Alright so we are we are discussing meromorphic functions okay and we were looking at meromorphic functions on the extended plane in the last class and in the last lecture and you know and we proved that the a function which is meromorphic on the extended plane is none other than quotient of polynomials okay namely a rational function okay. So what I need to what I want to tell now is about the collection of meromorphic functions on a domain okay you take a domain in the extended plane and look at the set of all meromorphic functions defined on that domain okay then that set has a has a nice structure in fact algebraic structure it is a field okay and in fact it is an algebra over the complex numbers okay and so the so it is a field extension of the complex numbers and the and the properties of this field extension algebraic properties of this field extension they have got they capture a lot of topological and geometric properties of the domain okay. So this is how there is a link from you know from the complex analysis side to the algebra side okay so you know geometry involves an interplay of various areas of mathematics. So studying something geometrically will involve studying it from the analysis view point okay and studying it from the topological view point also studying it from the algebraic view point but when you are looking at a particular nice object okay when you study it analytically it will have some properties okay some special properties and then when you studied algebraically it will have some properties special properties when you study topologically it will have some special properties and the fact is that these properties are interrelated it it is there is some beautiful relationship hidden relationship between these the analytic the algebraic and the topological properties of a nice object and that that relationship is what you may call as geometry, okay. So if you want to really understand the geometry of an object you have to analyze it using all the three you know viewpoints algebraic, analytic, topological, okay. So in that sense you know how do I do geometry on a domain in the complex plane or in the extended complex plane, okay. What I can do is of course the analysis is there the analysis will worry about what kind of functions you can define on the domain, what are the holomorphic functions or analytic functions on the domain, what are the meromorphic functions on the domain and so on that will that will be the view point from analysis but then how do you go to algebra, okay. The point is that you take the set of meromorphic functions that forms a field, okay and that is a field extension of the complex numbers and you study the properties of this field extension, okay. So in field theory you have lot of you would have come across in a course in algebra in field theory that field extensions have there are of so many types, okay there are algebraic extensions, there are transcendental extensions and then there were there are normal extensions, there are splitting fields, there are Galois extensions, okay. There are of course separable and non separable extensions and we study all these things and of course the most important thing here in general is the study of the nature of Galois extensions because that connects up with group theory which is it connects up with the so called Galois groups so you see the moment you look at the field of meromorphic functions you get an extension of the complex numbers and then you can do algebra, okay and somehow these things are all connected and I will try to give a couple of examples, okay. So first of all let me begin by first saying that if I take a domain in the extended complex plane then this set of all meromorphic functions on the domain is actually a field, okay. So let me write that down the field of meromorphic functions. So let D inside the C union infinity be a domain so you are taking a domain in the extended complex plane C union infinity. So in particular mind you it is a non-empty open connected set, okay and the advantage of taking a domain in the extended plane is that you can also look at the neighbourhood of infinity, okay that is the advantage. So you are also including the point of infinity, right. So let so here is the notation I will put script M of D be the set of meromorphic functions on D, okay. So what is this script M of D? Script M of D is the collection of all meromorphic functions on D and you know what a meromorphic function is. We have defined a meromorphic function to be a function which is analytic at all points but except for points on an isolated sets which have to be pole singularities, okay. So it is analytic except for poles and the moment you say analytic except for poles it means the singularities can be only poles and that in particular means that the singularities can only be isolated because poles are isolated singularities by definition, okay. So you take all the meromorphic functions on the domain, okay. Now the fact is that this is a field, okay. See so let us see that let f and g be meromorphic functions on D, okay. Then you see then you can notice the following things, number 1 if lambda is a complex number, okay then lambda f is also a meromorphic function, okay. Meaning a meromorphic function by a constant is going to keep it meromorphic, okay. Of course if the constant is 0 you will get 0 and the 0 is a constant function and of course you know when you say meromorphic analytic is also included, okay. So the definition for meromorphic is that it is analytic except for poles that does not mean that it has to have a pole. It can be it can have no poles and it can be analytic everywhere. So holomorphic functions are also included in the set of meromorphic functions, okay. So you know so the first so this statement is obvious if I take a meromorphic function multiplied by a constant if the constant is 0 of course I am going to get the 0 function which is holomorphic which is analytic because it is a constant function, okay. But if lambda is not 0 lambda times f will also be meromorphic and it will have the same poles, okay by multiplying by lambda you are not going to change the poles and you are not going to change the order of the poles. Essentially you are just multiplying by a constant, okay. So this is one obvious thing then the second thing is that if you take the sum of these two meromorphic functions this will also be a meromorphic function, okay the sum of f and g will also be meromorphic why because you see the fact is that f is meromorphic so it has some fine it has a collection of poles, okay at an isolated set of points. Then g is also meromorphic so it has also poles in another isolated set of points and then you take the union of these two isolated sets that is again an isolated set, okay and these are the only points where f plus g will have problems, okay. So at a point where f does not have a problem and g does not have a problem f plus g will not have a problem that is at a point where f is analytic and g is analytic f plus g of course will be analytic, okay. So the only problems for the function f, analyticity of the function f plus g will be at the points where f and g have problems, okay and it is possible that some of that could be some cancellations okay. So you know for example f may be 1 by z minus z not g may be minus 1 by z minus z not so if I take f plus g I will get 0 which does not have a pole at z not okay. So some poles see some poles can cancel out also and sometimes the order of a pole can come down okay when you add of course when you when I say add it also includes subtraction because subtraction is just adding with minus 1 multiplied by the second function okay. So the moral of the story is that sum of 2 meromorphic functions is again a meromorphic function it could very well be analytic okay some poles may cancel out all the poles may cancel out for example if you take the function you take its negative and add it you will get 0 and that is clearly holomorphic it is a constant function. So sum is meromorphic so you know the moment you look at the first 2 things this will tell you that you know M of D is a vector space over complex numbers see because it is you see so there is a scalar multiplication if you think of complex numbers as scalars then there is a scalar multiplication and there is addition so this becomes a so M of D is C vector space so you get that immediately okay. Now let us look at f times g look at f times g see f multiplied by g will also be a meromorphic on D this is also very very clear because it just from the fact that you know what are the problem points the problem points are the points where f has problems and g has problems okay so if you take out those problem points then f times g will be analytic so at a point where f is analytic and g is analytic f times g will be analytic and the only place where f times g will fail to be analytic is probably on the set union of the set of poles of f and poles of g okay and so you see and of course if you want you can write out always the principle parts and see that you know if 2 functions have poles at the same point they have common pole then if you multiply the product function will have a pole with higher order in fact it will have order equal to some of the orders that is obvious if you write out the principle parts okay so in the Laurent expansion alright so I mean the point is that you know all these algebraic operations of adding, subtracting, multiplying by a constant and just multiplying and of course we are going to see division all these things they do not change the meromorphic nature okay so by adding or subtracting or dividing or multiplying or multiplying by a constant you cannot change a meromorphic function into a non-meromorphic function you know if you are only working with meromorphic functions you will get back again meromorphic functions okay so fine so you have f and g are the product f times g is also meromorphic of course by product one means point wise product okay so f g is a function which at each point z is defined by f of z times g of z alright then of course I can say the same of f by g this is also a meromorphic function provided g is not identically 0 okay of course I should not divide by 0 so the fact is that see when I take f by g okay what are the problem points the problem points will be poles of f poles of g and now you will have extra problem points at 0s of g because they are 0s of the denominator they become the 0s of g will become poles of f by g they are likely to become poles of f by g and of course you know it might happen that some 0s of g may cancel out with some 0s of f because the 0s of f are on the numerator the 0s of g are on the denominator some 0s might cancel but the set of problem points are just the poles of f the poles of g and the 0s of g and these are this and you know the 0s of an analytic function are also isolated you know that that is a theorem okay in fact that is a that is another version of the identity theorem if you have seen it in a first course in complex analysis so therefore the set of points where f by g will have problems is still an isolated set of points okay and at each of those points you can only get poles you cannot get anything worse okay so it is so therefore f by g is also meromorphic in particular I could have taken 1 by g I can