Hello everyone, welcome to Options with Doug. Streaming live daily on Bookmap Discord and the Bookmap YouTube channel at 1.30pm Eastern Time. Before I get started, I need to go through the Disclosures. General Disclosure, all Bookmap limited materials, information, and presentations are for educational purposes only and should not be considered specific, investment advice, nor recommendations. Grist of Scotcher, trading futures, equities, and options involves substantial risk of loss and is not suitable for all investors. Past performance is not necessarily indicative of future results. Here's my contact information. The best way to get in touch with me is through Discord. My name on Discord is Doug Pee. Also in Bookmap Discord, there's an options-doug-chat channel that's a great place to post questions, comments, and content related to the topics of my presentation and the topics of the channel which I'll go through in just a moment. I'm also on X, formerly known as Twitter. My name there is at Doug Pless. And just a note about Bookmap Discord that is free and available to everyone whether you are a Bookmap subscriber or not. There's a lot of great content here, including my channel, of course. All right, so again Bookmap Discord free and available to everyone. The focus of my presentation today and the focus of the options-doug-chat channel is options, order flow, the impact of options markets on stocks and futures, and the influence of market maker hedging flow on price action. I have a two-step process for trading and the first is planning, and I use positional analysis. I look at how traders and market makers are positioned in the options market and how those positions change from day to day to develop a thesis regarding the expected trading range and volatility for the day as well as the directional bias. And the second step in my process is execution. I look at real-time order flow in Bookmap and real-time market maker hedging flow in SpotGamma Hero to confirm my thesis and for setups for entries and exits. And when I talk about setups today, I will be talking about setups in an underlying asset. For example, setups in the SAP 500 can be taken with ES futures, SPY shares, SPY options, SPX options, or even ES options. Questions and comments are welcome, and I will be watching both the chat and options in the options-doug-chat channel as well as the chat and YouTube. So please feel free to post your questions and comments. I'll do my best to answer your questions. And hello, trading for a living. Welcome. Glad you're here. All right, my agenda for today, Tuesday, October 31st, Halloween day. First of all, I'm going to go over news items, economic data, events, and earnings for today as well as the rest of the week. Then I'll go through my positional analysis. Then I'll review some setups from this morning. And then I'll take a look at the live market. All right, let's get started with news. So first of all, there was, I guess, a minor data release this morning. Mid-medium impact consumer confidence at 10 a.m. did come out greater than expected, and also lower than the previous number. And then also AMD reports earnings after the close today. All right, for the rest of the week, of course, the big event for the week is the FOMC meeting tomorrow. Before that, at 10 a.m., there is PMI data that comes out. But again, the big event, the 2 p.m. FOMC announcement, and then the press conference begins at 2.30 p.m., just as I'm wrapping up, I will be wrapping up my webinar at that time. All right, let's take a look at what the market expects for tomorrow. This is the FedWatch tool for the November meeting. So this is the November meeting tomorrow. The current rate is 525 to 550. And right now there is about a 97% chance of holding rates steady at the current target rate. And there's actually just under 3% chance of actually reducing rates. And according to this, there is no expectation for a rate increase at this current meeting. So really, I think the market will mostly be concentrated, focused on the statement, and then the press conference, what Dronpal has to say at 2.30. All right, so that's the expectation for tomorrow, near 90, over 97% chance of holding rates steady. All right, so that is the, that's tomorrow. Then on Thursday, Apple reports earnings after the market closes. And then on Friday is the jobs report. That is the monthly jobs report for October that comes out on the first Friday of the month. That's 830 a.m. Eastern time. There's also more PMI data that comes out at 10 a.m. All right, I'm going to start with my positional analysis now. And TK, hello, welcome. Glad you're here. Ask, could I take a look at ES and HERO and discuss any potential setups? I'll be glad to do that. I do that every day. All right, so let's get started with positional analysis. And when I get to the setups, I normally start with ESB 500. All right, so let's, speaking of ESB 500, let's take a look at the levels in play for today. So this is the ES futures and book map. Before I take a closer look at this chart, I'm going to take a look at a larger time frame. I'm going to go to SPX in a 30 day one hour chart. This is showing price and key levels again for 30 days of data. Let me point out a couple of key turning points. First of all, this is the October 6th jobs report for September. Gamma squeeze after that level. Here is the October monthly options expiration. And so that was on the 20th. And that was a very put dominated negative gamma environment. The expectation was for a put banner rally that lasted for a couple of days. And then I think the issues with treasury rates and geopolitical events moved the SPX lower. And then last Friday, this is the 4100 put wall. I'll talk more about that in just a moment. And that level held as support. It did its job. And the SMB 500 has rallied since then. And let's take a look at one factor in play for that rally. Let's take a look at a chart of VIX. So this is last Friday, this dark area here. As traders were buying puts for hedging for weekend risk. And that did not work out. That there was no, you know, of course, there's a very dire geopolitical situation in Israel, Gaza, but no really impact on the market. So VIX has been dropping. That is a tailwind for equities. VIX continues to drop. That's been a pretty good trade the last couple of weeks selling premium on Friday into that spike in volatility. So there's the drop in VIX starting late day on Friday, and then the corresponding rally in SPX. And of course, you know, the I guess risk of FOMC tomorrow is likely putting somewhat of a damper on a stronger rally, although SPX seems to be doing pretty well today. All right, so those are the, that's the price action key turning points. Let's take a look at some levels now. First of all, the dash purple line showing the lower and upper weekly expected move. They're kind of a cluster of levels right there at 4,200. That is the both the right around the upper daily and weekly expected moves right at 4,200. That's based on the options market. And that is set once a week, then the lower and upper daily expected move again the upper daily expected move right around 4,200. And that changes every day again based on the options market the expected move. All right, there are also some spot gamma levels on this chart. Let me point those out. These are spot gamma proprietary levels available to spot gamma subscribers. These are based on gamma weighted open interest. The first key daily level is 4,000. That is the absolute gamma strike. That's a strike with the largest absolute positive and negative gamma. Above that is the put wall that an active support last Friday. That's a strike with the largest net negative gamma that can be expected to active support. The next level up is the 4,280 level up here. That is the volatility trigger that a spot gamma proprietary gamma flip level. Below that level, market makers position on the gamma curve is negative. In a negative gamma environment, market makers have to trade with price to hedge their dealt exposure. And that tends to enhance or increase volatility, leading to wider trading ranges and more or less stable trade. So SPX is trading well below that level. And then up above is the call wall at 4,500. That's a strike with the largest net positive gamma that can be expected to active resistance. Not really in play for today. All right, let me talk about shifts in levels now. So the volatility trigger did shift slightly lower from 4,300 to 4,280. And the call wall also shifted lower from 4,600 yesterday to 4,500 today. So again, well out of play and not significant. All right, so again, the SPX well below the volatility trigger in a negative gamma environment with market makers trading with price to hedge their dealt exposure. All right, that is the bigger picture. Let's take a look at the an SPX chart for today. Just to show a closer focus on the levels that are in play for today. Here's the put wall down at 4,100. Excuse me. And then really the levels up above, right around 4,200. Upper weekly expected move, upper daily expected move. And also there's a combo level there that combines SPX and spy. Gamma weighted open interest into one level, and then also an L2 level. So an important level at 4,200 that was noted as resistance in the spot gamma AM founders note. All right, let's go back to book map now. All right, in book map, I have my own cloud notes so I can show a variety of levels. The levels are shown in this column. And I'm showing SPX levels, spy levels, and also other relevant levels for ES. So here's the spy 4,17 volatility trigger that did act as resistance earlier today. That was right around 8,30 a Eastern time before the cash open. Here's the SPX 4165 support level noted as support in the spot gamma AM founders note. And in this consolidation period, just afternoon it did act as support and kind of a launching point for the 1pm rally that actually started a little bit early. All right, so those are the levels in play for the SP500. Again, I've got spy levels, SPX levels, the spot gamma levels are shown with white lines, white labels, red text, and just the round number levels shown in yellow there. And then there's the upper weekly expected move for ES. All right, let me get to shifts and levels for spy. So for spy, just like SPX, the volatility trigger did shift lower down to 417 now. That level has definitely been in play for today. And also the call wall actually shifted higher. It was kind of a strange level at 414 yesterday shifted back up to 440 more in line with the SPX. All right, so that shifts and levels for the SP500. Let's take a look at NASDAQ now. Let's go to the NQ futures and book, ma'am. Before I take a closer look at this chart, I'm going to take a look first of all at a QQQ chart so we can isolate the QQQ levels in play for today and pretty similar to the SP500. Here's the volatility trigger at 350 acted as resistance before the cash open. And now QQQ trading above that level. All right, so that's QQQ levels in play for today. And let's take a look at NDX. And here's that combo level that was on the right around the 350 level combining QQQ and NDX, gamma