introduce some rather abstract objects and today I want to try and get some very computational and concrete methods for getting at these things in very explicit ways. So first of all I want to review some of the very nice constructions that I first learned from Kassan Wickelgren, Wickelgren about Euler classes and local indices, very explicit formulas for these Euler classes. So you remember we had a vector bundle on some say smooth scheme X and with zero section as zero then we have the Tom class and we have some co-mology theory E. Let's work over field K and we have some Tom class in theory Tom space so R here is the rank and of course the Tom the Tom space is just V modulo V minus to zero section so this is really the same thing as the co-mology with supports along the zero section on V and I forgot there's some kind of depending on the type of theory there's some kind of orientation information if the theory is oriented then this funny squiggle isn't there so the squiggle is equal to the empty set in the oriented case it could be the inverse of the determinant of V in the SL oriented case and it could be minus V in the general case. So that's the Tom class that lives there and the Euler class is just the pullback by the zero section of the Tom class okay and this lives in the theory on X again with whatever the orientation is okay now suppose instead of this section S zero we have just some arbitrary section S well then we get the same thing you can always do a A1 homotopy between S zero and S by just multiplying S by T and letting T go to zero so that says this is also the Euler class is also equal to the pullback of the Tom class in here but see when you pull back the Tom class it pulls also pulls the support aspect so if you let Z the S inverse of the zero section then you actually get a sort of a Euler class with supports on Z but this this will depend on the section S because if you move the section S it'll move the zero locus Z so you can't expect that to be independent but in any case so we put it in the notation and this lives in the co homology on X with what are their orientation with supports on Z okay so let's we'll be interested in the special case where yeah the rank of the bundle is equal to the dimension of X so let's just call it D and in that case a general section would have only zero dimensional zero locus so let's assume that let's assume that the dimension of Z is zero but we won't assume that the intersection the pullback of the zero section is transverse so let's just say that Z is some finite set of points and there might be some kind of multiplicities along these points and these Euler class with support is kind of telling you about what the multiplicities are so since it breaks up into pieces this this this guy here this co homology with supports also breaks up into pieces so it says this Euler class with supports on Z is just a sum of pieces Euler class with supports on each of the Z eyes okay and so for simplicity we can just let us assume we can localize around each of these points let's just assume Z is Z a single point just for the purpose of discussion and we want a formula for this local Euler class okay so well the first case let me say I won't say too much about this but the first case is when E is just a typical homology and then this 2dd part so D is equal to R so 2dd part is just the child group of co-dimension D cycles on X which is just the zero cycles because X has dimension D and then if you put the supports condition in you have a purity theorem on Z this is just child zero of the point Z which is just the integers times Z and what you're getting then this local Euler class is just equal to the some multiple of Z and what multiple is it is just equal to the intersection multiplicity of the section S of X with the zero section at the point Z times this single point Z okay so that's sort of the classical case let's do something a little more complicated second case is we take e to be k theory so this is the spectrum representing usual k theory and then the e2d we have a bought periodicity the 2dd doesn't do anything this is just k naught of X if we look at it with supports well I'm gonna do something a little perhaps unsuspected here it's k naught of X is really k naught of perfect complexes on X and k naught with supports is k naught of perfect complexes on X whose co-homology is supported at Z and on the other hand then you have some kind of resolution theorem which tells you that this is the same thing as k naught of the category of modules over the local ring of finite length in other words supported at Z okay and then you have a Davies-Age theorem which says well if I have a module of finite length I can find the filtration so this is this is like the resolution theorem and I have the Davies-Age theorem which says this is the same thing as k naught of just the residue field kz k naught of Z and of course this is just the integers by rank times the class of the residue alright so the reason I went through this is we want to calculate what is this e z we asked well remember this is I take s star of the Tom class of V where I'm taking the Tom class in k theory and remember how we got the Tom class in k theory the Tom class in k theory is the kazoo complex so I have this kazoo complex for the vector bundle V with respect to its canonical section and so what is that that's you call that's a p star v dual going to o v by the canonical section what's called t dual about the zeros of this is just the zero section so that's the structure sheet of the zero section and then you have the wedge two of that etc all the way up to the top wedge okay so when you take s star of that well now the point is that let's say we're working near work near z and let's take a local basis for for v so basis of sections you want to ed and then we can write our section s as some s i e i and the fact that the zero locus is just concentrated at z means that the this set of elements s one up to sd a sequence of elements is a regular sequence in the local ring o x at z and when you have a regular sequence the kazoo complex for that regular sequence remains exact and that's what happens when you pull this back so when you take s star of the kazoo complex or the monocle and use this basis this is just the kazoo complex for o z o x z to the d with respect to these elements s one of the sd so that's what you get it's the same thing as what I wrote here except it becomes a little more concrete it's just that we have o x so we just call o x o just o to the d going to all the old mod the sections we call that thing old bar and this is just the map you know where the bases on the e i goes to s i and then you have the various exterior powers on it all the way up to the top exterior power and so that remains exact okay but since it's exact so here we're over here now where it's a perfect complex with cohomology supported at z but since this is this is a resolution of all bar that says by the resolution theorem we're really getting just a class of all bar in k naught of modules of finite length over the local ring and then let's the day visage would say well I have something of finite length and then I take filtration so that the relative pieces are actually vector spaces over the residue field but to make it a little easier let's just assume that the residue field is contained in the local ring that's not really much of a restriction because if I want to make the computation I can always increase the base field to do that necessary or pass to a completion in any case when you do that when you apply day visage what are you getting then you're getting just the well on one hand you're getting the length of all bar as an all module that's the multiplicity times the class of the residue field and this is just equal to the dimension of all bar over the residue field as a subfield times the class of KZ okay so I'm sure you all know all about this it's just going over in some gory detail so this is the that's the formula for the local Euler class in that case now what I'm really interested in is the case of the Hermitian K theory so there again we have the member of the Tom class of V was the same casual complex together with the quadratic form and the quadratic form was I took the term in degree I so I think of this is V star V shifted by I it's in homological degree I and then gets paired with the term in degree D minus I of the dual sorry then just the wedge product goes into the determinant the top wedge will be determinant of the pullback of the dual so that's just the pullback of the inverse of the determinant of V but in degree D so induces the map of this Q which you think in just the casual complex answer itself into this determinant V shifted by D okay so that's the that's the casual complex with its quadratic form and then this local gadget here is just s star of this pair and where does this live well this is supposed to live let me just write the formula supposed to live in a not to be with supports on Z on X so what is this well these are quadratic form so let me call that GW growth in the bit of the perfect complexes whose comal co homology are supported on Z let's say over the local ring oh but the values are not in oh or in hey or whatever they're in this element this complex determinant inverse of the V shifted by D so that's where you end it's so this is the analog of K naught instead of K naught you have GW of the same kind of object and the value of the quadratic form is this inverse determinant in degree homological degree D so all right yeah that's a that's a question for you are you gonna Q&A I can't see the answer yes that's right how's that that better we'll be able to see it for a while okay so that's the that's where it's living and then we have the resolution theorem which tells you that this is the same thing as GW of modules of finite length over all with values in the inverse determinant now what I'm not telling you here is to get the GW to get the GW you have to say what the duality is and here there's a little technical point that the duality is given by X so we're not going to talk about that in any details so let me just hide that under the rug and then you have the Dave visage part which tells you that this is the same thing as GW of the residue field with coefficients in the determinant inverse of V but the fact that this X group you remember in sort of local growth and de-duality the X group corresponds in some sense to the top wedge of the ideal sheaf defining Z so you have to tensor with that or the maximal ideal of my local ring modulo it's square that turns out to be where it lives so there's a sort of purity twisting going on here so this goes to something so we have a bunch of rather complicated gadgets and this goes to something in here this is a fairly explicit gadget I mean these are just quadratic forms on kz vector spaces with values in this line this is this is a this is a particular one-dimensional kz vector space just to make things canonical and we should be able to write down a formula so there is a formula for this this is the formula just do a classical grand really building on some work from the 70s from Shaya and Schwarz so that's what I want to tell you about okay so we're trying to try and compute this thing okay so