put f equal to 1 I will get 1 by g is also meromorphic and that means so what will I get I will get 1 by g is meromorphic if g is not identically 0 so that means every non-zero meromorphic function every meromorphic function which is not identically 0 has an inverse okay and that is what you require for a field okay a field should be basically a group under multiplication if you throw out the 0 element from the set okay so well if you look at all these things these things will tell you that M of D is a field and you know you put it together with this fact that we saw earlier we have seen just above that M of D is also a complex vector space of a field extension with a field which is also a vector space is an algebra okay so basically you can very well see that M of D contains complex numbers because the complex numbers sit as constant functions okay you take any complex number lambda you think of it as a constant function lambda constant function lambda is analytic it is defined everywhere so it is analytic on any domain okay and it is meromorphic because mind you when I say meromorphic I am allowing also analytic or holomorphic meromorphic means that it can either be analytic and if it is not analytic that is if it has singularities the singularities must be only poles that is what it says so meromorphic does not say that it should not be analytic so in particular M of D contains the complex numbers as a sub field you know the complex numbers of course form of field and therefore M of D is a field extension of the field of complex numbers so M of D is a field so let me write that M of D is a field extension field extension of the field C of complex numbers and the beautiful thing is that the geometry on geometry on the domain D is done by the lot of topology of the domain D is connected to and a lot of analysis on the domain D namely the behaviour of existence and behaviour of meromorphic functions on D is connected with the algebraic properties of this field extension okay that is the geometric content okay so I mean this goes back to the work of the classical giants like Riemann and Clifford and Weierstrass and Abel and Jacobi and you know all these people who developed theory of Riemann surfaces okay of course principally from Riemann so let me write that down so let me write it out as a diagram so you have M of D and this is over C so I am using this I am using this field theory you know notation you write a field a bigger field on top and you put a smaller field in below and then you put a vertical line saying that the thing that comes above is a field extension this the things that come below are subfields okay so this so the algebraic properties of this field extension the algebraic properties of this field extension they are connected so there is a so the analysis on D which is existence of special existence and properties of meromorphic functions on D that is the analysis on D okay and this part is algebra on D the algebra on the domain is actually studying you may think of it as studying the this field extension and then there is the topology of D topology of the domain D and you know so I am not trying to be very particular or trying to go in detail but topology at the minimum for example D is of course connected but one of the simplest things that you can look of look at is whether D is simply connected or not okay and then or if it is not simply connected you can see you know if it is multiply connected and how many holes it has and so on and so forth okay after all D could be something like a something like an amoeba something that looks like an amoeba okay with some holes okay after all an open set can look like that and then the topology worries about whether it is simply connected if it is not simply connected how many holes are there and so on and so forth okay so all this topological information for example is encoded in the in the fundamental group of D and so on and so forth okay and in fact more precisely I should say that you have to study the theory of covering spaces of the domain D that is what the topology of D means so let me write this here the theory of covering spaces of D related which is actually related to the fundamental group in fact subgroups of the fundamental group of the fundamental group of D of course you know the so you know at this point let me tell you that if you have done a decent second course in topology see there is something called covering space theory which takes a topological space with decent properties for example something that is a house of locally house of connected locally path connected locally simply connected okay and then you study what are called as coverings of topological coverings of D and then there is a there is a Galois theory of coverings which says that you know there is a Galois correspondence between the coverings and subgroups of the fundamental group of your topological space okay and in fact under this under this Galois correspondence the you know the so this Galois correspondence is an analog of the Galois correspondence that you have in field theory see the Galois correspondence in field theory gives you a correspondence between field extensions of a given field and subgroup of the Galois group subgroups of the Galois group okay and there is an analog so the Galois correspondence in field theory is a correspondence on the one side between field extensions and on the other side between subgroups of a group and in this case it is the Galois group okay so it is a connection it is a connection between field theory and group theory okay and it is very useful because lot of field theory problems can be translated to group theory problems and lot of group theory problems can be translated to field theory problems. In the same way the covering space theory is very very similar what it does is it translates topological coverings which is topological data into subgroups of the fundamental group. So it also connects to topological side the topological side to the group theory side okay so that you can use some algebra in your topology okay so that is why usually this is a part of usually a first course in algebraic topology okay. So of course all this is very uninteresting if D is simply connected because if D is simply connected then your you know your the fundamental group is trivial okay but then it is still not so easy in fact that is the you know I will explain why you see see this whatever I have written here the algebra the analysis in the topology of D I have written it for a domain D in the extended plane okay but I am what the philosophy is that this is this holds for any Riemann surface okay now there is something called a Riemann surface a Riemann surface is something that locally looks like the plane okay but globally it may be a different surface. So for example it may be a cylinder in 3 space okay it may be it may look like a torus alright or it may look like an n torus so it may be it it might look like several tori which are stuck together by removing discs and open discs and you know pasting the boundaries of those open discs okay. So these are called Riemann surfaces and these are studied by Riemann okay and Riemann was fascinated to note that on these Riemann surfaces you can put many complex structures there you can put non-isomorphic complex structures and you must think of a complex structure as a structure which allows you to decide whether a function on that surface is holomorphic or not okay. So Riemann found that you know see Riemann try to do what we do in complex analysis on the plane on the plane what do we do we take a domain okay and ask when a function is analytic at a point okay and if it is not if it is analytic then of course you know if it is not analytic then you see whether that point is an isolated singularity and so on that is how you do the analysis. Now what Riemann wanted to do was he wanted to do this on a surface so he wanted to say that suppose I have now a function on the on a torus okay or say even an open subset of the torus alright when I say open subset you take the induced topology from R3 in which the torus sits okay and then suppose I have a function which is complex value to define on an open subset of the torus when can I say it is holomorphic when can I say it is analytic okay. So you are trying to study when a function defined on an open subset of a surface is analytic the answer to this is that you should define what is called a Riemann surface okay and there are different Riemann surface such as you can put and Riemann found that he was fascinated by these different Riemann surface structures and the most beautiful theorem in moduli theory is that you actually take the set of all these Riemann surface structures that itself becomes a nice object it becomes an analog at least on an open set it becomes an analog a higher dimensional analog of Riemann surface which is called a complex manifold and of course it could have a boundary which could have some singular points but it is a very beautiful object okay. So the moral of the story is that I am trying to say that whatever I am writing here for D a domain the extent plane it also works for it also works for a domain it also works for a Riemann surface okay and so for example in that context you know it is really it is really amazing that you get lot of so you know if you ask this so let me ask you a fundamental question the fundamental question is suppose you have a simply connected Riemann surface okay so the moment I say simply connected the topology seems to be very trivial because in the sense that in the fundamental group is trivial so you do not expect anything special but then you can ask how many simply connected Riemann surfaces are there which are not isomorphic to each other okay. Now you more or less know the answer partially because you know the Riemann mapping theorem tells you if you have seen it in a first course in complex analysis which you should have done that you know any simply connected open subset of the complex plane which is not the whole plane has to be holomorphically isomorphic that is biholomorphic to the unit disc okay so if you take domains in the complex plane okay simply connected domains in the complex plane there are only two types up to holomorphic isomorphism one is the whole plane the other one is unit disc okay so now it is an amazing fact that well even before that let me say look at the Riemann sphere okay which is you know we use that to study the point at infinity where the stereographic projection the Riemann sphere is also a nice surface of course and is compact okay and you can actually make it into a compact Riemann surface okay now the fact is that it is also simply connected the sphere is simply connected so that is also another simply connected Riemann surface now it is a very deep theorem that you take any simply connected Riemann surface it has to be it has to be isomorphic holomorphically isomorphic to one of this one of these three any simply connected Riemann surface has to be either it should either look like the whole plane okay or it should look like the unit disc or it should look like the Riemann sphere there are no other possibilities it is a very deep theorem it is called uniformization so even in the simply connected space even in the simply connected case you get a very deep theorem