Okay, very good morning to you. I hope you had a fantastic weekend It is Monday the 19th of July so as per normal gonna Encapsulate some of the major news from this weekend where we can talk about things like the OPEC deal between Saudi predominantly in the UAE Can have a quick check-in on oil prices. Let's talk about the week ahead as well We've got things like the ECB meeting on Thursday and we've got the flash PMI data is coming out on Friday So amongst other things there's some of the major things that I'll touch upon but I'm actually off today and tomorrow, but Fear not I'll continue to live deliver the briefings as per normal in the morning I know it kind of helps set out the stall for the day ahead So more than happy to jump on each morning and deliver these so please do like and subscribe to the channel If you're watching this on YouTube, I'd really appreciate that But otherwise look let's get straight to it and talk about the charts this morning And there's a little bit of risk-off trade being observed Or be it very moderate so at the reopening of electronic trade last night and generally in Asia Pacific session equities have traded south and so index futures seen lower across the board if I just check in here technically from a Chart perspective Excuse me the DAX has seen quite a nice technical setup Where you've just had the previous areas of support and resistance or an inflection point for price over the course of the last What two weeks or so we're just coming up finding a bit of a cap to that at the moment as we get underway in European trade So a bit of consolidation here in the DAX for the moment You've got the S1 setting just above the bottom end of the range There was printed overnight the APAC session and that upside level around 15 447 as upside near-term Resistance from those previous lows that was seen back on the 9th and then high and low seen back before that and so at the moment Index futures they said a little bit lower the NASDAQ pretty similar in its setup for the time being following the Late rollover and price that we saw going into the close on Friday A generally negative finish to the end of that week. And so now we've got a bit of a short term upside area of resistance from those previous lows and highs here at 14 656 as we just come up to test that at the moment With a bit of a recovery Just as european players start to come in In terms of the age of PAC session from equity perspective That's leading to some of that movement The Nikkei was pressured early on by haven flows into the yen albeit i'd say that has moderated More over the course of the last couple of hours. It was more at the initial open on sunday night virus fears Still in focus as well first covid 19 cases confirmed that the olympic athletes village just days before of course the start of the tokyo 2020 games The hangsang shanghai con were also negative just more unrest happening with frictions amid the us-china ongoing Or somewhat lack of dialogue after the bind administration issued hong kong related sanctions And highlighted the growing risk to china's democracy crackdown in hong kong Despite threats they made those comments by china to retaliate to such action Now also losses exacerbated particularly more evident in the hangsang by the tech sector Still somewhat weighed upon by lingering concerns of tighter regulation by beijing and then on friday as well china's military also conducted a beach assault drill As a warning to us and taiwan so although that has Things like that have happened before it's kind of a number of things to explain why there's some under performance still ongoing in china at the moment But overall other asset classes what's going on With the dollar as a touch firmer and i do want to talk about the dollar specifically this morning from a technical perspective Cable just finding a bit of a short term Top on price recovery seen late in apac trading hours and around the low that was seen I'm just going into the close on friday around 137 62 here in the sterling future of course the so-called freedom day today and we'll talk about that and Some uk politics over the weekend, which i'm sure you read about the p.m. Now self self isolating But we'll touch that in a moment. You're a dollar slightly negative But not really too much dramatic and fashion at the moment from a technical perspective that really stands out for the time being as you can see Having generally been slowly trending lower here over the course of the last two or three weeks From a from a currency perspective the one thing I did want to touch upon with the dollar was this And this is taking a look at non-commercial traders That have been positioned short on the dollar against other major currency since the kovid crisis hit and us interest rates fell toward those in other Countries remember it was kind of this ongoing dovish mantra from the fed that created that short positioning In what we've been seeing in cftc data But now as you can see these these kind of green wave patterns here Is the short positioning is getting less and less and less and now with the fit the fed started to hint about tapering to come Traders have basically reduced their positions so that we are essentially Neutral and again, this would be the first time since basically the onset of the pandemic now I don't find this particularly surprising as we Kind of tiptoe towards then getting to the point of tapering and certainly Although powers generally holding the line and we are moving ever closer towards having those more detailed discussions about tapering It's not a matter of if but when so this was always going to happen Um, but this has coincided with the slight rebound in the dollar And it could be a sign of things to come you would expect then that to overall switch and Just given the situation where other central banks like the european Central bank are so far behind the curve Um, then we could well see start to see a bit of disparity where dollar positioning starts to switch to long Which could be a weighted factor just generally speaking if we're looking at the dollar index It is kind of Poised for a bit of a break You can see we've been here The acceleration in the dollar price really kicked off After the hawkish f.o.m.c. We saw back on the 16th of june And since that point we've just found a bit of a hiatus and we've started to consolidate just under the 93 level Which does kind of come in with this trend line from september November April or march april and then where we're at at the moment So if it did break higher a quick run up towards that What would be the late march 30 first high About 93 44 and then back up a key level there would be here You can see the initial pandemic low that we saw on the onset of the pandemic before the safe haven rally commenced That coincided with around that top of price recovery that we saw back in september So 95 generally is the big target there Going further forward Certainly a lot of other things to monitor but but definitely quite interesting for the medium term perspective Generally the dollar then are touched stronger this morning. So precious metals just suffering a little bit Just right on cue gold just breaking down through the technical levels of support. You can see here going back to What was What last wednesday session and seeing off the floor of price Points down on friday sessions. So a bit of an extension of Weakness here in gold on the break break of that technical move with some dollar strength this morning Just probably targeting down down to around these lows that were seen midpoint of last week at 1805 So down about nine bucks there in the yellow metal silver as you would imagine Also precious space following suits just printing session lows as we speak Otherwise the other thing or major things to talk about from the weekend, which you probably caught was OPEC headlines. So OPEC and its allies struck a deal Saudi Arabia meeting the UAE basically halfway in its demand for a more generous output limit They basically had a very quick Convened meeting on sunday ahead of the Islamic Long holiday happening this week for Eid Allowing for monthly supply hikes of 400,000 barrels per day. So before I get into some of the details here The overall impact on price was absolutely minimal WTI crude I was at my desk last night because I just wanted to check in at 11 p.m To see the initial opening print and there was a little bit of volatility the opening price was actually If I go back to this candle Here on it. Excuse me here This candle was the opening price for WTI crude last night was here And it was really tame to be quite fair As we went into the overnight apac session away from the OPEC deal generally sentiment overnight a little bit negative A lot of focus on the covid front in the far east So therefore perhaps just impacting a little bit on demand sentiment We just drifted a little bit south but you can see a lot of that very short lived And in fact, we've we've seen a bit of recovery and just pivot providing a bit of near-term Resistance on that that price reversal from the overnight lows. So overall very minor impact to The the latest OPEC comments. I don't really want to go into it too much And largely that's because the 400k and the agreement that did come is all pretty much expected You remember what I was talking about a lot of briefings over the last two weeks or so since the inability to make a deal I've always had the view that the UAE are just make they're just taking advantage of the situation Looking to leverage that to make a better deal for themselves And so whilst that they can the price trading where it is It's time to call some headaches basically and look to extract a little bit of a better Outcome for yourself and you could argue that they have done so The UAE's level will be increased now to 3.5 million barrels per day That is below the expected 3.8 million it was demanding when it blocked the OPEC deal earlier this month But it's an increase from 3.17 million. So Hence the kind of art of negotiation, right? You put it out there quite a big figure at 3.8 You're currently at 3.17. They meet you halfway at 3.5 job done You've just earned yourself a couple of extra hundred thousand barrels a day So I don't think it was that surprising to be honest. The numbers are in fitting The next meeting is Well, nothing is said in stone is what some analysts were saying OPEC plus will still continue to hold talks every month Including a review of the market in December It could adjust the schedule of required according to the Saudis The next meeting is on the 1st of september. However, the baseline adjustments Won't alter the pace of the 400,000 