I'm not gonna show you how you'll go from here to here I'm just gonna show you what the answer is okay so I'm gonna leave this up here and so remember we had V what do we have we have V and we made a choice of a basis this is our local basis and we had our section s which in terms of this local basis we can write as si yeah okay and then we also have our local ring O which contains this maximal ideal and we'll make another choice we'll choose some generators for the maximum ideal we're talking about a smooth variety x over k really over kz and we'll choose we'll choose parameters t1 up to td okay so now of course the fact that the s that z is the zero locus of s just says that all the si's are in M so it says you can write si as a sum a i j e j so these a i j's that's essentially the partial derivatives of s with respect to the t's that's all that is all right and then great so now we have this element epsilon which is just the determinant of this matrix but I'm gonna view it in old bar remember old bar was old modulo the s's and it's not unreasonable the a i j's are not gonna be the ideal generated by the s's unless something funny happens in fact it's not because we divided by the t's all right so we've got that and now what do we do with that we choose so there's a choice here a kz linear map trace from old bar kz which is normalized by saying that this trace of this epsilon is one it'll turn out that the eventual the class of the quadratic form we get will not depend on the choice of this trace okay so once we've done that what do we do then we have a bilinear form kz the shea storage bilinear form where x y is just equal to the trace of x times y and we have the quadratic form q of x is just the trace of x square just what you get from the bilinear form okay so we made some choices we made choices of the e's and the t's and this is a quadratic form in kz so now I want to get a quadratic form not in kz but in this funny line and that will get rid of the choices so I just so the class of this the answer is the class of this local spoiler characteristic is exactly this quadratic form multiplied by well it should be in the inverse of the determinant so I take e1 wedge up to ed maybe it's ed up to e1 I think there's a sign issue here so that generates the determinant so it's inverse or if you like the wedge of the dual basis generates the inverse of the determinant and similarly if you take the images of the ti's there in m you look at them in m mod m squared and take this then this lives in a z tensor the inverse well it has values it's a map the quadratic form mapping o bar to a z tensor the inverse determinant of v tensor the inverse determinant of m mod square and you it's not hard to check that that's independent of the choices and that's the answer so that's a very nice result of Cass and Luekelgren so let me just make this even more concrete let's just take d equal to one and forget the bases just get the form so let's say s is just a unit times t to the n so then all bar has basis one t t to the n minus one call that the one up to the end and then okay we write s it's u times the use a unit this is u times t minus to the n minus one times t so this is our a it's a 11 if you like and then so the epsilon is just equal to ut to the n minus one bar in old bar in other words modulo which is all modulo t to the n all right and now if we take we can take the trace we make our choice of our trace by just saying trace of vi is equal to zero if i is not equal to n and let's see it should send this this epsilon is of course just u times the n so it should be one over you if i equals n okay and now we can just write down what our form is the bilinear form let's let's do that then we get a matrix this thing of x y is just equal to the trace of x y and the matrix in our basis is just one over you and this is so anti diagonal matrix okay now of course in quadratic formland one over you is up to a square that's up to a square is the same as you so this is isomorphic if you think about it you have you can pair the one with the one and the other one with the other one etc so you think about it a minute if n is even this is just n over two copies of the hyperbolic form just by pairing the corresponding off diagonal elements and if n is odd you have the middle element which gives you a one over you which will call a u just for a simplicity of notation and then you have oh sorry minus then you have the remaining off diagonal elements which contribute hyperbolic form and there's an even nicer way to write this this is just the one dimensional form u this is the form x goes to u x square that's the rotation times this form n epsilon so this is the number n viewed as a quadratic form and what does i mean by that i mean n epsilon is the sum of ones but it's the sum of minus ones minus one to the i i equals one the n minus one so because the hyperbolic form is a one plus a minus one you can see that this gives you exactly the same answer and now if we take a general d but assume well another case let's assume that our s is a sum u i t to the n i times e i then well i mean you can see what the computation is this quadratic form q s will just be the product the one dimensional form product of the u i times the product of the n i so epsilon that's an easy computation it follows from this and finally the the other simple case if it is if s is just sort of the non-degenerate case it's just gotten by a scalar matrix so here the let's assume that the a i j is symmetric and non-degenerate then this chaise-storch form is just equal to the determinant okay so and you can i mean in principle you if you have the the section explicitly you can find the form explicitly so it's it's nice and explicit all right so um let's see it's time for some further applications so i want to see how this works in a geometric situation and this is what we call the riemann provitz formula which is also can be viewed as a calculation of Euler characteristics using Morse theory all right so um let me just mention the theorem suppose we have field k x smooth objective over k c of some dimension d c smooth objective curve over k everything let's say irreducible and f from x to c um just some subjective morphism okay and let's suppose so this will have critical points in other words a general point of c let's assume that subjective and separable so it means that a general point of c the inverse image the fiber over c will be smooth and then there'll be some places where the fibers is not smooth let's assume that um f has isolated in other words finitely many critical points the critical points in other words are just where the differential is zero so let's say those are points x1 xn and a technical assumption we don't really need this but uh just to make the formula simpler let's assume that the field extensions are all separate so for example if k were a perfect field that would not be so then we have a formula for this quadratic Euler characteristic minus one to the dimension of x times the Euler characteristic in the quadratic sense you know this lives in gw okay there's the following so you have the sum over the critical points of what well you have these um ah i forgot an assumption we assume further assume awesome sorry we have an isomorphism row of the um utilizing sheaf we're just the one forms on c isomorphic to the square of some line sorry that's an important assumption i forgot that in okay so then when you do that this d notice that um df is a section of not just it's not a section of omega on x it's a section of omega tensor f star of omega c inverse so this is where the Euler classes would normally live but we're going to put them inside of this guy here by a row tensor l to the minus two or f star of l to the minus two so i've kind of run out of room with my explanations so remember we're thinking of df is living in this vector a section of this vector one and so if we do that we get the sum i equals one to n of the we take the trace of this quadratic form from k i extend onto k of the local Euler class for this df living in omega x tensor f star of the minus two so that's the that's the contribution for the critical points and then we have sort of the general fiber also contribute something so minus the degree over k but now this is all numerical information we take the usual first turn class of my l pull back and then times these are the usual first usual turn classes so in the chow chow ring if you like the omega x over k times the pullback of the first turn class of the omega c over k the i minus one is number times the hyperbolic one sorry for the formula giving a little messy so so why is this um so that's the so this is uh if you take the rank so the rank applying the rank gives you the classical Riemann-Hurwitz formula and if you assume that k sits inside of R and you take the associated signature this is just Morse theory this gives you the Morse theory computation for the Euler characteristic of the real points but this is sort of a quadratic lifting of both of those computations okay and let me make um perhaps one little remark from this formula um let's suppose that the dimension the dimension of x is odd well everything is hyperbolic here this part is already hyperbolic but we already know that a um Euler characteristic of a smooth projective odd dimensional variety over k is hyperbolic so that gives you a relation it says there's this funny relation on the on this term it says that the sum of these traces of these local shortness here is also hyperbolic so um I think it's I don't know what to do with it but I think it's an amusing and perhaps surprising relation and if you make you can even make this quite specific uh if you maybe a little exercise for yourself is to take the case where x is just a hyperbolic curve mapping to p1 ramified at some set of points say 2g plus 2 uh say a subs quo reduced closed subscheme of p1 of degree 2g plus 2 over the base field by taking the square root of the defining equation and then you get a funny relation on the um corresponding differential so if you take that case it says if you take p2g plus 2 in a x so degree this that degree with um which is a separable polynomial so all the roots are different in the algebraic closure then if you take um x tilde to be the zero subscheme then you have a tracemath from uh the Groten-Dieg-Wittring that this will be let's assume that this is uh their base field again is uh separable so this is a well separable polynomial so it's a separable extension of the total quotient field this might be several pieces and then it says that if you take the trace this tracemath apply to the one-dimensional quadratic form two over the derivative of p evaluated here this is hyperbolic probably easy to tell but it's just a special case of this formula which then generalizes a higher dimension okay so i want to turn to um so that's a little bit about Euler classes i want to turn to um Euler characteristics for computation okay so i'm going to remember recall this theorem um with raxit that says if we have x over k smooth objective then this Euler characteristic has an explicit form it's equal to this um sum of the hp x omega q's shifted by q minus p together with this um cup product form composed with the trace map so x dimension d this will