Hi, I'm Zor. Welcome to Unizor education. We continue talking about probabilities today. This is about conditional probability and this lecture is part of the advanced math for teenagers course presented on Unizor.com and that's where I recommend you to to go to listen to this lecture. It has notes and notes are very very useful. So you might actually choose to read the notes before you listen to the lecture and then listen to the lecture and then read the notes again. Anyway, so let's talk about conditional probability. I have this plan which I'm going to use and here is my first statement. If you have certain random experiment and you have absolutely no knowledge about results of this experiment, you did not conduct this experiment in the past. So you don't know the frequencies of occurring different different elementary events. You don't have any knowledge about the particular features of this experiment. So you cannot really predict with any certainty the one result over another. So you must assume that if you are really assigned the task to set the probabilities of different elementary events, which are the outcomes of this experiment, you basically have absolutely no choice but to assign them exactly the same probabilities. That's the start, right? So if you have a random experiment with n different elementary events, which are the results of this experiment and you have no reason to believe that one is is tending to occur more often than another, then you obviously have to assign the probability of every elementary event equals to 1 over n. All right. Now that obviously happens with flipping the coins, rolling the dice, shuffling the deck of cards, etc. All right, fine. So now we have actually spent some time and decided to put it on a little bit more formal, more mathematical standpoint. And our approach was as follows. The set of elementary events, which we called sample space, we have basically made an equivalency between the sample space and a set. Well, this is a set of elementary events. Set can have many different types of elements. So why not elementary event to be an element of this set? So that's one thing. The second thing is the probability of this elementary event, which happens to be 1 over n, if there are n elements in the set, n elements, is basically a measure. A measure introduced on the elements of this particular set. In as much as we can measure the area or the angle or we can measure the speed or anything, whatever we can measure. So the measure is a numerical weight, if you wish, of each elementary event. So basically we have associated a number, number 1 over n with every element. We also consider every random event, which is basically a combination of elementary events. Well, what is a combination of different elements in the set is a subset, right? So any random event is treated as a subset of our set. Our set or is basically an equivalent of the sample space, which is a set of all the elementary events. And what's important is that the measure should be additive. Which means, if you have certain random event, which contains certain elements, then the measure of this random event, the subset is equal to a sum of all the elements, which comprise this particular subset. And if you have two subsets, if they do not have common elements, then you just add the measure of each of them to get the combined subset, which we have introduced as an operation or, for instance, or in some other cases you can manipulate in a similar fashion with the measure. So measure is additive. It means it can be added up in exactly the same fashion as you are adding up areas of different figures. If you put them together, if they don't intersect, then the measure of the result is the sum of the measures of the components. All right. So I will use this measure theory equivalent or formalization of the theory of probabilities in the course of explanation of what exactly the conditional probability actually is. Maybe not in this lecture, but definitely in some other lectures dedicated to conditional probabilities. All right. Next. What's next is the following. You have assigned initially the probability of each of the n elements of each elementary event. The probability would be equal to 1 over n, because you don't have any knowledge about what's going on. But let's consider you do. Let's consider that, for instance, you can conduct this experiment time after time after time and you accumulate the statistics and you see that one elementary event occurs more often than others for whatever reason. Well, that's the reason for you to reassign the probabilities slightly differently. So instead of getting one nth for each elementary event, you might redistribute this probability and put a little bit more into the elementary event, which occurs more frequently. Now, there might be even some cases when certain elementary events don't occur at all. Well, then that's the reason to assign the probability of zero to these elementary events. Let me just give you an example. Let's say you are rolling the dice and you have decided that, well, you don't know how the dice will fall. So you have decided to use the first 10 digits as the result of this experiment. Yeah, but yeah, you can tell me that there are only six sides on the dice and there are numbers from one to six. Why did I include this? Well, number one, because I have the right to include anything I want into my set of elementary events. But now if I am a reasonable person, I should assign the probability of one six to this, one six to this, one six to this and zero, zero, zero and zero to all these guys, which basically never occur. So it's not wrong to include certain elements in your original set of elementary events as long as you know what's the probability of these. And if they are impossible, well, I mean, there are certain cases, maybe rolling of the dice is not such a good example because it's obvious that seven, eight, nine, ten cannot actually occur. But in some other cases that might not be such an obvious case. So you are free to basically assign the probabilities to any events as long as these probabilities correspond to either your experience, how you observe in the past, or your knowledge about the internal structure of your experiment. So your knowledge about the internal structure of the dice is that it has only six sides. So it cannot actually have seven, eight, nine and ten. So if somebody wants to include these elementary events as the results, that's fine. They are free to do it as long as they assign the probability of zero. So that's another point. So the probabilities are changing. That's what's most important. As long as you know something about your experiment, your probabilities might deviate from the original equal distribution among the elementary events to something else. That's a very important observation, actually. Now, I actually would like to use my equivalent with a set theory to put it in some kind of a more picturesque format. How can you actually graphically represent what I just said? Well, for instance, you can do it this way. Let's say you have certain area divided into certain parts. Now, each small area represents an individual outcome of your experiment. So in this case I have whatever, 12 different areas. So I can assign the probability of one-twelfths to each of these little squares. Now, there are 12 squares, so the total probability of all this is equal to one. Now, each square represents an elementary event. Now, the combination of the elementary events is basically some kind of random event, right? So I can say that the combination of this, this, this, this, this and this, it represents the event. Now, some of these numbers, which are inside, that's one-twelfths, how many? Seven times, represents basically the probability of the event itself. Because as we know, the probability is an additive measure. So it's exactly like the area. So the area of this figure is equal to some of areas of little squares inside, right? So basically this type of representation of the probability is very useful because it actually implies this additive feature of probability. As long as you know how to draw some picture which represents the results of your experiment and you assign the probability of each individual elementary event, then by combining elements into some kind of figures, you can always come up with a picture representation, the graphical representation of the event you're interested in. And the probability of this event is very easily calculated basically exactly like the area of something. Now, if for whatever reason the probabilities are changed, so instead of one-twelfths to each one of those, I have, let's say, 0, 0, I have 0, 0, 0 and 0 and the rest I have one-eighths, one-eighths. Now, I have exactly the same event as before, but now the probability is 0, 0, 0 and 3 times 8, so it's 3-eighths instead of, as it was before, 7-12th, right? So the graphical representation is very convenient because it really reflects how every event has the probability equal to the sum of the elementary event's probabilities which comprise this bigger event. So I'm going to use this graphical representation in the future. So I did not really touch the conditional probability yet, but I'm still kind of trying to introduce you to certain concepts which would lead me to introduction of the conditional probability, all right. So we have actually touched a particular case when from the initial distribution of probabilities among elementary events, we can switch to something new based on the knowledge or frequency or whatever else which we have. And let me just give you an example of certain case when something becomes impossible, okay. Let's consider we have a special dice. A special dice, when it's rolled, it shows only even numbers, 2, 4 and 6. Now, the odd numbers are not actually shown. And therefore, my distribution of probabilities instead of 1-6 to each should actually be switched to 0, 1-3rd, 0, 1-3rd, 0, and 1-3rd. So I have three possible outcomes, 2, 4 and 6, and three impossible, 0, 3 and 5. That's why I have assigned the probabilities of impossible events to 0. Now, it doesn't really matter whether I can make such a dice or not. This is a purely theoretical consideration. So that actually is an example of how exactly the red distribution of probabilities occur based on some knowledge. So what's my knowledge? And here is the link to the conditional probability. What is my condition? Condition is, or my prior knowledge, if you wish, about the experiment, is that this is a special dice and it falls only to show even numbers on the top. So under this condition, the probabilities have different values, as you see. So this is basically an example. Now, another example, a little bit more practical, because you can say that, well, it's probably impossible to have this dice, et cetera. But here is a practical example. Let's say you are playing Blackjack and you're not the only player. There is a player before you, so you're number two, let's say. So dealer first gives the card to the number one and then gives the card to the number two. Now, if you just don't know anything at all, what's the probability of getting a king? Let's say you're playing with one standard deck of cards or 52 cards. So you have 52 cards, and as far as the kings go, there are four different kings and four different suits. So if you don't know anything about anything at all, you would consider that the probability to get the king would be for 50 seconds, because there are four kings out of 52 cards, so the probability to get the king would be this. But what if you observe that the person number one, who actually got the card before you, got the king? Well, under this condition, there are only three remaining kings, right? So this particular condition that the first player got the king makes your probability instead of four, 350 seconds, right? So again, the prior knowledge or certain condition which you might not actually be aware of in the beginning, but now if you are aware of these conditions, that changes the probability of the event which you are interested in.  I mean, as long as you know that the first guy got particular card, that changes completely the probability of everything else for you and for all other players if they are there. But again, if you don't know anything about what exactly that first person received, you have absolutely no information about getting anything and you've probably decided that your probability is still 450 seconds. Alright, so that's just examples of how knowledge about something or statistical experience or whatever else, which basically represents certain condition your experiment is actually conducted. How that changes the probabilities of elementary events and any random event. So these are just examples. Alright, next we will do a little bit more precise one particular example and I will use the graphical representation which I have. Let's say you are playing with two dice and I will represent graphically the results in this way. 