barrel per day monthly output increases when they take effect next year So, yeah, overall, I would say not I don't think a particularly Big deal comparative to market expectations But definitely I'd be interested just to see how Crude settles as we go into the north american session today to see how our friends across the pond really digest that news as well When they come in later The other thing from a news perspective, you probably read plenty about because of the Absolute debacle between three of britain's most senior cabinet ministers In a so boris johnson and russia schoonack the chancellor are going to be self isolating As of today after sigi javid the newly appointed health secretary tested positive Despite being fully vaccinated on on saturday. He came out on that and then there was this dramatic row over self isolation because Johnson and sonac were trying to get away that they were part of some pilot program Which meant that they wouldn't have to self isolate and they basically you turned I was looking at the alerts on my phone from sky news and it was Pretty much within an hour the amount of backlash that those guys got sonac came out and you turned and stealthy would Said he was self isolate so away from the kind of political headlines couple of things to be aware of for one some 530,000 people were forced to isolate last week after being contacted by the authorities by this kind of pinged by test and trace and You know, why is that an important thing? Well, I do think it is quite important because Even where I am I live very close to a school And there's a whole the whole street I mean any one of the certain age will will know that people move within catchment areas of schools and so on So everyone on the road is kind of their kids goes to these schools And everyone's at home can't go to work because a people's had it an entire year has had to self isolate impacting then All of the families and households of that of the street pretty much and a lot of that's being echoed across the country at the moment And it is creating quite substantial labor shortages And so factories pubs restaurants public transport This 10-day isolation rule is having quite a big impact. I mean even a pub. I was at the weekend Um, it was literally there's no food because there was no chefs because because Because they're all just from different pockets of different pubs that they work at where various self isolations were happening And they couldn't run it. So I do think this is quite a big thing because the government is set to exempt the double jabbed from self isolation If they're contacted by test and trace, but that's not going to kick in for about another month until august 16 So in the meantime, according to the adam smith institute 5.2 million people could be self isolating By that point and that's up from the current 1.73 million And obviously that will have a degree of economic impact, of course So just something to be aware of I don't think it's necessarily a big sterling negative this morning But there's a couple of things here that I am monitoring and for one is well You know, what is the covid situation? Well, we kind of know this the uk is heading into this latest reopening At a point where case numbers are increasing rapidly at this point in time. I think we're up at the high 50 thousands at the weekend um And that number is expected to head up to 100k And then as far as hospitalization is concerned again This looks very different to the case chart because still comparative to the beginning of the year Thankfully touch wood the number of hospitalized Patients is particularly low And hence the reason why the uk government's pushing ahead with that strategy, but I would say three or Yeah, three things I'm looking out for really that could well start to influence the pound in a potentially more negative way So for one this test and trace How many people is it affecting because it it could well Lead to a degree of labor shortages. So again the kind of reference numbers are That was about 530,000 last week were forced to self isolate And then the number being put out there by the adam smith institutes 5.2 million by mid august so again, how far or not do we exceed these types of numbers as the The the kind of self isolation numbers start to accumulate over the coming month And then the other thing is I would say a hundred thousand cases. I know I say this with Full understanding that this is a hugely Kind of negative development. This is a really testing and challenging time for many people in their lives But I'd say a hundred thousand cases is very much baked into price as is a thousand hospitalizations So for me tracking those numbers I don't think it's really relevant to really track them until they start until we start modeling them out And they start to show that they're going to exceed those numbers And if that is the case, I think then you could see some negativity translate into the sterling currency So for now, it's too early really to say But that's how I'd be looking at it And then before I get into the the kind of week ahead and what the economic calendar looks like We do have earning season started to pick up a little bit of pace. So this 79 s and p 500 companies reporting this week including nine of the dowl 30 components And the highlights here just running through them. So aftermarket today, you've got IBM Tomorrow aftermarket, you've got the lights of netflix Their shares have been obviously on the recovery of late Given the fact that as well they had a little bit of volatility on that whole gaming headline We had last week, but be interested to see how their subscriber numbers hold up And then wednesday pre-market coca-cola j and j verizon asml from the chip maker side on thursday TNT pre-market intel twitter snap some of the social media names aftermarket closed then on friday You've got amix chalmage honeywell kimberley-clark some of the larger cap names to be on the lookout for So not until the following weeks when we start to get some of the mega cap tech names And then for the calendar for today, it's super quiet There literally is really nothing of greater magnitude happening today And so really we start to look ahead towards the week And it is a very quiet week in fact in terms of how data is concerned tuesday tomorrow You get you know, june housing starts from the us But really the main events are twofold ecb meeting thursday flash PMIs on friday So what are we to expect from the ecb? Well, it's just two weeks after agreeing on its first strategic overhaul in almost two decades In what promises to be potentially a lively debate among members of the guidance that we give on the path of interest rates And its bond purchases Sources over the weekend said the ecb reportedly disagreed on stimulus guidance drafts for this And a july 22nd meeting while talks on bond buying coincidences sources Will reportedly not happen until the september meeting and i definitely think i agree with that latter point I think it's way too early to give More kind of detail on the on the bond buying side So again, i'll i'll be back by then and more detailed preview our issue On the morning of thursday and i'll be covering that all live of course And then on friday probably the main data then is the eurozone flash PMIs The june figures. So again, it's july figures. We're going to be getting so last month's june figures Which are shy of euphoric Is what the analysts at i and g are terming them and the question though now is whether the extreme Positive outlook of the economy has been maintained Despite the delta variant taking hold in the eurozone, of course, which we know now Restrictive measures being reinstated supply side problems continuing for manufacturing As well So how high is that kind of confidence remained amongst these person managers about their general outlook given those circumstances We also get the uk PMIs and we also get uk retail sales for june also due on friday as well But that's pretty much it we're going to end it there. I'll let you guys get on with the day As i said, i'm not around For the next two days, but i will continue to deliver my morning note and this briefing Just so you're all set for the day. So otherwise i'll catch up with you guys on on wednesday All right, good luck today and for the week ahead. Thanks for my friends. Thanks very much. Excuse me


Let us continue with our discussion of projective varieties alright so what we will do is recall from the previous lecture that we are taking K to be an algebraically closed field. So we always at least when I define the projective space I do not need this and in principle according to general the most general version of algebraic geometry which is scheme theory in fact for defining projective space K need not even be a field it can even be a ring but then let us not go we are not working in that generality but we assume that at least we are working with fields and since we are going to do geometry we are working with algebraically closed fields K. So at least for the initial part K just need be a field it need not even be algebraically closed so you know we have projective n space this is projective n space n space over K and what is this this is it is a quotient of it is a quotient of the affine space the affine n plus 1 space modulo and equivalence and what is the equivalence the equivalence is you identify points on a line of course I do not want to I do not want to I want really on 0 points so I throw out the origin and then I go modulo and equivalence relation and what is this equivalence relation this is the equivalence relation that identifies two points if they lie on the same line passing through the origin in other words this is just the space of lines in K n plus 1 and so you have a natural map K n plus 1 to P n K n plus 1 minus 0 this is the punctured affine space to projective space there is a natural map which is the quotient map okay you can think of so you can think of every point being sent to its equivalence class and equivalence class can be thought of as a line okay so you send a point which coordinates lambda not lambda n to the line passing through to its equivalence class which is the line passing through lambda not etc lambda n and the origin okay this is the same as the equivalence class of that point lambda not etc lambda n where the square bracket denotes equivalence class and this is also the same it is also written in this form it is written as lambda not semicolon dot dot dot lambda n but this colon is to be thought of as a ratio and we put square brackets to say that this is a set of n plus 1 scalars which have this given ratio okay so in other words what