and the theorem is very hard to prove because you have to use lot of techniques from analysis to prove it okay so it involves lot of analysis it involves study of harmonic functions meromorphic functions etc okay and it has it involves lot of reasonable amount of functional analysis and measure theory okay you have to do you have to do all this to get that theorem okay so anyway so the fact I want to say was that now given all these three aspects of view putting them together is what geometry is all about okay so let me write here so geometry of D is you know the interplay between these three the geometry of a domain is actually the interplay between the analysis on the domain the algebra on the domain and the topology on the domain and I have given you a rough idea you know the analysis on the domain is the complex analysis part okay the algebra on the domain is to really study the field of meromorphic functions the algebraic properties of the field extension given by the field of meromorphic functions of the domain and the topological part is to study the covering space theory of the domain okay and it is and it so happens I mean as the great classical giants like Riemann and Clifford and Abel and Jacobi and we are stress have found and Clifford for example that you know all these properties all these various points of view they are all interrelated okay so it is an amazing fact it is an amazing fact and discovering that is what doing geometry is all about okay so you should not think that you should not think at high school level the geometry is just about drawing triangles and circles and you know measuring of angles and arcs and things like that but it is really higher geometry in the higher sense is actually looking at the interplay of all these things okay so I well now you see I want to give you a couple of examples so the so here is the first example so here is an example you take the domain to be the extended plane itself you take the domain to be the extended plane itself okay so after all we are studying domains in the extended plane so take the whole extended plane that is also domain alright it is in fact it is simply connected because it is you know it is homeomorphic the Riemann sphere the Riemann sphere is simply connected so it is simply connected it is compact okay so it is a very nice thing now what are the field of meromorphic functions on D okay what are the field of meromorphic functions on the extended plane so you know this is an extension of the complex numbers as we have seen this is an extension of complex numbers but you know what is it that we proved last time a function which is meromorphic on the extended plane is none other than a quotient of polynomials it is a rational function okay and therefore this is exactly equal to the this is exactly equal to C round bracket Z this is the algebraic notation C round bracket Z and the C round bracket Z is actually the field of fractions or quotient field of C of square bracket Z and C of square bracket Z is standard notation is the ring of polynomials in the variable Z with complex quotients okay and C of Z is the field of fractions which is quotients of such polynomials and so you take quotients of polynomials but of course you do not put in the denominator 0 anything other than 0 you put okay so the moral of the story is that you have very nice description of this field extension in the in the case of the C union infinity which is extended plane and usually you know extended plane is thought of as a Riemann sphere you know they are isomorphic but you can make them also isomorphic in a holomorphic sense by giving the Riemann sphere a Riemann surface structure okay so often people do not use if you see the literature you will see that people often use C union infinity instead of C union infinity they keep saying Riemann sphere all the time okay. So now you can see that what are the properties of this field extension you see this field extension is actually it is purely transcendental and has transcendence degree 1 okay it is purely transcendental and has transcendental degree 1 well the transcendence degree is actually the number of algebraically independent variables that generate the bigger extension okay so the bigger extension C of z is generated by a single variable z and that is the only algebraically independent variable okay that one variable is enough so the transcendence degree is actually 1 okay and it is purely transcendental because there is no element in C z which is algebraic there is no element in C z which is not in C and which is algebraic or C and that is you know why that is because complex numbers are algebraically closed they are all algebraically closed so this is an exact so field theoretically this is so this is called this is what is called as a function this is a simplest example of what is called a function field in one variable okay and the beautiful thing is that now if you take any compact Riemann surface okay then if you take the field of monomorphic functions on that compact Riemann surface what you will get is a function field in one variable but the only thing is that it may not be purely transcendental above the transcendent above there may be an algebraic part okay so it will be first a transcendental extension purely transcendental extension of degree 1 just like this and then be and above that you will have an algebraic extension which will be a finite extension okay so that is how it looks in general okay and well I will give you another example for that so let me write this here this is a purely transcendental extension of transcendence degree 1 okay so see the so the picture that is that is associated with this is the Riemann sphere okay so this is the picture that is associated with this and for all practical purposes you think of the extended plane as a Riemann sphere okay. Now let me let me tell you more generally what is it that happens with something so let me give you an example of a more complicated case the so called the case of so called elliptic functions or W periodic functions okay so here is so here is what I am going to do what I am going to do is I am going to take so I am going to define what a W periodic function is see a W periodic function so this is the topic of what are known as elliptic functions so this is also an example okay W periodic function is a function f of z with f of z plus omega 1 is equal to f of z and f of z plus omega 2 is equal to f of z where omega 1 by omega 2 is not real and of course omega 1 omega 1 omega 2 are non zero okay so I am just defining what a W periodic function is so the see the definition is very very simple for example sin theta you know is periodic with 2 pi because sin of theta plus 2 pi is the same as sin theta so the idea is that to the variable of the function or the argument of the function you add the period the function value should not change okay so what the first equation says is that f of z plus W1 is equal to f of z actually tells you tells you that W1 is a period okay and the second equation f of the second requirement f of z plus W2 equal to f of z tells you that W2 is also a period so W1 and W2 are periods and the fact is that these we want these periods to be linearly independent over R in other words what you want is that if you take these two complex numbers W1 and W2 then you join the origin to them okay namely you take the vectors that they represent in the plane then these should be different these vectors should be linearly independent they should be in different directions you know they will be linearly dependent if and only if the quotient W2 by W1 or W1 by W2 is a real number okay so this condition that W1 by W2 is not real is just to tell you that these two vectors are two different vectors they will form therefore a basis of C over R okay C the complex numbers over R is a two dimensional vector space and they will form a basis so this is equivalent to saying that W1 W2 form a basis for C over R okay and we are putting this condition in order to make sure that essentially you have two distinct periods okay which are in two so it is periodic in two directions okay the fact that f of z plus W1 is equal to f of z tells you that you know if you translate along the direction of W1 by integer multiples of W1 the function value does not change okay so you must remember that when I say f of z plus W1 is f of z it follows that f of z plus n1 W1 is also f of z for all integers n1 okay because I can just use induction f of z plus W1 is equal to f of z so f of z plus 2 W1 is f of z plus W1 plus W1 which is f of z plus W1 which is f of z and so on and so forth okay so what it tells you is that the moment something is a period then all its integer multiples are also periods okay and similarly you also have for the other so but what is adding W1 adding W see addition of a complex number is this translation along the direction along the vector that is represented by that complex number okay so you know basically if I have a point z what is what is z plus W1 it is actually this vector so this will be z plus W1 I am just translating z by the vector W1 okay and then similarly what is z plus W2 I am just translating z by the vector W2 alright and that is if I add W1 but if I add minus W1 you know I am translating in the other direction if I add minus 2 W1 I am translating in the direction opposite to W1 two times and so on and so forth okay so these are all so the moral of the story is that you know I basically the function values do not change if you translate along two different directions okay that is why it is called W periodic it is periodic and the period there are two different periods okay and such functions are called actually now you have to put some more condition on these functions the condition you put on these functions is that you know to make them very interesting these points W1 these points which are given by integer multiples of W1 added to integer multiples of W2 okay that they will form a lattice a grid in the plane okay and the function becomes very interesting if the function is meromorphic exactly at those points okay and such functions are called elliptic functions and believe it or not they are exactly the functions which are the functions meromorphic on a torus at a single point okay and this is the beginning of the so called Weierstrass Fe theory there is something called the Weierstrass Fe function okay which is a fundamental model of this kind of function and the beautiful thing is that every torus the complex structure on any torus can be controlled by prescribing such a function okay and so the Weierstrass Fe functions completely give you so if you want to study the various complex structures you can put on a torus what you will have to do is you have to study various W periodic meromorphic functions which are otherwise called elliptic functions the reason they are called elliptic is because this is beautiful this complex the moment you put a complex structure on the torus it becomes believe it or not it becomes a cubic curve it becomes a cubic curve and therefore it becomes an algebraic geometric object okay so algebraic geometry also comes in geometry also comes in algebra comes in a beautiful way okay and and this is also part of a very deep theorem which says that you know you take any compact Riemann surface it is