weighted open interest into one level here converted to an equivalent NDX price. All right, let's go back to book, ma'am. So there's the 350 level QQQ 350 and that combo level just below it that did act as resistance earlier today. And then support right around the more or less the 347 level QQQ 347. So just like the SMB500, I have my own cloud notes. I'm showing NDX levels, QQQ levels, and also the zeros in the 50s for looks like I'm missing some levels here for the NQ I need to update my spreadsheet. Move it lower. All right, so those are the levels of play for NASDAQ today. And we'll talk about setups in a few minutes. So there were some minor shifts in levels for the NASDAQ. For NDX, the volatility trigger actually shifted slightly higher. And for QQQ, the volatility trigger shifted lower. And note the volatility triggers are pretty volatile. They shift almost every day. But otherwise for NASDAQ, the put walls, call walls, and absolute gamma strikes all held steady. And again, we'll talk about setups in a few minutes. All right, so those are the levels of play for today. Talked about shifts and levels pretty minor. So really not enough shifts and levels repositioning in the options market to make any judgment on a directional bias. And really the dynamic in play for today has been that continued drop in the and volatility in the VIX and as well as drops in the treachery yields. But the tenure is dropping today. All right, let's take a look at gamma notional now to see how market makers were positioned on the gamma curve at the beginning of the day. I like to look at this every day. This gives me a sense of how market makers may need to react to changes in price and applied volatility. So first of all, the beginning of the day, here's how they were positioned on the gamma curve. I'm looking at gamma notional for SBX, SPI, NDX, QQQ, RUT, and IWM. So the SB500, NASDAQ, and the Russell 2000. And for NASDAQ, I pretty much focus on QQQ and for the Russell 2000, I pretty much focus on IWM. Let's start with the SB500. Note these numbers are all negative. For SPX, minus 1.12 billion. For SPI, minus 1.76 billion. This indicates at the beginning of the day, market makers position on the gamma curve was negative. This means that traders are long puts, market makers are short puts, and they have to trade in the direction of price to hedge their delta exposure. And for QQQ, also negative. And the same for IWM. And note for the SB500 and the NASDAQ, gamma notional, although still negative, became less negative than yesterday. And for IWM, stayed about the same. Maybe slightly less negative. All right, let's take a look at the VANA model now so we can see a graphical representation of what this means. I'm going to start with the SPX. This is the VANA model. This chart is showing market makers delta notional. That's their delta exposure on the vertical axis, price on the horizontal axis. There are two curves on this chart. The first, the light gray curve, shows how market makers delta notional will change with changes in price only. And then the purple curve adds implied volatility to the equation. That shows how market makers delta notional or delta exposure will change with changes in price and implied volatility. And that change in delta with the change in applied volatility is the VANA effect. VANA is the second order Greek. All right, let's take a look at some prices on this chart. First of all, for the SPX, the low of the day was right around 4153. Somewhere right around here. And let me check, see where SPX is trading now. Right around 4188. So from this point to this point, I'm looking at the purple curve. So what this is showing as price increases, market makers delta notional will decrease, has been decreasing, and they can buy back short futures. So we know that VIX has been dropping also. So following the purple curve, we know that price is increasing, implied volatility is dropping, market makers delta notional is dropping, and they can buy back short futures. And that is a put VANA rally. All right, so that's a very important dynamic for price action today. All right, that's SPX. Let's take a look at SPI next. Go to SPI, very similar curve. SPI, low of the day, right around 414. Just a little bit above 414. So right around here, SPI currently trading at for almost 418. Same dynamic, price increasing, implied volatility drops, market makers delta notional is dropping, and they can buy back short futures they use to hedge their delta exposure. And then let's take a look at QQQ. QQQ, low of the day, we saw support was around 347, right around here, and QQQ currently trading over 350. So we're around here. So the same idea, price increases, implied volatility drops. In this case, market makers can buy back short NQ futures. They used to hedge their delta exposure for options trades in QQQ and in DX. All right, let's take a look at some setups now. So based on this, my thesis for the day was really looking for still higher volatility, watching VIX carefully to see if this continuation of this put Vanna rally would continue based on the drop in implied volatility after the weekend, as traders loaded up on puts on Friday, implied volatility continues to drop, market makers can buy back their short hedges. All right, so let's get to some setups now. I'm going to start with the SB500. I'm going to first start with the hero signal, hedging impact real-time options to see what options traders have been doing today. So this is from Spot Gamma, available to Spot Gamma subscribers. Again, it's hero hedging impact real-time options. All right, I'm checking for questions. Truman asked, is that Gamma chart this week's expiration upper left corner? So Truman, I'm not sure what chart you're referring to. I was looking at the Vanna model before this. All right, so this is the hero signal. Again, hedging impact real-time options. So everything that we've looked at so far other than book map has been static data from Spot Gamma based on Gamma weighted open interest that's updated once daily, sometime overnight. Now we'll get to real-time setups. So first this is the hero signal for the SB500. White line is showing price. Purple line is the hero signal showing hedging, options trades and hedging activity for a combined signal for SPX, SPI, XSP and ES futures. All right, let's zoom in on this chart and I'm going to focus on setup in the morning right around 10 a.m., somewhere between 10 and actually around 950 and 10.05, something like that right around here looking at this setup and note that the hero signal was trending up. So there's somewhat of a divergent setup right here. Price was moving lower, hero signal moving higher, traders taking positive delta positions. All right, Chat Picat asked, how do you know the bias trend direction of ES? So I'm talking about that right now. All right, so this is the first clue. I see that options traders are taking positive delta positions. So when traders take positive delta positions, market makers take the opposite side of that and they have to buy ES futures to hedge their delta exposure. So this is the hero divergent setup around 10, 10.05 this morning. All right, let's go take a look at book map. I'm going to focus on this, oh I'm in, sorry, still in in queue. It's going to be a very similar setup but let's go to ES. All right, Chat Picat, I'm watching, when I'm trading, I have two screens. I'm watching hero on one screen, book map on another screen and putting together a narrative to understand what's driving price and potential setups especially on a day like today where I'm, I am expecting based on this drop in VIX a slightly bullish bias. That was my thesis for the day. All right, so then I'm watching hero. I see that traders are taking positive delta positions, watching book map for signs of a move higher and that comes right around 10.05, watching order flow here. These volume dots show market buy minus sell. Magenta dots show more sellers than buyers. Green dots show more buyers than sellers. Kind of a false start here before 10 and then the real move right around 10.05, aggressive buyers. You could have waited for the break above 4.15 that acted as resistance before and then there's a quick move up to, first of all, this liquidity right around 41.92 and then up to 41.96. This is the heat map and book map showing a history of limit orders in the order book and these were limit sell orders in the order book. Stayed in there, the darker the band, the more orders there are in the order book and these definitely acted as, first of all, a target for price and as resistance. All right, so it took some patience today watching hero and seeing what options traders were doing watching order flow and book map with the expectation also watching VIX with the VIX falling and also in the sub chart here you can see the the sharp move higher and all these lines, the pink line that is cumulative volume delta CVD it changes from magenta to dark blue and taking a shift from negative to positive. The yellow line is showing stop orders that's buy stop orders helping to fuel the move higher that's also shown by the on on chart indicator with the small green dot there and then finally this rising light blue line showing iceberg orders those are orders that larger traders use to hide their size those are buy buy iceberg orders it looks like there's a series of smaller orders this icon is kind of obscured here but that is showing 2,320 contracts and that definitely helps to fuel the move higher then buyers are temporarily exhausted reverse and then aggressive buyers start to command again right around the 415 level back up to the liquidity the resting liquidity right around ES4196 let's zoom out all right so overall that move higher has continued today so the initial move higher right around 10 a.m deep pullback consolidation period final test of 4165 support level and then price moves higher for the afternoon rally all right let's go back to hero and see what options traders have been doing zoom out making sure make sure i'm looking at the entire day here all right so this is the this is the cash open there's that initial move higher note that hero up until about 11 o'clock was making a series of higher lows then traders options traders often in this environment will have kind of a mean reverting action on price as price reached a new high options traders were taking negative delta positions and then right just before noon they shifted higher again started taking positive delta positions and so far that continues so they got started a little bit early on the the 1 p.m rally let's just take a look and see a closer look at what they're doing today so the i'm separating the output and call transactions the orange line is showing calls very close correlation between calls and price action so what this is showing right around just before 11 traders started selling calls buying puts when the both lines move in the same direction that's a very powerful signal and then just right around an hour later sorry about that nothing i did so then just right around 12 o'clock they stopped buying puts note the blue line flattens out resume buying calls price moves