Hello, welcome to next lecture that is on circular cube. So these are the learning outcomes. At the end of this session, the students will be able to illustrate the circular cube in data structure. So now we'll start with the circular cube. Previously what we have studied in the video lecture was implementation of linear cube. As you know that Q will follow first in first out linear data structure. But even if you have the memory space in the Q, we cannot insert the new data into the Q. So that's why that is to make the effective utilization of memory. We are going to see the circular cube. So first we are going to see the algorithm for the insert operation. So in the same manner what we have studied in the linear cube that is for the insert operation we are going to check the full condition. In the same manner if you are going to delete the atoms we are going to check the empty condition. So why we are going to for circular cube? As we know that in the linear cube if you delete all the elements that is 10 20 and 30 now this is R now this is an F. If you delete all these elements the F will come here but the R will be at this same position. We are not going to connect this to the first position that's why the memory utilization is not done properly in the linear cube means what to make the effective utilization we are going to connect the last end of the Q to the first end for proper utilization of memory means we have a space then we can reinsert the data. So that is known as an the circular Q. Logically we are going to connect that is this end to the this end here. So we'll start with the an algorithm that is an insert operation. So this is the condition to make the full. So this is the full condition. So this condition is appear so you can write Q is overflow. So if it doesn't happen so we have to go for the next step that is an F whatever you're going to conditions are there. Now suppose we are going to take the size is equal to 3 here. So now I'm going to draw the circle because we are going to connect logically that is last end to the first end here. Now suppose this is a of 0, a of 1 and a of 2 here. So now before inserting the new elements every time we are going to check the full condition that is F is equal to equal to 0 that is ending with R is equal to size minus 1 or condition either this condition or either this condition is satisfied you can say the Q is full that is F is equal to equal to R plus 1. Now we are going to insert the new element that is an 10. So whenever we are going to insert the new element initial F and R is equal to minus 1. So they are pointing towards the minus 1. So this condition is satisfied when F is equal to equal to minus 1 then we can have F is equal to and R is equal to 0 means both the pointers are going to be increment by plus 1 here. So that's why it is F is equal to equal to minus 1 means F is equal to equal to 0. Now this condition will appear only once when we are going to insert the first data in the circular cube. If this condition is not satisfied so the normal condition will appear that is R plus plus. Now if you want to reinsert the data means you are going to insert the new data so at that time you can say that is R is going to incremented by 1. So this will be F and now this is an R. So this condition has been incremented because this condition is then R is equal to 0 this is most important condition. So from this condition only we are going to connect this end to the next end here. So this will happen. So now suppose one more data if you want to insert that is F that is 10, 20 and 30 so R will be at this position here. So now this condition is full. How you are going to check this if you want to insert 40 you cannot insert that data that is 30 this is R, this is A of 1, this is A of 2, this is A of 0, 10, 20 and F. Now going to check F is pointing where here 0 and size minus 1. Size is what we are going 3, 3 minus 2, 3 minus 1 so it is pointing so R is actually pointing towards 2. So this condition is satisfied this if doesn't happen so this is known as an R condition. So this condition is satisfied so this is in full condition so we cannot reinsert the new data. So when this condition will appear that is R is equal to equal to size minus 1 so when you are going to delete this element so F will come here. So what will happen if you delete 10 so what happened your first now F will be incremented by this position 1, 20 and this will be an R. So in linear queue what happened if you want to reinsert the data that is not possible because we have not connected logically last end to the first end here but in circular queue by writing this condition if R is equal to equal to size minus 1 we have seen this in the year that is R is now where 2 that is size minus 1 so what we happen we can insert the new data that is an 40 so what will happen this will be at this position R is incremented by plus 1. So in the same manner if you want to do one more F is deleted so the space is available you can reinsert the data that is effective utilization of a memory here. In the same manner we are going to see the algorithm for the delete so in the same manner delete operation we are going to check the empty condition here so what we have taken that is the size we have taken 3 so already what we have seen 10, 20, 30 this is F this is R so if you want to insert the data R is incremented by plus 1 if you want to delete the data that is you are going to increment