algebraic it can be it is just given by a common 0 set of a bunch of polynomials okay and that is an amazing theorem okay. So I have so what I want to tell you is that I have given an NPTEL video course on Riemann surfaces and all these things are explained in detail throughout the course you can you can when you find time you can have a look at that and the other thing that I want to tell you is that there is this book that I have written and it reads an introduction to families deformations and moduli this book is basically available as a free freely downloadable copy in the form of a navigable PDF file and it contains a lot about the geometry of Riemann surfaces so at least the first chapter so that is also something that can be advanced reading material for people who are interested in pursuing this so let me write let me continue so I have also f of z plus n to w2 is equal to f of z for all n to in z so in totality what I will get is I will get f of z plus n1 w1 plus n2 w2 is equal to f of z for all n1, n2 in z if I put both these together okay and what are these points n1 w1 plus n2 w2 they are the vertices of a grid of parallelograms okay. So in fact you know if you draw this if I draw a diagram it is going to look like this so I have this this is my complex plane and you see I have w1 here I have w2 here okay and you know then you know if I draw this parallelogram then you know pretty well that this is w1 plus w2 okay by the parallelogram law of additional vectors if you want and then you know if I extend this parallelogram below then I am going to get this point is going to be you know it is going to be w1 minus w2 okay and this point is going to be minus w2 and if I extend it like this this point is going to be w2 minus w1 and this is going to be minus w1 minus w2 and this is going to be minus w1 okay and more generally if I draw if I look at all these points that go on that I get as the vertices of the parallelograms that I get by simply starting with this parallelogram and simply displacing it by either plus or minus w1 or plus or minus w2 okay that is by translating it with plus or minus w1 or plus or minus w2 I will get so many the whole plane is covered by these parallelograms okay and the vertices of the parallelograms are precisely the points which are of the form n1 w1 plus n2 w2 okay and that is called a lattice okay and the fact is that you see where is the you know just to give you an idea of what is going on where is the topology coming in here. So the fact is that what you do is you divide by the equivalence relation equivalence relation Z1 is equivalent to Z2 if and only if there exists n1 n2 such that Z1 is equal to Z2 plus n1 w1 plus n2 w2 okay so see this is the plane this is the complex plane and I am defining an equivalence relation on the plane the equivalence relation is two points are equivalent if one of them is a translate of the other by one of these grid points okay and what is the advantage of this advantage of this is that if two points are related like that then the w periodic function will have the same value at both points because f of Z1 will be equal to f of EZ2 plus n1 w1 plus n2 w2 but that is also equal to f of Z2 so f of Z1 will be equal to f of Z2 okay because when I apply f to this equation okay on the right side I will get f of Z2 because of periodicity of f so I will get f of Z1 equal to f of Z2 okay what it means is the function does not the value of the function does not change if you change the point by a translate by a vector which belongs to one of the grid points okay so if you divide by this equivalence relation what you will get is you will get the torus you will get a beautiful torus and you can see that very very easily you just take this fundamental parallelogram okay this fundamental parallelogram will if you take the interior every point in the interior will be a unique representative in the in its equivalence class but for points but then you will have to only identify the boundaries see if you identify the top boundary with the bottom boundary you will get a cylinder okay and then you will have to identify these two which will in the cylinder look like circles so if you identify them you will get a torus okay so the moral of the story is you take the plane divided by a lattice like this you get the torus okay and all the points in the grid including the origin the origin is here they all will go to a particular special point on the torus okay and the beautiful thing is that the function that you defined on the complex plane will go down to a function on the torus okay and if the function is meromorphic exactly at all these grid points it will go down to a meromorphic function on the torus and you know you may wonder why should I worry about why should I not consider holomorphic functions of the torus and you very well know the answer there will not be any non constant holomorphic function of the torus because the torus is compact since the torus is compact if you define a holomorphic function on the torus okay you are going to get something that is holomorphic you can use Liouville kind you can use the Liouville theorem if you have a holomorphic function on the torus if you compose it with this map that is that goes from the complex plane to the torus you will get a holomorphic function on the plane but since it is defined on the torus which is compact its image is compact therefore the image is bounded so I get an entire function which is bounded and that is going to be constant by Liouville theorem okay and this picture also explains why the only functions on the torus are exactly the functions on the plane which are W periodic with respect to the periods W1 and W2 and the if you take this unique point P which is the image of the grid the meromorphic since you know holomorphic functions are not available they are constants the only things that are available are the meromorphic functions and then if you look at meromorphic functions on the torus at the point P you will get the they will be the same as W periodic functions okay and therefore the moral of the story is that you know if you look at the if you look at the field of meromorphic functions on the torus okay they are the same as the collection of meromorphic functions on the collection of W periodic functions with these two periods okay and that is a field of course okay mind you what is the domain now the domain is the whole plane okay and I am looking at meromorph I am looking at functions which are meromorphic with poles at points of the grid okay possibly at points of the grid alright and then what I get is I get a field and what is that field that field is nothing but this field of meromorphic functions on the torus which are meromorphic at a given point okay and so let me call the torus as T okay mind you the torus it depends on the choice of W1 and W2 okay and it is a different story that there is a lot of geometry there but what I want to tell you is that well I want to tell you the following thing simplest meromorphic function function on C that is W periodic that is W periodic with respect to W1 and W2 is the Weierstrass phi function and this is the phi function so there is a pretty symbol for that very special symbol so phi of z is so there is a formula for that basically it is a formula that will tell you that it is a meromorphic function which has a double pole with residue 0 at each of those points of the grid okay so you know what you are going to get let me write that down here you can find this in any standard book for example Alphos book on complex analysis which is a classic so it is 1 by z square plus summation over n1, n2 belonging to z minus 0, 0 z cross z in fact so I should write I should write it carefully n1, n2 ordered pair belonging to z cross z minus 0, 0 it is 1 by z minus n1, w1 minus n2, w2 the whole square minus 1 by n1, w1 plus n2, w2 the whole square so this is the expression for the Weierstrass phi function which was discovered by Weierstrass and of course if you go through in detail the lectures of my video course you will see how this comes about but you can see something immediately you see that this 1 by z square is the principle part at the origin and that will tell you that you know origin is a double pole and residue is 0 because there is no 1 by z term okay and then you look at each of these other terms 1 by z minus n1, w1 plus n2, w the whole square tells you that n1, w1 plus n2, w2 is a point of the grid is a general point of the grid okay and if you when I write 1 by z minus that point the whole square actually I am looking at a pole of order 2 at that point okay and again the residue there is 0 alright. So as a result this already gives you a you know it gives you a meromorphic function which is having a double pole at each of these grid points okay and this extra term that is added here is for convergence because you know I have added infinitely many poles okay I have added poles at every point of the grid I have made every point of the grid into a double pole and I am getting a huge series I want you to converge and it is only for this convergence that this extra constant term is being added okay and therefore I get this fair function and here comes the amazing here comes the amazing theorem. The amazing theorem is the following that if you take the complex numbers and you take the meromorphic functions on d on the complex plane with respect to w1 and w2 okay you look at the meromorphic functions which are w periodic at with periods at w1 and w2 okay and mind you this is the same as the meromorphic functions on the torus okay which are meromorphic at that unique point which I will call it star which is the image of the grid the whole grid the whole grid goes to a single point on the torus because all the points in the grid are equivalent to each other okay and they all define a single equivalence class so it is a single point on the torus that mind you the torus is a set of equivalence classes okay topologically and you give it the quotient topology alright. Now the beautiful thing is that what is this set of meromorphic functions you know it is a field what is that field you know you know what that field is that field is just the field gen it is just the field of fractions of Phi fz and its derivative it is beautiful and how does this extension break up it breaks up as the first Phi fz this is again a transcendental extension is a purely transcendental extension of transcendence degree 1 and then from here to here from here to here this is an algebraic extension this is an algebraic extension because the derivative of the Phi function Phi prime satisfies a polynomial relation with respect to Phi and that is expressed as a differential equation it is a very famous differential equation and that differential equation interestingly it comes from analysis but it tells you that the torus is algebraic it tells you that the torus is nothing but a cubic curve okay which is an amazing illustration of the fact that in general a compact remand surface is given is algebraic it is given by algebraic equations okay so all these details you can have a look at in detail in more detail in my video lecture course but this is to tell you that a lot of geometry is involved by looking at the field extension given by the field of meromorphic functions okay so I will stop here.