higher so it really helps to get uh a more uh comprehensive picture here by separating outputs and calls there's one other way to look at this we can take a look at next next expiry and that would be the options that expire today for the smd 500 and that is showing uh about a third of the notional value today so not um you know just playing a part uh zero dt options traders are playing a part in the move today but uh two thirds of it are options that expire um further out than today that show by the purple line the zero dte trades shown by the next expiry all right so this bullish move continues higher as traders are taking positive delta positions all right alexis ask do you stop orders detector and how please if you use it yes i do so i showed it uh when we were taking a look at um at es just a few minutes ago that it's d the yellow sub chart indicator let's go back take take a quick look at that back to book map so that is the yellow sub chart indicator i have this in some mode that's what i like to see that's the yellow line so that's showing that a good bit of this rally today has been fueled by by stop orders and that is uh the stop indicator uh is a pretty good uh often a very good indication of uh direction of price and this move higher uh also starting in the afternoon fueled by aggressive buyers that shown by the cumulative volume delta there and you can just see all the green volume dots and we just saw with hero that uh options traders were taking positive delta positions also helping the fuel this move higher all right so there you go alexis again in the sub chart i'm using the um stops indicator in some mode i also have the on chart indicator so i can see the individual events and that's actually uh somewhat of a an aggregation in this zoom level of the chart so if you trade futures i think the stops and icebergs are an essential part of analysis for futures and i have both here they're part of the mbo bundle uh that's available in the book map marketplace and the stops and icebergs do work for uh cme futures only all right let's take a look at nasdaq now all right so very similar price action now nasdaq trading above the 350 volatility trigger let's take a look at hero and see what options traders are doing first of all i'm going to go to the nasdaq signal this is a combined signal for ndx qqq a little bit more choppy today here's this morning setup not quite as clear as the sb 500 but traders start taking positive delta positions right around uh 9 59 55 and nasdaq rallies let's take a look at one other thing this is a new indicator a new um uh symbol for hero called magnificent seven and this paints a much more clear picture what this is showing is options trades and hedging activity for a combined signal of stocks that are known as the magnificent seven apple amazon google meta microsoft nvidia and tesla and these stocks make up a very large component of the nasdaq 100 so let's take a closer look at this chart and this after this chart this indicator the hero signal has been rising after about the first 10 minutes of trade this morning traders taking positive delta positions so today they are selling puts taking advantage of high implied volatility they're not shown by the rising blue line they're also buying calls that shown by the rising orange line notional value for today both of those are positive so this provides a very clear signal for the for the nasdaq today that traders are taking positive delta positions in these seven stocks when that happens market makers take negative delta positions they are selling calls buying puts and they have to buy stock to hedge their delta exposure right so this is a great new symbol that's been added to hero that provides a lot of clarity for both the sp500 and nasdaq but especially for for the nasdaq all right so let's go take a look now at so tollman number 12 asks when i say traders are buying calls does this mean that market makers are short calls yes so i'm talking about options market makers they take the opposite side of customer trades and then they hedge their delta exposure in the case of stocks with the shares of stock and the case of qqq options for example they will hedge their delta exposure within q futures most likely right so tollman that's kind of a basis of my my approach to trading is that options traders and market maker hedging activity are a key driver of price for equity index futures and many stocks and certainly all these magnificent seven stocks have very active options markets and are heavily driven by options trades and market maker hedging activity all right so let's go back to nasdaq now all right so here's the reversal at right around 10 am right around the qqq 347 level and after that order flow becomes a little bit more clear green volume dots aggressive buyers multiple pack pullbacks as nasdaq makes a series of higher lows and makes a final launch for the afternoon rally right at the summer between the 348 level and the indiax 299 zero gamma level and now trading above the trading above again the qqq 350 level right tollman asked if market makers are short calls they are interested for the stock price to go down or am i wrong well here's how here's what market makers are doing all they're doing is making markets so they want to they want to make markets that's how they make their living and then they're hedging their delta exposure so they don't care if price goes up or down they don't care what they're praying what price they're paying for stock all they want to do is hedge the risk and make markets so they are not interested in the stock price to go up or down right so i hope that answers your question again market makers all they're doing is making markets that's how they make their money and hedging their delta exposure so they remain close to delta neutral all right so that's nasdaq let's take a look at some stocks first of all again amd reports earnings after the close so let's take a look at amd we'll see how traders are positioning ahead of earnings the earnings report this afternoon so there's amd and book map let's see what options traders are doing so let's go to amd all right so traders are taking positive delta positions in amd ahead of earnings and mainly call buyers driving price higher that's shown by the rising orange line put the blue line is fairly flat so called traders are buying calls ahead of the earnings report this afternoon very steady uptrend and tollman number 12 you're welcome hope that helps all right so there's amd traders buying calls ahead of earnings this afternoon all right let's take a look at invidia gonna zoom in on this let's zoom in on the morning so in the morning invidia was trading below its foot wall that was expected to act as support and right around 940 945 traders start taking positive delta positions note the floor alerts here indicating significant options trading activity and hedging activity and price moves back above the foot wall it did act as resistance temporarily and one more time and then price broke higher then acting as support as expected and so the 400 level very important in in invidia today let's see what traders have been doing so they're actually they're selling puts today that show them by the rising blue line they're also buying calls but the the blue line the put line really driving price higher traders taking advantage of low price advantage of lower prices and at high implied volatility to sell some premium in invidia today all right let's go take a look at book map remember the 400 level so right at the open invidia broke sharply below the 400 level let me just tone down these volume dumps just a little bit so here's the 400 level price broke lower right after the cash open acted as resistance and note the liquidity at that level there were some sellers at that level that's what the heat map is showing price broke through consolidated for just a few minutes moved higher then that level acted as support and now invidia is trading higher note there are a lot of aggressive buyers today you can just see all the green volume dots there right so that's invidia let's take a look at tesla tesla has really been beat up lately all right let's see what options traders have been doing with tesla right note for tesla the 195 level that's the foot wall did its job as expected tesla did trade slightly lower but as tesla was testing that foot wall traders came in started taking positive delta positions note the floor there there again signaling significant options and hedging activity and the hero signal indicates that traders are continuing to take positive delta positions in the case of tesla they are buying calls and selling puts shown by the rising orange and blue lines notional value positive for both and 200 is the key gamma strike and that's also the hedge wall that's shown here so the gamma levels on this chart 200 key gamma strike hedge wall and the 195 foot wall those are the spot gamma levels in play for today let's go back to book map oops back to book map so the 195 let me tone down the volume dots just a little bit here so we can see price action a little bit better all right so what the stock is volatile is invidia just like tesla just like invidia support and resistance levels may not hold to the tick there's the 195 foot wall support initially the 200 level acting as resistance and now price trading above that level note a lot of aggressive buyers today all the green volume dots very few magenta volume dots that's also shown by the rising cumulative volume delta just a steady uptrend of aggressive buyers coming in taking advantage of lower prices all right let's get back to the well first of all is anybody have any stocks they want me to take a look at please let me know if you have any stocks you want me to take a look at i'll be glad to do that otherwise we'll go back to the sb 500 first so it looks like it is finding some resistance at spy 418 the sb 500 all often reacts at the spy round number levels and there it is the 418 acting as resistance let's see what options traders are doing now go back to the sb 500 so here mean reverting options trades taking negative delta positions then a few minutes later price starts to move lower at that potential resistance level at the spy 418 all right let's take a look and see what they're doing with the magnificent seven very similar options traders taking their foot off the gas price moving lower nasdaq also similar traders starting to take negative delta positions fading that move to uh to highs of the day let's go back to book map we'll take a look at in q all right so traders starting to take have been taking negative delta positions looks like some aggressive sellers coming in with the magenta volume dots here showing that and alexis back to your question about stop orders you can see the buy stop orders helped to fuel that move higher and then that line levels off no more buy stop orders fueling the move higher aggressive sellers start to come in we know that traders are taking negative delta options positions and prices starting to move lower so i'm using stops as part of a narrative for this we don't know if it's a pullback or a start of