it means that is that you know if I multiply if I multiply this by any non-zero element of K okay which means I multiply each entry by the same non-zero element of K then the line will not change alright and the equivalence class will not change okay and this also will not change namely there will be the same scalar multiplied through each of these entries and the rule is that the idea is that whenever you have this whenever you have this colon you can cancel off the common factor so this is the same as t lambda 0 t lambda n for every t not equal to 0 in the field okay so that is because if you take t lambda 0 to 3 to through t lambda n that also lies on the same line joining lambda 0 lambda n through the origin it defines the same line so it is the same equivalence class and it is the same point but the idea here is that whenever you have a whenever you have ratio whenever you write a ratio as a is to b you also write this t a is to t b so you so the ratio a is to b is the same as the ratio t a is to t b okay and you can cancel the t off from the t from the ratio t a is to t b to think of it as a ratio a is to b that does for this is for the case of two ratio of two numbers but then this is a ratio of n plus 1 numbers okay and that is the meaning I mean that is the reason for this notation these are called homogeneous coordinates okay they are called homogeneous coordinates because you can cancel out a common you can cancel out a common divisor alright so well I have now the point is that we would like to do geometry on the space so first of all you know the starting point is always to first make it in a topological space and the two ways to do it okay so I will tell you one way the one one way is of course use the notion of quotient topology which comes from the from topology itself that is because the space above is actually the affine space affine n plus 1 space which is punctured at the origin okay and this is a Zerski topology okay so this is a topological space and this is a subjective map from a topological space on to another set okay and then you can make this into a topological space by giving this what is called the quotient topology for this map okay what is that quotient topology it is the open sets here are precisely those for which the inverse image under this map are open sets here okay and similarly for closed sets okay so what we do is that P n can be made into a topological space in two ways number one use the Zerski topology on a n plus 1 minus the origin okay a n plus 1 minus the origin which is the punctured affine n plus 1 space is an open subset of the affine n plus 1 space and any subset of a topological space automatically gets induced topology okay so this has an induced topology from the bigger space which is the whole affine space okay and so this has that topology and you can use the Zerski topology to give the projective space the quotient topology topology via the map you think of pi as a quotient map because it is quotient by an equivalence relation the after all what is projective space is the space of lines what are the lines they are the equivalence classes under this equivalence relation so this is a set of equivalence classes and you always call a map that goes from a set to a set of equivalence classes a quotient map and you call this set of equivalence classes as the set of you call this the you call the quotient of the you call this the quotient of the given set by the given equivalence relation. So this affine space here is being this projective space is here being thought of as the punctured affine space modulo going mod this equivalence okay and going modulo this equivalence is geometrically the same as looking at lines okay and every equivalence class here is a line passing through the origin okay so of course when I say space of lines in KN person I should say through the origin that is important if I simply say space of all lines that is not correct I meant space of lines in N person affine space through the origin alright and okay so this is a standard thing in topology whenever you have topological space and you have an equivalence relation on that then you can go to the set of equivalence classes that is a natural subjective map which associates to every point it is equivalence class and then you can always give the quotient the set of equivalence classes a quotient topology using that map okay and we can follow that and what is that that is a subset W in P N is open and respectively closed if and only if pi inverse of W is open respectively closed in the space above because in the space above which is a punctured affine space you know what closed or open means okay open means it is a complement of a closed set and a closed set means it is an algebraic set so a closed set here is an algebraic set in the affine N plus 1 space with the origin removed okay that is what closed sets here alright for the induced topology so this is one way you can give this quotient topology and you see in this quotient topology always in the definition of quotient topology automatically it is a it ensures that the quotient map becomes continuous because you are saying that a set here is open if and only if the inverse image is open so automatically a set is if a set is open then its inverse image becomes open that is the condition for verifying the continuity of a map okay so when you give the quotient topology automatically the map becomes continuous okay the what is the other way of doing things the other way of doing things is you try to indigenously define the Zariski topology on the projective space okay and how do you do that you imitate what you did for the affine space okay for the affine space how did we define the Zariski topology we define the Zariski topology by defining closed sets and whatever closed sets they were sets of common zeros of a bunch of polynomials in the right number of variables. Now what you do is you just adapt the same definition okay but now you say that you look at common zeros of a bunch of polynomials but not just any polynomials but polynomials are homogeneous okay because if a polynomial is not homogeneous then it is not it does not guarantee that if it vanishes at one point of a line it will vanish at other points of the line passing through the origin okay so you know what I am trying to say is that when we defined for example closed set in An or An plus 1 of course this sitting inside An okay I mean An plus 1 and how did we define a closed set here a closed set here was well we took a polynomial in N plus 1 variables alright and then not one of not one polynomial but several polynomials a collection of polynomials and look at the common zeros and define such sets to be closed sets here okay but the problem is that if you take a polynomial okay in N plus 1 variables trying to say that it vanishes at a point here is a little tricky which means effectively you are trying to say it is a you must think of it as a polynomial okay which vanishes on a point here corresponds to a line here a line in the affine N plus 1 space passing through the origin you want a polynomial to vanish on a line okay now that would not happen unless the polynomial is homogeneous okay so at least I should say that at least if the polynomial is homogeneous you have hope of the property that if it vanishes at a point on a line passing through the origin then it will vanish on the entire line passing through the origin okay so that is the reason that when you define the Zariski topology on projective N space you look at homogeneous polynomials and you imitate the definition of Zariski topology on affine space by in affine space a closed the closed sets are given by common 0 loci of a bunch of polynomials in projective space the closed sets or the algebraic sets are given by common 0 loci of bunch of homogeneous polynomials okay so here is the second definition define the Zariski topology on PN to be the one for which the closed sets are given as the common 0 common 0s of a collection subset of homogeneous in the all the you know functions on the affine space above which is just identified with k x0 through xn so you can define another topology on the projective space this is the Zariski topology on the projective space which imitates it is also given by common 0 loci of a bunch of polynomials but this k but in this case we consider only homogeneous polynomials okay and mind you homogeneous polynomial means that in the variables if you multiply each of the variables by t then that is if you substitute for each variable a constant multiple of that variable okay the same constant multiple then that constant comes out with the power it can be factored out okay and that power that comes out is called the degree of homogeneity of the polynomial okay. So the fact is that actually I think it is a good point to tell you one more thing there is in fact one more way of getting the topology on PN okay so I will tell you what that one more way there is yet another way so there is yet another way of getting the topology on projective space okay and of course you know what I am trying to tell you I am just going to tell you that you define there are three ways of defining the topology on projective space and they are all the same okay they are all going to give you they are all going to give you the same result alright. So what is this yet another way this is this is the standard way in which you think of a projective space as being gotten by gluing N plus 1 copies of affine space okay so you see I will tell you well let me put it as 3 in continuation with this via the gluing of N plus 1 copies of an now so I have to explain what this means so what we will do is let me do the following thing so you take projective space okay and you know the points in the projective space are given are written in terms of homogeneous coordinates like this okay and then what we will do is let us look at for each of the for each fixed position look at all those coordinates okay for which that particular coordinate does not vanish okay so what I mean is you define Ui to be the set of all points with homogeneous coordinates lambda 0 through lambda n such that lambda i is not same you define Ui like this okay this is for for i equal to 0, 1, etc up to n okay you define these sets alright and well you know you must guess that the sets are going to be open sets because you know they look like for example U0 is first coordinate first homogeneous coordinate not vanish okay U1 is second homogeneous coordinate not vanishing Ui is ith homogeneous coordinate not vanishing and whenever your coordinate does not vanish that should be an open set okay so you must you know you