go from h d x all right um so the thing is this is um mostly hyperbolic so we already know that if the dimension is um odd then everything is hyperbolic but you can see that because if the dimension is odd then you're always pairing an hp q with an h d minus p d minus q and those are never the same group so you're always pairing one group with its dual and that's automatically hyperbolic and so if you throw away those parts you see the only thing this is this is equal to some hyperbolic part plus the form just in the middle dimension so let's assume that d is equal to 2m then this is just equal to the cup product form on h m x omega m into k which is just saying x goes to the trace of x squared so that's a very explicit just one piece now since you have such an explicit form what you can do with this is this this enables um sort of uh twisting constructions so let me just say a word about this let's just take the case sorry mark just real quick kirsten says that you have too many m's in your formulas oh too many m's yeah yeah there we go hey yeah thanks thanks kirsten so this uh too many m's just thank you i see that so this enables twisting so let's just take d equals 2 first non trivial case and then we're just looking at a quadratic form on h1 x omega x going by this k x goes to trace of x square now if we we're working over a field k but if we go up to k bar we know that h1 x k bar is just the base extension here and we have the same form right we have the exact same form here and so the the h1 x sits inside here as a k subspace so explicitly what it says is you can take the form so for q you can take this let me just call this form q you can take q for x over k bar and then restrict it to this guy here this will land in k and this is just equal to your original form q so now usually what happens is you might have a natural basis for the thing over the algebraic closure where you can easily compute the form and then you just have to see i have to find now a k basis of this k subspace and that will tell me how to compute my original q so for example one way you do this so example if uh pg of x is equal to zero then this h1 x k bar or even over k is just equal to k bar times let's say and the q is equal to zero q is a different q so let's say h h1 of x o x equal to zero then this thing is just k times the group of x over k bar and you the this form here q x k bar is just equal to the intersection form on pic extended k bar whoops extended k bar linearly so that's often something you can compute explicitly and then you have this question of thin binding this k vector space in here and making a basis so that's the that's the twisting construction and i just want to remark is you can't do that in the golden digvid ring the golden digvid ring doesn't have descent but if you have an explicit form then you can do this twisting construction okay so i won't running out of time let me not do an example of that but instead let me turn to another explicit case explicit case of hyper surfaces so uh one simple case is a diagonal hyper surfaces so in other words generalized for my hyper surfaces so that's you have a hyper surface given by a sum of like this ti to some power e so this is sitting inside of projective space of dimension n plus one so this is a hyper surface of uh degree e dimension n and let's assume n is even all right so um in that case the remand let me just tell you what the answer is the remand herwitz formula you can apply it to the following thing you can take the map you have a rational map of x to p one which is just sending x zero to x n plus one to the last two coordinates of course that's not defined when the last two coordinates is zero that defines a hyper surface z it's uh co-dimension one sub-variety of x z which is just another diagonal hyper surface in a projective space of um smaller dimension so if you blow this up then you have a morphism here f you can apply the remand herwitz formula to this guy plus uh induction for the z plus uh a blow up formula you have a formula for the blow up let me be interested in time let me just say that or the Euler characteristic and that leads the following formula that um in this diagonal case the Euler characteristic um it's quite simple it's one dimensional form e plus some multiple depending only on n and e of the hyperbolic form this is again if e is now if e is odd the odd degree case is simpler and the even case you have an extra term times again the product of the ai plus a multiple of the hyperbolic form this is for e so for example if if e is two if you have a quadric then uh this so in other words then you're just taking a quadratic form every quadratic form can be uh t can be diagonalized so this is some quadratic form q and this product of the ai is just the discriminant of q and so you get the um right the Euler characteristic of the quadric x is just equal to two plus minus two times the discriminant and then plus i guess n over two times the i so quite explicit and you get this well maybe a little disappointing you just get this rather elementary invariant of the quadric in the Euler characteristic all right so um that's nice but not every hyper surface is diagonal so what can you do for a general hyper surface well there's there's a way to do that also let's uh make the general hyper surface so you let's take x inside of t n plus one over k uh smooth and defined by some degree e polynomial does the question mark uh what is a and e it's a number so a and e yeah thanks a and e is a number um i mean you can write down what it is it's um the number that it is let me put it this way you can find it's not hard to find a formula for the Euler the topological Euler characteristic of a hyper surface of degree e in p n plus one let's say you know that then the rank of this quadratic Euler characteristic is that number on the other hand the rank of the h is two so you have that equation to solve in other words the this a and e in the odd case is the topological Euler characteristic of x minus one then divided by two and in the even case it's the topological Euler characteristic minus two divided by two yeah thanks for the question okay so um back to the general case uh let's say we take our um hyper surface defined by um degree e polynomial and we assume it's smooth and let's assume that e is prime to the characteristic of k then we have the Jacobian ring so this is uh Jacobian ring um this is defined by taking modding out by the partial derivatives of f and under this assumption that e is prime to the characteristic the Euler formula tells you that the original f is in this idea and what that means is of course f being smooth means that f together with all of its partial derivative have no zeros in the projective space which means if you look in the affine space this thing here has zero locus um this ideal has zero locus just zero in the affine space which means that the Jacobian ring is a finite dimensional a algebra and more more over you have n plus two variables and you have n plus two equations it's also a complete intersection the idea complete intersection okay so this uh the general theory of such rings tells you that it has a one dimensional socket in other words you look at palm of k which is in other words you take k modulo the variables mapping into this ring there's a one dimensional image of that and so it has a socket one dimensional and it's generated by this shear storage element so in other words you write the partial of fi with respect to that's partial of f with respect to ti you write it as a sum the aij times tj when you let epsilon be the determinant of this aij the bar so this is in jf then the socket is equal to k times epsilon so we have a quadratic form we have the shear storage quadratic form from jf or let's say and going with the quadratic form or the pairing maybe i just went down the okay so um let's see i need the trace form but i'll say what that is in a second you have to choose a trace form i choose the trace form since jf is graded see these are all all these partial derivatives are all degree e minus one so jf is a graded ring so i define it this way i take x yj and if these are homogeneous then this is equal to zero ah so what's the i see what's the degree of this thing this is in degree to count let's see the derivatives have degree e minus one that means these coefficients have degree e minus two and there are n plus two of them so this is in degree e minus two times n plus two and so for homogeneous elements x and y the pair will be zero if degree of x plus degree of y is not equal to this magic number and it's equal to lambda if x times y is equal to lambda that's absolute in other words the trace form is just the one which says epsilon to zero and all the other graded components in jf to zero okay so what does this have to do with anything well on the other hand we have this really beautiful theorem of clemens and griffiths arising from griffith's study of rational integrals back in the seven days which says that there's an isomorphism from hq x omega p isomorphic to the certain graded piece in this jacobian ring and it's in this degree q plus one times e minus n minus two so this is a generalization of the classical theorem on residues which tells you that the you look at the case of sorry this is not quite right I have to put a little prim here and this is for p plus q equals the dimension n so I'll tell you what the prim means in a minute it usually doesn't mean anything but yes so if q is zero and p is n this is just this is just the canonical this is just the global sections of the canonical sheaf and we know what the canonical sheaf is by the adjunction formula so this is just the generalization of that to all the cosmology so and what's the primitive guy hq x omega p primitive is equal to um zero if p plus q is not equal to m it's equal to the h the whole thing if p plus q equals n and p is not equal to q in other words they're both different from this half and what is it in the interesting case well if p equals q equals half the dimension so this is m which is n over two then we have the class of a linear section so this x is in a as a hyper surface in a projective space and we have the class of a linear section of dimension let's see equal of co-dimension n over two this lives in h m x omega m and we have our quadratic form the cup product followed by the trace map so this is equal to the perpendicular with respect to l okay so you're losing exactly one class it turns out that the p plus q is not equal to n everything in the co-homology of x in degrees different from n comes from the projective space so it's not so interesting all right so that's the clemen griffith's theorem and now we have two things we have let's see we have of course and this is perpendicular with respect to the intersection form which defines our quadratic Euler characteristic then this j f is telling us should be telling us something about the remaining part of the quadratic Euler characteristic right we know what the quadratic Euler characteristic is sorry we know what this q on l is it's just equal to the degree of l squared which is equal to my e so we know what that part is and this is telling us what the the remaining part where p plus q is not equal to n this just contributes a