a move lower so far maybe just a pullback uh potential support level right now 350 and then this 364 combo level that acted as resistance earlier all right again uh last call for stocks anyone have any stocks they want me to take a look at come back to the sb 500 potential support at 417 the volatility trigger that acted as resistance before let's go back take a look at hero see what options traders are doing go back to the sb 500 i'm going to go to a shorter rolling window period just to see if we get any additional clarity so it looks like so now i'm just looking at the last 30 minutes of data rather than the entire day so in this with this viewpoint options traders continue to take negative delta positions all right shane asked about tlt i think you asked about that yesterday is that correct so we can take a look at it again i don't have tlt in um the book map we can take a look in hero here so not a very active options markets market in tlt so mixed picture overall um slightly positive for the day traders taking positive delta positions but up and down i hero very very thin options market in tlt at least according to the notional value here let's see what they're doing in the magnificent seven so again i think it remains to be seen whether this is just a pullback or the start of a move lower all right let's end up with book map so now we're watching for the um potential support at 417 let's go check nasdaq so nasdaq now breaking below uh trying to break below the 350 364 level all right my time is up um so if anybody is looking for for an entry i suggest watching this um 417 level um and that's also right at the es 4200 level so easy to remember all right again my time is up want to thank everyone for watching thank you for your questions and comments and remember tomorrow uh first of all uh well for today amd earnings after the close tomorrow pmi data at 10 expect kind of a quiet market after that uh up into the um fomc meeting again the announcement at two press conference begins at 230 and i'll drop whatever i'm doing at at two and so we can watch the live market all right thanks everyone again thanks for watching thank you for your questions comments and i will see you tomorrow bye


Hi, this is Don. Santosh asked me to make a quick little video showing you how I get set up in MyStatLab to export this data that we've been using to accumulate the total time and the assignment times as well as the grades. Okay, here is MyStatLab. And of course we go down to Course Tools, Gradebook. And when the Gradebook opens up, we want to click on Export Data. And we want to use the advanced version. I'm just going to type in the name there, put appropriate names so you can recognize your file, Export Type. I want the assignment results. I'm going to go to the MyLab XL, which is the XL format, and then leave the defaults at first, All Students, All Dates, All Categories, All Assignments, All Chapters, Included, Attempt Only, Set the Best Score, Export Scores as Percentage and Decimal Format. These are all the standards. Get down to the bottom, and I want to click on Other Data, Time Spent on Assignment so that we get the time spent on each assignment, the date worked on the assignment, Overall Score, and I like to get the study plan points and time spent as well just to get additional information. And then just click Request Export, and then you will be sent the email. Really that's all there is again. Just a quick look at the file that you will get. Here I've deleted the student names, email and login information, and you should do that too before you share it with other people. You have to do that in two places down at the bottom. If you've clicked on the study plan, you'll have their names as well there, but there's the data for the study plan. And it gives you, of course, the Overall Score, Total Time Spent in MyStatLab, and of course, the Median Time Spent. As you scroll across, it will have all of the assignments, quizzes, homework, midterm, and final exam, and of course, it's got the date and the time. Hope this helps.


Hello, everyone. I guess we'll get started here. My name's Steve Wall. I'm from the ECS team, now a part of CGI. My partner here is Steve Greenberg, and he is with Resilience Scale. And we're going to be discussing active multi-cloud today. And it's ephemeral cloud foundry instances. So there's been a lot of talk in the conference about multi-cloud, and that's running cloud foundry on many different clouds. And what we're going to be talking about here is active multi-cloud. So that's running many instances of cloud foundry, and them all appearing as one. So at the heart of multi-cloud is Bosch. Bosch is really the enabler. So Bosch allows you to deploy cloud foundry to a variety of different eye-asses. So with Bosch, you can deploy cloud foundry out to several different eye-asses. And from an end user perspective, they don't know what eye-ass or service provider that cloud foundry is running on. So that's kind of a multi-cloud situation. So you've got cloud foundry running in multi-different clouds. So active multi-cloud is where you have your multiple instances of cloud foundry running on multiple different service providers, but from an end user perspective, it just looks like one. So what are some use cases that enterprises or companies would want to use of active multi-cloud architecture? One of those use cases is resiliency. So if a service provider had a hard outage, so say Amazon went down, it had a regional outage.  You could simply stand up another instance of cloud foundry in a service provider that was fully healthy. Now, you might see some small degradation in service, but your service wouldn't go away. And then you quickly stand up another instance of cloud foundry in that service provider that was fully healthy. And then you are at full capacity once again. So you have good resiliency when you go to active multi-cloud. Another reason that an enterprise may want to go to active multi-cloud is geography. So say an enterprise had customers distributed globally. They may want to stand up a variety of cloud foundry instances globally. So that they have cloud foundry instances that are located close to their consumers, and they could do something like latency-based routing. So your application is globally distributed, looks like one application, but from an end user perspective, wherever they are in the globe, they get the same type of response from your application. So another reason that an enterprise may want to go to an active multi-cloud architecture is workload distribution. So say, for instance, that some service provider was running a special on the cost of their VMs. You may want to shift your workload over to that service provider for a period of time. Or if a service provider had some high-performance compute you'd want to take advantage of, you may want to run more workload on that service provider than on another service provider, but you still want to have your application distributed on multiple clouds for the other reasons, like resiliency or geographic reasons. So how do you actually achieve an active multi-cloud situation? So in how to describe that implementation, I'm going to turn it over to my partner here, Mr. Steve Greenberg. And he's going to go into some details there. Thanks. All right, so a little reality check. This is not a trivial problem. Certainly not a trivial problem at 5.30 when we're standing between you and getting out of here for the day. But our intention is to give you kind of enough information, talk a little bit about how we've actually solved these problems, and make it easier for you to actually get there if that's what you need to do. All we're really doing, the secret, is applying things like 12-factor app principles, microservice architecture principles. But we're trying to apply them at the Cloud Foundry level. So let's start with a little bit of baseline, and probably one of the most important things when we're trying to actually create an active multi-cloud. The good news is, all the building blocks are there. We've done this for customers. It's actually not that hard. First thing we need to consider, though, is separation of concerns, the power of isolation. We know the power of isolation when we're building applications. We know why that's so important. If we can apply those same principles to our Cloud Foundry instances, we can end up creating ephemeral Cloud Foundry instances, which allows us to create an active multi-cloud pretty easy. So let's start with the really, really simple problem. In this scenario, we've got multiple Cloud Foundries running. We've got users throughout the world. And they want to go to one location. Routing, how do we actually route traffic into these different Cloud Foundry instances? So some user gets on. They go to cloudfoundry.org. It's actually pretty trivially easy using regular internet available technologies to figure out where that user is and route them to the appropriate location, based on latency. There's a lot of solutions for this. Route 53 is a really good one. There's latency-based load balancing solutions. That's the really, really simple use case. So where does this get to be a hard problem? It's all about the data. The data is the hard problem. Our applications can be stateless, but if they don't use data, what are they doing? So let's take a look at what data sources we really have to contend with if we're going to do an active multi-cloud. And then maybe we can reason over some of those data sources. So there's the Cloud Foundry data sources. So things like the Cloud Controller, maybe the Blob Store, probably UAA, where all of our users and roles and orgs and spaces and stuff are. Those are all of those Cloud Foundry type data sources. There's also app-specific data sources. So obviously, our apps have to have data. If they don't have data, they're pretty much not doing anything. But let's actually go back to Cloud Foundry and start talking about some of the things that we've seen. The reality is, if you go into any enterprise customer, we can take a look at our little friend UAA over here. And you're going to be hooking up at least part of that to some enterprise system. And usually, that's authentication. That's going to be LDAP. It's going to be Active Directory. It's going to be a system that already exists. The other mistake that we've seen customers do is they take the UAA that's in Cloud Foundry, and then they use that for tokens for apps. And they apply it to their application level security. So all of their end users are registered in UAA. And all of their tokens are coming from that same Cloud Foundry UAA. It's actually a really poor separation of concerns. And it will lead to very, very significant challenges if you actually want to do this for real. So we recommend separating out those identity services. And then if you're building a microservice architecture, if you were in the last session, and you've probably heard some stuff about things like service discovery and service registries, that's actually part of that app portfolio too. And the reason that we apply Cloud Foundry on one side and we look at the app