can see that these are going to be the open sets okay but then let us not worry about them as open sets you know well in fact I can say I can say that you know if you take the inverse image of this Ui okay under this under this map then I am going to get the complement of the 0 set of xi the ith coordinate the in fact I should call it i plus 1th coordinate because I am starting with 0 my numbering starts with 0 so you know if I take this Ui and take the inverse image above what I will get is I will get all the points with the i plus 1th coordinate not 0 okay and that is the I will get that set inside the punctured affine space and you know it is complement will be all those points with i plus 1th coordinate I mean with the with that coordinate 0 which is of course a closed set okay. So the inverse image of Ui under pi is certainly an open set and therefore you know this is certainly open according to the definition of the quotient topology okay I mean according to the definition of the quotient topology. It is also open according to the definition of the of this indigenous risk topology because it is actually the complement of the 0 locus of that coordinate where that coordinate is considered as a homogeneous polynomial of degree 1 okay see so note that note that pi inverse of Ui is actually z it is an plus 1 k minus z of xi intersection with an plus 1 k minus z in the origin okay this is what it is this is actually what this is just T xi it is this so the complement of the 0 set of xi is the basic open set defined by xi alright and that is an open set and if you intersect it with this subset it will give you an open subset of this okay so this is which is open which is open in the punctured affine space above of which the projective space is a quotient okay. So you know according to definition 1 this is an open subset of the projective space if you give it the if you look at the quotient topology then this suddenly an open set okay and what you must understand is that you see all these Ui's i equal to 0 to n there are n plus 1 of them and they cover the projective space because when you write a point with projective homogeneous coordinates all coordinates cannot be 0 because you thrown out this is the image of a point on the punctured affine space above okay so you are not so all the coordinates cannot be 0 alright so whenever you write homogeneous coordinates at least one coordinate is not 0 which tells you that therefore that point lies in one of the Ui's. So all the Ui's these n plus 1 Ui's they are a cover for the projective space okay and according to this definition if you give the map pi equation topology then these Ui's are open okay. According to this definition also it is open because you see note also that if you take away Ui from Pn what you will get is of course the 0 set in Pn of the homogeneous polynomial X i see you take the polynomial X i okay that is of course homogeneous polynomial it is homogeneous of degree 1 alright. So by definition it is 0 set is should define a close subset of projective space and therefore it is complement which is precisely Ui is open okay so this Ui is open according to this definition it is also open according to this definition alright. Now further all the Ui's union of all the Ui's i equal to 0 to n is actually Pn of course the projective space is covered by these sides okay but the third topology on Pn is something that I have not yet defined so but I need these Ui's to define it. So what I am going to do is now you take this Ui which is set of all homogeneous coordinates points in Pn such that the ith coordinate is not 0 which is subset of Pn and what you do is you define a map like this into An okay the point is that each Ui is automatically an affine N space each Ui is actually an affine N space okay and beautiful thing is it is complement which is Z X i is a projective space of dimension 1 less okay. So the beautiful thing is that each of these Ui's is actually an An okay and it is complement is a Pn minus 1 alright so let me explain that and that complement Pn minus 1 is called the hyperplanet infinity okay so let me explain that so you know this map is very easily defined so what you do is you take a point it coordinates lambda not etc so I will write this down there is a lambda i minus 1 there is a lambda i and then there is a well there is a lambda i plus 1 I will not write lambda i plus 1 because it might be the last then lambda n okay and you know what I am going to do see I know lambda i is not 0 alright and you know the homogeneous coordinate is not going to be affected if you know I multiply the whole set of homogeneous coordinates by non-zero number and what I am going to do is I am going to just multiply it by 1 by lambda i which makes sense because lambda is not 0 alright and then that will give me a 1 in the i th position okay in the place of lambda i will get a 1 so what you must understand is that this is the same as well let me write this lambda or not by lambda i lambda 1 by lambda i blah blah blah here I will get lambda i minus 1 by lambda i and I will get a 1 and so on I will get a lambda n by lambda i these two ratios are these two are one and the same point on projective space that is because this multiplied by this same lambda i which is a non-zero element of k okay and but then you see this point 1 is I mean this 1 is redundant so what you do is you get only the remaining n distinct scalars and these are the n scalars to which I mean these are the coordinates to these define the coordinates of the point which you are going to send it so what you are going to do is you are going to just send it to lambda not by lambda i comma lambda 1 by lambda i and lambda n by lambda i where you know the way I have written it is you omit lambda i by lambda i which is 1 okay so this is n plus 1 entries I omit that 1 and I get n entries okay now because of the way I have defined it you can check that this is a bijective map this is a bijective map that is the reason is because you know the moment one of the coordinates becomes 1 you see then the other coordinates are unique for that given line because if you try to change it by a scalar okay then this you cannot change it by a scalar without changing the 1 to something else okay so if you freeze the ith coordinate as 1 then the remaining then for each line for is ith coordinate is not 0 okay if you if you if you rationalize so that dieth coordinate becomes 1 for a representative point on the line then the remaining n coordinates are unique so the fact is that this is a bijective map so I know let me call this as phi i phi i is a bijection this phi i is a bijection and of course you know it is a bijection why is it surjective well phi i is injective it is injective and surjective for the reasons I just said but anyway let me if you want you can try you do not have to big deal you know if you have a phi i of one point given lambda 0 through lambda n is equal to phi i of another point mu 0 through mu n then so this will imply that lambda 0 by lambda i etc lambda n by lambda i is the same as mu 0 by mu i and so on mu n by mu i okay and what this will tell you is that it will just tell you that lambda not so it will tell you that lambda not is lambda i by mu i into mu 0 so each lambda j is lambda i by mu i into mu j and this lambda i by mu i is 1 the same it is fixed i is fixed okay lambda i is fixed mu i is fixed okay therefore this quotient is a fixed set scalar and what you are saying is that every lambda is a fixed set scalar times every mu the corresponding mu and that means that these two points define one and the same line that means they are one and the same point on projective space so this implies that the these two points are the same okay this is just injectivity I mean and of course surjectivity is very very simple again why is that so you give me a point in affine space a point etc xn in An is phi of well I can write this point you know what I am just going to put yeah I put x1 etc and then I put a 1 in the ith position and then I put of course I put same I have to put colons here because this is just homogenous coordinates so if I take this point this point will go to the point I started with so it is surjective so if you want a map from here to here you simply write these coordinates in the same order as n plus 1 entries homogenous coordinates but put do not fill the ith position the ith position you put 1 the remaining n positions you put these in the same order that is the that is the map in the reverse direction therefore you know these phi i's are all bijective maps the phi i's are all bijective maps and the fact is that now what I can do I can do the following thing. So you know in very much the same way in which you get a manifold by getting local charts into Euclidean space you think of trying to make projective space you know in a sense like a manifold by taking these as local charts which identify these essentially open sets with affine spaces so you have so you think of so in other words I am saying that you know I am saying use these phi i's which are bijective maps to transport the topology on the affine space back to ui see each affine each copy of affine space has a topology it is a Zariski topology so I can simply transfer the topology here namely what is it you give me a subset here subset here is open or respectively closed if and only if it is image here is open respectively closed for the Zariski topology on these affine space okay so with that definition also I can give you a topology on each ui okay and the ui's cover the ui's of course cover the affine space and the only thing that you will have to worry about is whenever there is an intersection the topologies agree okay. So that will happen okay that is analogous to the compatibility of charts which involves the transition functions being good okay and in this case the transition functions will be just xi by xj okay and therefore what will happen is that you will get a good all these topologies on each ui which are transported via the phi i so each of these topological spaces ui they will glue well on the intersections ui intersection uj to give a unique topology on pn okay and that is yet another topology on the projective space okay which comes from which comes via the gluing of n-toson copies of affine space is n-toson copies of affine space are given by the ui's which have been identified with the corresponding affine spaces via phi i's okay. So there are and the big deal is that the big deal here is that you know that all these topologies agree on the intersections okay that is a very important thing. So you know if you have a topological space okay and suppose you