hyperbolic part in the Euler in the quadratic Euler characteristic so we might expect that this quadratic form here this j is storage quadratic form on these this part of the jacobian ring tells you exactly the Euler characteristic and that's almost right so what the theorem is is essentially due to clemen's griffith's although not quite but more or less is that the j is source form on this direct sum over q of this jf q plus one where is it q plus one times e minus n minus two is equal to well let's see you have to multiply by minus e and then of course we have to add in this e due to the l and then plus the hyperbolic form and i think it comes out to be um let's see what is this n times the hyperbolic in over two yes once in over two times the hyperbolic form is equal to this okay so you have a really very explicit way of calculating these things and you can even use this to check the previous answer on the diagonal hyper surfaces and the answer is yes they agree okay so i think um i'll stop here i wanted to uh say a few more examples and also talk about the theory of characteristic classes for um for bundles in uh bitcoolmology but i absolutely don't have time for that so thanks very much okay yeah thanks mark bill thank you for a wonderful talk i particularly enjoyed all the examples um and i want to ask if you think there are other types of examples one should try to work out oh yeah there are lots of examples so i mean one example just yeah i mean it's hard to tell exactly what kind of what kind of examples you might be interested in but just to test your computational powers um that there's this i think it's called this uh quine cubic in the surface case we look at this equation nice cubic surface and then uh what's the Euler you can use this they can i mean it's something you can do sitting in front of the tv you have nothing better to do than the Euler if this is the defining equation for x then calculate you get a 25 we know the rank has to be um what does it have to be has to be nine because that's the Euler characteristic of a cubic surface and this five is a little surprising but it's not so surprising because it means that this thing is actually singular mod five but it's not singular mod three so even in the characteristic so the theorem doesn't work in characteristic three but since you can specialize this it says that this form really does calculate the Euler characteristic also in characteristic so can write down your favorite examples can we compute the Euler characteristic of k three surfaces well i mean you could do it for quartic surfaces in p three certainly by this method you could um write down any equation and it's probably not so hard to find the answer for any specific equation it just um i would think if you you could might even be able to get an answer or say the general cubic equation but the formula would probably look rather complicated and mysterious too many too many coefficients but um yeah i what about relating it to a degree six curve over which the k three is branched off um yeah because then it would require a modification of this method so this is really based on the this Clemens Griffiths theorem telling you how to compute the um hodge cohomology in terms of the jacobian ring and if you take a branched cover well it's in a it's in a um weighted projective space naturally so you have to um modify that but i'm pretty sure that that should work there should be a way um to make that work it's not really special for projective space you need the basic theorems you need some bot vanishing theorem the difference in weighted projective space is um you have to be careful about the singularities in the weighted projective space so that's some technical problem but if you stay away from the singularities i'm sure that that would work um uh following up on that you for your riman herwitz you have this multiplicativity of Euler characteristics that um uh you know it's great it doesn't hold generally but in the case of a branching over a curve you show that um the whole the Euler characteristic of the whole total space is the base Euler characteristic of the base times the Euler characteristic of the general fiber minus these correction terms yeah and in general we don't have this multiplicativity that you could ask um you know maybe in the case of uh k3 branched over a degree six some nice multiplicativity for the p2 and then uh uh i see uh huh um i mean if you take if you well if you take it over if you take a cover branched over a curve then you can try and do it by the riman herwitz formula also just by taking a pencil in p2 and um counting you you get um if you take a general pencil then you'll i mean that might be a way to do it you get a general pencil and you'll have the um the singular will be ordinary double points so it should be fairly explicit to calculate that i mean it's a finite map it's not a pencil of curves um well i was thinking about the k3 surface the case yes finite map to p2 and but it's branched on a degree six curve right it's i guess it's the two to one case is that which k3 you're talking about yes yes yeah so then if you take um you take a general point in p2 and project to p1 from that point then that will if you blow up your k3 at the two points lying over the general point then you have a morphism of the k3 to p1 and the singular fibers of that will be when your line is tangent to the degree six curve great so if you take a general point you can arrange that so it's just tangent in an ordinary it's you know an ordinary tangent and that will just give you an ordinary double point a quadratic singularity on the k on the fiber over the k3 and um you know it'll just be a single point so there'll all be rational points so you don't have any trace to compute so it should be fairly explicit you need a you you'd get something in terms of this cover i mean i guess you can figure out how many you know how many double points are there how many how many tangencies are there for degree six curve you have that extension of the base field on the degree from the degree six curve the the sink the ramification points for that projection to p1 that's some fairly large extension and presumably the the trace the trace form for that extension then suitably modified by the value of your local local indices at the double points upstairs on the surface will give you the answer but how explicit that is i don't know i i think i see what you're saying that sounds really fun i i have a um more basic question too it had if could you go up to where you had the um the the form on the jacobian ring this one um no farther up before the yeah right so see how you have degree um uh x plus degree y is not equal to the the degree of epsilon in there is it a fact that um the uh the um the that in the jacobian ring all the terms of this degree are a multiple of epsilon so are you saying that the way jacobians rings work out it's one-dimensional that's i mean that it's also the degree not this is thought the sockle is in this degree but the sockle yeah i didn't say this this is also equal to the jacobian ring yeah thanks of this degree we should have seen that great thanks thanks thanks for pointing that out yeah that's exactly right the sockle is equal to that graded component that degree component in the in the jacobian ring yeah thanks good point okay any other questions i don't think so then i want to thank mark again thanks for a wonderful talk


Good afternoon everyone and welcome. This is Melissa Armo with the stockswush.com. I'm here today to review the QQQs. The reason that I thought it was a good idea to review the QQQs today is because the market did actually have a bearish gap down on Friday. Would I have shorted this? No. If you shorted the market on Friday, did you make any money? Yeah, a little bit of smidgen. Not really anything of any significance, really went nowhere. Why? This is not a good bearish gap to short. High of the market up here was $109.30. Low is $108.60. It's not even a buck. And this is assuming you would have even shorted the high, which why would you have done that? I know that a lot of people are looking at this market and they are thinking and predicting and assuming and guessing and believing that the market is not going to go higher and they are believing that the market is lower. However, it is not. The market is going to continue higher, is going to get up over this area and over this high and go higher. It's more clear to me, it's clear on both charts, the QQQs and the SPI, but it's really, actually no, there's different reasons for why I'm saying this, but both the QQs and the SPIs are look higher. Different reasons why though? Because they're different charts. Why? Because the QQQs actually have not made an all-time new high yet, whereas the SPI has. So they've been lagging from the SPI market if you look at actually the way everything's rallied for the last few years since the rally back up and began. But the high over here is like 120 something that happened back more than 10 years ago, 15 years ago. My prediction for the end of the calendar year 2015 is that we still get up over this number and we still get up over this level. And at this point, the last high we had in the QQs was 111.16. And so for us to get up to 120 would be a 10-point jump. I think we go further than that. I'm going to say 120, 125, 130, 135. The dream target really for the QQQs this year is 150, and I'm not saying it gets there by the end of the calendar year at all. I'm saying that would be the dream target if it would. I really think 135 even. 135, 120, 125, 135 really let's say for the dream target this year for the QQQs. And I was really thinking 150, 175 when I first saw the market was higher this calendar year last October of 2014. But we really have been sideways now for mostly the spring, March, April, May, and June. And this again is another reason why many people are saying, gosh, we're having trouble going higher. We really are not doing it. We haven't made the move higher and therefore we're lower. We made a series of lower highs and lower lows. We can't get up over it. We're struggling. We've had these gap downs. We've had many, many red days in the last six weeks or two months even of trading. And I know what people are saying and they're even looking at the entire year and they're saying, well, from where we opened this year, let's just see the open of the year, the first day of the year here. From where we opened on the year was 103.76. We're still positive on the year. Okay, but not by a lot of as far as the entire overall game. It doesn't matter. We're still holding. We're still strong. We still have made a gain this year within that six month period, but really ever since March, March, April, May, and June, a lot of people are looking at this market as bearish. I'm not. In fact, I'm still looking as bullish. Some people would say sideways or chopping, but to be honest with you, we're not sideways. We're actually higher. We actually bullish. We are holding the bullish trend. We are higher. We are bullish. We are not sideways and we're not bearish. So if someone asked me what's the trend right now, the overall market and what has it been this entire year bullish? That's the answer. I'd say 100% conviction and that's it. Now, in