the F pointer here. So if you delete 10 so the new queue will be look like this F this will be empty 20 and 30 this will be an R so this condition is written here F plus plus here so when this condition will happen F is equal to equal to R then we have to point R is equal to F equal to minus 1 here so all this condition before inserting we are going to see that our empty condition that is R is equal to equal to minus 1 ending with F is equal to equal to 1. Now what we will see that if you delete this one more delete 20 so what will happen so F come here and R will be at same position that is A of 2, A of 1 and this is A of 0 so suppose now what happened this condition will be executed when F is equal to equal to R then F equal to so if you delete 30 so what will happen so if you delete this so both are pointing towards the same element so F in equal to re-initialized to R and F equal to minus 1 so Q is in empty condition so if you delete the element from the queue so this condition will be executed as F R and F is equal to minus 1 okay so when this condition will occur F is equal to size minus 1 so if F is at this position okay and R is at this position that is 10 20 and 30 here so if F is equal to equal to size minus 1 so F is equal to 0 if you delete this element that is delete 10 so what will happen so F will point towards this element and R will be at this position only that is in this position means by connecting last 10 that is A of 2 to A of 0 that is logically we have connected here so F is going to incremented so how we are going to do this so we are going to do this by the modulus that is an mod of this here okay so these are the algorithms used for and delete operations by using this an algorithm we have to implement the C program here so for this you should remember all these three conditions that is F is equal to equal to R then what we have to do if F is equal to equal to size minus 1 what we have to do and normally F is incremented by plus 1 here okay so now we will see the algorithm for the display here so now in display what we have to do that is we have to display all the elements how many elements are there which are the elements in the queue so these are the queue that is 10 20 and 30 this is A of 0 A of 1 A of 2 here so by deleting the elements okay so in the same manner if you want to bring the data so what what we have to check is the queue is empty or not how you are going to check this F and R is equal to equal to minus 1 if this condition is satisfied your queue is in empty so you cannot delete the elements from the queue so that's why so every time whenever we are going to display we have to check this condition that is Q is empty or not if Q is not empty then we have to go for an for loop that is I equal to F I is not equal to R and I is equal to I plus 1 mod of size so here size what we have taken is 3 so this is most important as you know that in for loop this is for initial condition this is condition and this is increment or decrement here okay so now I is equal to F means F is where pointing that is F of A of 0 here so here I value becomes 0 okay I is not equal to where till where you are going to check this condition that is I is not equal to R so where R is pointing where R is pointing towards the 2 so that's why I is not equal to 2 so I 0 0 is not equal to 2 so that's why this condition is satisfied you are going to increment this so it will print 10 so we are going to increment the value of and this so I is what 0 plus 1 mod size of 0 3 so 1 3 0 divided by 0 that comes to be an 1 so this is incremented by plus 1 here same manner it is pointing towards the 1 it will be pointing towards the 2 so this way the condition is going to be incremented here so this is the most important thing that mod we are going to connect all the logical end from a of 0 a of 2 to a of 0 here okay so it will print all 10 20 and while leaving these elements we are going to print the Q of R that is 30 so this condition will satisfy till two elements are printed here and one more element if you want to print you have to print the Q of R while leaving the this for loop here so this is the regarding the algorithms for insert delete and display for the circular Q so now pause this video and answer this question if if it is a normal Q if we implement using an array of size that is max underscore size to get full condition so when what will be the condition for that so are the answer is rare equal to max underscore size minus 1 so these are my references thank you


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Well, in 40 minutes, it's a big task to deal with such a vast subject, but I will try and touch on the basic points. If anybody wants to do any historical reading on this, I would suggest Trotsky's writings on China in the 1930s. I would suggest Ted Grant's material after the Second World War, what he wrote in 1948 and in the 60s. And then, of course, more recent material, our document, the Long March to Capitalism, and other articles on China, available on Marxist.com. Now, what is the Chinese regime? It's a good question, and we have to be able to answer it concretely. Because you see, when you're struggling to change society, and you present yourself as Marxist or Communist or Socialist, you're inevitably going to get the question, do you want to create a regime like the one we had in Russia under Stalin? Do you want to create a regime like the one we have in China? If the answer to that is yes, I'm telling you, a lot of workers can say, I'm sorry, not interested. It would not enhance the cause of socialism, it would stifle it. And yet, we have to answer the question, what were these regimes and what is China? Now, recent little bit of news that came out was that the number of billionaires in China this year increased by 257. Now China has 878 billionaires. The United States has 626.  