themselves or their students and families. I interviewed 10 mothers and 10 teachers teaching the grades 4 to 6 across the greater Toronto area through snowball sampling and I invited participants to join me by circulating my posters on social media. Teacher interviews showed us what actually goes on. Teachers highlighted power relations in teachers' work, fear of repercussion from colleagues if they witnessed escalated discipline practices against students. Teachers said, I don't want to get into trouble. That is not my place. Maybe it is their professional judgment. It is power play. Who do I tell their friends with other powerful teachers? Or they, meaning other teachers with power, are in the principal's inner circle. And as this happened, I noticed another text coordinating mothering work. I noticed that although there is an Education Act and there is a parental engagement policy, there is also the Elementary Teachers Federation of Ontario's professional judgment guidelines. Even before this pamphlet was published in 2016, the professional judgment discourse has been strong. It stopped conversations between mothers and teachers as it came out in my research. Speaking to the principal did not make a difference for the mothers who talked about what they did not receive from teachers. Complaints from mothers about information gaps, assessment issues, etc. remained. None of these complaints go into teachers' records. The parental engagement policy, I call it a tattered text. Powerful insider mothers can get things done with or without teacher support or the principal. Marginalized mothers, on the other hand, are caught in the web of parental engagement discourse and the professional judgment discourse. So deepening the conversation. Although I did my Ahmed from York University in Toronto and my PhD from the University of Toronto, it is interesting to me that neither of these institutions have approached me to share my research. I did find a place at the School of Disability Studies at the Toronto Metropolitan University who invited me to speak at a public lecture. And I was able to disseminate my research findings there on April 13. I'm also planning to engage in community conversations with participants, both mothers and teachers. As I do my research in multiple languages, whichever is the language the participant chooses to speak in, I will continue to disseminate my research in multilingual settings where women receive their mothering work supports. At academic conferences, there are many challenges. One of these challenges has been surmounted by the initiative of Morena Tatari and therefore we are gathered here today and to you, Dr. Tatari, I owe a huge thank you. So back to my title from Mrs. To Ms. To Doctor. Institutional credentials and returning the gaze. It doesn't matter what my institutional credentials are. I am still an immigrant woman and I'm okay with that. I went from being married to being single to earning my PhD over a period of 10 years alongside doing my research and that has germinated into this title for the presentation today. Biometrics will textually track me as a global North researcher. My embodied presence though in social, academic and professional spaces is that of a global South woman. I am seen, heard and responded to as a third world woman. I am an immigrant woman, which is a labor market category. Saying, I see how you see me is a way of returning the gaze. Writing and speaking about how the personal becomes political is the way of institutional ethnography and activist research. Situating myself in this way in the paid employment and research problematic is the talk project of talking back to discourse. Researching at the margins, the price we pay. In decolonizing research methodologies, Linda to Y. Smith says, researchers who choose to work at the margins are at the risk. Are at risk of being marginalized themselves in their careers and workplaces. She suggests a strategy for overcoming this predicament by embracing the work and committing to building a career from that place.  Smith cites bell hooks and Gloria and Zaldua who assert this. Merchants are also sites for possibilities that are exciting and on edge. Here, cultures are created and reshaped. People seen as dangerous, unruly, disrespectful of the status quo and distrustful of established institutions innovate in these spaces. They, we are able to design our own solutions. They, we challenge dominant research and society to find the right solution. And as we have here, we reimagine new possibilities and new worlds from below. A heartfelt thank you to Dr. Morena Tartari for organizing this global conference. Reimagining our worlds from below at a time when other conferences are in a great rush to offer face to face initiatives and thereby exclude those of us who cannot participate in these initiatives. You have definitely shown the way. And for that, I thank you.