have two if you have a sub if you have just have a set okay and if you have two subsets and suppose on each subset I give you a topology alright then on the union of those two subsets this will give a topology provided on the intersection the topologies should agree okay the intersection a set in the intersection considered as a subset of one is open if and only if it is open with respect to when it is considered as subset of the other okay. So you know if you have two sets two subsets of a space of a set you give them topologies separately okay then when do you get a topology on the union you will get a topology on the union only when on the intersection the topologies agree that is a subset of the intersection is open respectively closed in one if and only if it is in the other okay this is the situation for two sets but if you have a collection of sets for example even a collection of subsets which is a cover for the given set on each subset if you give me a topology when will all these subsets glue together to give a global topology on the set that requires a compatibility okay it requires a compatibility that whenever you take a certain subcollection of subsets and you take their intersection a subset of that is open if and only or closed if and only if it is so in each topology coming from the corresponding members okay. So and the fact is that is a big fact the fact is that all these topologies on each UI okay which come from the topology on affine space they all glue together and give you a topology on the project space this is the third way of getting a topology on the project space and the fact is all the three are one and the same okay so there are three ways of giving a topology on project space and they are all the same okay. Now what I am going to do is so let me write that down so maybe I will write it here so I will write here Q each UI a topology by declaring PI to be a homeomorphism that is that is I subset W in UI is open respectively closed if and only if PIW in AM is open respectively closed. So if you give this topology to UI via PI then it becomes it is automatic that this PI becomes a homeomorphism of UI to affine space okay. So you know what you are doing is for every point on this UI you are giving me you are just taking n affine coordinates okay so each one is a coordinate map each one should be thought of a chart the analog of a chart for a manifold okay so each one is a coordinate map alright and what we are saying is that once you have a manifold structure each chart in the atlas if you take the corresponding coordinate map it is of course a good function it is of course an isomorphism okay. So in this case what is happening is we are gluing topological spaces all the UIs are glued together to give the topology on Pm so the topologies on each UI agree or compatible on the intersections and define a topology on Pn and for that topology on Pn when do you say a subset of projective space is open or closed it is open or closed if and only if its intersection with each UI is correspondingly open or closed in that UI and how do you check it you check it by taking its image under PI and see whether the set you get in the corresponding affine space is open or closed okay. So this has to be checked that you have to check that the topologies on each UI they are all compatible alright. So you know for example if I take two different UIs UI and UJ it should not happen that a subset of UI intersection UJ is an open subset of UI and is not an open subset of UJ such a thing should not happen it would not happen that is the compatibility you have to check okay. So in other words UI intersection UJ has a topological subspace structure from UI it also has a topological subspace structure from UJ and my claim is that these two are the same okay that is what has to be checked right it only if you check that then you are saying that these topologies glue together to give a global topology on the ambient space which is the full projective space okay and what is a big deal the big deal is whether you use one two or three the topology you get on projective space is one and the same and that is the that is the that is the topology on projective space okay. So let me write that down fact so here is a fact it needs a little bit of checking that it is pretty easy to do you can try it out fact or theorem if you want or each of these ways each of these three topologies on P n is the same as the others so the so what you must understand is that all these phi i's are actually homeomorphisms so if you think of projective space with the Zarisky topology if you think of this as an open subset and you take the induced topology then each phi i is automatically homeomorphism okay by this fact by this theorem alright. Now so this is something that I leave it to you to check but probably portions of it we will check as we go ahead now what I what I want to do is I want to you know the purpose of of course this course is to translate from geometry to algebra and back okay. So all this that we have been saying is more or less completely geometric and topological but then to go to the commutative algebra side we will have to worry about the polynomials that are involved okay and you know that is where the second definition comes in the second definition which says that the Zarisky topology on projective space is given by close by taking the close sets to be common 0 low psi of a bunch of homogenous polynomials in the right number of variables the number of variables will be one more than the than the subscript the superscript of appearing in the projective space okay. So we have to study these homogenous polynomials so for the first question is why homogenous polynomials okay and then that will lead to looking at the theory of homogenous ideals okay. So the fact is when we did just like when we did translation from the Zarisky topology on affine space to the polynomial ring we were worried about ideals okay we will when we do studies Zarisky topology on projective space you will also have to translate everything to ideals but the only thing is the ideals now will be so called homogenous ideals okay and one way of thinking of homogenous ideals is that these are ideals are generated by homogenous polynomials okay. So first of all I wanted to realize that the homogeneity property it can be defined in more than one way and to explain that let us look at an example okay so suppose you take a suppose we take your point lambda not lambda n in projective space and let f x not through x n vanish on the line through lambda not etc lambda n which is actually this point okay the point in projective space is the same as its inverse image which is the line okay. So you know well the way I have written it is that when I say that this point on projective space corresponds to this line this line can also be thought of as leaving it is actually being thought of as leaving above so actually you know this the inverse image of this point is this line thought of as a line above okay here you are thinking of the line as a point here okay but if you think of it as a subset of An plus 1 okay then the inverse image of this is this line itself okay. So you know pi inverse of this point is actually the line through lambda not lambda n considered as a subset of An plus 1 right that is what it is so you know you get another nice picture of this whole of this map you get the picture of what is called as a line bundle a line bundle is something for which you have a you have a map from one topological space to another topological space for every point below the inverse image is a line above okay. So you think of this as a bundle of lines above this and what is every line above goes to the corresponding point it represents in the projective space okay so you think of this vibration as it is called as a it is thought of as a it is sort of as a line bundle you think of given a point here a point here corresponds to a line above a line through the origin above and what is that line through the origin above in terms of the point here it is well if you take the inverse image of this point that is exactly that line above okay that is what I have written here alright. So you see if the polynomial vanishes on this line okay what it means is so this implies for every t which is non-zero scalar the field okay well f of t lambda not etc t lambda n should be 0 this should happen right because you see I have told you that the so you know if I draw a diagram here is my projective space okay and I have taken a point here with homogeneous coordinates lambda not through lambda n and under this map pi I go to the affine punctured affine space above what I get is well I get the line through lambda not etc lambda n minus of course the origin okay so if I take pi this is what so pi inverse gives me this inverse image do not confuse pi inverse for inverse map I get this line of course the origin is thrown out because I am considering it as a subset of the punctured plane then you see when I say the polynomial vanishes on this line it means that it should vanish at every point on that line and every point on that line looks like this okay but then you write this out you write this out what it will tell you is the following yeah so I should here also when I say pi inverse of the point is this line I should say this line minus the origin inside affine inside the punctured affine space because pi is not defined at 0 alright so this is the right way to write it but you do not worry about the point at 0 which has been intentionally removed okay so well so in any case now you see you write this polynomial f of x not xn you break it down into its various homogeneous components you see a polynomial in n variables can be broken up into pieces a sum of pieces each of which is homogeneous of a particular degree so there is a degree one term which will be linear expression in the x i's okay then there will be a degree two term which is a linear expression in product of two x i's namely it will involve x i squares and you will get x i x j okay and so on you will get so the polynomial breaks down into f not x not etc xn this is this degree 0 so this is actually a constant this is the 0th this is the constant term of the polynomial which is what you will get when you put everything 0 okay plus I will get f of f1 which is degree one term which is degree one this is the linear term then and I will get it and I will go on like this and I will get fd which is the degree d term this is the degree d term and of course when d is one it is linear d is two it is quadratic d is three it is q b k and so on and so forth and so on you go on up to f to the degree