And this year, they earned more than ever before. Now, if we were to believe some of today's Stalinists, apologists, let's say for these regimes, we'd have to believe that they're building socialism in China. But what I just said seems somewhat to contradict that. And yet, I've actually heard Chinese Stalinists arguing that we're building socialism through capitalism, you see, which is an interesting way of building socialism, i.e., by allowing the profit motive to be unleashed. Another question is a genuine socialist regime oppressing nations, national groups such as the Uyghurs, or unleashing the brutal oppression of the movement in Hong Kong. Is that socialism? Is that what socialism is about? If you argue that, you're welcome to go and try and... I haven't got time to go into the details of it, but what is socialism? Well, let's put it this way. What is... It must be based, one, public ownership with the means of production, planned economy, but that's not enough. It also must be based on democratic workers' control and management of those means of production. It must be based on the workers' rights to assemble, to debate, to criticize, to elect their leaders. It must be based on the principle that the people who are elected can be recalled by the workers who elected them at any moment. And, furthermore, they must not earn more than the average wage of a worker. Question, do we have that in China today? I don't think you need to worry too much about looking this up on the internet to get the right answer. You don't have to do a Google on that one. I think it's abundantly clear that that is not what we have in China today. We actually have not just these billionaires, they're members of the Communist Party. They use the Communist Party to promote their interests. They're tightly connected with the bureaucracy of the Communist Party. Now, in 1949 there was a revolution which abolished capitalism, and landlordism, feudalism established the economic, let's say the economic premises for the creation of a worker state. But never, since day one, were there the elements of workers' democracy. Right from the beginning, that regime modeled itself on the Soviet Union, but not the Soviet Union of 1920 or 22 or 23, the Soviet Union of 1949, i.e. a Stalinist dictatorship. So, right from the beginning, it was already what we would define as a deformed worker state, i.e. you have the economic base for a worker state, i.e. you have the nationalisation, you have the public ownership, you have planning, but it's not under the control of the working class. And if it's not under the control of the working class, somebody else obviously is running society, and that somebody else is a bureaucracy standing above society, above the working class. Now, I haven't got time here to go and quote Trotsky, but Trotsky pointed out that once you have such a deformation, in the case of the Soviet Union, a degeneration of the revolution and the rise of a bureaucracy, a privileged bureaucracy earning above the average wage of the workers standing above society, implicit in that is the danger of capitalist restoration, because the bureaucracy would not be satisfied simply with being a privileged bureaucrat who's coming off some of the wealth produced in the plan, but would want to become owners of the wealth, direct owners, direct owners of the means of production, and to pass it on to their heirs, to their offspring. Trotsky makes that point. And when Trotsky made that point, he was attacked by the Stalinists, had you attacking socialism, communism, all the rest of it. History now has come full circle was Trotsky correct to raise that point. Where did capitalism come from in Russia? Who promoted capitalism in the Soviet Union when it collapsed? It came from within the Communist Party of the Soviet Union. You look at even Putin used to have a Communist Party card in his pocket. And in China, what we have is a process whereby from within the bureaucracy, you've had the promotion of capitalism. Now, after 1949, there was a development of the Chinese economy. Between 1957, 1970, it was growing by something like eight, nine percent a year. Just to give you an example, in 1952, China was producing 1000 tractors in this huge country. 