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6. So I have two dice and the result of my experiment would be in these squares and the square has row and column. Row means the first dice result and the column means the second dice result. So if I throw a dice and I have a combination 3, 4, so it would be 3, 4, that's the square which represents the result. So the results of my experiment are in this 6 by 6 matrix, 36 different squares and if I don't know anything about how the whole experiment actually is conducted, then I can say that 1, 36 is the probability associated with each particular result. So each square has an area of 1, 36. I have 36 squares so all together we have 1 as it's supposed to be and now if I would like to know let's say what's the probability of having let's say even number on a second dice. So the second one which is the column should be even. Alright, so it's and I don't care about the results of the first one, right? So this square and this square and this and this and this and this represent the results of the my experiment if the second dice is 2. Now these are, the result is 4. The first one is irrelevant and this is the result is 6. So all these squares which are marked with a point represent an event. It's a combination of elementary events. So each one of these is elementary event when the first dice falls whatever it wants and the second one is even, right? So that's what I have. Now if I would like to know the probability of the second dice to be even, I just have to add together the measures of each of the pointed square and there are 6, 6, 6, 18 pointed squares. So the probability is a sum of 18 by 136, so it's 1836 which is 1 over 2. So the probability of 1 half.  Alright, fine. That's unconditional. That's when I don't know anything about the dice. Now let's consider a slightly different case. What if my dice are special and they are so special that their sum, so the dice number 1 plus dice number 2 always comes to 6. Now, by the way, it can be actually practical. I mean you can say that this is impossible to have such two dice that will always fall in the way that the sum is equal to 6. But I can make a different experiment. I can just completely disregard all the events or all the experiments when the result is not 6. So I'm throwing the dice, a pair of dice, time after time, if the sum is equal to 6, I consider this to be my experiment and they don't count anything else, right? So basically my experiment is only this. So I'm on the condition that two dice always come up, always sum up with the sum of 6. Now, what does it mean? Well, it means that I have to completely redistribute my probabilities. Instead of 1.36 to each one of them, I have to redistribute it differently because there are only how many? 1.5 gives me 6 in sum, right? 2.4, 3.3, 4.2 and 5.1. So only these results of two dice give me a sum of 6. So only these 5. So what does it mean? It means that I have to assign the probability of 0 to anything else and the probability of 5 of them, right? So 1.5 to these combinations. So 1.5 should have the probability of 1.5, 2.4, 3.3, 4.1, 4.2 and 5.1. These are only possible choices which I have. Everything else is impossible, which means everything else is 0. So all elementary events which are not marked with 1.5 actually have the probability of 0. This condition actually completely changes the distribution of probabilities among events. Without this condition that the sum is equal to 6, each of these squares have the probability of 1.36. Now, under this condition the probabilities are changed and these guys have 1.5 and everything else has 0. So what is the probability of having even number as the second dice results? Well, let's just see. I have this one, 4.2 and I have this one, 2.4. That's the second one is even. All other cases are either impossible, I mean this, this, this, this, this, this, this, this, this and all these are impossible. They do have zero probability, right? So if I will start adding, this has 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0. If I will start adding all elementary events which comprise my, the result which I'm interested in, that the second dice is even. I will have to really add all these which are in blue but these are zeros, right? So only these are important. So what's important is this one and this one, right? Out of these which are my result which I'm interested in, only these two have non-zero probability. So I have only 2 out of 5 and my conditional probability of having the second dice as an even number, my conditional probability under the condition that the sum of two dice is 6 is equal to two-fifths, not one-half as used as was before. So as you see, my condition which I have imposed on my experiment has changed the all elementary events, now all elementary events are either one-fifths or zero and as a result it changed the probability of the event I'm interested in. So the second dice shows the even number. So that was my illustrative example of how condition changes the probabilities associated with the results of experiment, alright? So let me just summarize this lecture. If you don't know anything about the random experiments which you are conducting then you are forced basically to assign even probabilities to all the outcomes and if you have n outcomes you have 1 over n as a probability of each one of them. But if you have certain knowledge, like for instance knowledge that these are special dice which sum always to 6, that knowledge completely changes the probabilities of elementary events and as a result it changes the probabilities of your events which you are interested in. So the conditional probability is this changed probability based on the knowledge about certain conditions which you have. And graphically it might be represented as a table or in next lecture when I will be going into more details it will be actually represented in the language of the set theory which I was trying to just touch once before. Well that's basically it for this introductory into what conditional probability actually is. Conditional probability is the changed probability of the events based on certain knowledge, certain conditions which you impose on your random experiment. And this is an example of how to apply this particular conditional probability, how to calculate it. Well that's it. I do suggest you to read again this lecture on Unisor.com in the notes for this particular lecture. And that's it. Thank you very much and good luck.


NATO flexes its naval muscles in warning to Putin. Britain's HMS Prince of Wales has been pictured leading a NATO fleet as the military alliance carries out its largest joint drills in decades, according to the Daily Mail. It is noted that the Royal Navy aircraft carrier led 14 vessels in formation through an exercise in the North Sea as part of the Nordic Response 24 drill set on training the alliance's ability to reinforce Europe's northern frontiers. The 65,000-ton ship was seen flanked by the American destroyer USS Paul Ignatius and Spain's ESPS Almiranti-1-Duborbon frigate as Swedish and Finnish fighters flew overhead. Operating around the icy fjords of Norway, the carrier strike group will practice defensive manoeuvres and amphibious landings to recapture Allied territory. Exercise Nordic Response 24 is part of the broader exercise Steadfast Defender 24, the largest NATO exercise since the Cold War, bringing together more than 90,000 troops from all 32 countries to train in unison as war edges ever closer to Europe amid Russia's ongoing invasion of Ukraine. The North Atlantic Treaty Organization is no longer in hiding. Over the three decades of military retrenchment, following the fall of the Berlin Wall in 1989 and the break-up of the Soviet Union two years later, NATO has decided to boost its capabilities in response to Russia's special military operation in Ukraine and the threats made by its President Vladimir Putin to countries on Europe's Eastern flank. On January 24, the alliance launched Steadfast Defender 24, its largest series of military exercises in Europe since the end of the Cold War. Scheduled to last until May 31, these large-scale manoeuvres bring together some 90,000 troops from 32 countries and no fewer than 1,100 armoured vehicles, including 166 battle tanks. The exercises are taking place mainly on the plains of Poland and Norway, but also in Germany, the Baltic states, Romania, Finland, Slovakia, Greece and Sweden. Russia fuelling US border problems to distort presidential race in US. There is little doubt that the main issue in the presidential campaign in the US this year is immigration according to Forbes. Current discussions on Capitol Hill are focused on persuading House Speaker Mike Johnson to bring the Senate-approved immigration bill to a vote in the White House or on gaining sufficient support for adopting a discharge petition to circumvent his obstruction. It appears that the discharge petition to sidestep the speaker is the more likely option. In the meantime, the presidential campaign is in full swing and Biden and Trump are racing across the country talking about immigration to garner supporters and votes. Given this delicate situation regarding adopting an immigration solution, it is disheartening to learn that, once again, Russia is trying to meddle in the US presidential election by promoting discord over immigration matters. A recent AP article pointed out in recent weeks, Russian state media and online accounts tied to the Kremlin have spread and amplified misleading and incendiary content about US immigration and border security. The campaign seems crafted to stoke outrage and polarisation before the 2024 election for the White House and experts who study Russian disinformation say Americans can expect more to come as Putin looks to weaken support for Ukraine and cut off a vital supply of aid. The article continued in social media posts, online videos and stories on websites. These accounts misstate the impact of immigration, highlight stories about crimes committed by immigrants and warn of dire consequences if the US doesn't crack down at its border with Mexico. Many are misleading, filled with cherry-picked data or debunked rumours. Polish troops would never leave Ukraine. Russia is technically prepared for nuclear war. Putin, any attempt to send Polish troops into Ukraine could end in a long-term occupation. Russian President Vladimir Putin has warned in an extensive interview with Russia One TV broadcaster and Ria Novosti, if Polish troops enter the territory of Ukraine in order to, as they say, secure the Ukrainian-Belorussian border, for example, or in some other places to free up Ukrainian rear military units to participate in hostilities on the front line. Then I think that Polish troops will never leave, Putin said, according to excerpts from the interview. The deployment of NATO troops to Ukraine amid the conflict with Russia is not unthinkable. Polish Foreign Minister Radislav Sikorsky claimed last week he was commenting on a statement by French President Emmanuel Macron, who said late last month that he cannot exclude the possibility of soldiers from the US-led military block being sent to aid Kiev. Putin believes that Polish officials dream of returning those lands that they consider historically their own and which were taken from them by Joseph Stalin and transferred to Ukraine. They certainly want them back. So if official Polish units enter there, they are unlikely to leave, he claimed. Putin said that Russia is technically prepared for nuclear war, and if the US sends troops to Ukraine, it will be seen as a significant escalation of the war. From a military technical point of view, we are ready, of course, said the 71-year-old Putin in response to a question about whether the country is truly prepared for nuclear war. Putin said that the US understands that if they deploy troops to Ukrainian territory, Russia will consider this a step toward intervention. In the US, there are enough experts in Russian-American relations and in the field of strategic deterrence. So I don't think everything is rushing headlong here, but we are ready for this, Putin said.