f this is the highest degree of the polynomial okay it is a degree of the highest weight monomial that monomials that occur okay and the point is that each of these guys is homogeneous of a corresponding degree so if you take fd this homogeneous of degree d f1 is linear it is homogeneous of degree one alright and now you know let us look at this condition f of t lambda not through t lambda n is 0 for every t non-zero so what it will tell you is 0 is equal to f of t lambda not t lambda n that will be that will tell you that you know when I plug this in I will get well this will just remain f not and t lambda not t lambda n plus when I plug t lambda not through t lambda n here then t will come out because it is degree one similarly when I plug it in fd t power d will come out because it is homogeneous of degree d so the expression will look like t times f of lambda not etc lambda n plus t squared f of lambda not etc lambda n and so on t to the d f of lambda not etc lambda n plus blah blah blah you will get t power degree f times f of lambda not etc lambda n this is what you get you get for every t for every t not equal to 0 in the field okay you get this expression sorry I should put sorry you are right I should put f1 should put f2 fd f sub degree fd yeah thanks these are not they should not be f they are the corresponding homogeneous complex thanks so this is t times f1 t squared times f2 t power d times fd and so on okay now you know I want to I claim that this implies that you know I claim that this implies that f each fd of lambda not etc lambda is 0 for every d greater than or equal to 0. I claim that each fd each fd vanishes and so fd of t lambda not etc t lambda n also vanishes for all t in a scalar which is non-zero this is my claim okay you know see therefore what I am trying to say is that if a polynomial vanishes on a line then each of its homogeneous components also vanishes on that line okay each of its homogeneous components also vanishes on that line okay that is what I am trying to say and this is the key to defining a homogeneous ideal alright and why is this true why this is true is because you know let me call this as let me call this scalar as a0 after all this is scalar of course here this is independent of t because this is the first term it is constant so let me call this so what I will get is I will get a0 plus ta1 plus t squared a2 plus t power d ad and so on up to plus t power degree f a sub degree f is 0. So I will get a polynomial in one variable okay with these coefficients k coefficients and what I am saying is that that polynomial identically vanishes on the affine line minus the origin okay but you know if a polynomial vanishes on an open set it vanishes everywhere okay if a polynomial is 0 on an open set it is 0 everywhere that is just because of continuity of the various interpolation therefore and of course mind you here is the first place where you are using k is algebraically closed because I want k to be an infinite field okay so probably I do not okay I need k an infinite field alright for that. So I am just saying the fact that you have a polynomial in one variable you know if it vanishes on an open set okay then it is identically the 0 polynomial so therefore this polynomial is identically the 0 polynomial therefore every coefficient is 0 that is exactly what I have written here okay. So the moral of the story is I want you to remember is that this is the key fact defining what a homogeneous ideal is. The fact is if a polynomial vanishes on a line then every each one of its homogeneous components that also vanishes on the line and in particular it is a non-constant power I mean the constant term is 0 in particular the constant term cannot be non-zero the constant term has to be 0 okay so this is the point I want you to remember when we go to the next lecture right so I will stop here.


Welcome to Toffee TV, this is the Everton News Daily. Marco Silva has given his press conference before the Derby and he's confirmed that everybody is fit for the 232nd Masyside Derby at Anfield on Sunday game which kicks off at 4.15 of course on Skysport. Silva said that he's told his team he wants them to play with our fear, he said fear is not something we have in our dressing room, it's not a word that is allowed in our dressing room. He wants us to play our normal game and these are all the right things, you know, too often we change to try and accommodate things and some might say it's naive for us to play the way we play but how else are we going to get better? How else are we going to realise? How else are we going to learn? Because you can't always play against the top sides and hope to keep it tight and nick something because if you want to try and improve yourself as a team that's not going to work long term. So, you know, it's all very well having to say it in game plans and I'm sure he will make slight adjustments but ever and I've got to do whatever they've been doing which is playing their own game. That's what's been getting them results. Marco Silva did say that, you know, he's got loads of respect for Liverpool. She had clopsed on an excellent job, they're the R atop team and they are atop side.  You know, we've got different weapons this year that can cause them problems in the past. It's been one man. It's been, if we can get the ball to LeCarco, if Ross Barclay turns up, you know, and this time I think there's four or five players who can hit them as well as the bench and so that should give us some confidence. And you know, when Marco Silva was asked, is it a specific, you know, are you doing a specific game plan for this game? He said, why would we, you know, the challenge isn't just to say enough for one game, to do a game plan for one game. He said that so we can compete against the top side every time, every time we played them, not just on one day and that's absolutely the attitude we want to hear and that's absolutely the outlook we want to hear and the ambition. We're trying to build a team that can compete with these teams regularly, not just as a one-off. So get a thumbs up from me. Jürgen Klopp has praised Everton in his pre-match press conference, said it's the best Everton side he has faced while being Liverpool manager. He said they have a proper squad now. He's very, very impressed with Charlotte and he said what a player he is and he said it's going to cause Liverpool problems. There's no question about it and he's expecting a really tough game. So, yeah, and I think he's right. I think Klopp is right. It is the best Everton side he will have come up against, whether that plays out results wise. We'll have to wait and see but certainly with a team that can go to Anfield and give a really good account of ourselves, really play, get on the front foot easy for you to say and play well and see what happens at the end of the day. That's all you can do. If Everton go out there and play the football he's been playing of late, I will be very happy because I think A, it'll cause Liverpool problems and B, it'll show me that we do have that confidence in each other and in what we're doing to go to Anfield and to take the game too. We've had difficult away games so far this season and we played well in all of them, certainly in patches and we've still got another couple to come. We've got this on Sunday in Manchester City next but let's try and perform for us and let's show Jürgen Klopp that he was right to have some concerns about us definitely in our squad. It's 12 months of the day since Sam Allardy's became Everton manager. I'm just glad it's 12 months later. There's a real compact, I've wrote a little piece and wrapped it up in some stuff from Skysport as well on the website. Go on and check that out and feel the fact that there's silver under Marko Silver or something like that. I did write it on. Go on and have a little check of that out if you've got some time because there's a lot of stats in there from Skysport which highlight the difference between Al Grafibolt and Marko Silver and all the people telling us what more do we expect. I'm me and Perth have also done a video on that probably going to be out on YouTube later. It's already been out on Patreon but I think we'll be putting it out on YouTube as well so check that one out. Everton have been told that Tangoid Ndombele will cost around 70 million euros if Everton want to sign from Leon. 21-year-old Miffiel player, friend Miffiel player who was watched by Everton on Tuesday night I think it was when they drew to Manchester City as well as a host of other clubs. It was good as well, 70 million euros. I can't see what's paying that but interesting that it's rumbling on and it probably will because he probably hasn't pressed Marcel Brands but Marcel Brands knows value for money and I can't see him. I want nothing to pay anyone near that amount for an unproven player in the Premier League. Everton's public consultation regarding the People's Project and Bramley Mordock now closes on Monday 3 December. The final one on Monday is going to be in Crosby. It's going to be in Sainsbury's car park in More Lane. You'll be able to still fill the survey in online up until 3 o'clock on Monday so if you can't get down to that consultation in Crosby fill it in online but make sure you have your say because once it's shut it's closed on Monday that is it. Do you think they've already extended it from the original date which would have been Saturday? They've already extended it till Monday so try to make sure you get your point across if you feel like you need to make some.  I don't know what the fuss is, it's our building, we own them. We had the live a bird and it was blue long before Liverpool were a thing and if we want to light our own building up blue we will do because we own it. In all seriousness it was very good, it's great to see where the club's going with it. Them images of the fireworks and it all in blue and everything was fantastic and I'm sure everyone who attended that event had a great time. So well done Everton and the people who didn't attend it or didn't get invited or however it was paid competition wise don't be jealous just accept it's good, it's good. The club are trying to reach out for these things but it was fantastic, it looked fantastic and it just wet the appetite doesn't it for Bramley Mo and what kind of fanfare will go on for that when we eventually get in there. That is it for the Everton News Daily, if you haven't subscribed please do so, it only takes a second if you want more videos, get over and join us on page. And thanks for watching, have a fantastic weekend and here is to Everton ending the Derby who do at Anfield. See you soon.