With what you've seen out of him in a couple of days, what do you like from what you've seen out of him? And when do you make the decision that of trying to promote him and possibly have him play Sunday night? All those moves can go down tomorrow. So I would imagine that those will go down. There'll be some moving parts with the roster and all that will happen tomorrow when John and I discuss them. But he's come in very professional, excited, studied, worked with Tony, him and Deontay. Him and Deontay, I told you, that's what Dantro did last week. So that's what they did. And then we're just trying to get him brought up to speed. A lot of the guys have said he's brought a degree of energy for you as a coach watching this. Do you feel like he's been able to charge you guys a little bit this week, so to speak? Yeah, we walked through on Thursday, which was real exciting. On Wednesday, we walked through on jog through. It was even more energetic. And then the Friday was really good. How are you going about what they said? I wasn't accusing you of anything, Tarana. I just was telling you that we had an exciting walk through as we possibly could have had on Wednesday. And even more ruckus Thursday jog through. I thought today was very good. I know we're beat up, but I was very encouraged by what I saw today just from a focused standpoint. Guys out there moving around. So I think that that's important. Tony kind of talked about needing to get Adrian and Foreman out there and see what they could do in the jog through, walk through thing. How much can you see what they did? Did you do anything extra with those guys padded up? Yeah, they did. They did. And really, that's just making sure that these guys are conscious of the ball security and have the pads on. So I thought that was important to use our padded practice for one day. As far as distribution periods, I mean, how much depends on just how they do, whether it's on Adrian, Yonte, Jeremy, just production, going to lift them early? Yeah, I mean, I think that production is obviously going to be key with whoever's out there. I also think that being conscious of where they're at, conditioning wise and trying to find roles for everybody. Is it allowing you to be creative with different guys in different situations now that you've got three or four guys that you're going to be choosing from? I mean, I think we're just trying to call whatever. I don't want to, like, OK, this guy is going to only going to run these types of plays. That really doesn't really work. It's hard for a play caller. But just trying to get them all brought up to speed. And we know that Jeremy's got a great grasp. And Don trail's got a week here. And then we'll see where Deontay and Adrian are. Fied with as much energy sets as a grand flight to run. What does that kind of put, what kind of strain does that put on?  You're right. It's the smartest thing you've said since I've been here.  You know, they mix in quick game. They mix in, drop back, you know, they'll chip to gain some extra time with guys on the edges. Everybody can go in any of those spots across the field. So whether, you know, there's a running back at number one or tight end or receiver at number three, they all run the same route, you know, whatever route they call that just they go out there and they line up in different spots.  It's kind of been, you always kind of kept an eye on, you know, if this ever was to happen, Derek, that you thought about a good fit? Um, no, I mean, not me personally. You know, I mean, that's, that's, you know, I'm trying to coach the football team that we have. And, you know, I think that John and his staff probably have, you know, a list of guys that they target and track. I don't think that that would be necessarily, you know, something that I would want to do until the situation arose. We're, then we go over the list that they have compiled, you know, so that's really kind of how that goes down. But, you know, aware of his career and what he's done, obviously.   Jay, on progress, guys, is there a way to confirm? Good, good, yeah. So I would say, you know, hopefully, you know, we can start to consider getting Jay on back in there soon, real soon. Have you checked in with Derek again recently this week? How's he doing coming off surgery? He's doing well. He's doing well.  And he stopped in and, you know, he'll continue to be around us more and more. Any questions? Evans, Davis, Blowsing game.