perspective on the other side is the scope is different. We want to keep our Cloud Foundries isolated from each other, but our apps actually span all of those different Cloud Foundries. So if we look at it from that perspective, then we can start to tackle what this data problem is. Everybody's probably heard of consistency, hopefully an eventual consistency, right? A lot of people think they're kind of diametrically opposed. A lot of people think that a lot of systems have to be eventually, or they have to be consistent. Eventual consistency can't work. The reality is there's far more use cases where you can have eventual consistency and weigh the risk and the reward. So the good example I always use is most of your financial transactions, if they're under a certain amount, say $10,000, are actually eventually consistent because your bank would rather pay the penalty of a difference than pay for the solution to make them consistent in scale. So keep that in mind. In modern systems with eventual consistency, the window of risk is actually really small. It used to be there's large network delays. Used to be that it was very, very hard to send data because maybe the pipe was small. But that's not true anymore. The reality is that in most modern systems, that window of risk is really small. And if we couple those things with other good best practices, things like doing blue-green deploys, things like doing non-breaking schema changes, then eventual consistency allows us to actually gain consistency because that window of risk is so small. So we can gain consistency by using eventual consistency and dramatically simplify the problem at hand. So let's go back to our data picture, cloud foundry and apps, right? If we're trying to apply principles of separation of concerns and isolation, all of these things can actually become eventually consistent. Everything in cloud foundry can be isolated and we can use automation to actually make them consistent. Our app picture, so our service registry also can be eventually consistent. Hopefully you've heard about the container-to-container networking going on within cloud foundry, but we don't necessarily want applications reaching outside of cloud foundry and knowing about other applications because that becomes a harder problem. So we can actually make that eventually consistent for most use cases as well. And that really leaves us with two challenges and they're not always challenges. Our identity service, so think tokens for apps and think about app data. Not every single use case does app data have to be consistent but a lot of them do. Not every single use case does a token have to work in different cloud foundries, but in a lot of cases it does. So let's tackle eventual consistency first. How do we deal with this in isolated cloud foundry instances? Reality check, you don't push apps to anything that's not dev. If you are, you're probably doing it wrong, right? Who does? Concourse does, right? Automation does. Automation should be deploying to everything else other than your dev environment. And we can leverage that to our advantage if we understand that. So how do we do that? We can use automation to provide eventual consistency across different cloud foundries. And maybe that involves writing a YAML file that specifies what orgs and spaces and apps and users and roles exist or should exist across our cloud foundry instances. Maybe we do it in YAML because CF doesn't have enough YAML and we just miss it. But we can feed that into something like concourse and allow concourse to actually keep these things in sync in an eventual consistency fashion. How often are you adding users? How often are you updating roles? You're certainly pushing apps, hopefully a lot. But again, with blue-green deploys and non-breaking schema changes, that window of risk becomes really, really small. The important factor here is that this source of truth, this document that tells us what our world should look like has to come out of a single declarative source. I don't care what that is, it could be GitHub, could be Minio, I don't know, whatever. But if it comes out of that single source of truth, we can use automation to actually make these things look the same. So that just leaves us with the other problem, right? Our identities, our tokens and our application data. Now our application data is gonna be very specific based on what our apps are doing. I can't tell you what your apps are doing. I can't tell you what data sources they should be using. And identity doesn't always have to be consistent, but let's assume that it does. So what most of our customers do initially, when they go to an active multi-cloud, is just declare one as a master, declare one as a failover. And leverage the multi-cloud capabilities at the application level. That's okay, but it's not ideal, right? We obviously have single points of failure, we obviously have failover scenarios, and you pay a pretty significant price, especially if you're dealing with geography, when you're accessing data from very, very far away. But you can introduce caching strategies and you can kind of live with this and you can make things work. What we really want customers to do though, is actually make better life decisions. We want you to think about your applications and what data they're using. We've both had customers that literally just picked a SQL database because their DBAs knew it and it wasn't Oracle. And then they did it for everything. And that's really hard, right? Because there are technologies that actually make this a lot easier. And you should probably go check them out. Hopefully some people went to the Google Cloud Spanner talk. There's other technologies that are there to actually implement distributed resilient data stores. And you should leverage those technologies. Don't just pick MySQL, don't just pick Postgres, don't just pick MongoDB or whatever. Really understand what is it that this piece of data is and then find the best solution for it. Everybody thinks it has to be a relational database because we know them. Yes, there's always relations between data but in microservice architectures there's very rarely lots of joins going on in databases. So do you need a SQL database? No, you really probably don't. You might be able to use something like Redis and go talk to the Redis Labs folks about how to do this at scale. Or use one of the cloud providers like Google or even AWS or RDS that are providing very high level service capabilities for those backing data services. So we've actually done this for some customers. Again, it's not trivial. It's very specific based on the use case of the applications in the portfolio that you're trying to run. It's also very specific because sometimes you're trying to run the same applications on every cloud foundry, sometimes you're not. And those are very different scenarios. But there's a couple of best practices that in lessons learned that we've figured out kind of the hard way and by working with customers. First one is understand why you're doing this. If you're doing active multi-cloud because it's cool you should probably stop. If you don't have a real business use case to try to tackle these problems, you should stop. Because cloud foundry is actually really resilient on its own and most of the cloud providers are really resilient on their own. And so do you really need this? The second thing is weigh that risk reward for that eventual penalty. If your apps can be eventually consistent like in financial services, your job becomes so much easier. If you can introduce caching and they have to be consistent, your job becomes so much easier. It's a really hard problem if you don't. If you're told that it's a requirement for consistency. In most cases it's not a requirement and you need to push back and you need to challenge that requirement. And the last thing is you have to automate everything. If you don't automate everything, and I mean everything, I mean your application life cycles, I mean your deployments of cloud foundry, I mean your updates to cloud foundry all the way from dev through production, this is impossible. So make sure that you automate the life cycle of your environments, make sure you automate the life cycle of your applications, make sure you automate the declarative source of truth that's providing you with that eventual consistency. If you do those things, this actually becomes a pretty easy problem. It's really not that hard. It's just hard to figure out initially what our apps are doing on a case by case basis. And we can go through and we can evaluate what we're actually trying to achieve and we can make good decisions based on those microservice principles. So think about things like separation of concerns. Think about things like 12 factor apps. And apply those to cloud foundry to create ephemeral cloud foundry instances. Hopefully we're all here and we love cloud foundry, but we need cloud foundry to be able to die. We need it to be able to scale up, scale out, fail. Maybe the underlying infrastructure needs to be able to fail. That's the whole purpose of this active multi-cloud. If you keep these things in mind, it's actually not that hard. And I know that was quick, but we wanted to leave some time for questions. Are there any questions? Yeah, sure. Yeah, so the question was which type of data source gives us the most problems? And I mean, it's basically the SQL data sources. I mean, that's all it is. And so if you can avoid using those, that's great. But again, it depends. I mean, if you're building this large, geographically distributed multi-cloud, that's very different than let's say just building a multi-cloud in the US, right? So those are the hardest. If you can pick something else, then do it. Just might be new skill sets. Any other questions? Yeah, so we're not talking about using concourse. The question was concourse replicating data. We're not talking about using concourse, let's say to replicate application data. What we're talking about is making two cloud foundry environments look the same. So the same apps are deployed. The orgs and spaces exist. The users exist. The roles exist so that our automation can work. That should be driven out of concourse or some automation tool. You should really be using concourse. Any other questions? So, sorry? Is Google Spanner available in the marketplace? So the marketplace is different depending on your distribution. So it depends on what distributions you're using. I actually don't know. I don't work for Pivotal anymore, so I'm not sure. Part of what you'll find though with this is that you're not using the marketplace for production services. You might use it to spin up the initial service, but if you're doing active multi-cloud, you're not sharing those brokers necessarily between different cloud providers. And actually, all of your production data sources would be attached through a user-provided service instance. Any other questions? All right. Well, thanks for hanging in there. I know it was the last session of the day. I know everybody's tired, but yeah, we appreciate it. If you have any other questions, feel free to find us. Thanks.