1970, 690,000. So there was development. There was a step forward for society. China was actually on average growing faster than the capitalist West in in that period. Of course, that was starting from a very, very low level of economic development. And therefore it had a lot to do in terms of catching up with the advanced economies of countries like the United States or Western Europe, Japan, etc. But nonetheless, there was a development. For example, what I was talking about privilege in 1976, an industrial worker working at about 48 hours a week was earning about $12 a month. Professionals, on the other hand, were earning $120 a month. That's already a difference of 10 to one. So we have the danger of the capitalist restoration, i.e. that privileged layer has an interest in first maintaining that position, i.e. privilege, and then finding ways of guaranteeing that privilege is preserved. Now, I haven't got time to deal with a lot of the questions which I'd like to deal with, but we cover a lot of this in our written material. Let's look back a little bit at the history. In the 1970s, it was clear the Soviet Union was in severe crisis. Its rate of economic growth had slowed first down to two or three percent, i.e. slower than the capitalist West. And then eventually it hit zero growth, and that's where you had all the scenes of shops that were empty, lack of goods, etc. In the meantime, in China, you had the conflict within the bureaucracy, which is basically between the Mao wing, you could say, and what was to become the Deng wing of the party. The gang of four after the death of Mao were trying actually to launch a second cultural revolution, but with the Mao gone, the bureaucracy felt far more confident and stopped them, arrested them. And then by 1978 you have Deng Xiaoping at the top of the party, who begins the process which led us to where we are today. That process was opening up China to foreign capitalist investment. He began by promoting four special economic zones, not by chance they were around Hong Kong and Macau, which were pockets of capitalism, let's say, in China, in Guangdong and Fujian. We would argue that when they started on this road, there was no preconceived plan. It's not as if Deng said, right, we're going to launch this plan, and in 40 years time, China has gone to be capitalist. That wasn't the way things developed. It would be too much to think that that was how far ahead he could see. We would argue there was a logic in what they were doing in the sense that there's a certain similarity here with the NEP, the new economic policy that was launched in the 20s in the Soviet Union, i.e. In the backwardness of the economy, in order to attract foreign investment, foreign technique and technology, you can open up to these private investors in an attempt to advance further your own economy. Also, allowing sectors, such as the small farmers, etc., to sell their goods in the market. Now, it started off as a way of trying to stimulate growth and develop the economy, but step by step it moved closer and closer to a moment where the economy actually was transformed. For instance, in 1983, the state-owned enterprises were allowed to hire workers on a contract basis for a temporary period. Thus, the old guaranteed for life jobs began to go. More and more, the profit motive was introduced as a criteria for the functioning of the economy. In the mid-1980s, however, was still within the sphere of using market stimuli, but within the plan, where the plan still dominated and the state sector still dominated. But already you see how the terminology begins to change. In 1984, for instance, at the 12th Congress of the Party, they started talking about a planned commodity economy. In 1987 at the 13th Party Congress, they developed the idea of an export-orientated economy. You see if you have an export-orientated economy, that means you're orientating your economy to sell goods in the world market. So you start to begin to function according to the needs of the world market. And capitalism was still the dominant mode of production on a global scale, which is an important factor in determining the way things turned out. Now, this led to a big increase in the importation of machinery in order to develop the industry and a sharp rise in China's balance of trade deficit in that period. And in 88-89, we had the emergence of inflation, 18% a year, for example. And we had what in a planned economy is an anomaly, is the first recession in 1989, which comes at the same time, not by chance. This is what is the background to the movement of Tiananmen, student protests, but also, very importantly, the working class began to intervene in 1989. We published something interesting recently written by some Chinese lefts who written some very interesting material about what happened at that time and about the role of the working class. IE, the working class began to move independently in an attempt also to build its own organizations. That's why the bureaucracy clamped down so hard on Tiananmen. They could not tolerate an independent movement of the working class. Ten minutes ago? Put yourselves in the shoes of the bureaucrats in China. You have Tiananmen. You have the potential, the potential for a political revolution in China at that time. You look across the border in the Soviet Union and in Eastern Europe. And in 1989, you have the movement beginning in one country and then spreading right across the whole of Eastern Europe, finally with the collapse of the regime in Romania in a very violent way. The whole of Eastern Europe, the regimes collapsed and began to revert to capitalism. Within a short period of time, 1991, the Soviet Union collapses with a huge collapse in production. Just the collapse of the Soviet Union as a default workers state, the collapse of the plan, the Soviet Union breaks up into its component parts into the 15 republics who become independent. And the power of the old Soviet bureaucracy enormously weakened. The Chinese bureaucracy observing this before it happened, the crisis and the slowdown of the Soviet Union. And what we see is the bureaucracy clumps down in 1989, 1989 slows down the process of reform for a short while to try and restabilize a situation. And then out from 1992 onwards, the process starts to accelerate again. And for instance, for instance, at the 14th Party Congress, they came up with the term a socialist market economy with Chinese characteristics. What is a socialist market economy, the two things are in complete contradiction. And it is they began a process of privatization. 