Welcome to the Chrome Enterprise Technical Community Hour. Today, we'll be talking about Policy API on Chrome OS. My name is Rich, and I'll be your host for today's presentation. Joining me today, we have our speaker, Ankur Mathur, who's a product manager for Chrome OS. For today's agenda, I'll start with a quick introduction of the Chrome Enterprise Recommended Program and the Technical Community Hour. After that, I'll hand it over to Ankur, who will cover today's topic and our call to action for our Chrome partners. Today's Chrome Technical Community Hour is brought to you by the Chrome Enterprise Recommended Program, which is Google's partner program for third party solutions that are optimized for Chrome OS or integrated with Chrome Browser. This webinar brings you the opportunity to engage with our team about new features and updates, enterprise development best practices, and our enterprise strategy. Now without further ado, I'll hand it over to Ankur and he'll kick us off. Ankur? Awesome. Thanks, Rich. Let me switch over to my view and we'll get started. Hi, everyone. As Rich mentioned, my name is Ankur Mathur. I'm a product manager here at Google since August 21. Prior to Google, I've spent time at Microsoft and Amazon and have experienced with a wide range of products. Today, I'll give you a quick overview of the Policy API and show how you can supercharge your apps and relationships with your and your customers Chrome OS fleets. The Policy API empowers administrators to manage Chrome OS devices. As next steps after this webinar, I'll show you how you can get some resources, documentation, and reference implementations to get started. Partners can utilize this Policy API to manage their customers' fleets. The Policy API by itself is very powerful. But when you combine it with some of our other APIs, such as the Admin SDK and the Telemetry API, you can do a lot more. The Admin SDK as a quick overview gives you the ability to manage, change your use, provision, deprovision devices, browsers, and printers. The Telemetry API lets you monitor the health and operations of the device. So putting all three of them together gives you a very powerful application at the end of the day. As I mentioned, the Policy API is for use internally within a Chrome customers' organization and then to build a commercial API client with it on top of this API, you'll need to submit a partner application, which you'll have done already, and then you can work with our partner escalation desk to get help in getting started with building your applications. So going into the API itself, I'll give you some of the key advantages and some of the key benefits of using this API. There's compatibility within the Admin console. So any changes you make within the API are reflected in the Admin console UI, advice versus any changes you make in the UI, and we'll update when you recall that API again. Any changes within the API are logged within our standard Admin console audit logs so you can track any changes that are being made. And then finally, we've taken a data-driven approach to building this API. So policies are represented as data elements and aren't referenced directly by the API. This lets you automate API clients for consumption of future policies without additional development. And we're constantly adding in new policies both from the Admin console UI and through the API for better benefits for our users. As a quick overview on how to use this API, there are a lot more details that we can go into through the partner escalation desk. You need to first create a Google Cloud project, enable the policy API as one of your endpoints, create credentials, you can test your app in the all-off playground and then verify that your app is trusted as you're going into production. Next, I'll go through some sample code requests and responses with the policy API to showcase its power and the capabilities that you can use with it. First up, let's read a policy. Let's read a policy for printers and see which users are allowed to use a particular printer. If you notice on the left-hand side, I've highlighted the additional target keys where I'm specifying a specific printer ID in this request that I'm making. I can also specify an organization unit or a group right before this to say, to really dive into where I want to see which users are allowed to use a printer or not allowed to use a printer. In my response, you'll notice the source key which again specifies the org unit or group value from where the policy comes from. And then here we have a couple of different possibilities which is really cool. If the source organization unit is the same as what was given in the request, it means the policy is locally applied to this organization unit. If it's different, it means the policy was inherited from this new source OU that's being provided. If there's no source key or the response is empty, it means the policy has not been set by the customer and has the default system values. For groups, the source key will always be the same group that was submitted in the request. This is an example for an org unit. A group request would be the same and instead of it being a org unit in a target resource, it would be a group instead of that org unit there. Next, we have the ability to modify the printer policy. So as you'd seen in the policy response before, the allowed for users had one field named allowed for users. This field in this example is a Boolean. It can either be true or false. In this specific case that I'm showing here, we only have one field. We only have one possible field for this policy. Other policies could contain multiple fields and you could set multiple policy items within it. In this example, we're setting the policy to false to prevent users from accessing this printer. And again, you can update for a specific org unit or for a group. Next, we also have the ability to control apps that are installed on a device and within an org unit. So this is an example of how you can force install an app on a specific org unit. Again, in this example, I'm specifying a unique org unit. You can specify a group here as well if you choose to. And you can provide through this additional target keys, the app ID and the specific app ID that's listed here on what you want to install. We can choose to force install this app as in this example in the second highlight or we can delete an app. We can get a list of all the installed policies for an app or all the apps within an org unit that are installed. And we can also clear auto launch kiosk apps as well. So there's a lot of control that we have over apps through this API. Next, another big ask and big task is controlling and defining networks. This API lets you define a network or a certificate, remove those networks or certificates or interact with saved networks to set which network a client and a device should be using. On the left-hand side in this example, you can set an Wi-Fi network and provide details in terms of the SSID, the security encryption and which devices and users are allowed to connect to the specific Wi-Fi network. On the right-hand side, we're showing how you can define a specific certificate to make it easier for a device to connect to that network and have greater authentication and greater control over it. I'll hide some of the plans for 2023. We've added in support for groups, which I was mentioning earlier, and added in some network policy capabilities that I just highlighted. We'll continue to enhance group support later this year with user groups and not just device groups as coming soon. We're also working to improve the schema improvements overall, which will provide you default and allowed values. So you can first retrieve what is possible with the API, what is the default and what is the allowed value. And then make another API call to set the policy correctly. So this gives you the ability as we add new policies to understand what's already there and then what you can actually set with it. So this is gonna be really powerful and really helpful as we continue to add more functionality within the API and to make your integration in your lives easier. And then finally, we've launched our partner escalation desk, which I mentioned earlier. This can be your first stop for questions. They can provide you with best practices. Support resources, documentation, and reference implementation. For more details, that's another starting point. You can take a look at the Chrome Policy API overview guide. You can use your, you have a QR code here and we can share out a link to this as well. It's on our public documentation site.  As next steps, you can reach out to our partner escalation desk and they can help you with questions. They can provide you with best practices, support resources, documentation, and reference materials. I'll now pass it back on to Rich for our next steps. Thank you, Anker. In closing, please visit the Chrome Enterprise Developer website for additional information that will supplement your learning. This concludes today's presentation. We look forward to seeing you at the next webinar. Thanks for joining and have a great day.


Hello everyone and welcome to the LGBTIQA Plus and Mental Health Collab Lab. My name is Erin Halligan. I'm the Director of Mental Health and Suicide Prevention at the LGBTIQ Plus Health Australia. My pronouns are she, her, and I identify as someone with lived experience of mental illness and suicide. My background is mainly in mental health and suicide prevention in both government and community sectors. I'd like to begin today by acknowledging the traditional owners of the land on which we come together. I'm meeting from the Wuradjuri Nation. I'd like to pay my respects to elders past, present and emerging. And I'd like to recognise that sovereignty of this land was never ceded and always was and always will be Aboriginal land. And so welcome everybody. I hope everyone's well and they're different places of the country. We've got a bigger audience here and I know you're all from different areas with different disciplines and roles. This is excellent because the spirit of today's session is to learn as much as possible about interdisciplinary collaborative care. And by the end of this session we'll hope to have both an increased confidence to participate in interdisciplinary collaborative care when responding to LGBTIQA plus mental health presentations and a better understanding of how interdisciplinary collaborative care can contribute to better outcomes for LGBTIQA plus community members who are experiencing mental health issues. Today we'll be working in three parts. In part one we'll be together in this main meeting room. I'll provide you with an overview of the field of LGBTIQA plus and mental health with a particular focus on why it's important that we talk about mental health within our community and particularly the most recent impacts of the COVID pandemic on our community and how it has exacerbated the vulnerabilities to mental illness and suicide. Part two is where, as they say here, the fun begins. You'll move to moderated breakout rooms where there will be a task to collaboratively develop a mental health plan for a vignette developed specifically for this activity. You'll be in safe hands. The moderators with selected for this task I'll introduce to you shortly and I'll also be dropping in and out of the rooms to touch base and see