So, let me thank the organizers again a little bit more loudly. And so, I will give four lectures, and the topics in each lecture will be quite different. So, if you don't understand something in this lecture, you can come to the next lecture and hopefully, maybe it will be better, or maybe not. So, I'll try to illustrate various techniques that appear when you try to solve counting problems. And I will try to illustrate them in the simplest examples, but try to illustrate the ideas. And the examples that I'll try to work out, I think, contain the main ideas that appear in much more complicated situations. So, in particular, if you have questions about anything that I am explaining, you should not hesitate to stop me as I'm explaining. And if I haven't defined some object that you're not familiar with, I'm happy to define it here. So, yeah, so the goal is for people to try to understand what I'm actually saying. And to begin, I'll start with probably the simplest sort of classical problem, which is called the Gauss-Circle problem, and it's concerned with the following question. You have Z2 in R2, and you'd like to understand how many integer points there are in the bowl of radius R. So, Br is X is less than R, or X is in R2. And the question one would like to understand is the number of integer points in this big disk. And the fact is that, of course, the number of such integral points is proportional to the area. This is the area of, maybe I'll put this here, the area of Br by R squared. If I move this, I think it will be better. Okay, so the area is quadratic, but what I want to discuss in this lecture is how to deal with the error term. So, the easiest error term is just linear, so you can get a linear error. And how do you see this linear error while you put a square of size 1 around each lattice point? And you can see that if you change the radius by, let's say, 2, then the area that you get here is an undercount for the number of lattice points. And if you change the radius by plus 2, the area will be an overcount for the number of lattice points. So, getting this linear term is quite easy. So, this was essentially known to Gauss. And what I want to explain today is how to obtain a stronger error term, which is, so this is due to Sierpiński, and it wasn't done today, but in 100 years ago. So, the error term, so you can improve the error term to R to the 2 thirds. And the conjectured error term is, exponent here is 1 half plus epsilon for any epsilon. But, and there have been some improvements to this 2 thirds, but there hasn't been, the 1 half hasn't been reached. So, as I said, I'll explain how to obtain this error term today. But the techniques that are used are actually the kind of basic steps that will appear, are useful in other general contexts, and hopefully in the next lectures, I'll start connecting these kinds of counting problems with dynamics on homogeneous spaces. And there, you can also get error terms of this type. So, this type of 2 thirds, so there's even a name for this kind of 2 thirds type estimate in other contexts. And the techniques are essentially the same. So, if you understand how to do it in this simple case, you can do it in these more general contexts of semi-simple league groups and so on. Okay, so what is the plan for proving this? So, the plan is essentially the following. So, we're going to use Fourier transforms. So, we're going to take, so we're going to define the function f of r to be this. Sorry, let me introduce some notation. So, define f tilde of r of x to be 1 if x is in vr. And 0 otherwise. So, this is just the indicator function of this big ball. And then, we consider the projection from r2 to r2 mod z2. And we just push forward this function. This is going to be our function that we'll work with, f of r. So, you have this big ball and you push it to this fundamental domain. And you see how it overlaps. And the quantity that we're interested in is we want to understand fr of 0. The value of this function at the point 0. And this is exactly the quantity that we are interested in. So, it's the number of lattice points in the ball of radius r. Okay, so this is the object we're interested in. And we are going to proceed as follows. We're going to estimate the freer coefficients of this function. Then we're going to regularize f. So, what is push forward? So, you just... Okay, maybe let me... I'll define it in one second. I apologize. So, what I mean by push forward is you just take the function and you kind of throw it down. So, if you want to evaluate the function at a point here, you take all the pre-images under the projection map and you sum their values. So, I'll write it formally. Let me just write this. So, forward of f at x is defined to be the sum. So, x is now an equivalence class of points. So, it's the sum of... This is f tilde of x, of x prime. Oops. So, this is x prime. Where x prime is x plus some v and v is in z2. So, you sum over all the values that you see above you. And again, this kind of process where you have your kind of uniformization, you have this universal covering type object and then you have a function defined on that and then you push it to some compact manifold or some finite volume manifold. This is a standard step. Okay. So, are there other questions about the objects I've defined so far? So, I'm trying to explain the plan of how we're going to do the estimates. So, we have this function f of r that we've pushed forward. We're going to estimate the free acquisitions. Then we're going to regularize f. So, what do I mean by regularizing f? Well, f is not really even a differentiable function because of the edges. The function is one here and zero here. So, this is not a smooth function and things don't work so well for non-smooth functions. And then we're going to... After we regularize, we're going to estimate again what happens and we'll see that something good will come out of it. Okay. So, this is the plan. So, now we fail. So, the first step is to estimate the free acquisitions. So, I have to remind you what the free transform is. So, if you have a function, the free transform is just the integral. So, we're here t2 of f of x e to the negative 2 pi i cx dx. So, we're going to work with free transforms either the torus or r2. So, c is in here z2. But if you take c to be in r2, then you're integrating over r2 and x can also live on r2. Okay. So, there's this basic free theory and the kind of formula we want to use is free inversion which tells you that you can recover the value of a function if you know it's free i series. So, it says that f of x is the sum of over xc and z2 of f hat of xc times e to the 2 pi i. So, this is the free inversion formula but the problem is that it's not true in general. It doesn't hold for any function. So, this holds if f is, for instance, smooth. So, if f is a smooth function, if this formula is true, our function is not smooth. So, we'll have to deal with this problem later. So, what is the main technique for dealing with non-smooth with functions free transform? So, I'm going to give you an exercise which I'll essentially solve later. I'll tell you the main idea but the exercise is the following. So, if f, if, let me call it eta, is ck, so has k derivatives, then you have an estimate for the free coefficients. So, if you take the norm, the absolute value of the free coefficient, then, so I'll explain in a second, plus an eta ck times c to the minus k. Okay, so what does this say? It says that, so let me explain the notation first. So, I'm going to use this less than or equal sign, so a less than or equal b, and here we have x, if exists a constant c which could depend on x, such that a is less than or equal to c times b. So, what is this exercise? It says that if a function is pretty smooth, if it has k derivatives, then the free coefficients decay up to some constant which depends on the k we're talking about and the norm of the k derivative, they decay like a power, and the power is the power here in the derivative. So, this is a standard fact, and as I said, I'll essentially give you the idea of the proof of this. Okay, so I said we have to compute the free coefficient of our function, so we have r of c by definition. So, if you unfold that projection, it's going to be the integral over rn of this after all the r of e to the minus 2 pi i c x dx, which is simply the integral over the bowl of radius r of e to the negative 2 pi i c x dx. So, the function is 1 on this bowl and 0 outside, and also notice that, so here we have the dot product, right? We have the dot product of two vectors and the bowl is symmetric, so we can rotate. So, rotating, so this integral is rotation invariant, so it means that we can take c to be 1 0, sorry, not 1 0, but absolute value of c 0. So, we can rotate c to our free coefficient to whatever value we like, and we can normalize it to be this. So, let me draw a picture of what integral we're computing. We have this sphere and we've normalized our direction c to point in this direction, and you see this function does not depend on, so let me write this here. So, c times x is c absolute value of c times x 1, so if x is x 1, x 2. So, it doesn't depend on the second coordinate, so we can integrate out this second coordinate. And how large is it? Well, if this is x, and this is r, then this is square root of r squared minus x squared. So, this integral is just going to be the integral from minus r, so we have minus r here and r here, then go from minus r to r of e to the minus 2 pi i c x, or absolute value of c x, times square root of r squared minus x squared dx. This is the integral, and so we're going to factor out the r squared here. So, I'm just going to do a manipulation to normalize it. Can people see here? No, okay. Can you follow what I'm saying, or is it okay? I'm working with general what? Yes, r2, yes. Everything is in r2. Thank you. Okay, so let me write the integral. So, we have an integral from minus r to r of e to the minus 2 pi i c x. So, I lost the factor of 2 there, but it's not too important. 