reference to what the market's going to do, I'm still saying higher, higher yet even this calendar year. And not only that, it's going to blow. What does that mean? It means when the market does make the move, it's going to be so unexpected. And this I did predict actually in October, 2014 and December, when I sent the email out to everyone on my list, I said, be prepared for a volatile market in the year of 2015, which means there will be days when the market is red and there will be days when the market is green and there will be volatility in the market of 2015 like we've never seen. And this kind of has already started because we've seen sell-off days in here, but yet it hasn't gone anywhere to the downside. And we haven't seen really the big, big, big green days that everyone would have thought we would have seen if we were going to be higher in the upside. And yet I'm telling you, I see signs of strength all over this market. I know what weakness looks like and it isn't in this market. I know the market's higher from what I know based on gaps. And again, it's because of my gap analysis and I'm able to predict what something's going to do in the future. I look at the gaps in the QQQs and I determine from those gaps. If they are golden gaps that rate 20 points or more, per my 26-point rating systems, I determine if the bullish or bearish gaps are going to follow through. That's how I know. That is how I know to predict the move that the market is in fact higher. So given that we're in June, it's mid June, really it's almost like basically exactly half the year is over, almost. And so given that, I'm saying we're still going to blow. We are still going to blow. So really, I'm going to say 135, 135 by the end of the year, 130, 135. I think 150 is a stretch now. Still could be a dream target for the market this year, but I really think dream targets for the QQQs this year is 130, 135. We will get up on over 130 this year. That much, I know. So there you have it. Many people believe the market isn't going to make another high and many people are even thinking really now that it's definitely lower, but it's not. In fact, not only is it not lower, we will not even go anywhere near this area here again. We're not even going to go anywhere near it. That's how strong we are. That's how strong we are. Okay. And we won't even touch this. And many people think we will touch that and break that in more. So it's all about my gap analysis and I'm able to predict these things. If you go back and look at how I've been predicting the market for the last several years, I've been correct. And I will continue to just look at everything and double triple check myself and what I know. Again, it's based on my gap analysis. That's how I'm able to do it. So I'm able to see these things and that's why learning my class and taking my golden gap rating system is so powerful. It is challenging to predict what things are going to do in the longer term. It is much less easier to predict what something is going to do in a short term. In other words, it's easier to predict what something is going to do in a one minute chart rather than even an hourly chart or a day chart or a monthly chart, an annual chart. It is easier to predict something in the short, short, short term than is in the long term. And that's what I find so hilarious about the fact that many people think that day trading is so hard and they want to do swing trades instead of day trading. It's easier to predict what somebody is going to make a fast move on the day in a day trade and you're less at risk than if you are in longer term trades. It's harder to predict what somebody is going to do longer term because you can't predict the timing of it exactly when it goes to where it's going to go. And that's why these numbers have changed here given the fact that it's taken us months now to even continue to get going. And I don't know when we get going again. I really am surprised that we haven't gotten going over the high yet. I thought it would be April, May, now here it is June. I don't know if it'll be June, July, or August till we get up over the high again. I don't know the timing. I know it's coming up. That's all that I can tell you. It's coming up. It could happen any second, any day, any day we could do it. This is Melissa with thestockswish.com. If you'd like more information or if you want to learn how to trade and if you want to take my Golden Gap class, the next class is June 20th and 21st that's next weekend. It is an incredible class. It teaches you how to predict what stocks and the market are going to do before they do it by reading and analyzing gaps. And quite frankly, that's the only way to predict what something's going to do. All this other stuff, looking at pimmits and determining trends with that and moving averages or higher highs and higher lows and lower highs and lower lows. None of that stuff works with any level of consistency. That's how many people are going to get tripped up here in the market because I'm telling you this is the last thing I'm going to say, did I get market when it makes the move is going to blow? And I did predict that last year. I did say it would be one of those years where the market would blow higher. I didn't see when exactly. Again, the timing of that would be, but I knew when I saw it that it would be that the market would make an unprecedented move. That's exactly what I said in the email. I have to go back and read it, the email I sent to everyone on my list. I said the market would make an unprecedented move at that time in my mind. Hold on. I'm just going to write this down. In my mind, the unprecedented move that the market would make, I thought was based on the number. It still is going to be as far as the number of the market is going to hit as far as the target. But really what's going to end up being unprecedented about this year of 2015 is the percentage of the jump of the time that it's going to make it in the time that it's going to do it. Does that make sense? I'm going to write this down so don't forget this. So when I saw what I saw, when I made the market prediction I did in October of 2014 and then in December of 2014 I sent the email, what I saw would happen in the US stock market in 2015 that would, number one, it would be bullish the entire year unless there was a war of calamity, which there hasn't been so far. Of course that could always happen. I also predicted the market would make an unprecedented move. At that time, when I did, I thought in my mind it would be the target. It is going to still be the target because it's going to be a big target that we're going to hit. But really the unprecedentedness of what is going to happen is going to end up being the percentage of the move of the jump that we make at the time that we're going to do it. In other words, we're going to make such a massive move like a blowout in a short period of time that is what will be the unprecedented part about it. In fact, I'm just going to look here at something and see if anything has ever happened like this, so I can find it. Let me just look at something here. I'm going to go all the way back. I just want to see if we've ever done something like I'm seeing, I'm seeing, I can see it. I just want to, I just, I don't know if we've ever done something like I'm seeing, in my mind I'm seeing it. Let me just look here. Like I can see in my mind what it's going to look like. I just want to see if we've ever done something like this. We'd have been around this time if we had, but I'm not sure if we even did here. Let me just look. We're, we're never going to have seen anything like what we're going to do. That's, this is what's so amazing about what I can see this stuff. We never, we're going to do something like we've never done before. This is, I don't even know if I can explain it right. Well, I just did, but the way I'm seeing this, like if I would draw, I should just draw it, just draw a piece of paper, but I'd have to scan it and show everybody what, the gist of what I'm trying to say, I just wanted to look at that. Sorry, I just had to look at that for a second. The move that the market's going to make, I'm saying it's going to blow, but the move is going to be so aggressive. It's almost going to look vertical. That's the best way I can describe it. I was trying to find anything, the markets that were done even back when it had made all-time highs back in 2000. I was trying to see if there was anything that we bet, any section or piece of time in there that I could even say, yeah, this is what I'm saying. It's going to be like this is going to do like this. I don't, I don't see it at all. And I way passed the time of that. And even in this recovery in here, it didn't do that. So it's, that's part of what is going to happen that is going to be unprecedented. Okay, so there, I just, I'm predicting it now. The move that the market's going to make, the aggressiveness of the move of it is almost going to look like vertical where if you don't, if you don't, if you're not in it, you will completely miss it. You will, if you're not in it, if you're not in the market long, you will completely miss it. Like there will be no entry. You know how everybody likes to buy and pullbacks? There will not be any pullback. The, the move, I'm not saying it's going to go right now and go vertical and never look back and go straight up over 130, but I'm saying that it, I'm saying that it could basically I'm just, I'm just going to die that it could whenever it does and it's going to blow and it could, because I'm seeing that the move that it's going to make the way that it's going to shape itself. And this is all just, it's just, I'm so good at reading money. I'm just so good at reading money and I'm telling you what it's going to do. It's going to make a move that is so aggressively higher, that's going to happen so quick, so fast, so unexpected, it's going to blow and the percentage of the move that it makes and the time that it moves it, that it makes it will be unprecedented. That is what is going to happen this year. So I'm saying with 100% conviction and no chance of failure, the market and the QQQs will make a new high this year. I'm giving a dream target of up over 130, 135-ish, but over 130 for sure before the end of the calendar year. And I can even put out their dream target 150 because I have no idea really the percentage of the blow that we're going to make. It's going to be big. It's going to be something in a format and a fashion here that I can't even find that's ever happened. And I really saw that last year. I really saw that last year, although I thought it would just be the way that we'd rally all year. But here it is, it's June and we haven't done it really yet at all this year. Even though we have and what I'm reading that we have and we're going to and we really haven't, it's setting up to do it though. And that's what's so amazing. And that's what's so amazing. I'm just going to say one last thing. The market isn't doing anything wrong. It's not doing anything bearish. The market is still higher. How do I know this because I know how to recap? So if you want to learn how to trade for me, you can email me at Melissa at the stockswush.com. Have a fantastic weekend, everyone. Good luck.