what's happening for us to all have come together talk at the end. And that leads you to the part three where we'll turn to the main meeting room and share our learnings and insights about the challenges, merits and hurdles to engaging with collaborative care in our community. I might get Emily to jump in now just to talk through how we can interact with the technology. Thank you so much, Erin. Welcome everybody. So just a reminder in part one and three, you can find the chat box located to the right hand side of your screen. And this can be used to engage with your other delegates and peers. From here, you can also send your other delegates direct messages. Certainly as well, please pop into the chat or the QA box if you have any technical issues and we'll be here to support as well. Just a reminder, your chat box is located on that right hand side of your screen. It should be that first icon and the Q&A is located just below that. Throughout the session, you may also be asked to complete a poll and that's the fourth icon down as well. Again, if you have any trouble locating these or have any tech issues throughout, please don't hesitate to pop it in the chat and we'll be there to support. Thanks, Erin. Great. Thanks, Emily. And I'll put my hand up to ask for your support if anyone else has any issues in the day. I'm mindful of the large numbers that we have for the interactions in the beginning and the end sessions as we'll all be coming together. And this will be limited to the chat feature. We won't be fielding content questions from here. You could send them via direct message or to an individual delegate. And again, if you want any advice from Emily, you can also put that in the chat function as well. In the part two areas, there'll be a lot of interactivity and each moderator will negotiate how this happens directly with the breakout rooms. So one of your first tasks, make sure your camera's on, make sure your microphone's muted and follow the lead of the moderator to establish how you're all going to work together. Great. So I'd like to begin by providing some context to why it's so important to talk about mental health and suicide for the LGBTIQ plus community. And also reflect on the particular adverse impacts COVID has had on our community. So next slide, please. And I'd also like to begin by acknowledging people with lived experience. The individual and collective contributions of those with lived experience essential to co-designing and delivering the work that we do. And every individual journey is unique and is a valued contribution to our commitment to the mental health and suicide prevention programs and advocacy. And I'd like to recognise and thank everyone here today who might be, you know, involved or have experiences or lived experience in however many ways, shape or form with mental health and suicide. So next slide, please. So why is it important we talk about LGBTIQ plus mental health? Although many lesbian, gay, bisexual, transgender, intersex, queer people and other sexuality and gender diverse people live healthy and happy lives, a disproportionate number experienced poorer mental health outcomes and have a higher risk of suicidal behaviours compared to the broader population.  The adverse health outcomes can be seen often directly related to stigma, prejudice, discrimination and abuse experienced due to being part of our diverse community. Next slide, please. So here are some sobering facts for you to have a look at. As you can see, the rates of mental illness and psychological distress, suicide and suicidality are exponentially higher than the general population. This information I have here has been collected by LGBTIQ plus Health Australia and in its Suicide Data Snapshot and consolidates evidence that we've put together in our private lives report with the Trobe University. It's important to note in that this is privately, independently collated data as rates of mortality is currently not reflected in ABS causes of death data nor in any other health data such this places significant limitations on any real trends and subsequent funding and investment into mental health and suicide prevention programs and services and our understanding of the circumstances and impacts on the community more broadly. Next slide, please. And yes, so why are there worst outcomes for LGBTIQ plus people? LGBTIQ plus people are exposed to many experiences and factors that might not exist for non-LGBTIQ plus people. Basic experience that tend to increase a person's vulnerability in society are often more present. This can include homelessness, financial and security, poverty, workplace distress, unemployment, family and domestic violence, social isolation, clinical mental illness, diagnosis, poor access to primary health and poor access to mental health care. These vulnerabilities are compounded in times of heightened environmental risk and disasters where the experiences of these risks are escalated. Next slide, please. So I'd like to talk to a study that LGBTIQ plus Health Australia has recently done in partnership with the Australian Research Centre in Sex, Health and Society at the Trobe University. And it looks at the impact of the COVID-19 pandemic and its impact on LGBTIQ plus people in Australia and their mental health as well as rates of family and domestic violence and economic situations. So next slide, please. So the report highlights the challenges and health disparities of the that are already experienced by LGBTIQ plus people prior to the pandemic that have been further exacerbated during the pandemic in the face of the new challenges. These findings will be used to address the unique and challenging needs of people going forward and now in future crises. We often reflect on circumstances with flooding and climate change and broader social and environmental impacts that, you know, can be identified with the same level of touchpoint for the vulnerabilities of our community. And this report is looking to kind of inform, you know, inform our responses going forward. Next slide, please. So as you can see from this data, almost two thirds of participants felt that their health and well-being had gotten worse since the beginning of pandemic of these respondents who had received a previous mental health diagnosis. Seventy one percent recorded their condition worsening. The group of participants highlighted that the changes in their financial situations, their social social isolation and unsafe living environments were often the driving force for a decrease in their health and well-being. More than half of the participants experienced some form of change to their employment circumstances during the pandemic. Next slide, please. Almost one fifth experienced violence from an intimate partner during the pandemic and more than a quarter experienced violence from a family member. Most participants reported less social interaction with the families of origin, chosen family and friends. But seventy five point eight percent also recorded an increase in the use of social media and online platforms, which is a good buffer for this social isolation, given that we already know that is a significant part that already impacts the mental health and suicidality of people in the LGBTIQ plus community. Additionally, the use of tobacco, alcohol and other drugs increased with over eighty three percent of participants who consumed alcohol and consuming more during the pandemic period. So what I'd like to recognise again is this data is obtained through independent research. LGBTIQ plus data continues to remain uncaptured in Commonwealth health data and as such, there continues to be limitations in our understanding when being able to apply any of what we know and what we understand to proper investment and service provision for our community. The government policies consistently reflect LGBTIQ plus people as a priority population. So there obviously is broader recognition by government that this is an area that needs to be addressed and, you know, we need to have a priority for more work to be done on this, particularly data to resolve this issue. Currently, the National Suicide Prevention of officers developing a new National Suicide Prevention Strategy, which we are closely working with them on to make sure that we include some of these recommendations and the findings of this report. If you go to the next slide, I have put in there some brief detail on our website where you can find the report, but I'm also happy to circulate the resource information, as I can see that coming up as a question for people more generally on our website and on Lautraub's website. So that's my background to the presentations. So I thought I should now introduce you to our moderators. So we have Nova Delaney, who is a consultant psychologist and a board-approved supervisor that operates exclusively in the field of eHealth telepsychology provision in Australia. Emma Love, who works with Diverse Voices, an LGBTIQ plus peer helpline. Emma's experience in mental health and well-being with the LGBTIQ plus community. Brent Mackey, who is the director of policy strategy and research at the AIDS Council of New South Wales. He has experienced an expertise in communications, population health, sexual health, drug and alcohol and mental health policy and research. And Dan Maida, who has extensive experience in narrative practice and currently works with the Dullet Center. So welcome everybody and you will obviously have more detail to talk to about your respective careers and expertise in your breakout rooms. So I'll leave it there. And I would like to get perhaps Emily to jump in to get you out all into your groups, to pop into each of your groups. And it seemed like everyone was really engaged and dynamic and I've just noticed here that there's a comment about it being the most emotional session. And I really just want to pick up on that because talking about these particular case studies and vignettes is a really kind of an intimate experience. And it's always important when we talk about mental health that we do check in and just reflect on sort of individually stuff that we've talked about and how it may impact us as people with lived experience or people who might just be newly absorbing information. And so, yeah, so I just reiterate just being able to have some space for yourself to regroup if you need it. Having said that, we are needing to push on the time a little bit. So I managed to pick up on some great sort of topics just off the little bits that I've heard and what's been really interesting is picking up on some of the issues around intersectionality and the collaboration across services and picking up on where existing services can be better connected with the community and how we can better support them to be able to engage. But I will throw to my team of people who, well, not my team, the team of people who are conducting these sessions. So Nova, would you like to start? You're on the top of my list, so apologies. Sure, so I think the group with my vignette, we really sort of really tried to explore and understand how to check our assumptions, understand that we may not or accept that we may not know what we're doing or be as culturally competent as we need to be and be as client centered as we really can. So going from this bottom up approach of how the individual is interpreting themselves, their world, their priorities and where they're