Hi everyone, Joe for Jasby's casebreaks.com coming at you with a fun little break here. Three boxes of a 2022 Fanatics under wraps 2.0, autographed baseballs. One spot gets you nine players. It's a pretty big list, but a lot of great names on there and one spot gets you nine. And there's a any and all autographed baseball, so there'll be no randomizer if there's a multi-auto ball. That'll go to one person, so that's a pretty good spot right there. So one spot gets you nine. So let's just use the spreadsheet here to make sure I get all this right. So there's already, there's one already. There's two, three, four, five, six, seven, eight, and nine. So there's 171, nice. All right, so I got all the players on the list here. Let's get everybody's names in here. And we'll use the first tab right there to put in the randomizer results. Randomize both lists three and a one four times. Good luck. One, two, three, and a one fourth and final time. Got Riley down to Jeremy. Three and a one four times for the list of players. One, two, three, and a one fourth and final time. All right, there you go. Jim Rice all the way down to Trevor Story. Might change positions, Mark was saying earlier. All right, now first I'm just gonna show you the list as is and then I'll alphabetize by your first name so you can see all of your players grouped together. I just wanna get this on tape, on wax as they say, on the record. And obviously when this video uploads you just wanna, oh, look at that. Jeremy with the any and all multi-signed baseball spot. I don't think I've seen one of those yet, so it'll be interesting. Now let's sort by your first names right here. So Aaron Miller, there's your group of teams or players right there. Another big group of players right here. And another, and another. There you go. Annie Owens, there's your group of players right there. David has a big group of players right there too, including Lou Pinella. Jeremy, there's your group right there. Do you need anything? Oh, hey, thanks, see ya. Jeremy, there's the rest of your teams right there, including the any on a multi-signed baseball spot. There are your last spot mojo players. It's a good group. Justin with those players and those. Nicholas, Riley, that might be it, right? Yeah, and Riley. There you go. I'm gonna pause the video. We're gonna see if there's any trades. And when we come back, we'll have that three box break. Stick around, B, R, B. All right, welcome back everybody. No deals were done. So let's pop this fresh case open. Let's see how am I here. See it on this can. Next break, we'll just grab three random boxes, but just cause it's the fresh case, we'll do some dice roll and see which stack of three we're gonna do, which group of three we're gonna do. It's an 18 box case, so six stacks of three. And you can see, oh, actually that camera is turned from the previous break right there, so that actually helps with this break. So we'll roll the die, and we'll just from left to right, it'll be one, two, three, four, five, and six on the far right over there. All right, good luck boxes. It's five, one, two, three, four, five. So it'll be this stack right over here. So I'm gonna move this stack, all the stacks on my table. And here we go, and put your thinking caps on.  Let's see what we got here. There was not, I actually forgot to do that. I'll do it after this. All right, who am I? Active baseball player, 34 years old from Visalia, California, righty. Lifetime 297 batting average, 1,576 hits, 107 home runs, 581 runs batted in. Played for three teams my entire career, three-time All-Star, three-time Gold Glover, three-time Wilson defensive player of the year, two-time Silver Slogger, a two-time batting champion, and a fielding Bible award winner. He's currently on a, he's currently on a playoff team right now, although I think he's injured. Who am I? While you're thinking, who this goes to? He's a current Yankee. That's right, NJ's got it. DJ LeMayhew, take this little tray out. You've got the DJ LeMayhew card down there. There's his autograph right there. He's a pretty good hitter. There's the MLB hologram sticker. Is there a fanatic sticker somewhere? Maybe not, just the hologram sticker will do, that's fine. And Justin Gutting, with the DJ LeMayhew. Congrats right here, random player number seven. Next box, and we got another three box of this loaded up. These are pretty fun breaks to do. So get in on it. One spot gets you nine players, which is about $3 a player, if you kind of think of it that way. Let's see who we have here. Ooh, it's a gold baseball. That ball is gold. All right, who am I? 24-year-old from Kenosha, Wisconsin. I'm one of those bat left, but throw right kind of guys. Has a lifetime, I have a lifetime 253 average, 18 home runs and 105 runs batted in play for one team my entire career. And led the NL this year in triples with seven. No guesses here. Was drafted in the first round by the Dodgers. First round Dodger 20th overall in 2016, signing a $2.3 million bonus as well. It's got some speed. He should be able to hit the ball pretty well. I feel like his hitting talent is better than what his average shows. Got to work on the defense a little bit, but he'll play, think of Dodger's second baseman, shortstop, sometimes outfielder. I'll stick him in left field sometimes. No, no guesses. It is, yep, Jilo's got it. Gavin Lux, that's pretty cool. Gold Dodger baseball. Kind of take it out of here a little bit. That's pretty awesome. There's the MLB sticker right there. Got the Dodgers logo on there. Gavin Lux's autograph. Got the card in there as well. Nice, and that's Aaron Miller with the gold Dodger ball. Gavin Lux took a nice step forward in his career this year. I think this was sort of a, sort of a make or break year for him.  Possible trade candidate or something like that. The numbers aren't quite there. I think you would hope the numbers. I don't think he's ever gonna be a big home runner here, but we hope that the average keeps going up. But he led the league in triples. All right, third and final. So we had two active players and the third player, can't read upside down. Third player is also another active player. All right, so the final round of who am I? I am a 32 year old starting pitcher. Born in Newport Beach, California. Have a lifetime 130 wins to 71 losses. A lifetime 323 ERA and almost 2,000 strikeouts, 1,930 and a 110 whip. That's pretty good. Played for three teams my entire career, a five time All-Star, an A-L-E-R-A leader and a two time MLB strikeout leader and an A-L wins leader. In spite being born in, and then I think he went to UCLA. In spite being born in Southern California and playing baseball at UCLA, he claims that he's a Yankees fan. It's kind of weird. Yep, NJ Card Collector's got it. It's Garrett Cole. All right, so this one also has a fanatic sticker in the MLB hologram sticker. There's his autograph right there. Garrett Cole. Garrett Cole card down there as well. And that will be for, that's for Justin Gutting. Justin with two of the three baseballs. Sometimes these breaks work out like that. And there you go, gang. I'm Joe for jasphyscasebreaks.com. Thanks for watching. Thanks for breaking with us. And I'll see you next time for the next one. Bye-bye.