2500 local smaller state on enterprises were prepared for privatization and about 100 century run state on enterprises. By 1998, all of these had been privatized. In time, the Chinese bureaucracy were not going down this road to open up China completely to multinational corporations and to imperialist domination by the US, Europe, Germany, Japan etc. They held on to about 1000 of the major corporations. They kept them as state owned, they closed the less productive sectors, they've used companies together, they invested huge amounts of public money. The plan of course is to create competitive Chinese corporations that can hold their position in the world market. Now, by the end of the 1990s, the state owned enterprises were employing something like 83 million people, which was about 12% of total unemployment. We're still talking about a kind of economy with a large rural sector. In the cities, around 30% of the workers were now working in the state sector. But compare this to 1978, when 78% of urban employment was in the state sector, you see the shift that's already taken place. And I have quotes here from some sources. This is from a book called The Exit the Dragon privatization estate controlled in China by Stephen Green and Guy S. Liu. It says we see how around by this period, around 40% of industrial output was privatized. Together with this, we have the private sector that developed outside the state sector, both front and indigenous, i.e. you have privatization of some of the state owned enterprises. The promotion of homegrown capitalist companies and the coming in foreign companies, which at one point represented something like 20, 25% of GDP in China. So we see a gradual decline in the percentage of the economy, which is in the hands of the state sector. And then in September 1999, we have what they call the let go of policy. I was adopted, which is further loosening up in the medium and small state owned enterprises. Between 1990 and 2000, between 30 and 40 million state jobs are destroyed. The so-called rust belt of the Northeast where households of industry were closed. And the growing capitalist sector within the economy. This eventually led to a situation where something like 60% of GDP was being produced by private companies. The fact that 450 of the top 500 multinationals were operating in China is an indication of how far the process had gone. Now, the calculations of how big the private sector is depend on who's making the calculations. But all of them indicate that a majority of GDP is produced by private companies. The Chinese trade unions and the Chamber of Commerce in China even say that something like 60% of the economy is in private hands, if not more. However, we still have the big state sector, state owned companies, which are being promoted and defended by the state. And now it's because of this big state sector that there's confusion on the left, i.e. Because there's so many state-run companies, China must still be somehow communist. Well, you know, it's not just the number of state-owned companies which determine the nature of an economy or a regime. In 1934, for example, how many of you know that 70% of the economy was in state hands? That wouldn't you say, well, it must have been a worker state, but it was fascist Italy. It wasn't obviously a worker state, it was the state intervening to save capitalism, taking over companies and running them with public money. Even in the 70s, Italy, 50% of the economy was in state hands. And countries like Britain, France, Germany in the 1970s, 30 to 35% of the economy was in the hands of the state. You know, in Britain, British Leyland, cars were being produced by the state, coals, gas, electricity, water, all of these sectors and others. The railways, they were all state-run. Italy had five-year plans for decades. It didn't make it a worker state or socialist economy. It's not the first time that we've seen capitalism being developed from the state, you know, the backbone of the economy was actually the state itself. We had it in Japan in the 1800s and in Germany under Bismarck. What I'm trying to say is state enterprises alone does not equal socialism and even a degree of planning by itself is not socialism. We have to look at the overall functioning of the economy. So what we have is the building of a market economy through the state. We reached a point where at its peak, I think foreign investment represented something like 28%, 25% depending on the sources you look at. And the indigenous private enterprises were something like 22% about 20 years ago. Now, this growing private sector is something you need to look at as Marxist. We don't limit ourselves to saying, yes, it's capitalist, no, it's socialist. You look at a process and you see what direction it's going in. Now it's 40 years that they've been on the capitalist road or building socialism through capitalism. As someone would say, if you continue on that road, you're going to get from A to B, you're no longer going to be at A. Eventually you're going to be at B and you're going to go beyond it. At a certain point, there's a qualitative change that takes place. And what we saw was at a certain point in the 2000s, when you start looking at the figures, looking at the facts of what's been going on, you find that the bulk of the economy is being produced privately. The state-owned enterprises are not functioning as companies within a planned economy, but have been given the right to hold on to the profits. They run according to the profit motive. And you have a greater and greater connection with the rising capitalist class. 