2 r squared minus x squared to the 1 half dx. So, I'm going to write this as integral from minus r to r, e to the minus 2 pi i c r x over r, times 1 minus x over r squared to the 1 half dx over r, and r squared in front. Okay, so this is just algebra. Hopefully, it is correct. So, now if you call the new variable x over r, so this just becomes the normalized integral, integral from negative 1 to 1, e to the negative 2 pi i, your absolute value of c times r, times x dx. Oh, sorry, I forgot the main thing, that's 1 minus x to the 1 half dx. So, we have this integral. So, let me try to draw a picture of what we're looking at. We have something that looks like this. Okay, there's a 2 that I keep forgetting. So, we have this function from negative 1 to 1, and we're integrating it against this sign, which oscillates more and more and more. So, we're trying to integrate something like this, and we want to say that there's a lot of consolation, because you see this integral, this sign is oscillating. So, we want to say that as this number gets larger, then we have lots of consolation. And this was the type of problem that is also illustrated in that exercise. So, if you want to say that it's a function of smooth, then you have some consolation. So, let me make the claim that this integral, so I'm going to just call this c times our parameter. So, integral from negative 1 to 1 e to the minus 2 pi i. So, I'm going to get rid of the minus, because the minus is not relevant. kx, 1 minus x squared to the 1 half dx, equal to O of k to the 3 halves, negative 3 halves. So, k is some positive, large positive number, a real number. So, assuming this claim, so assuming the claim we found our first estimate, so we get that the free coefficient, the r free coefficient. So, what is our k? Our k is c times r, and we had an r squared here. So, we had r squared, and then r times c to the negative 3 halves. So, we get that this is less than or equal up to some constants. So, less than or equal to what? To r to the power 1 half times absolute value of c to the power negative 3 halves. Okay, so let me write this first estimate. So, we'll have it in the future. f r hat of c, oops, less than or equal to r to the 1 half, e to the minus 3 halves. So, I want to, okay, so let me make a few comments about this estimate. So, the claim here which implies what we want is, so if you look in standard textbooks on this material, they'll tell you lots of things about Bessel functions and lots of words which somehow essentially mean this. So, you have an oscillator integral that cancels out a lot, and you can in fact have much more precise asymptotics. You can not just bound it, but you can say it's equal to something plus some error, but this is the only thing you will need, and you should know that these kinds of, these things are called oscillator integrals, and in this particular problem, something that comes up a lot is Bessel functions. So, you get an estimate of this type. Let me remark the following. This estimate is kind of weak. So, if you look at this formula, this free inversion formula, if you try to apply this estimate, you're going to get that the function is not even summable. Okay, so this series does not even converge if you put absolute values with that estimate, and that estimate is essentially accurate. So, now we're going to prove the claim. So, we want to estimate this integral. So, we have negative from negative one to one of, where one half e to the two pi i kx dx. So, the way to estimate all of these integrals is the same, namely, you integrate by parts. So, integration by parts will give you that it's the integral from negative one to one. So, what do you do? You differentiate this, and you have to integrate this. So, you get negative x one minus x squared to the negative one half, and here you have e to the two pi i kx over two pi i k dx. This is just integration by parts, but you've gained something. You've gained this one over k factor. Okay, so you've gained. So, this is the key point. You've gained already a k to the minus one in this decay, but this is still not too good. So, I want to emphasize one thing, which, so when you have integration by parts, you have a boundary term. And here we didn't have a boundary term because the function ventures at the boundary. Here, this function already blows up at the boundary. So, you cannot integrate by parts again. You cannot integrate. So, this function one minus, so negative x one minus x squared to the negative one half at plus or minus one is equal to infinity. So, you can't integrate by parts again. So, if you had a smooth function, you could do the same trick, and you would do the exercise. You could do it as many times as you want, and you would get as high a power of k as you ever wanted. So, here you can't do this. So, what can you do? What you can do is the following. So, you have your minus one one, and what, here zero. So, you will fix an alpha. So, fix alpha, which, so, and consider, and you break up the interval, interval into x is less than one minus alpha. So, you set this parameter to be alpha, and absolute value of x is bigger than y minus alpha less than or equal to alpha. So, let me call this interval a, and this interval b. So, now you have to play a little game, because you cannot integrate by parts. You have to do what is possible. So, you can integrate by parts on this interval. And here, so, one thing I didn't mention is that this integral is bounded, right? This function, even though it's infinity, it's integrable at the edges. Okay? So, since this function is integrable at the edges, this integral is finite. So, if we could just estimate this integral by absolute value, you would get a k to the minus one. But we want a k to the minus three-halves, so we're going to do this trick. So, on interval a, so, on interval a, use integration by parts again, and you'll get a boundary term. And on b, you can just estimate by absolute values. So, you can say it again? On b, yeah. So, b is this interval. So, this is b, this is a. Yeah? So, the absolute value of this term, where's the typo? Is this better? Thank you. Is that the only complaint? Okay. So, the point is this. On this interval, we can gain something by using integration by parts. On this interval, we cannot gain anything, so we just estimate by the absolute value and we'll have to figure out what alpha is. So, because I want to actually finish this proof today, I'll leave the actual integrations by, you know, I think it's clear what you have to do and probably the effect of the coffee is wearing out. So, I'll just tell you what the estimates are. So, okay. So, what is it? Yeah. So, on b, so, okay. So, on b you get, so, the integral is o of alpha to the one-half and on a, on a it's, the integral is o of one over k alpha to the minus one-half. So, you see that this kind of makes sense in that if you make alpha to be very small, the boundary, the b term goes to zero, but this thing blows up as you would want it. And so, you have to optimize and it's clear. So, if you do the optimization, you get that the best choice is to take alpha to be k to the, was it negative one-half? Yes. Negative one-half and you'll get another factor of one-half which will, so you'll get that power saving and this power saving is essentially the key point where you get this minus three-thirds and three-halves and this is the, what will eventually give the right error term. If you want in the exercise session I can actually do these integrals a little bit more slowly. Okay. So, next, we, sorry? Yeah, yeah, yeah, sorry, minus one, you're right, yeah, yeah. Alpha to the one-half is k to the minus one-half, that's right. Okay. So, the next step is to do this smoothing. So, as I said, this estimate is not even good enough to sum the series. So, what you have to do is you have to smooth out the function. So, smoothing, you have, so you'll pick a bump function, so you'll take eta on R2 and B1. So, just take a function which is of integral one and you define eta epsilon of x to be one over epsilon squared eta of x over epsilon. So, this is a bump function which is getting narrower and taller and the normalization is such that the integral of N, that the integral of this gadget is one. So, now you'll convolve these, what we're going to do is we're going to smooth out the edges of our function. Sorry, I'm going to erase this side of the board so that I don't have to write on wet again, so we're going to smooth out the rough function fr on which we have our estimates and then we're going to estimate the new free coefficients of the new function and hopefully this will give us the result. So, the new function is going to be f of r epsilon is going to be defined to be fr convolved with eta epsilon. So, what is the convolution of two functions at x is the integral of f of x minus y g of y dy over, it doesn't matter here, it could be Rn, Tn, your favorite