Good afternoon sir. Thank you sir. How are you? I am good sir, how are you? Yes sir, we have a midterm break going on. Midterm break for how many months? For a week. For a week? We are completing our training also. It's going on. Training is for how many months? So it's for like 11 months. 11 months, then you have break. Then we will go for the district practical training. Which is your category? West Bengal sir. After this break will go? No, so I will go to West Bengal in December. Okay, now. I will go straight away to Hyderabad. So how many months training you are completing? Around 6. And you have Bharat Darshan also? Bharat Darshan is in the phase 2, so I am currently in the phase 1. Okay, phase 2 is how many months? It's around 2 months. So how is physical training there? Exhausting sir, it's difficult. Tough, no doubt. But you don't have this physical training in your career. This is there and that's all. That's why you find Otwele, persons, even IPS officer. But a lot of us are actually adopted in our daily habits. So many of us are still running in our midterm break. Because this is something which has become a routine. No, it's okay. Training time is okay. You are forced to. Till your second phase is there, you are forced to have that. Training afterwards. I hope to remain the same. I want to have it as a good takeover. What are the views that during career also like in military they have this endurance test? Should it be there in police office? I think it should not be made compulsory. Rather people should try to inculcate this in their habits. Try to, how many have been invulcating? These observations are being made even by the course also. You find Otwele, inspectors and others. And many of the IPS officers also. It doesn't happen in army. But they have mandatory endurance test after few years. They have to maintain their health, otherwise they are out. Right, so even in the paramilitary forces there is the endurance test. Even in police in some states there is an endurance test. But no one is permanently retired. But people are posted to positions which are non-field positions. Having said that, if I take the case of IPS per se, I will say that largely we will be occupying the decision-making positions. Which are going to be really hectic. So making a test mandatory for people would be too rigid. Decision-making positions is okay. But don't you think when agencies need, they have to go to the field also? Yes, no doubt sir. They have to walk also. Can it become very difficult for them? Sir, I think if I want to be a good administrator, if I want to be a good police person, I should be fit. This is something which should come- This is all by personal level. I am telling whether it should be mandatory or not. Unless it is mandatory. There is no guarantee. And especially in the cladders of North India, where they are in the habit of enjoying life and all, which is not there in South India. So it will be like that, which is there now. Sir, I feel, I am sorry sir, I am back to differ. I strongly feel that these things should not be imposed on some person. Rather the person should inculcate these things in themselves. Because if I want to be a good administrator, I have to be fit. So it means the officers having hot bellies, BGEs and all, they don't want to be good administrators. Probably they have different priorities. Many of them. In fact, my director is a DG rank officer. He runs every day. This is something that should come from within. No one could teach a DG rank officer. Okay, okay. Now police reforms are there as per that. And now selections of DGEs in the state and of police policies made by the officers. Do you think this type of system is required or it is against the spirit of federalism that Chief Minister doesn't tell me powers? Whereas this selection was previously being done at the state level and it was working there. And I don't think it is making much difference. Because then they have to send a panel of three officers. It must be also the same thing. Okay. So firstly UPSC is a constitutional body. It is like a federal commission. So it represents both the union as well as the state characteristics. So I won't completely agree that the powers have been taken from the state. Secondly, there is a panel of three. So the Chief Minister still have a prerogative on who to choose. Having said that, if you say that the judgment is not being implemented in its spirit I will say that policing is something that has to be said. I am not talking about the truth is not being implemented. Simply talk about this. Right sir. I didn't talk about other reforms. I am just mentioning only this. Because this is causing delay. Yes. Not a delay is being caused. You can state government can select a chief secretary. Why not a DGP? Sir, in this case I would like to cite what Supreme Court said. And it made a clear distinction between how the post of chief secretary is different from the post of DGP. DGP heads a police force. And a police force has to be necessarily separated from the direct control of the political executive. Because eventually it is a force. It commands violence and therefore it has to be impassionate at times. So a DGP should be someone who is politically neutral. Having said that, it is a political executive that should guidance, supervise it. But eventually it should have certain autonomy. So that condition is being put in after U.S.C. election? To some extent, sir. We have a number of cabinet committees. We have a powerful group of minister. What is the difference between them? So there are, cabinet committees are more permanent. So they are over a particular sector. Say a cabinet committee on security or a cabinet committee on infrastructure. Political affairs. Yes, political affairs. On the other hand, EGEOMs or Empowered Group of Ministers is basically for a certain subject. It is temporary. And it is for a more specific understanding and analysis of a particular issue. It might be a bill. It might be a specific reform that the government is targeting. So it is temporary. Once the task is done, it dissolves. What is the major difference? While the cabinet committees are permanent, the EGEOMs. These committees are the mandatory values. Whereas the cabinet committees are decision-making. Right. Decision-making. Okay. Roads of UPSC. One role will be discussed. Tell me other three, four roles. So one is of course the selection of, the recommendation of group A officers to be selected by the government of India. Then it has a role in disciplinary matters. In case any group A officer, some disciplinary action has to be taken. The advice of UPSC is taken. Then UPSC could also perform the recruitment functions for state governments. So wherever there is no state public service commission, UPSC could perform that function. That's it, sir. Anybody's biodiversity? Sir, biodiversity is the total accumulation of all the flora and fauna in a particular ecosystem. How much we have in India? Sir, you are... Are we reaching biodiversity? Yes, sir, we are a mega-diverse country, so... Approximately how many species of plants and animals in India? I am not sure, sir. I am not sure. Do we have good shampoo products? Yes, sir. How much? So the number is around 23%. I am not sure of the categorization person. You heard of one principle of equity and common but differentiated responsibilities and... Yes, sir. Where it is being applied? So this is being applied in the conference of parties of UN framework on climate change. So common but differentiated responsibilities mean that both the developing as well as the developed world have common responsibilities of preventing catastrophic climate change. But at the same time, the responsibilities are differentiated. Oh, okay. Please tell me. You are founding member of computer society, isn't it? No, sir. It should be founding member of the finance and economic society at college. I was an executive board member of the CSI. Prishan. Sir. Do you read newspapers? Yes, sir. Whenever I get time at the... I came up with an interesting concept today. It was basically saying that there is a difference between anthropocentric jurisprudence and biocentric jurisprudence. Can you think over this? The basic difference I see is the jurisprudence which is largely defined by natural causes versus what is defined by human interventions. So I am not very sure on that. Is our judiciary more in favor of anthropocentric factors? Does it also take into consideration the biocentric factors which surround us? It does, sir. Is there a bias towards anthropocentric law? To the contrary, I feel our judiciary has been really proactive and it has... India's jurisprudence is actually very proactive when it comes to biocentric jurisprudence. Can you give me some biocentric laws or... Sir, if I take one example, it would be the MC Mehta case where the Supreme Court banned the refineries near Taj Mahal. And so in that case, Supreme Court said that even though they are refineries like the Mathura refinery, they are causing pollution and they are harming the ecosystem around the Taj Mahal. So... You have got West Mughal Canada. That should be a difficult place to put its sufferers towards looking at what is happening in Mughal. It will be an interesting place for sure. I am not sure about being difficult. Is there a politicisation of politics and the worst? That's something which I'll say is happening across the country. It's something which will happen is always a fear. So... If you have to summarize the four major reforms that are required in politics in the country, what are they? The first one, of course, is the condition of the constabulary. In my interactions at various paramilitary forces as well as the state police forces, I've found that not just the constabulary suffers from the problem of low remunerative incomes, they also suffer from the problem of mental and physical health issues. The second one, of course, would be the reforms at the top level. That might be the selection of the post of DGPs or the district superintendent. There should be more autonomy. There should be a security of tenure. The third would be greater