20 minutes. Yeah. I don't want to quote Trotsky here because I've already said what he was the famous quote about how the bureaucracy would eventually, he says here, he says the victory of the bureaucracy in this decisive sphere, i.e. this growing privatisation, would mean its conversion into a new possessing class, i.e. the bureaucracy itself begins to transform itself into a property class. And that's what we're seeing in China, I have seen in China today. I'm going to have to skip a lot of the stuff I have here because of the time. Now, the question of the direction.  Because the question was being posed. Are they moving towards capitalism? This is how he posed it. Of decisive importance in evaluating the nature of society is the following question. Is the society evolving in the direction of equality or in the direction of privileges? The answer to this question does not leave room for any doubts, whatever. And then Trotsky added, 20 years after the revolution, the Soviet state has become the most centralised despotic and bloodthirsty apparatus of coercion and compulsion. The evolution of the Soviet, therefore proceeds in complete contradiction to the principles of the Bolshevik program. The reason for it is to be found in this, that society, as has already been said, is evolving not towards socialism, but towards the regeneration of social contradictions. That was the process. Hadn't completed yet, it remained still a planned economy. But the direction it was moving in was the point that we have to stress. And Trotsky reiterated the point. He said, in 1937, he said, nobody has ever denied the possibility, especially in case of prolonged world decay, of the restoration of a new possessing class springing from the bureaucracy. Well, this was a brilliant analysis on the part of Trotsky. And I think that what has happened in China, particularly confirms what he was saying. The fact that we have all these billionaires now is is a confirmation of a process indicates where we have reached on the question of growing inequality. In China, it's abundantly clear to anybody that looks at the facts and figures that China is more and more unequal, not more equal. Inequality means privilege and means the tendency towards class differentiation. A report in 2008 by the Boston Consulting Group concludes, thus, wealth in China is highly concentrated. Less than 1% of households in China hold more than 70% of the nation's personal wealth. Even amongst these households, the richest 0.1% of the nation, with financial assets of more than $1 million, control approximately 45.2% of China's wealth. How more unequal do you need to get before you start to think and realize this is not the commonest paradise that some people try and approach. The fact that the flag is still red and has a hammer and sick and it means nothing in terms of the nature of the regime in terms of the inequality that exists. There's lots and lots of facts which I haven't got time to go into or lots of them about the degree of inequality and the poor getting poorer, etc. Although, obviously, there's a layer that's got richer and there's a middle layer which has risen up with economic development, obviously. Right. Sorry, I'm trying to skip forward. Now you see, we, as an IMT, looked at all these facts and figures after 2000, and it was becoming clear that this was no longer a transitional regime from the previous deformed worker state, how your planned economy still, let's say this is the economic base of a worker state, albeit without the worker's democracy. It was in transition. And sometimes you have to maintain the position that it's in transition. It hasn't completed the process yet. It's still fundamentally of a particular nature, i.e. it's still a planned economy. But we know in which direction it's going. At a certain point, Marxists have to look at a process and say, has the process stopped? Has it reversed? Has it moved on? It was clear by the 2000s that from a whole number of statistics, facts and figures, the process had moved further. Now you can never actually say exactly when something changes into something else. You know, night is night and day is day. The dawn is a moment of transition between the two. When exactly is it daytime and when exactly does nighttime cease? I don't think anybody can actually say it's that precise second, but they can say the night was the night and the day is the day. And we've gone from one to the other. And you have to acknowledge the fact at a certain point that that transition has taken place. And that is what we have in China. It's a peculiar regime because it's still the old bureaucracy. You'd say, oh, still the same people in power. There was no revolution. There was no counter-revolution. There was no violent overthrow of one or the other. But even Lenin accepted the idea that the danger of a capitalist restoration could come from within the Bolshevik Party itself. He