Hi. Welcome to Succeed as a startup PM when you've only worked at big companies. I'm Justin Neckhouse, and I'm going to walk you through this. First, a little bit about myself. I am a product leader, startup advisor, angel investor. I started my career as an engineer and then moved over into product. I've been doing product management and leadership for about 20 years at companies such as Amazon, Airbnb, CBS Interactive, and a variety of startups as well. Currently at Amazon, working on bringing podcasts to the Amazon music app. I'm based in the Bay Area. All right, let's jump into it. Before I tell you best practices for how to migrate on that path from a large company PM to startup and find success, I want explain some of the high-level differences. These are just a couple of the thematic ones. Of course, there's a million little things, but hopefully this gives you a big sense of is this the right path for you? First of all, product market fit. At a big company, unless you're starting something brand new, likely you have found some product market to fit. At a startup, depending on the stage, you might not have any product market fit. You might not have launched your product at all. This can be a pretty big difference between iterating on something that's working to optimize it versus starting with a blank piece of paper. Next up, scope. At a large company where there's a lot of product representation, maybe you're working on just one feature. Maybe it's a big feature, but still a feature. Sometimes it's a group of features or a small part of a really large feature. At a startup, especially early stage, likely your scope could be the entire product, the entire company, depending on, again, the stage that this is in. Scope can vary a lot and could seem overwhelming, but also could be fun. Impact. At the big company size, you may be focused on one or just a few metrics. At a startup, you may be responsible for the success or failure of the entire product or company. There can be a lot on your shoulders, but also you have the ability to create huge impact in what you do on a daily basis. Process. Larger companies have probably gotten to where they've gotten because they have some established process. As you scale, process becomes more important and people more and more want to throw process on there and quickly can become a large part of your daily job. At a startup, you typically start with no process. You really have to choose what is the right process. How much do we need? Can we go with less? It's rough and it's minimal. A lot of that, frankly, is by design resources. I know at big companies, we all feel like we never have as many resources as we need, but in comparison to a smaller startup, you have a lot. You have people potentially to look at BI and to do project management and to do all different facets of product, whereas a startup, you have very few and oftentimes really have to be a jack or jill of all trades. Data is the last piece that I want to talk about on the data front at a large company. You usually have a lot of data to go on. Maybe you have bad tools or it's hard to get at, but normally because you have customers and because you're more mature, you're tracking many different things and you can report on many different things. At a startup, if you haven't launched, you may have no data. Maybe you have some user research, but typically you have to get very creative on how you go about using data to make decisions or just making decisions without data. Now that we understand some of the key differences, let's talk about how to be successful in a startup environment. Product fundamentals and hopefully find this a bit reassuring. Product fundamentals for a startup are really the same as for a large company. What you are going to do to find success with what you already know is to really shift where your priorities are, where your time is spent. So for example, on your mission, your vision, your strategy, these are really worth your time. These things are super important and are going to really set the stage for success for everything that comes after that. What comes after that is your tactics, your roadmap, your prioritization, your metrics. So again, you're going to spend time on this, but probably a lot less than you typically spend at a large company because your tactics are a best guess based on hypothesis or based on small amounts of data or no data at all. And so we want to make sure that we're not spending a lot of time on these things and rather really setting ourselves up to sort of find that product market fit and try a bunch of different things out. But remember to sort of lean back and remember that you have spent your career at a large company and learned a bunch of product fundamentals that will apply, just need to shift how you apply those. So what's your job as a startup product manager? A couple of different things I want to say here. So we throw around this concept of CEO of your product a lot. You hear that written in articles. A lot of people reject that. And I think at a large company, it definitely is not exactly the right metaphor to use because a lot of times at a large company, although you are responsible for your product or your feature, so much is out of your control. So much is handed down to you. You don't have the ability to impact resourcing decisions nearly as much. At a smaller company, at a startup where maybe you're the only PM, this can be a lot more true. And I think it's important to embrace this and feel like the success or failure of the company is your job, not just the feature, because oftentimes you're a one product company and oftentimes a one PM company. And so to start with, really think of everything that determines the success of your product as your job. So you can't just build a good product. You need to either be the product marketer or work very closely with the product marketer to figure out what is the pricing and positioning going to be like. You need to validate that sales is on the same page with you about how your product is going to be discussed with customers. What is the value prop they're presenting? Is that resonating? Can you help adjust that if it's not working out? I encourage you go out sales as much as you can, both to sort of see how sales is presenting things. And if that aligns with what your vision of the product is, but also so that you can get close to the, so we talked about product marketing and we talked about sales research, oftentimes you don't have a researcher and it's sort of up to you to find the research that you need to do need or go out and do the research if it doesn't exist. BI is again another thing that sometimes you have, sometimes you don't, sometimes you just got to get some cheap tooling in there to do it yourself, expect to write sequel in the early days. You may not have someone to do that for you. You'll go for partly a joke, but partly it's true. I can't think of a startup that I have been at where I have not gone out and bought lunch or dinners for people to continue forward momentum. Recruiters are the last thing here that constantly sort of being evaluating, do you have the right team to make your product successful? And if not, even if you're not the CEO or you're not the recruiter, get involved and figure out how do we get the right people on. That's the right sales team. Do we hire a consumer sales team when we really need an enterprise sales team? Are these people not the right people to represent the product? Is the designer delivering what we need to be successful? So really, I want you to take on a lot more than you typically do in product at a larger company and this is why you need to really adjust your time investment with other things like the tactics on the roadmap where you might otherwise invest significantly more time today at a larger company. So continuing on that theme, you want to keep things light, keep the process light, but don't skip steps. So we talked about this a bit before, but I just want to emphasize and sort of explain more about what that means. So product process is product process, right? You start with your mission, vision, strategy, objectives. Again, to emphasize, you really want to spend time there because these are important and this sets you up for success in the rest of the process. And you want to run the process all the way through QA and release and looking at success of the product once you release it and iteration, but you're going to spend your time a little bit differently. So your tactics and your roadmap look differently from a large company, especially if you don't have product market fit. And what does this mean? This means that, for example, on your roadmap, you want to spend some time on it, right? Because this really can inspire the organization and set up a North Star for where you went ahead and may totally change very often. So what I recommend is spending a fair amount of time on the current or next quarter up and doing a full year roadmap, but doing it at a more thematic level after the first quarter, just knowing like you want to set a direction of where you're heading, but you don't want to spend huge amounts of time in detail going quarter by quarter because likely that's going to shift a lot. So I also want to emphasize that you want to plan for success. So don't hedge too much, but that you don't want to overinvest. Again, so keep in mind that a lot of the tactics or items on your roadmap are going to fail or they're going to have minimal impact. And so you don't want to overinvest and assume that everything's going to work there. You really want to embrace the MVP, the Lean startup methodology. If you haven't read the Lean startup book, I recommend highly you read that and think about what is the least that you can do to test your assumptions, your hypothesis about your product, about a feature that you're going to release and set that up so that that works as a test, but is not sort of a fully functional feature. You're just looking to sort of say like, how do I test my hypothesis here? And this is often a very uncomfortable place. I encourage you to find the MVP and think again, what can I do that's even less than this? Ask other people for input there because it's uncomfortable, but almost always you can pare it down more and more to really answer the question of, well, is this finding traction? Is my hypothesis true or not? Had a big company, I know this can be very uncomfortable to do, but it's really sort of a mind shift that you're going to have to make to be successful at a smaller company.  And this is a product that is certainly about finding product market fit and success with customers, but it is about learning every time you do a release and adding to what you know, which will add more certainty to prioritizing the roadmap. A framework that helps a lot with this is called GIST, which I've linked here and really helps build this sort of learning framework. So every time you're releasing, you're set up to learn more to make future investments more certain. Prioritization is a bit different as well. So what you want to use is a lightweight prioritization framework because you are not going to start with a lot of data, a lot of knowledge about what's going to work. I like rice. And if you do have really some data, rice is pretty nice. It looks at the reach and impact versus the effort of any given feature and helps you prioritize in sort of an ROI type of way. If you don't have any real data, then any sort of numerical-based framework can be difficult. In that case, I recommend a consensus-based framework. Pandora invented a pretty unique one where you basically have a team come together and put money down that corresponds to the resources you have to choose what the most important thing to work on is. So take a look at that. So on actual sort of day-to-day tactical approach to things, agile works. So you're probably doing agile now. Hopefully you're not doing waterfall. How you do agile maybe is a little different. Maybe you're doing scrum-based agile today. That can work. But again, make sure you keep it light. Evaluate do we want our sprints to be three or four weeks instead of two weeks so that we have less overhead. Another framework that I like that's part of agile is called Kanban, where you really don't have sprints and it allows you to sort of really shift a lot.  