autonomy. By autonomy, I do not mean less accountability, rather greater autonomy for police chiefs to exercise power. There should be lesser political interference and more of political guidance. The fourth area would be in the criminal investigation. I think there's an urgent need to separate the law and order and investigation duties. We have at NPA a lot of simulations where we try to learn how a good investigation is done, how it is done in countries outside India. And I realize that in India, police people are so burdened with law and order duties and the forensic collection which are very important for high quality investigation are missing. You have been under training for 6 months. What is it that they don't teach you at the academy? What should they be teaching you? That's a difficult question if I say something. In fact, we have given a lot of feedbacks and the academy has been really accommodative. So we talked about having an election. The previous batch talked about having an election attachment. For this batch, they ensured that we have one. Having said that, I think because of COVID, we missed out on a lot. We could not get a lot of practical insights into how Athena works, how different functions, how police commissioner rates and police districts basically manage complex functions like traffic and other things. We used to have such attachments. We are not having them right now because of COVID. You have mentioned the classic fiction and contemporary Indian nonfiction. What do you read in contemporary Indian nonfiction? So a lot of stuff, things about India's history, things about... I think we have to say one book that you have written. So that's not a purely contemporary Indian nonfiction. I read Perilous Interventions by Shri Hadip Singh Puri. Can you tell us a little bit about the book? So it's basically about the role of UNSC and how it impacts countries in general. So there's in fact an element on India also, how UNSC is a very political body and many of the interventions it makes in countries including its standard in countries like Libya, Syria and even in Kashmir were highly political. So they were basically influenced by the opinions and whims of the P5 countries which is the permanent members and ignores the actual needs of the country. Is it time to dissolve UN? Does it have any contemporary elements? Sir, the book? UN, United Nations. No doubt it has. Do you think it is playing its role in that? Sir, to a good extent UN has played an important role. I agree that there's a need of reforms. The present day UN doesn't reflect the realities of our times. But this doesn't mean that UN doesn't have a role in peacekeeping or the UN forces in very turbulent areas of the world. The UN agencies which are doing really good work. So I think if there is any need, there's a need to make UN more proactive, more robust and more reformed. What is your take on encounters? Sir, by encounters I understand that you are referring to extrajudicial killings. So I think any kind of killing is not justified, especially if it is done by the police force. So there is only one provision in the Indian Penal Code which allows such an action, but that is an exception to the rule. So I do not, I think all the convicts, all the accused have to go through the trial. So there is no point of an encounter. How, when it comes to police reforms, why governments are always half-hearted in implementing the reforms? What do you think the probable reason? Sir, the reason which I feel is of course that governments exercise a lot of power through police and they do not want to give away that power. And so there is a vested interest involved and therefore police reforms are slow. That doesn't mean that they are not happening. So what is the security arrangements in coastal areas and what is the role of police? Sir, by coastal areas do you mean the coastal policing or the regions person? There is a recommendation by Cargill committee. How coastal areas should be strengthened and clearly demarcating the role between different forces, right? Can you elaborate on that? So I am not exactly sure of the facts. I do understand that there is a jurisdiction of state police to some extent in Indian Coast Guard and a lot of reforms have been made in coastal policing after the 2008 Bombay attacks. I am not sure of the exact facts. Do you know something about NADGrid? What is NADGrid? So NADGrid is like a master database that collects inputs from all the intelligence agencies. It collates them and it uses a lot of big tech like big data analysis and provides the same feed to 10 intelligence agencies in India. What is the due process and process laid down without? Sir, this differentiation was made by the Supreme Court in the Main Ka Gandhi judgment. A procedure established by law or a process laid down by the law is basically that there could be certain restrictions which could be placed as per the law. It is laid down by Supreme Court in Main Ka Gandhi's case or it is in the constitution itself? The word due process, sir? No, the procedure. It is in the constitution Article 21. The differentiation between the due process and procedures established has been made in the Main Ka Gandhi. Due process would basically mean abiding by the principles of natural justice. What are the? So there are two basic principles. One should not be condemned unheard and no one should be a judge in their own cause. Enough of police, I think. Let us go to some international relations. I am going to NP anyway, so there will be a lot of police there. I know, but you will be grilled on policing mode. That is why everyone of us are trying. What is your take on India's position on RCEP? So in 2019, India decided to leave the negotiations for RCEP. So as the things stand right now, the 15 members, the 15 countries are going ahead with it and India is out of RCEP. So if you are asking about my take on it, I think trade deals are something which is desirable. But we should not enter into trade deals only because we have to enter into them. We have to enter into a club of nations which are trading with each other. It has to be in our terms. The questions are minuses of the RCEP. So RCEP as it stands right now is going to benefit countries which have a strong manufacturing ecosystem. So of course there will be tariff relaxations, there will be greater trade facilitation between those countries. India's stand has been that it should include share for services and it should include items where India has a good export basket. Over the years, in recent 6 or 7 years, there is widespread feeling that India is not able to maintain its relationship, good relationship even with its neighbors. Do you agree with this statement? Sir, I feel that India has made the best efforts to have good relations with its neighbors.  Even those smaller neighbors have their own national interests and therefore they keep on changing. But India has always made the efforts. Hi Nishant. Hi sir. So I heard about peacekeeping missions. Yes sir. So where does India played a key role in peacekeeping? So there were of course many theaters. One that I fondly remember is Cyprus, where in fact India lost the retired general. Apart from that there was Kosovo, there was Sri Lanka also. So there were many theaters. And there are many countries in Africa where India, Indian forces were a part of peacekeeping missions. Why do you mean to say that those countries who are responsible for peacekeeping? And we have been called for peacekeeping. Internal security in the concerned countries threatened. Is it so? Sir I think the very idea of peacekeeping is that the force that is deployed should be neutral. Many of those countries were fighting civil wars, where both the parties had legitimate claims on the government, claims on the power. And therefore in order to ensure peace at first and in order to ensure that women and children and all the vulnerable sections of those societies are protected, there was a requirement of a neutral and efficient police force. And that is the reason why UN has been deploying those UN peacekeeping missions. Okay. So there are still internal security problems in India, like nexalism. When we cannot able to control our own problems, who do we get into peacekeeping missions like other countries? Sir there are two reasons. Firstly to the first statement on not being able to control, I was recently attached to the CRPF and we went to Jharkhand and saw the nexal issue. I think our forces are doing really good when it comes to solving the nexal problem. We went to a lot of areas which used to be liberated zones and India has done a very good work all thanks to our security forces because of which development is reaching now. And when I say that the role of security forces is this important, it is important to train them and we have to train them in different theatres. So if they go in civil war regions where the problems are actually complex, they are going to learn a lot from them. So I think it is a very good training experience for those forces, an experience which they might not always get in India. So of course one because of training, second that is because of India's commitment to being a responsible international player which is always committed to ensure international peace. What is your opinion on community policing? Have India suffered into community policing so far? Sir there are some states which have done fine when it comes to community policing. Kerala has in fact implemented a model of community policing. Then there is in Maharashtra, Mahalla committees were experimented. A very good concept in my opinion and it has actually led to a good number of cases being resolved at the community level only. My last question, what is a new development bank and what is it created? So a new development bank is a product of BRICS and it is basically created to assist the developing world in various infrastructural and developmental projects. The larger reason why it might have been created is to create a developmental bank ecosystem apart from the existing institutions which are largely west dominated. Say the boy bank in the IMF. Okay, good.