quoted a professor who was boasting about that possibility. And he could see it. And yet today's so-called Marxist Leninist can't see what Lenin could see as a possibility. And even spite of everything that's happened, in spite of the return to capitalism and the Soviet Union in Eastern Europe and the building of capitalist market economy in China in the hands of the old bureaucracy, of course still in power and still have the leaves of the state in their hands, but they have consciously guided a process towards the development of a market economy. Now I'm going to jump forward because of lack of time. There is one positive aspect in all of this from Marxist point of view. From the point of view of the regression to a market economy, it's a step backwards, in spite of all the propaganda that the market is the best thing since sliced bread. Look around the world today and you'll see that's no longer true. Look at the situation in America. Look at Europe, the growing unemployment, the growing levels of poverty, the wars, the barbarism across the whole planet. This is capitalism, not the capitalism which gives everybody a nice standard of living and peace and tranquility and you can get on and enjoy your lives. We're in a very critical moment, but in them to go back to China, it's a regressive step and it will be abundantly clear in the near future how regressive that step has been for China. But the one positive aspect of it is this, the enormous industrialization of China and the enormous transformation has taken place, something like 20 million people a year moving into the cities from the rural areas. It's an unfinished process of course that because there's still large areas of rural China, which is still pretty backward, a large section of the population is still rural. It's not at the level of the United States or a Europe where the overwhelming majority of the economy now is industrial and the peasantry has been reduced to a tiny or they've actually disappeared. It's now industrial farming in the West. In China, it's gone a long way, but not completed.  Working class, far stronger, numerically speaking than it was in 1949. Very powerful working class, in fact, the biggest industrial proletariat in the world, in fact, something like 200 million workers. Now, that is where the danger for the bureaucracy lies. And it going back to Karl Marx, they have created their own grave diggers, this massive proletariat. And in the period in the last decade, the last 10 years, the early part of this decade, 2011, 2015, 16, we saw a massive increase in strikes. Protests, etc. Year on year on year on year. And we published material on this in over the years. If you look at the statistics, you see this growing level of militancy of the working class. Now that partially was cut across by the bureaucracy, forcing some companies to make concessions, pay back wages. They needed to calm this situation down because it was getting out of control. It also explains another aspect of Chinese society today. There was a time when you had this so-called collective leadership at the top of the Chinese Communist Party. And the idea that every secretary would be, for one term, and then would concede to the collective leadership at the top, and somebody else would come in as the secretary. You notice that Xi Jinping has changed that. He's centralising power in his hands, actually. He's emerging as the strongman at the top, the bonaparte. This is combined with greater and greater repression. For instance, what they're doing to the Uighurs, the level of surveillance and the methods they use there are in effect experimental and practicing methods which they use more widely. They've massively increased control of the internet in an attempt to stifle discussion, stifle debate, stop young people and workers in China from looking for ideas outside China. And in spite of that, the other day we published an article which showed that there's a mass radicalisation taking place amongst the youth, a questioning of the regime. And in spite of all the measures they take, the young people find ways of getting around that and making news circulate and discussion take place. But this increased centralisation of power is an indication that the bureaucracy at the top are conscious of the dangers implanted in the situation. Their legitimacy is guaranteed by the ability to develop the economy to be able to say, look, we've created a powerful China, we've modernised, we've increased the standard of living for a significant layer of the population. All of this is what they need to legitimise their own regime. But a crisis of capitalism means a crisis also of China. China has been slowing down in the recent years. Over the last 10, 15 years we've seen China going from extremely high levels of growth, over 10%, 13%, 14% in some years. Before the coronavirus crisis hit, they were hovering around the 6% mark, which would be amazing for Italy or Britain. For China, because of the ongoing process of urbanisation, they always said they required 7%, 8% growth just to maintain stability, i.e. to create the jobs necessary to absorb the migrant labour coming in. They've reached a point where they're now facing crisis and instability. So what we have is a Chinese economy tightly connected to the world market, a large of the majority of the GDP produced in private hands. The economy functioning according to the laws of the market, the profit motive, although