And so at any time you can sort of reprioritize that. So Kanban is it's worth looking at to find a framework that works to allow you to really shift and pivot a fair bit without incurring a bunch of sort of debt or overhead. So take a look at Kanban data. So I mentioned that data is very different in a startup. You may not have much at all. So what do you do if you have little to no data of your own? This is very often true of a sort of pre-launch startup. You may have some research. If you don't, I would say let's go find it. So where do you find this? You can go and look for market data. You can look at competitive data that's out there that you can sort of just Google for that maybe is findable on YouTube through presentations that people have posted from conferences. There's often a lot of data there. Talk to customers, right? So customers are an amazing source of data about needs and wants and behaviors and motivations. So you really got to make your customers your friend and you should be talking to them on a regular basis to really find out what's happening in the market that they know what's working and what's not working for your product, what new problems have they started encountering with their job that maybe you can go out and figure out how to fix. The flip side is that your non-customers may be more valuable than your customers. And so again make your non-customers those who have said no we're not buying your product even better friends and figure out why not. And is there some point that you can build in the value that they need that will convert them to be a customer? And so important to sort of really pay attention who has to sort of convert a specific customer and more to really find value that's going to apply to a bunch of different customers or a new segment of customers. So you're going to find that A-B tests are usually not possible in the early days of startup. This is especially true in enterprise businesses. So what do you do? Well, you got to look at other ways to test. So we do this thing called pizza testing or what I'm calling pizza testing. And that's where you literally bring in friends, family or potential customers of the product and give them free pizza in exchange for sitting down and walking through some user testing. So build out some scripts, have them test your product out, see what's working, see what's not. If you're pre-launch another thing you can do is do test your competitors products. There's a great way to sort of learn about what's working and what's not working with their product and find potential opportunities or things that maybe you don't need to build in your product. And so you can do that again with sort of the pizza testing. You can also use online testing resources. So there's a lot of online testing available that is a very sort of inexpensive way and quick turnaround way to get user testing. Another thing that I think is worth looking into is how you track within your product. Tracking is very different on these early launched products because you don't exactly know what to track. So we definitely put in some tracking, right? Put in Google Analytics or one of the other analytics software that is cheaper, free, and get you to the basics and track things that you know, but also realize that you're not going to really know what to track on the early stages or it's going to be very expensive to track every single thing that you want to. So what I recommend is user session recording software. There's a lot of fairly inexpensive good ones such as InSpecLit or Full Story that will actually sort of track the user session as a recording. And so you can see everything that happens and get a sense of how people are using their product, your product, where they're getting frustrated and both take that in to iterate on your product, but also to instrument tracking later based on the pieces of your flow that end up being important. So smoke and mirrors and not scale. So scaling obviously is something that you're going to need to do to be successful in your startup, but don't get ahead of yourself.  So you want to spend your time where you create differentiation and don't prioritize work that you can cheaply buy or fake with people. And when I say do things that don't scale, what does that mean? That means early on your job is not to scale is to find product market fit. So talk to customers one by one. Don't send mass surveys and expect great data back. Talk to them one by one. Do stuff manually. And this means you may want to automate every part of the app or whatever you're building, but do you really need to do that? Or can you do some stuff manually until you've proven that you found product market fit or the feature is going to work? Buy stuff. So we'll talk about this more later, but think a lot about the build versus buy decision. As you're doing things that don't scale, one thing to be careful of because you're going to be out there talking to customers is don't lose focus and build a feature that's just for one customer to solve one customer's problems without evaluating if it has wider appeal. So this is a trap many people fall into because you're talking to a customer, they have a real problem, and you're like, oh, I could definitely solve that with this. But if that's not going to be valuable to a large segment of your customers, then you're going to start building custom software and that is never going to scale. So really be careful here and think about can I generalize this problem into something that applies to many different customers. You're going to want to spend your time wisely. You're going to want to focus your time on things that people can see. Think about how do I build delight for my customers? And so what does this mean, things that people can see? This means your UI for sure. So investing in great design is super important. Great design builds trust with customers, even if you haven't really earned it. Customers see something that looks great on the front end and they have trust that your back end and the rest of your organization is built in in such a careful way as well. So really putting out a product that looks great is super important. So don't skimp out there. This also means that you need to spend time building where people can see on the back end too. So don't spend a lot of time building, say, some kind of a package that you could just use Excel for in the short term. If that is not important for your customer, customer isn't going to see that. You want to focus your time on table stakes, for sure, if you're entering a competitive space, but on the table stakes that matters, you want to focus only on the table stakes that matter so you can spend more time focused on the differentiators that are going to truly find a place in the market for your product. So table stakes, what do I mean by focus on the table stakes that matters? That means that a lot of your competitors may have a wide feature set, but their product may have been first released 10 or 15 years ago. And so maybe a lot of those quote unquote table stakes are not table stakes anymore. And so you really need to figure out which of those are true table stakes and which of those you can leave out or shift even how they work because the way they were thought about a decade ago maybe doesn't apply anymore. And this is where that user testing of your competitive products can really help. Buy, buy, buy. So early on, you don't want to build anything that doesn't add value to your product. You don't want to build commodity things. You want to build where it counts. And so software is actually pretty cheap for startups most times or even free. There's a lot of startup packages or credits that you can get as a startup. And so really take advantage of that and use external products to integrate your product to bridge functionality rather than building it yourself. Maybe later on you'll replace it or you'll build a better version of it. But really think about what are my differentiators? Spend time, build those and buy or integrate anything else that you can on the early days when you're really trying to find product market fit and figuring out how do we sell this thing and how do we build a product that customers want. And then finally I want to talk about doing things manually and specifically ops teams. And ops teams typically can be fairly inexpensive especially if you find them in lower cost labor markets and they can fill in a lot of product holes. And so you definitely want to look into can we use people to do something that maybe will automate later when we're looking for scale but is less important early on when our goal is to find product market fit or test a hypothesis about a feature. Until you're really certain you don't want to invest in total automation here when ops teams can potentially do it with the current scale that you're at. So those are some of the high points of how to be successful as a startup PM when you've only worked as a big company PM. Thanks so much for your time. If you have any questions feel free to reach out to me on LinkedIn or my site below. Thanks so much.


Thank you very much. So let me just say that it's a great honor for me to speak at this conference. And like everything, pretty much everything in geometry and physics in the last 20 years, this talk will be greatly influenced by work of Maxime and Jan. So this is about connection between Donaldson Thomas invariance and character varieties. So to be more precise, especially the motivic Donaldson Thomas theory developed as I said by Maxime and Jan, and a conjecture formulated by Haussel, Rolette Lié, and Rodriguez Viegas on the cohomology of character varieties for smooth projective curves with marked points. So for the first, I don't know, 15, 20 minutes, I'm going to just set up the stage and review this conjecture and try to write down the formula they proposed, which is a truly beautiful and remarkable formula. So let's set the stage. So C is a smooth projective curve over complex numbers. The genus is going to be G. We're going to have two marked points on C, P and infinity, P, of course, these things marked points. This is not a serious restriction. This is mostly for presentation purposes. I don't want to get tangled in too much notation. And gamma, P, and gamma infinity are the corresponding generators of the fundamental group of the punctured curve. And then we need to specify some more data. So we're going to have to fix a diagonal matrix. It's all complex numbers. I'm not going to keep writing complex numbers. Diagonal matrix and invertible diagonal matrix. Mu is going to be a partition of R. This is the partition that determined by multiplicities of eigenvalues of lambda. So we determine a partition of R. And now, finally, an integer E co-prime with R. So E is just an integer co-prime with R. So what is the character variety then? The character variety is going to be, roughly speaking, the motorized space of conjugacy classes of representations of this fundamental group into GLR. So we set C e lambda to be, as I said, the motorized space of homes from the fundamental group of the punctured curve into GLR, sending gamma P to the conjugacy class of lambda, and sending gamma infinity to a central element, 2 pi square root of negative 1 e over R. And all of this is modular conjugation. So this is a major object of study. The topology of this character variety is very, very interesting. Now, we have a theorem here. So in order to make the notation short, I'm going to refer to the work of Hauser-Lettelier and Rodriguez Viegas as HLRV, saying that for sufficiently generic lambda, C e lambda is smooth, is a smooth quasi-projective variety, of course, over complex numbers. And furthermore, a