Today is a trade deadline and the Leafs have already made two moves the day before Yo, what's going on leaf nation sleeper back again and welcome back to the channel today We're doing a shootout commentary and today we're gonna talk about the moves the Leafs have made And they're pretty big ones. So switch over here. Let's hop into a shootout But guys the Leafs have made two big moves The day I'm recording it. It was yesterday. They've made these two moves on Sunday First off before we head into a shootout. Let's just talk or talk about what the trade was So I'm gonna pull it up. So I don't get the details wrong I don't think I will but better to be safe than sorry. I'm gonna pop it up on screen, too I'm sure you guys have heard of these trades already But the first trade the Leafs made yesterday was acquiring Nick Felino and Stefan Knowson Hopefully I said that right from these San Jose sharks. It was a three team trade deal So the Leafs acquired Felino and Knowson They gave the blue jackets our first round pick in 2021 and a fourth round pick in 2022 and the sharks acquired a fourth round pick in 2021 so I'll give my opinions on the trades Further into this video, but that was the first trade the Leafs made the second one that they made late last night It was acquiring gold tender David Riddick big save Dave from the Calgary Flames in exchange for a third round pick in 2022 so The good things about these two deals are the Leafs did not have to get rid of anyone on their team They didn't have to get rid or yeah They got rid of someone on the team future potential in the team But they didn't get rid of any current players on the team No prospect that they have no one on the current roster, which is absolutely huge So you guys are probably wondering how the hell did the Leafs do this with the cap hell that they're in if you Want to call a cap hell it's cap hell They not too long ago acquired Riley Nash from the blue jackets and put him and Frederick Anderson on long-term IR Freeing up about just over five million cap space so with that being said the I know the flames retained 50% of David Riddick's contract and I think the Leafs are only paying 25% of Nick Foligno's contract something like that So they're not paying full contract, which is nice So that's how they were able to get that done Here we are with Ilya Mikaev We're gonna start right off the bat with a datu flip here. Let's see if this works out Almost okay first things first. Let's talk about the Nick Foligno deal Did the Leafs overpay for Nick Foligno? 100% especially after seeing what Taylor Hall went for which was a second round pick and York I think it was from the Bruins 100% the Leafs overpaid a first-round pick for Nick Foligno is an ideal, but what I will say is Nick Foligno makes the Toronto Maple Leafs a better team Plain and simple Nick Foligno makes the Toronto Maple Leafs a better team The Leafs gave up a lot of draft picks today They gave up four draft picks and what I'm gonna say to that is The Leafs don't need to be drafting players. They have a pretty strong draft pool if I do say so myself To name three guys right off the bat Rasmus Sandin, Nick Robertson, Timothy Lilligren Three elite top tier prospects Plain and simple. This is coming, you know a little biased obviously But you got to be real all that could have worked out You got to be real those are three great prospects that the Leafs have in their pool and they have other great prospects too I'm sure I'm forgetting some names With that being said Let me make the safe here real quick Beautiful. With that being said Nick Foligno makes the Leafs a better team Now the way I look at it He's probably gonna be playing with William Nylander and John Tavares Which makes your top six better and your top nine even better the way I'm gonna look at it here is I Don't even know what I did there the way. I'm gonna look at it here is it sucks for gouch He was he's been playing great He played top line minutes the other night with Matthews and Marner and played absolutely amazing But it looks like he's gonna be playing on either the third or fourth line Which is fine. It makes our depth even better, which we like to see but I Think the first line is gonna be Hyman Matthews Marner great first line second line being Nylander Tavares gouch or not gouch Foligno and Then we're gonna score here and I can explain these third or fourth lines Oh, that would have been beautiful. The third line will be Kerfoot, Thorton, Spetza. I'll let me make the save I'm forgetting what the fourth line might be. Oh, and we take the L. We take the L And then the fourth line being Kerfoot, McKay of Galley. Now the way I'm looking at it Here is the third or fourth line aren't really set So maybe that Kerfoot line will be the third line and the Joe Thorton, Spetza line will be the fourth line Like these are good problems to have you don't know who's gonna be on your on your foot on your forwards All great players that deserve to be on the team, but all in all given the lines you're gonna have to wait seven days For Nick Foligno to quarantine and come on to the loose roster William Elander is still out with COVID protocols. He doesn't have it But he's got a quarantine because he came in contact with someone who possibly had it I believe at the time I'm recording this they haven't given any updates on William Elander So we're still assuming that he's out as well But I like this trade around Kyle Dubis he pretty much said listen, we're we're we're going for it That's that's what this trade means 100% you're giving up all those draft picks and pretty good draft draft picks I'll first round overall pick in 2021, which again isn't overpayment But I'm fine with it Nick Foligno makes the least better and yeah I mean Kyle Dubis a plot to you now. Let's talk about the day verdict trade There's there's some things I want to talk about here I'm happy that Dave Riddick is a charming belief. You're only giving up a third round pick in 2022. I believe which is okay I'm 100% okay with that the one sign that this day verdict trade tells me is that Frederick Anderson is hurt and Longer than they expected So you have a tandem of Jack Campbell and Dave Riddick if what we're thinking about Freddie Anderson is true They haven't given any update about Freddie the time I'm recording this but You got to assume he's out for longer than they expected and this is why They acquired big save Dave And you got to look at it this way if this is true You're going into the playoffs with Jack Campbell as your starter and Michael Hutchinson as your back-up goaltender God forbid knock on wood that Campbell goals out hurt in the playoffs You're gonna be starting Michael Hutchinson as you're starting goaltender Let that sink in because I don't think you're winning a playoff round with him as your goaltender. I didn't go in That's crazy. So I like this trade by Dubas Dubas. You know what again? The deadline still going on. He could still possibly make moves and I hope he makes at least one more move We'll talk about that as well, but big ups to Cal Dubas. I like both trades. You're getting rid of draft picks Which it's not a leaf worry right now. They're not looking at draft young players They're looking to win now with the core that they have and this is the stuff you got to do to try to win and Yeah, I like it big up to Dubas and I hope these trades work out where we don't regret it I don't think we will Felino makes a team better Dave Riddick. Definitely will be making the team better Not having to see Hutchinson net again, which is great with that being said I know I'm gonna pop up a tweet if I find it Dave Riddick has a losing record with the Calry Flames in a 904 safe percentage if I'm not mistaken If we're gonna be honest a 904 with a losing record with the Calry Flames who haven't been too great It's pretty good and I like to think the Leafs will play better defensively in front of them So you're gonna get a better Riddick in front of you like a bet the best version of Dave Riddick, which I'm excited to see and the good thing About Dave Riddick is he can possibly be playing tonight. I don't know how that went in But David it could possibly be playing tonight. He doesn't have to quarantine. He's coming over from a Canadian team The Leafs have a back-to-back today and tomorrow. They're playing Montreal tonight Calgary tomorrow Calgary tonight Montreal tomorrow So maybe you give Campbell the start tonight Riddick the start tomorrow or vice versa. So Yeah, I like this trade from the Leafs. Don't got to worry about quarantining, which is absolutely huge Cal Dubas Two thumbs up from this guy on the two trades that you made now that being said before this shootout ends I think the Leafs are gonna go out and make one more move. I know they have just over two Mill maybe two point one mill left in cap space. What are they gonna do? You might be asking I think they're gonna go out and try to acquire a defenseman whether that be a top six guy or a top six guy Like someone to play on the third pairing where Dermott's your depth guy or a depth guy You look at it this way. We're gonna that's who flip to end it beautiful If you look at it this way, let's say God forbid. I'm gonna knock on wood again Someone goes down in the playoffs or in the regular season like a Riley a Brody a Muzzin a Bogosian a Dermot your depth is Rasmus and Dean, which isn't bad. Who's after him? Who's after him Timothy Lilligan who's barely played any NHL games? So I think the Leafs are not done. I don't think how Dubas is done I think he's gonna make one more move to acquire that depth defenseman whether it be a guy who can play was a Bogosian or a guy that You know sits and comes in if someone God forbid gets hurt But yeah, I don't think the Leafs are done only time will tell it's 1046 while I'm recording this There's still a full day of trades to be To happen so it's gonna be an interesting deadline day. I don't think Dubas is done I think he'll make one more move. We'll see what happens But guys that is gonna be it for today's video Hopefully you guys did enjoy it if you guys did be sure to drop a like on it if you guys are new around here and make sure to hit the subscribe button down below and Hopefully We have a good deadline upon us Um, let me know let me know what you guys think about the deals that happened already the Taylor Hall trade the move that the The senators have made on the key goose of signing with the Florida Panthers after getting his contract bought out The Panthers are looking scary. Let me know what you guys thought have been thinking of all the deals that have happened so far and Yeah, that is it for today's video. Thank you guys very much


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