Given the immense scale of suffering in Gaza, there must be an immediate ceasefire. For at least the next six weeks, which is what is currently on the table. You just heard remarks from Vice President Kamala Harris in Selma, Alabama. And as you saw, she actually called for a ceasefire for six weeks. Now, if she just stopped after saying ceasefire and didn't say the six week part, I'd actually commend her for saying exactly what the American people want to hear. Having said that, though, even a temporary pause in violence is objectively good. And I would absolutely welcome that. But the ultimate goal is a permanent ceasefire, not a temporary one. So the administration really shouldn't co-op that language if they're not going to implement the policy associated with that call for a permanent ceasefire. But merely using the term ceasefire in and of itself is a pretty significant change in tone from the same administration that called ceasefire calls repugnant at the beginning of all of this. But we don't need a change in tone or a change in rhetoric. We need a change in policy and actions speak louder than words. But let's hear the rest of her speech. This will get the hostages out and get a significant amount of aid in. This would allow us to build something more enduring, to ensure Israel is secure, and to respect the right of the Palestinian people to dignity, freedom, and self-determination. Hamas claims it wants a ceasefire. Well, there is a deal on the table. And as we have said, Hamas needs to agree to that deal. Let's get a ceasefire. Let's reunite the hostages with their families. And let's provide immediate relief to the people of Gaza. So if you agree that six weeks isn't enough, well, rest assured because the administration is going to use that time to work towards something more enduring. Now, the problem is Netanyahu has already made it clear that a ground invasion of Rafa is inevitable at this point, and a ceasefire will merely delay that initiative. So the real question is, what's the Biden administration going to do to reign him in? Because talk is cheap.   People see through your lies. And that's the thing. Their words have no teeth or meaning until we see an actual change in policy. And at this point, we're just not seeing that. We are, however, seeing an increasingly desperate Biden administration try to placate Democratic Party voters by basically doing the diplomatic equivalent of throwing shade at Netanyahu. As the Hill reports, Harris, along with other Biden administration officials like Jake Sullivan and Anthony Blinken, are going to talk to Benny Gantz, an opponent of Netanyahu without the approval of Netanyahu. And this comes after the Biden administration subverted Israel by air dropping aid to Gazans who are currently starving after Israel has refused to let aid in. But 66 pallets with 38,000 meals isn't going to feed 2 million starving Palestinians. So it almost feels cruel in a way to not even do the bare minimum here and basically force Palestinians to fight over crumbs. I'll accept that 38,000 meals is better than zero. But if we have the capability of delivering more, then we should be doing that immediately if you're not going to tell Israel to stop. But the fact that the Biden administration had to airdrop food in the first place just really demonstrates how stubborn Netanyahu has been.  Israel should be isolated from the rest of the world and sanctions should be placed on him and his cabinet. But Netanyahu hasn't listened to the concerns that the United States has continued to vocalize because the United States hasn't issued any threats to cut off material support for them. They still haven't allowed a single ceasefire veto to pass at the UN Security Council. So I'll say it again uttering the word ceasefire is very different than actually implementing the policy of a ceasefire. But I won't pretend like this is an a drastic shift rhetorically because it is. And it comes at a time when we're reaching a breaking point. The uncommitted vote I think really scared the Biden administration and a lot of Democrats. And that coupled with terrible polls for him and constant disruptions of Biden events to the point where he's now having to basically hold campaign events incognito. It kind of goes to show you that supporting Israel and conditionally is politically untenable for the Biden administration. And they now know they're losing Arab Americans, young people, black voters. And if they want to actually win this election they have to recourse now and try to save this campaign.  Take it from Senator Chris Murphy who said just ignore the polls. What we saw in Michigan were over a hundred thousand Democrats went to the polls and voted non-committed. Obviously a protest vote to the president's handling of the situation in Gaza. How concerned as a political matter should Democrats be? I don't think we should be concerned about this as a political matter because this is such a critical issue relative to America's national security and the security of the Middle East. I would hope that the president doesn't make decisions about what to do in Gaza or the Middle East based upon how the votes line up. Listen I think it is time for the president to use all the leverage that he has to get a long-term ceasefire. I think if that ceasefire doesn't come it's in Israel's interest for them to pause military activity to solve the humanitarian crisis. But to the extent the president is using additional leverage on Israel he should do that for national security reasons not for political reasons. These issues are too important to be dictated by the polls. So on one hand he's saying that Biden should use his leverage to stop Israel. Awesome I agree there. But on another hand he's basically saying that it's okay to ignore the base of your party. That is something you don't want to do.   But he has been ignoring his base at least up until recently. He hears us now. He's trying to placate us now. Will he actually do what we want. That's another question. But I mean this is one example of a person that the president listens to.  In a New Yorker piece about Biden's campaign his chief of staff boasted about how he seeks advice from people like Larry Summers, Thomas Friedman and Mitch McConnell with the logic being that you know he's pressure testing his decisions by getting the input of his critics. But I mean they're not helping you. You need to listen to your base. Don't just try to placate your base actually implement the ceasefire policy okay. You can do so by cutting off weapons. By actually using your leverage. But in conclusion I welcome the Biden administration's shift in tone on Israel. But actions speak louder than words. So prove it. Don't just tell us. Show us.


Actually, I'm pretty impressed by what this does Hey, what's up guys, it's Josh here today. I want to review this Calvin Klein escape from men I've spent a lot of Calvin Klein fragrances They've always been a very general likable smell, but I haven't really found one that's really wowed me a really one that I could recommend highly on my list could have sworn someone suggested that I pick up CK escape for men So I'll go ahead and search my comments and see if someone did suggest this to me And I just decided to pull the trigger. This is actually a pretty old fragrance I was actually surprised that I haven't heard of it or if I have smelled it I don't remember that I have I think it came out in 1992 if not, I'll correct it down below So there is a lot of notes to this fragrance, but overall this fragrance is somewhat simple in my opinion It's really not as complex as it sounds here, but the notes are melon eucalyptus Mango grapefruit juniper bergamot sea notes birch fur Cypress rosemary sage oak moss vetiver sandalwood amber and patchouli to me I get this banana musk kind of smell and it's not like a musk like an Abercrombian fish fierce like dirty like Strong style musk. I don't even know if it's technically musk what I'm smelling But just a general manly backbone musk type fragrance that kind of dances around what smells like to me is Banana and it's some CK escape. So I when I think of escape I think of tropical and I haven't really smelled too many things with banana I think poppy oh fresh and maybe had a little bit of a banana vibe But besides that I haven't really smelled banana and too many other things and again banana isn't listed in the notes here So I'm guessing it's kind of like the melon and the mango that I'm smelling What I smell is musk might be the fur the oak moss things like that It's got a very masculine backbone, but it's probably been reformulated quite a bit I think this came out in like 1992. I mean as you know if a fragrance has been out for like five years It's been reformulated for sure my opinion the pros are it is somewhat different because to me it smells a little bit like banana It smells a little bit like something I haven't smelled before so it's gonna be better in the hot weather. It's got a nice tropical smell It's cheap and affordable. It's just overall pretty likable and pretty different Maybe you guys have smelled it a lot again It's been out since 1992 so a lot of people have had a lot of time to smell it Actually was really surprised how likable it is how different it is and I really liked that It has kind of like a banana style musk because I don't really have anything with that tropical kind of smell in there for the Collins It's a little bit simple and it has average longevity and projection But maybe you could even say it's not as good in the winter besides that it really doesn't have that many weaknesses The projections average and it is a tiny bit simple But overall I've been kind of impressed by what it does and then also I think I wore this it was either in the high heat or the high cold and it had what I smelled as more of a Objecty sandalwood style vibe I definitely had a little bit of another vibe besides the banana musk that I was smelling I did smell a little bit of a robust sandalwood kind of smell Which I also did enjoy that I didn't get right off the bat But overall really not too much bad things to say about this This is an average projector and if it projected a little bit more this actually might be in my list and again It's not a bad projector, but when I have so many good fragrances that project pretty strong and pretty high Little things like it having just okay Just slightly above average projection kind of knocks it down a little bit But if I were to give this a smell rating guys, I'd give this a solid eight Actually, I'm pretty impressed by what this does like it has a little bit of a different kind of tropical smell It doesn't really smell very cheap It is kind of light and likeable, but also when you get more into it It does have a little complexity. Like I said, it did smell a little bit different when I think it was in the cold And I smelled like kind of this object like sand wood vibe that I didn't get any of the other times I was testing it. I tested this for about five days or so, but the smell would give it an eight out of ten Likeability, it's probably its strongest point is it's like ability. I'd give it an eight and a half out of ten I like it. I think most people would like it A lot of people I've talked to kind of like lighter smells and they don't want an overpowering super colonial kind of smell So this one is just lighter easy to like just overall likeable really good eight point five and then longevity and projection I'd probably give this a seven to a seven point five. It's right down the middle to me It doesn't seem bad, but it doesn't seem great either But just overall right down the middle seven to a seven point five I'm on the fence of whether I should recommend that you pull the trigger on this right now If you like it what I'm saying I do recommend trying it out because it is pretty nice with that banana style musk. It is different It is cheap. I forget how much I paid for it online, but I think it was about $20 so super super affordable So let me know what you think of CK escape Let me know what your favorite Calvin Klein fragrances and let me know any other cheap fragrances I should pull the trigger try out right now and get a review out there We're on the road of 25,000 subscribers We are almost there and it's all because of you guys. Thank you guys so so much I'm having a great day out here. Hopefully having a great day at home. See you all in the next video. Peace