consequence or empty, possibly empty. But if not, empty is smooth and quasi-projective. And if you compute this dimension, you find out it's some formula where it depends only on the partition mu. So it has some formula I'm not going to write down, but it's only dependent on the partition mu. All right, so now what are we going to study in this context? It's some kind of weighted Poincare polynomial of this variety. So what I want to study, I mean, what I would just say, HLRV studied, was they looked at, since this variety is non-compact, it has a non-scrivial weight filtration and comology due to Dulin. And they studied the, so we have the comology of this variety. C has a weight filtration, W. And you can define, I mean, I'm sorry, the compactly supported comology, I wonder, it still has a weight filtration. So we can define some kind of weighted Poincare polynomial by the following formula, p compact of this. It's a function of two variables, formal variables u and t. It's the sum over dimensions of the graded pieces of the weight filtration on HK compact weighted by this monomial, which looks a little bit funny because it looks like I need to take a square root of u. But then furthermore, but in fact, this turns out to be a polynomial. So there are further results, again, of HLRV, saying that this is in fact a polynomial of, so this PC, it's a polynomial of u and t. So only even i occurs in the formula. And furthermore, it depends, again, depends only on mu, so on the partition mu. So a lot of data that I specified before doesn't actually enter in the, I mean, there's no dependence on the specific values of the eigenvalues, for example. So we can call it, so we can denote it, we can just write p of mu u t without any ambiguity. OK, so now finally, let me get to the point of writing this conjecture. They formulated it. So the question was to determine all these polynomials, for all possible values of the rank and all partitions mu, and they wrote a beautiful formula, which I want to write down here and leave it on the blackboard for the rest of the talk. So this is the HLRV conjecture. So it's written in a very, which is in a way that's really very, very suggestive to a physicist. It's written like saying that the partition function of some system is the exponential of a free energy. Well, I'm using these names. They certainly did not use, you did not use this terminology, but for a physicist it's very natural. So first of all, let me write this here. So what appears in the exponential, it's a plateistic sum. So it's a sum over all partitions. It's a sum over k greater or equal to 1, 1 over k. And then we have what we have here. And let me write the sum in here.  So the k divided by 1 minus z to the k, w to the 2k minus 1, m mu of x to the k. OK, z, w and x are formal variables. x is actually an infinite set of formal variables. The m mu's are the monomial symmetric functions. So this is what appears in the exponential. Is the dependence on genus? Oh, I'm sorry. You're absolutely right. There is dependence of genus in here. Oh, no, it's because I just forgot the p. I apologize. Sorry, I need to have a genus. And the polynomial depends on the genus, certainly. I apologize. This was just an empty formula what I wrote there. OK, so yeah, so it depends on g. OK, so yes, it depends on g. OK, good. So this is one side of the formula. And then what is the other side? The other side, it's the way it's the monomial symmetric functions associated. m mu is the monomial symmetric function for the partition mu. And x is just an infinite set of variables. So you just imagine x is just x1, x2, dot, dot, dot. So it's all formal variables here. All right, so this is one side. The other side, quite remarkably, the other side is just a pure combinatorial object. It's a sum of generable partitions, but other partitions. I mean, well, the same partitions, but they have a different nature. So there is a rational function I'm going to write in a minute associated to any partition. And then it's weighted by modified McDonald polynomials. So I'm going to explain all these quantities here. First of all, this is easy to write. It's some rational function, which is a product over all boxes in lambda of z minus 2a plus 1, which a is the arm length of a box, minus w to the 2l plus 1 to the 2g. L is the leg length of a box divided by z to the 2a plus 2 minus w to the 2l, z to the 2a minus w to the 2l plus 1. So indeed, as you see, it's explicitly dependent on g. So let me just write this in words before I explain what these functions are. So h lambda twiddle is they are called the modified McDonald polynomials. Now, I know very little symmetric function theory, so I couldn't give you a definition of that function over there. Fortunately, I don't have to give you a definition in symmetric function theory. What I am going to do is give you a geometric construction for these functions, which is due to Heyman. So for our purposes, it's much more illuminating to think of them to employ the geometric construction for this modified McDonald's, which is the result of Heyman. So it involves the Hilbert scheme of points in C2. It's sufficiently generic here. How effective is it to complement finally many points or many points? There is a precise condition that I'm not remembering, but there is a very specific condition here. I think it might be a complement of high dimensional subspaces, not just points. Yes, it's explicitly given in the paper. OK, so now let's discuss a little bit the modified McDonald's functions, because this is going to be important for what I have to say next. All right, so now this is, as I said, according to Heyman. So we're going to look at the following diagram. This is the symmetric product S of C2. This is the Cartesian product, and this is an n factorial 2-1 cover. Now here we have the Hilbert scheme, hn of C2. And this is the Hilbert Chao morphism, collapsing whichever cycles there. And now we complete the square, and we get what's called the Isospectral Hilbert scheme. So this is the Isospectral Hilbert scheme, which is, again, it's generically an n factorial 2-1 cover, but something pretty bad happens over the non-trivial fibers of this morphism here. And in general, this is, as the scheme is not even reduced, but what Heyman proved, so there is a structure theorem, what he proved is that this H-twiddle and C2 reduced, if we take with reduced scheme structure, it's irreducible and Colin McCauley. And that was very important, because what he had to, he did next was to take a push forward of the structure here down here. And if you endow this with the reduced scheme structure, what you get down there, it's a nice vector bundle, which is known under the name of Prochese bundle. And that place, that's really the main geometric ingredient in the construction of the McDonald polynomials. All right, so now let's let P, P is the push forward of OH and twiddle reduced. So now this turns out to be a rank n factorial vector bundle on Hn. And not only that, but it carries a fiber-wise action of the permutation group of n letters. So it's an action of Sn by naturally, by construction there. All right, so now the other thing we need is there is not a natural action of a torus on the Hilbert schema point, which comes from the just the skating action on C2, the diagonal skating action on C2. And it's very well known that if we classify fixed points here, what we get is a bunch of monomial ideal labels labeled again by partitions of n. All right, so now to get this McDonald polynomial, what we need to do is look at the fiber of P over a fixed point there, and look at it. And there is a simultaneous action of Sn and an action of T, and they commute again naturally by construction. And what we get, what we learn then, is that if we restrict P to a fixed point, this decomposes into a sum of characters, obviously. The compose over a sum over partitions. Let me write this like this, and I'm going to explain the notation. These are the irreducible representations of the symmetric group. And these are just representations of T. So it has a decomposition like this, which is OK. It's all natural. All right, so now finally, what is the McDonald polynomial in this context? Let me write this is a theorem. Maybe I can write it here again due to Heyman. We have the following. I mean, this is a hard theorem. This is hardly non-trivial identity. Although me being a physicist, I'm going to take it basically as a definition for McDonald polynomials, in my case. So it's given by this following formula. q1 and q2 are the natural generators for the character lattice of this guy, of this torus, algebraic torus. This is the character, and these are the true functions. So as I said, this is a hard result. It has a complicated and hard proof. But this is, for us, the most natural definition for this. I mean, it's not a definition, but for the purpose of this talk is going to be a definition. All right, so it's very intriguing that indeed they appear, these functions appear in this particular formula. And I should say that the way they arrived at this formula was by counting. So what they did was to formulate the problem over finite fields, because the character variety is defined over finite fields. And they counted rational points. And what they did is that eventually using them, the real conjectures, they arrived at this formula. And they even proved a specialization of this formula, which I'm thinking sets z equals w inverse or something. So that is proven. That's a theorem. Just a second. If you can't write your final field, you can count on the weight, but you cannot distinguish between conmology. Exactly what? That's what they proved, the specialization of this formula. Yeah, specialization where z equals w inverse, I think. That's exactly what happens. A turn of the conmological degree. Exactly, yeah. And that's proven. But then based on some symmetries and some they conjectured this. All right, so now, as I said, me being a physicist, when I saw this formula, it looks very, very intriguing and very elegant. So I was thinking, how can I get this out of string theory? Is there any way of deriving such, not proving it, but deriving it or explaining it from string theory? And especially, how can I explain the occurrence of these guys, of the McDonald polynomials? And how McDonald polynomials are proved? Well, in their proof, what they get is a specialization of these McDonald polynomials. Because, you see, that's what I'm saying. They collapse with something much simpler. So only when you write the full conjecture, then you get this two-parameter family of functions. All right, so. OK, so the way this goes is, by first, employing another piece of mathematics which relates character varieties to the modulized space of Higgs bundles. And in this particular case, it's going to be Higgs bundles, parabolic Higgs bundles on the curve. It's not a piece of beautiful mathematics which goes back to work of Hitchin, Donaldson, Simpson, which tells us that there is a deep relation between character varieties and the modulized space of the Higgs bundles. And in this particular case, we get a special flavor. I mean, we get a flavor of Higgs bundle, which has parabolic structure. So I want to say a few words about to bring the Higgs bundles in the picture. And the p equals w conjecture. This is a conjecture formulated by Cataldo Miglorini. Oh, sorry, Halzel and Miglorini. But it really relies on previous work, as I said, of Hitchin, Donaldson, and Simpson, which tells us basically that as a real manifold, this character variety we had with defined, it's actually canonically identified to a modulized space of stable Higgs bundle. And in this particular case, parabolic Higgs bundles on C. So I'm going to call it H, let's call it H e lambda of C from Higgs. Hopefully it's not going to be too much confusion between this and this H and that H. Maybe I should call it Higgs to make sure that Higgs is the degree. I haven't told you the map between character variety that and Higgs bundle. But e is going to be just the degree of the bundle. So as I suspect, most people in the audience know a lot more about this than I do. Just to fix notation, what I'm talking about here is pairs e phi where e is a vector bundle on C and phi is a, so we're talking about pairs e phi where e is a vector bundle on C, phi is an endomorphism with values in a twist of Kc by the point P. And I'm just looking at a single point for simplicity. And what we need to know is that we have to specify, we specify a flag of type mu at P, sorry, in the fiber, in the fiber ep. And then we also have, we have some parabolic weights. Well, OK, I'm being a little bit sloppy here, but I just don't want to get bogged down in details. What I am going to do here, basically, is take a special, I'm going to take the lambda to be something like a diagonal of pure phases, e to the 2 pi i square root of minus 1 alpha 1 and so on. And the parabolic weights are going to be just these numbers that appear there. So it disqualifies as sufficiently generic according to the definition.  