wanting to work. It was a very complex vignette and I do thank the group for sort of sitting with the complexities and how there was just a lot of things to challenge and think about the humility that was within the room around just trying to do the best with what we have and what we know and being really quite humble within that. Yeah, great. Thank you. That sounds like a great outcome for that. Emma? Yeah, there was, I think a real sense of overwhelm in terms of the complexities felt by our delegates in my breakout room. There were, again, same as sort of Nova in that my vignette was quite complex and there was a lot going on. There's a lot felt in terms of the collaboration between all service providers and what might be available in a city versus regional as well and how to manage those sorts of things focusing on how to stabilize a person and all the different complexities and levels in which that would need to occur and the barriers that they face as well. So in terms of, you might have all of these excellent providers but they all have extremely long wait lists or available at different times as well. So yeah, so that was very difficult. We all sort of agreed that providing a wraparound support and safety net for the person would be very beneficial and that's where a lot of us would start but being able to effectively do that with the barriers and the lack of those providers being in place, especially ones that are trained and knowledgeable within the rainbow community were certainly felt, yeah. Great, thank you and Brent. Yeah, we had a great discussion in my breakout room and again, we really looked at a range of, obviously we had a, the person in the vignette had multiple issues and the intersectionality of a whole lot of various components of their life impacting on the issues for that person. So we really looked at ways that services code could become more inclusive and more understanding of the issues for LTPTIQ people but also because the vignette also overlapped into Aboriginal and Torres Strait Islander communities, how they can work with the person in the vignette to work with those communities in terms of putting a recovery plan for them, which is really heartening to see from the discussion in the group. We really talked about obviously how to prioritise what was important and how to work with the person in the vignette in terms of working on how to first, you know, developing inroads into providing services for them and getting them into services, which was, you know, it's a complex area, much like Emma and Nova's vignette. And so we talked around where we would go with that. But really, I think it's developing a care plan that looked at a multiple range of services engaging with this person, including mainstream services, but possibly specialist with LTPTI or Aboriginal and Torres Strait Islander services. Looking at a case manager or process, I think it was a discussion around a really interesting sounding service in Victoria around, I think it was called Model and Complex Needs Care Admission, but working around that management plan involving a range of different services to work with the person in question. Yeah, but also acknowledging, I mean, importantly, that these services have to be understanding of and inclusive of LTPTI people and other cultural and community factors that have come into play. Great. Thanks, Brent and then. Thanks, Erin. We were speaking about a non-binary neurodivergent young person of color living with chronic illness, which, you know, sounds like a lot of things. Also sounds like most of the people that I work with. So really a common combination of experiences and identities. In particular, one of the things we spoke a lot about was the intersection of transphobia and racism, the experience of experiencing racism within LBTIQ services and communities, the kinds of expectations that both mainstream and LBTIQ services might have around coming out that can be, you know, essentially like centering white experiences and maybe a really inappropriate kind of binary for people of color or people with cultural communities that are not from the white Christian Judeo complex. We spoke about the importance of having services that can support families, which again, when we have, you know, individualized support, that can be something that really falls short and can make a huge difference in terms of safety of trans young people in particular, but also well-being because, you know, especially when we're thinking about the pandemic and young people being at home with families who may not know how to use their preferred name and pronouns, that can be a highly stressful experience without the opportunity to move into community spaces where they are being referred to in their preferred ways and seen in those other parts of their experience. I think that really as well, we just acknowledged that the idea of trying to facilitate interdisciplinary care is not at all easy when we're thinking about this stuff. It's a totally different conundrum if you're in a metropolitan region versus a regional area because in regional and remote areas, the services just don't exist. The safety concerns are different and it can be really hard work trying to imagine who we might refer folks to and whether that's going to be a safe enough kind of service. So what's our responsibility as practitioners to kind of go above and beyond and do the work of finding out about what are the practices of these services that we're referring to? What is their level of skill and competency? What do they need to know about the name and pronouns that, you know, are used by the folks that we work with and in what circumstances in order to keep them safe? So, yeah, I guess the only one other thing I mentioned, which I think is really important to consider is considering sexual violence services and such. You know, obviously domestic violence services will fit within this as well, but services that are likely to be quite binary gendered centric and what it might look like for non-binary folks navigating those services, especially for folks who've experienced sexual violence from women or trans folks.  So if we are not ensuring that those services can not make assumptions about the gender of the perpetrator and things, we can make sure that they are more inclusive. Anyway, there's a lot of work. We've all got to do. Woohoo! Woohoo! Thank you. Thanks so much, guys. That was really good. And I think, I mean, I think so a few takeaways. Firstly, the word complexity and complex has obviously come up in each of your feedback. And I think that's a really important thing to pick up on because I think often in the world where the LGBTIQ plus people, you know, service providers and community workers, we kind of become a little bit lost in how complex the group is, our community is. And it is, we're talking about, you know, what they say, diversity within diversity. We've got in LGBTIQ plus people, we've got all these people from all backgrounds with all different needs and being able to come up with particular situations and vignettes where, like you said, Zan, although it sounds complex, this is sort of people that we engage with often and all the time. And these are the people who sort of present and, you know, first and foremost in needing support because of all these barriers. So I'm really glad that that was picked up on and I'm really grateful that everyone kind of was able to engage and, you know, feel compelled to contribute, even though there was that huge complexity there. And noting as well that the emotional toll that takes in those discussions. I think that the two bits that I can identify have come up with was the idea around wraparound services and taking person-centered approaches to care and looking at how to connect existing services and provide inroads for collaboration with stuff that's already happening, having care plans for people to be able to, like, move through the system and identifying where the barriers might be that we could improve the service system to help support the community better. I think the other area that we talked about a lot was the intersectionality of the community and intersectionality across different areas. I think we noticed people talking about racism and sexual violence and gender and then, you know, just you're saying then, you know, Aboriginal community groups and how the challenges are in being able to connect those levels of vulnerability that coexist already with the vulnerabilities that the LGBTIQ plus people already have. And these are big questions, you know, we don't have answers to these and so that's why these discussions are really good and important and I'm really, really motivated that everyone's kind of been out to have these, yeah, these meetings and talks. So, now, where are we at? So that's with some of that up. I think we've got a bit more time. Was there anything else, Emily, that we needed to touch base on? I know some people were doing some polls. Did they get reflected anywhere? Hi, everyone. Yeah, the polls in the breakout rooms and they're not currently in this session, but what we would love to encourage everyone to do is get started on the event survey. So that's just located on the right hand side of your screen and it's available there to start completing. But certainly, Erin, if there wasn't anything else, I can let everyone know what's coming up next as well. Yeah, look, I just wanted to again take the opportunity to make sure that we all take some space to reflect on things that we've talked about today, to be able to have some self-care and recognise that all these conversations are very challenging and confronting and we're talking about things that are very, distressing for a lot of people and distressing for people to hear about should you not have been exposed to these stories before and also people with lived experience of mental illness and mental distress and suicide and suicidality. Myself, as someone with lived experience, being able to engage in these conversations in a strong way can sometimes require some fortitude and so I'm really grateful for people with lived experience, particularly who've come today to be able to provide that input and to feel, I hope, supported and encouraged by these conversations. But yeah, I mean, I, if anyone has sort of further feedback that they would like to provide in relation to that by all means, you know, put something in the chat or reach out via one of the messages. And I note that there is a guided mindful session, mindfulness session, which will be beginning 15 minutes for 15 minutes at 5.15. So that's also a good opportunity for, you know, yourself as, you know, to personally kind of take some time to reflect. But yeah, I mean, I just, other than that, I'm really grateful and thankful for you all participating today and hopefully you got something good out of it and I'd like to thank Xan, Brent, Emma and Nova for all their work and development of the vignettes as well because they were quite comprehensive and like from what I heard in the little bits that I attended were very well managed and had lots of good conversations. So yeah, thank you everybody and if you want to finish off Emily about the surveys.


The study found that the SV had good face and content validity, as well as good reliability.  This article was authored by Shahzad Sanyari, Mohammed Reza Mohammadi Soleimani, and Afsane Karamat.