group. So, this is the convolution. So, what does the convolution do geometrically? Let me just draw the picture here. So, you have your function, this is Br, and convolution, better, okay, I'll draw the picture here. Okay, so the convolution does the following, it takes your function and it puts, so it takes this bump function and it moves it around and it puts it. So, you integrate your function, this bump function against this Br and since this has norm one, you see inside it's going to be always, this integral is always going to be one, so it's not changing the function. It's only smoothing it out at the edges. So, okay, so we have, let me write it again, of r epsilon as f of r convolved with eta epsilon. Now, eta epsilon is a smooth function, so eta epsilon of k of c, of a freia coefficient c is less than or equal to any constant absolute value, any k, let's say, eta ck, oh no, sorry, let me, this. So, you get an estimate of the following type for times epsilon to the minus k. So, you get an estimate for the freia coefficients of eta epsilon in terms of the freia, in terms of the ck norm of eta, which is a universal constant, eta is fixed, and then the epsilon gets in here and then you have the c, which gives you lots of dk, where as the epsilon, see the epsilon is small and it's bad for you because it comes with a negative power, the c is good for you, comes with a positive power. So, and we're going to also use two, so let me write this estimate here, because this estimate is a little bit, okay, so this is something that depends on k, and then c times epsilon to the minus k. And we're going to also need a very trivial estimate, which is that this is less than one, and this is true because the, this function has integral one, so this is a trivial estimate, but it's good in the regime, you see this number can be bigger than one when c is very large, this number is, this is a very good estimate, when c is small, this is not a very good estimate. So, we're going to use these two regimes for this estimate, and we're almost there. So, I need one other fact, which I forgot to mention, so f r of zero, f r hat, the first free coefficient is just pi r squared, because what is f hat r of zero, so we don't need to estimate this, this is the main term, this integral of a b r of just one, this is just pi r squared, this is the main term. So, now we can apply free inversion to our regularized function, so f r epsilon, we can write this as f r, sorry, we can write this, I forgot the main thing. Okay, let me write like this, f hat r epsilon c times eta, no, times e to the 2 pi i c x. So, we need to compute the free coefficients of this new function, but these free coefficients are actually very easy to compute, so again, this is an exercise that we can do in the exercise session, that if you convolve two functions and you want to compute the free coefficient of their convolution, then you just take the product of the free coefficients. So, free coefficients behave very nicely under convolution, so the free coefficients here are basically the product of these two guys and we have the following estimates, so we have the estimate on this guy, this kind of decay, and on this guy, we have this kind of decay. So, we have to estimate, we have to see what kind of information this gives us, and as I pointed out earlier, they're essentially the free coefficients of eta epsilon, they depend, we have two estimates for them, and the product of c times epsilon is one, less than one and bigger than one. So, let me write this, so f hat r epsilon of c is f hat r c times eta hat epsilon of c, so we're going to estimate each of these guys individually. For this one, we have a single estimate, we have the estimate when it's less than one and bigger than one. Okay, and finally, the zero of coefficient is just what, it didn't change, it's pi r squared, f hat r, sorry, now we have, oops, so now we're actually using the free inversion, so now we want to evaluate our function at zero, so we see that it's pi r squared plus the thing we want to estimate, some c different from zero, f hat r epsilon of c. So again, we divide this sum into two terms, we have a term a where c times epsilon is less than or equal to one and b when epsilon times c is bigger than or equal to one, depending on which estimate is better. And I want to, again, so you see that there's the same game that we played in the previous estimate, we have two positions, two cutoffs, and I haven't told you what epsilon is, so we're going to choose epsilon at the end, not now, and we'll optimize in terms of what we will get. So hopefully, so this is defined on the torus, so this function is so small that you can view it either on the torus or on r2, it doesn't really matter. So everything now is on the torus, but if you want, you can do everything on r2, but you'll just get integrals instead of sums. It's not, the issues here are the exponents, not the summation versus integration. Okay, so let me get the right estimates. Okay, so we have, okay, so summation a is, we have some zero less than c less than one over epsilon, and here we have r to the one half times c to the minus three halves. You see, so I'm using just the free coefficient that's less than or equal to one, the second term in this product. So this is a product, and the second term is just less than or equal to one. So this term is r to the one half times epsilon to the minus one half, and for integral b, so here, so the integral b is going to give us a little bit more savings, so one over epsilon is less than or equal to c. So we're going to have r to the one half, c to the minus three halves, then c to the minus k, then epsilon to the minus k. So you see now the epsilon comes in, and this is less than or equal to, so if you do all the estimates, it comes out to r to the one half times epsilon to the minus one half. So they're actually of the same order of magnitude, as it should have been in some sense, but the point is, okay, so what is the upshot? We're almost there. So the upshot is that our term is our, so we found the following fact, so we found that f r epsilon at zero is pi r squared plus o of r to the one half times epsilon to the minus one half. So it would seem that we can set epsilon to be, let's say, one, and we're getting the r to the one half, which would seem like it's too good, but the problem is that you see our function is a little bit smoothed out, so we have here b r minus epsilon, we have b r plus epsilon, and we have b r. So our function f of r epsilon really is an approximate count. So we have that f of r plus epsilon at zero, so let me get the exact inequalities because, okay, so the claim is that there are two claims. The first one is that f of r at zero, so remember this is an un-regularized function, this is bigger than or equal to f of r minus epsilon, epsilon at zero. So what does this say? You see here we're doing the pure count and here we have the regularized count and the support of the function which is in b r, so we're convolving this b r minus epsilon and we're smoothing it out and its support is contained in here. So this function is everywhere bigger than or equal to this and so this thing as we got here, so this is pi r squared, sorry, pi r minus epsilon squared plus o of r to the one half times epsilon to the minus one half and so this is pi r squared plus two pi r, sorry, minus to pi r epsilon plus this o to the r to the one half times epsilon to the minus one half plus lower order terms. So now again we have these two terms so we want to balance these two terms out so again you see we haven't fixed epsilon yet so we want these terms to be of the same order so that we get some information so we want again r epsilon to be approximately the same as r to the one half times epsilon to the minus one half and so what does this give us? You have that r to the one half is approximately epsilon to minus three halves so then epsilon is r to the minus one third. So if you plug it into here you get that f r of zero is bigger than or equal to pi r squared plus something that's so epsilon is r to the minus one third this is this term so you have r to the one half plus one sixth which is o of r to the two thirds and for the lower bound you play the same trick but you'll have that f r zero is less than or equal to f of r plus epsilon epsilon and you'll get the same type of estimate. Okay so we found so let me just finish and just say that we found this estimate the number of points, integral points involved radius r this is pi r squared plus o of r to the two thirds so before finishing I just want to emphasize what were the main ideas that made this proof work so first we used the spectral decomposition which is something that will have a general semi-simple e-group and this is going to work more generally and so we're going to be able to decompose our functions if you want to count in terms of this Fourier series what you want to do then is you want to smooth out your counting function because things usually don't work so well for these rough counting functions when you just cut off at straight r and then you want to have some parameters you have the smoothing parameter and then maybe some spectral parameters and you want to optimize them so that you get some kind of power savings when you've played your games right so I don't think I'll do any other kinds of epsilons and deltas in the next lectures but I think hopefully you got the idea and probably the simplest example so thanks