And from that, you should also order some terms of partition. You're absolutely right. I'm being really sloppy here. Yeah, what you have to do is take the first multiplicity of the first guy, and that's the dimension of the first quotient, and so on. The fastest one. Yes, so I'm being sloppy. I could go through the details, but yeah, I should say the type ordered mu with the natural order given by the order of eigenvalues. And then maybe that's it. And then there is a notion of slope, parabolic slope, and the notion of slope stability. It's all standard. And you get, you have slope stability for these guys. And you get the smooth-modelized space, which is this Higgs-modelized space, e lambda c. So this is, again, it's a smooth quasi-projective variety. And as I said, it's, as a real manifold, it's identical to the correct variety. But the complex structures, the algebraic varieties, a variety of structures are very different. And that leads to this highly non-trivial conjecture formulated by Cataldo, Hauser, and Migliani concerning what is the interpretation of the weight filtration on the Higgs side. So that's the issue. And again, without getting into too much detail, because this is really an embarrassing moment. I mean, most of the audience knows a lot more about perversives than I do. What they do is to define a perverse filtration on the homology of the modulate space of Higgs bundles using the decomposition theorem for the Hitchin map. So the story is that there is a Hitchin map to basically a vector space, which is obtained by taking the polynomial invariance of the Higgs field. So this maps to, by map H, to some vector space, which I'm not going to try to write in detail. And this is the map which associates the phi trace of phi to the k, a vector polynomial invariance for 1 smaller than k smaller than r. To e phi, but yeah, you just take the polynomial invariance of phi to the k. And in this complex structure, this is a holomorphic. It's an algebraic map. And using the decomposition theorem for this, we obtain a perverse, so-called perverse sheaf filtration on the homology of the Higgs and the modulate space of Higgs bundles. And this is going to be called P. And the conjecture is that, basically, Pi, if you set up your conventions correctly, which is a fairly subtle issue here, so set up your conventions, Pi equals W to I for all I. As I said, this was formulated by Cattadoe, Hauser, and McLean. And they proved it for rank two with no parabolic structure. So that's even there, which was a hard proof. So it's proven for rank two with no parabolic structure. OK, so these are the ingredients. So this is all a math story. So now what I want to do in the remaining, I don't know, most 15, 20 minutes maybe, is how this all ties together and all the pieces click together in the context of Motivic Donaldson-Thomas theory. So what is the plan here? Well, we want to have. Do they show that this conjecture implies the formula? The conjecture certainly implies the formula. No, no, no, no, no, no, no. No, I apologize. No, the conjecture doesn't imply the formula. The conjecture basically gives you another conjecture for where the dimensions of the W-graded pieces are replaced by the dimensions of the P-graded pieces. The conjecture doesn't imply the formula because this side of the formula came from, as I said, by counting points and then extrapolating from the point count. And on the Higgs side, it was checked for rank three by direct localization on the Higgs side, by just fixed point theorem on the Higgs side. It was checked. What was checked? What was checked was the prediction obtained from there for the Poincare polynomial. That can be checked. But for higher rank, it's getting really hard. So let me just, so what's the plan? We want to obtain some kind of derivation of the formula. Of the formula from, well, if you are a physicist from string theory, if you are a mathematician from Donaldson Thomas, or as you say, Motivic Donaldson Thomas theory. So Motivic DT theory. So I should say that from the outset, this is not going to be approved. But basically what I'm going to manage to do is reduce that formula to some sort of standard conjectures in DT theory. So I don't know how much time I have to explain, but that's going to be, so it's going to follow from some fairly standard conjectures in DT theory. But it's not, it gives you a framework for the proof. So basically it tells you when those are proven, then this is proven too. But OK. So all right. So now what's the first step? Well, it's just we're talking about Motivic Donaldson Thomas theory. So what we need is to find a collabial threefold lurking around here. So far, I've been on a curve. So everything was done on a curve. So where is the collabial threefold in this picture? So the first step would be construction. So what I want to do first, step one, we need to construct a collabial threefold x, for example, such that this modular space of Higgs bundles is going to be isomorphic to a modular space of Bridgeland stable, stable pure dimension one sheaves, pure dimension one sheaves on x. Now this looks kind of a loose statement, because I mean, how can you collabial? There are lots of collabial threefolds. But fortunately, there is here we can rely on some results obtained in the literature by, so this was born Greschenig, which basically tell us, give us a machinery of identifying parabolic bundles on C are identified with just ordinary vector bundles on a root stack, which I'm going to call C twiddle. So basically what's happening, informally, of course, you can immediately write down some rigorous formulas. You take this point P here, and you excise it. And what you glue in is a stack, basically, which looks locally like C mod mu L. You insert in a stack a point where L is just the length of the partition mu. How many parts you have in mu?   It's a smooth little amount for stack, which furthermore projects naturally to C by some map row. And this is going to be a coarse-modulized space. I mean, C is a coarse-modulized space for C twiddle. And then there is some work I don't have time to review in detail, which allows me to reformulate the modulized problem for parabolic Higgs bundles on C as a modulized problem for just ordinary Higgs bundles on the root stack. And from there, the next step is to come up with a collabial three-fold. So coming up with a collabial three-fold, it's fairly easy at this point. It's kind of disappointingly easy. What you have to do is to take a fairly naive construction, which is to set x is the total space of O C twiddle plus K C twiddle, which is, of course, I lied a little bit. This is not a smooth collabial three-fold.   And that's the setup. And then you can easily prove that. What you can prove, it's not quite that. What you can prove is that if you cross with A1, then the modulized space of Higgs bundles, it's isomorphic to the modulized space of dimension one sheath. But this is pretty harmless. I mean, this is pretty harmless there. All right, now, what we have is a geometric setup. So we lifted the problem from a curve to a collabial three-fold. So this is what physicists say, well, it just increased dimensions. So now what's the second step? Now, for physicists, I should say, once you get to this point, this modulized space of pure dimension one sheaths and so on, these are just d2, d0 bound states in this all-before background. So we managed to reformulate Higgs bundles to identify them with supersymmetric d2, d0 bound states. Now, the second step is to imply, now we're looking at the motivic dt theory of x, as I said, of Konsevich and Swabermann. And more specifically, we're going to look at specific objects in the derived category of x, which are the stable pairs considered by Pender, Rependi, and Thomas. So from Pender, Rependi, and Thomas, we know there is a counting theory for accounting theory for, let me write upwards, for stable pairs, which means our pairs o, x to f, where this guy is pure dimension one sheath, and s is generically surjective. I'm not going to, I don't have time to get in too much detail. The construction of Pender, Rependi, and Thomas produces a modulized space with symmetric virtual abstraction theory, so with symmetric, sorry, symmetric virtual cycle, and then you can get numerical invariance. But then using the machinery of Konsevich and Swabermann, we can actually get motivic invariance. We can get motivic invariance for such objects. And then from motivic invariance, I'm going to just specialize the polynomial invariance by taking this, so-called, a CERC polynomial. So this is, as I'm summarizing, a lot of mathematics in a few words, but I don't have time for saying more. I didn't want to erase this one, sorry. OK, whatever. So from Konsevich and Swabermann, what we get is we get a motivic stable pair invariance. And then you specialize the polynomial by taking the correct polynomial specialization, which is compatible with motivic decompositions. So we get polynomial invariance. And finally, we're getting some, we can get a partition function, which I can call Z, Pt. I should say that the polynomial invariance are called refined in physics. We also saw this refined BPS states occurred in Sergey's talk. Yeah, they are very much related. It's just the background. This is different. So we get some partition function like this. And yeah, so we get some generating function for such, for polynomial invariance. And OK, so now the main conjecture we formulate then. The main conjecture states the following, that this side equals this side of the HLRV formula up to a simple, a monomial change of variables. So it's not quite coming in W and Z. I have to take some combinations, but really easy combinations like Z squared and Z over W or something like that. Nothing complicated. So this is the main conjecture. This is one of the conjectures we have. Oh, man, I forgot. I should say this is all work with. We end Chuang, Ron Donaghi, and Tony Pantev. So this is pretty much, yeah, this is work done together with we end Ron and Tony. And one of the main conjectures we formulated is this one. It's to identify this side of this identity as a refined with the partition function. All right, now, what will this buy you? Well, if you combine this with the other identification of the modular space of Higgs bundles, what you realize then by staring at the formula is that, in fact, the whole HLRV formula, so what this buys you is that, so what we get from here is that if we get a diagram of our identities like this, this is the exponential of HLRV. And then we have a change of variables here and the change of variables here, the same change of variables, which identifies this side with a refined Donaldson-Thomas partition function here, or PT partition function. And this one, it's identified with a refined Gopal-Kumar-Vapha expansion. So this is refined Gopal-Kumar-Vapha expansion, which appeared in many places in physics. So this appeared in, well, in some ways, implicit in the original work of Gopal-Kumar-Vapha, Katz-Klem-Vapha, then it was written explicitly for Tori-Kulabi-Au-3-fold in Iqbal-Koskazin-Vapha, and more recently considered by Katz-Klem, and sorry, I forgot the full reference. So anyway, so it's considered in many places in physics. In this particular case, it's slightly exotic because we use this expansion for a non-Tori-Kulabi-fold. But anyway, the message here is that this reduces to, and this is one of the standard conjectures in Donaldson-Thomas theory. I should say that this is a, for mathematicians, this is a refinement of the strong rationality conjecture, conjecture of Pandare-Pandain-Thomas. So, OK, so it has a well, it has a good, it has its own place in Donaldson-Thomas theory, but to my knowledge, it's not proven, at least in the refined form. OK, so this is what you get from here, and then perhaps I'll finish in five minutes. The last thing I want to say is that what's the evidence for this conjecture we formulated? So there is two types. I mean, there are two things that physicists can do to get evidence for conjectures like this. One of them is the recomputations. And we were successful in doing some of the recomputations to check this for rational curves with a single marked point. So that was done by localization using, well, using further conjecture, further a k-theoretic index of Nekrosov and Okunkov. So that's one thing, so we can get evidence by direct computations. But we can do something perhaps more conceptual, at least from the physical point of view, which is try to use string theory dualities. So we can train this particular case, M type 2A duality. And this is a sequence of dualities which is now in the physics literature of Geometric Engineering. It's Geometric Engineering. So there is a long history in physics, which I don't have time to get into. But there is a mathematical formulation of this sequence of dualities which occurred in the recent paper of Nekrosov and Okunkov. I think, to my understanding, it's still conjectural. Part of the paper is still conjectural. But there is a mathematical framework proposed by Nikita and Andrei concerning about this duality for this duality. So how it goes is as follows. I'm just going to sketch it in a few minutes. I'm not going to give you the punchline. And the good news for us is that, although perhaps it's not completely rigorously constructed, it does explain the occurrence of McDonald polynomials. So it goes like this. So this is, as I said, using recent work of Nekrosov and Okunkov. So I'm going to state it in this particular case. Day work is more general. But I only need it in this particular case, which now we take a five-fold that's going to look like this. OC twiddle plus OC twiddle plus OC twiddle plus KC twiddle over the root stack C twiddle. And now this is a five-fold. And what they do is they conjecture the existence of a counting problem for membrane quantum states in M theory, which is a counting problem formulated on the five-fold. So there is going to be some counting, so-called M2 counting problem, which is in full rigorosity is going to be a very hard theory, I believe. But let's accept that that exists. So there is some counting problem. And what now they do, what we can do using this, we notice that there is two torus actions here. There is a C star 1 and C star 2, both of them anti-diagonal. So there is two torus actions. And what you can do when you have torus actions like this, you start with this complicated counting problem on the five-fold. And using localization, you can reduce it to a three-fold problem. And if you use localization for the first torus, what you get is the dT slash pT theory of our x. It's a long story, but I'm being schematic here. And now if we do localization for the second torus, well, what we get is something like pT theory on for a non-colabial three-fold, which is just C twiddle cross C2. So you get a pT. It's all equivariant. It's a long story. So pT theory for this non-colabial three-fold. And now if you analyze this theory to employing some physics tricks along the way, from here, with a lot of work, you can actually show that the partition function you get here, you get from this theory here, it's basically, again, given by this side of the formula up to change of variables. And the way this is done is by it goes, this arrow here goes exactly through to Heyman's construction for McDonald's polynomial. So from here, you get very naturally McDonald's polynomials using Heyman's results. But I really don't have any time to go into detail here. It's a fairly long computation. And it's quite remarkable that at the end of the day, going through this five-fold theory, that's kind of mysterious. Everything gets all parts clicked together. And I think you're convincing evidence for this conjecture. And I think it's a good time to stop. And thank you very much. And I want to wish Maxine a happy birthday. And the question, where is the character of a right? It's the last part of your talk. It's the one about Higgs bundles. Yeah, so it got replaced by Higgs bundles. And the W filtration got replaced by the perverse filtration. The filter level high. You're assuming Pw. I'm assuming. I'm sorry. I should have stressed it. Yeah, I'm assuming Pw. You're using that, actually. You know, I found another one. And how you use it? Well, when I. Your filtration has. Well, it's, yeah. Well, you see, there is further. OK. Yes, it's part of this refined Gopal Kumagwafa expansion. Because there is, even in 2000, when the Gopal Kumagwafa appeared, there was work by Hosono, Saito, and Takahashi saying that the GPS, the GV, BPS invariance should be identified with the dimensions of the graded pieces in the perverse filtration on the homology of Higgs. Well, so this is a natural interpretation. In this side, it's a natural interpretation for GV invariance. Of course, their claim is true, basically, for smooth spaces. But under the circumstances, this is a smooth space. I mean, the modernized space is a stable Higgs bundle. It's a smooth. So in this case, we can actually employ their definition and think of this as a mathematical definition for GV BPS invariance. It's very much in agreement with the intuitive picture of Gopal Kumagwafa. Because if you think how you compute the graded pieces here, you have this left-shits action by the vertical polarization. And that's really what it's in sync with physics. When you use necrosis, they have a very unpopularization. How are you comparing them? No, that's a further step. Now, at the end of the paper, they also talk about refined invariance. You have to upgrade. Yes, I gloss over a lot of stuff. I agree. But yes, it's kind of you have to, in particular, they have this definition of refined invariance as a case-erratic index. So it's not defined. It's identified with some case-erratic index. I think you have two more questions. All right, so that works. Thank you, Manuel. Thank you.


At Audiology Associates of Prestonsburg, you can live your life the way you want and find the freedom of better hearing. You'll experience patient care that is specific to you with exceptional follow-up care that ensures your hearing and balance needs are being met. Audiology Associates at 1428 Northlake Drive in Prestonsburg. One church of Kentucky is seeking volunteers for its annual Thanksgiving dinner after a one-year hiatus. Tuesday we spoke with Pikeville Free Will Baptist Church pastor Jim Patton about the dinner and the need for volunteers. We've got our annual Thanksgiving dinner coming up. Of course, last year we were not able to do it because of the COVID restrictions, but this year we're planning a big event. A total of 9,200 meals we're planning on putting out in the community. 800 of those will be on site, approximately 8,400 will go out and be delivered in the community. We have plenty of work available for folks that want to come by, spend a couple hours or spend all day with us however much time you want. You can come out at 7.30 on Thanksgiving morning and we'll be working until about 2 o'clock and there'll be a devotion in the middle of it. There'll be time if you want to eat or do something there on site. We just invite everybody to come out. It's a great time of fellowship and if you haven't been involved in it, we want you to come be involved. Be a volunteer. It's one of the greatest blessings I've ever had and I've been a pastor for a long time. I've been a ministry for a long time, but to actually feed the hungry, what God told us to do. He said, if you do this, it's the same as like you're feeding my son. If you'd like to volunteer or participate in the annual Thanksgiving dinner, you can visit the Appalachian Wireless Arena on Thanksgiving Day, November 25th, from 7.30 a.m. to 2 p.m. For Mountain Top News, I'm Joel Hodgell.


look at all those sleeping trams getting ready for a very busy day here at Hollywood Studios. Good morning star shines it is 5 20 in the morning on Tuesday January 7th and it is my first time attempting a riding the rise of the resistance. Let's go do this. And for it being 5 20 in the morning there's really not that many people here. I mean I've seen pictures where people were lined out all the way to the security checkout lanes and that was at like four o'clock in the morning so I'm feeling lucky. And then once we get in here you see the lines are starting to form. We may possibly let you into Hollywood Boulevard a little bit early but you'll only be going as far as the Charlie car because the park will not be open yet. What you can do is you could potentially get breakfast, you could get Starbucks at the Charlie car and take pictures of our beautiful Christmas lights but you will not be able to make those boarding groups until 7 o'clock. And just like you heard the park doesn't open until 7 o'clock. They are gonna let us into the park early and then we can try to get those boarding passes. I'm hoping that I can get one semi early. I don't want to like go here now and then come back four or five hours later so we'll see what happens. It is 5 44 now and let's just get an update on what it looks like here. And a lot more people have showed up. There's only about 15 people ahead of me and it's looking very promising. Oh I see the lights flashing they're letting us in. Some people are excited. It's 6 16 a.m. We are now in the park and I think we have 40 minutes, 40 minutes to wait until we can try to get those boarding passes. They've got most of the areas open just nothing open so that you can actually go into which is kind of cool because you get to just take a look at your surroundings and that's one of my favorite things to do in Disney is actually take a look at all the surroundings. So it is 6 56 and I think what I'm gonna do is just sit here and refresh like that and hopefully it works. If not then I am leaving the park a very disappointed man. Okay 6 59 guys we're gonna try this method out for a minute long. Yeah a minute long here. Oh it's 7 o'clock it's not worth it. Oh no guys. I got in. 37. We won. Wait why did mine not load? Already a boarding group thank you. 37. Wow really? We did it at the same time. Same time. Wow crazy. Where do you see that because I'm not seeing it on mine? My status. Wow. So that was really interesting the whole entire joining a boarding group thing. As you can see I tried doing the refresh method and it actually didn't work.  I still got boarding pass boarding group 37 which I'm kind of happy with but I probably could have gotten like in the 10s or the 12s if I just started out open it up and then click right on. You live and you learn. And since I'm awake and I already have a Starbucks coffee I think I'm just gonna wait it out. Wait till my party gets called 7 30 now. They're already on boarding group 10 so that means I only got what 20 27 groups ago. So I should probably get on there around 9 a.m. Hopefully fingers crossed. I guess I might as well take a tour on Star Tours to get myself all ready for the rise of the resistance. Thank you. What are your favorite glasses at Disney parks? I mean I'm very fond of these ones but my favorite are the bug eyes and it's tough to be a bug in Animal Kingdom. I really like those but let me know in the comments what's your favorite. I really enjoyed the new rise of the Skywalker scenes in there especially like those ghastly beasts that come out of the water there. They were really nifty. It is 8.04 a.m. and they're already boarding groups 8 through 27 so I only have 10 to go. I might as well head into Galaxy's Edge and just get ready. Wow I'm excited. I decided before I actually get to ride right of the resistance I better get myself some breakfast. I didn't get a chance to see what they have on the menu but I hope it's none of that. I decided I'm getting the Bright Sun's breakfast platter. I don't need egg though so I substituted extra sausage for the egg but it comes with sausage, purple potato hash and a cinnamon roll. I'm excited to see what this is like. There aren't many options for breakfast at Hollywood Studios and I've heard a lot of people talk about the Ronto Wrap but the line for the Ronto Wrap is like 10 times longer than here so that's why I decided to come in here and honestly I think it's a better value. I don't know how much that Ronto Wrap is. I know a lot of people love it but I mean the breakfast platter was only 10.99 and you get potatoes, eggs, cinnamon roll and sausage. There it is now we got to find an interesting place to sit and eat. Oh how about right there? Look at that though. That looks delicious. I also got the move juice which is right here and it's a simply fruit punch with simply orange with pineapple and chipotle pineapple. I'm excited to try that. I think I'm most excited for this move juice. It tastes very pineapple-y. Pineapple-y. But I can really get into that cinnamon roll.  The sausage is phenomenal. I really like it. It's like a cabassi-ish style sausage but the potatoes not that great to me. I don't know why. They're not crispy. They're kind of wet and soggy and they're purple. I like the color but I don't like the taste. Gore-Tex drink. I weeded after I ate my sausage and potatoes to get to this little guy here and that looks scrumptious. I don't like mixing sweet things with like non-sweet things so I treat this like a dessert and I wait till afterwards. But it's going in. That was a satisfying breakfast. The sausage was the best part and the move juice I kind of like it. It's grown on me a little bit and this is what I was talking about the line for the Ronto morning wrap. That is a very long line for a wrap. They are now boarding 8 through 31 so only six more to go. Not long. Not long I say. Until then I think I'll just stand here and stare. I just got the notification it is officially 9 a.m. and I had until 11 to get here. So not too bad. So what's that? 7 8 8 9 2 hours. So number 37 it took 2 hours. Here's some comments happening from my video too. Jennifer ever after love your b-roll. Hillary said I'm gonna be so sad if they rip up the fiber optics. Jerry Lupo said thanks for the update Nate. Very cool. Let's go do this. Gotta get the band out. I'm going right here. Thank you. All right good luck on your mission. Right all right. It's so crazy seeing this from the outside and now being able to walk in here. I have stayed clear of all videos of this attraction. I haven't looked at anything. I know that some of the big things but I've known those for a while. So I'm gonna be kind of surprised. Very pretty cute though I have to say. I like this. I probably shouldn't be touching everything like I am but I'm so fascinated. Can I do a little photo here? This is where we are. We have to be right here. So we're gonna take this route down and then go up here. I don't even know what I'm talking about. I like walking on this ground. It feels very fancy. That's kind of cool. Look at some blasters up there. One. Thank you for joining the calls. No need for safety here. Yes, I just want to be sure. Good. Yes. Where are all those Thai fighters coming from? Hiding. He just looked at me. He's not looking away. Oh, we're all clear. They get started. We're in our seats. I'm all seating up. I'm ready to go. Tell them it's a prisoner transfer. Did it work? Good. Now give them one. There's a clear path to the turbo lifts at the end of the hallway. Turn right.  Take those turbo lifts and stay out of trouble. Hey, you're not upright. Get out of here. Look good. Try to find another way down to the escape ports. The resistance wants them to rescue the prisoners. They have neither the resources nor the courage to engage us. You want the rest to meet their conviction? Raise the shields. Shields? I see no evidence. And the arm. All comes to those in that station. Sir, some prisoners have escaped. They're alive, but ultimately hopeless. And if you don't want them to run the street, the location of the secret base, and they're not just for you and the resistance. Rob's came in a little off target. Rob Cruise will make sure you're outside the wreck. It's not what she signed on for, but hey, your resistance now. I think I have that authority. Right back? Where's the lieutenant? I need eyes on Beck. Hey, Lieutenant Beck here. I should hope so. Chris Chopper recruits. Thanks to your heroism, the location of the resistance base is secure. Holy bread, son. That was amazing. I really, really, really, really, really, really liked it. I loved the theming, the animatronics. It was full of surprises and twists that I did not see coming. Wow. Seriously, I just can't wait to ride it again. And I don't even know when that's going to be. Maybe I'll come back tomorrow. I don't know. Well, I guess that's it. Like I said, I really loved the ride. I actually had to take a couple minutes to take it all in. I'm glad that I did video, but I did focus on the ride. So I don't know how that's going to come out. I don't know if I can put it all in or not, because, like I said, I wasn't paying attention. I was really just looking at everything around me. And seriously, guys, such a great ride. Such a great ride. So, love the life you live. We'll see you next time. Bye.


Item number, SCP-660. Object Class, Safe. Special Containment Procedures. SCP-660 may be safely stored in a standard locked safe deposit box, provided its mouth remains covered with an inert perspex disc and sealed with a fixed tamper-proof strips. SCP-660 may be checked out for testing and research with a written approval of staff of level 3 clearance or higher. Human testing of SCP-660 is discontinued. Description. SCP-660 is an irregular earthenware urn, roughly 23 centimeters in height and 28 centimeters in diameter with an opening 20 centimeters in diameter. The composition of SCP-660 is consistent with generic earthenware clays. The interior of SCP-660 is coated with dried mammalian uterine tissue, similar to human endometrial lining. DNA testing is inconclusive. SCP-660 is capable of indefinitely preserving all biological tissue placed within it, regardless of external conditions or the state of the sample, until the tissues are removed. Living tissue will remain viable and living organisms will remain alive, even when deprived of sustenance for extended periods of time. While the exact mechanism responsible for this is uncertain, isotopic labeling has shown that it involves the exchange of biological matter between the sample and SCP-660's interior coating. Cell cultures from tissues preserved in SCP-660 retain an abnormally high rate of cell replication. Several samples have tested positive for pre-cancerous markers and smear tests. SCP-661 was a human fetus, originally found in SCP-660, most likely having been sealed in the artifact for at least 6,600 years. However, SCP-661 immediately underwent massive transformation upon removal, additionally developing several consistent with anthropogenic progression. SCP-661 is most likely neutralized. Details are available to personnel with clearance 4 slash 660. Recovery SCP-660 was originally recovered from an archeological dig in where it was found in a purported gravesite, among a number of other mundane artifacts. A beeswax seal covering the mouth of SCP-660 was dislodged during the excavation process, revealing SCP-661. Data expunged, after which the site was secured by the foundation. The notes of Dr. W. Presum deceased, enabled the isolation of SCP-660 from the rest of the artifacts. Log of tests involving human experimentation with SCP-660. Lab report, 660A. Sample condition. Skin flap harvested from the left thigh of the cadaver of D-612. Donor was healthy in all respects prior to death. Sample use. Tissue sample left in SCP-660 for two months duration. Result. Daily biopsies indicate viability of sample. Cell samples taken from the tissue graft are cultured in vitro, revealing a predisposition towards pre-cancerous and cancerous growth. Lab report, 660B. Sample condition. Skin flap harvested from the left thigh of the cadaver of D-612. Sample use. Tissue sample left in SCP-660 for two months duration. Then grafted onto the torso of a D-class personnel with second and third degree burns. The recipient's anomaly testing record had no contraindications with SCP-660. Result. Tissue grafts takes without complication, healing according to usual steps of adherence, serum inhibition and revascularization. However, as is common with severe burns cases, the compromised skin barrier led to infection and eventual fatal septic shock. Postmortem biopsy and analysis indicates skin graft would have returned to full histological structure and function had the recipient survived. Lab report, 660C. Sample condition. Live member of mus musculos, house mouse, BALB slash C strain, acquired for laboratory testing. By use of SCP-614, the consciousness of D-614 was implanted into the specimen. D-614 was chosen for his demonstrated resistance to the negative effects of prolonged isolation and sensory deprivation. Sample use. Specimen left in SCP-660 for two months duration. Result. Unusually, SCP-614 was incapable of restoring D-614's consciousness to its original body after removal from SCP-660. A computer interface was developed to facilitate two-way communication with D-614. An interview was conducted once D-614 was familiarized with this interface. An excerpt is included below. Dr. Brown. I guess I'll have to take your word for it. And how did that feel? D-614. Suffocating and stretching. Dr. Brown. Could you be more specific? D-614.  Even mice need to grow and move and live. Every part of me was straining, but I was stuck. I thought I would explode. Full contents of this interview are classified. An autopsy revealed precancerous growths throughout the mouse's body. Lab reports 660D. Sample condition. Right arm lost in traumatic amputation just beneath the elbow. Sample use. D-778 was severely injured in an incident involving SCP- Lost limb recovered as part of cleanup operation. Subject would not have survived immediate reattachment surgery due to other injuries involving severe vascular compromise. Severed extremity preserved in SCP-660 and bid to preserve tissue viability. Result. Right arm successfully reattached to previous owner. Subject reports return of sensation in fingers of right hand. Contrary to poor prognosis due to severe tissue avulsion and nerve damage sustained in initial injury. Biopsy reveals little to no precancerous cell growth. Transplant ruled a tentative success. D-778 has been retained for further observation. Addendum. Human testing with SCP-660 is suspended due to the sudden death of D-778 from teratoma with malignant transformation. I've seen fetus in fetu before, but never have I had one try to grab my probe during an autopsy. Dr. Grayson. Lesson complete. If you missed the previous orientation, go watch SCP-659, communal avian intelligence, right now. Or for the complete course, watch this playlist.


You have the Holy Spirit, but it's in the surrendering to the Holy Spirit that what we have in us manifests around us. But it's untapped power that rests within you, and the only way you tap into that power is by surrendering to the Holy Spirit in greater depth. The Holy Spirit is all in. Are you?