it's distorted clearly by the state, promoting certain companies, forcing the banks to lend and to support certain companies, because they're building Chinese capitalist companies. They have no intention of giving up on that. But we have all the statistics about the growing levels of debt. China had no public debt in 1978. Now the combined total of public debt, private sector debt, corporate debt, local administration debt and household debt, the figure is something like 250% of GDP. The growing levels of bad debt in China, they've sustained it by constantly pumping in more and more money to sustain the economy, probably the biggest Keynesian package we've ever seen in history of public stimulation of the economy. But that cannot continue forever. In 2019, here's the figure, 245% of GDP, the total debt, and it grew by 6% year on year. But the predictions are that it's going up further. And the latest figures show they've gone over 300%. And a large jump in the public debt took place earlier this year as they spent massive amounts of money to buy themselves out of the crisis. Now they, they, everybody's boasting about China now is managed to manage the COVID crisis. They did that through stringent lockdown and tracking mechanisms. Hundreds of billions of dollars in major infrastructure projects were spent to fuel growth. The problem is that's contributing more and more to rising debt and bad loans. And this inevitably is preparing another crisis. And China is going to be a contributing factor to the world crisis capitalism. Back in 2008 2009, China provided an outlet for the capital, Germany, for instance, was growing on the back of exports to China. Now China has slowed down, even though it's recovered from the worst of the crisis earlier this year. But it's a capitalist economy. Now, to go. I don't want to repeat things I've already said. As I said, it's integrated into the world economy. It depends a large degree on the world market. It must export. It's behaving in a way, which is like that of an imperialist country, building the road network, rail network, sea routes, et cetera, to guarantee the export of goods and the import of raw materials. Massively investing in continents like Africa, building roads and railways, but grabbing mineral resources. And it's emerged as a major power on a global scale. But you would think that that would be good for world capitalism. The problem is, when you have massive overproduction on a global scale and massive over capacity, and China has contributed to that, building massive capacity on a global scale. This must mean conflict on a global scale. You know, it's not just the Cold War that produces conflict between Russia and the West as it was then. The First World War erupted at a time where there were no worker states or any kind of planned economy. The war was a war between powerful capitalist economies who had reached the limit of how far they could settle accounts on economic terms, i.e. through competition. They reached a point where the only way they could decide who was going to dominate was by going to war. Now we have a conflict between China, the United States, China, Europe, Russia and Europe, Russia and the United States. And these are conflicts of powers in a period of overall crisis of world capitalism, which explains the conflicts which are taking place. It explains the measures that Trump, for instance, is taking, although even the American ruling class is in conflict with itself on that one. Because measures you take to curb imports from one country inevitably lead to tit-for-tat measures, and you risk pushing the whole world economy into a global trade war, which has already begun. But we're only in the early days of that. So to conclude, what began as a process of stimulating the planned economy with market measures? Because it was done by a privileged bureaucracy and not a revolutionary communist regime with the workers in power, which would have been a very different thing, you have already a privileged bureaucracy. Take me into account what I said about Trotsky's analysis of that and what that would entail in terms of consequences. We ended up with a peculiar capitalist regime created by the bureaucracy of a deformed worker state. Now from Marx's point of view, the fact that capitalism has been created in China, not by a rising bourgeoisie like the French or the British, but by basically another caste of people in society shouldn't be too surprising. Because if you look at the way capitalism was created in Japan or Germany, you see elements of one class. Because of the global developments, and i.e. if a country is to remain powerful and to defend its national interests, where capitalism dominates, you must go down the capitalist road. That's what the feudal aristocracy did in Germany and that's what a section of the Japanese feudal aristocracy did in Japan. In China, a different version of that is the bureaucracy of China, seeing the collapse of Eastern Europe, seeing the crisis of the Soviet Union and the dangers that the alarm bells ringing there for their own system. They were determined that they were going to maintain themselves as a privileged elite and this could be done by adopting capitalist methods. Once you go from a certain quantity, you produce a qualitative change. What we've seen in China is the quantitative changes of a capitalist type over a period of decades has produced a qualitative change in the nature of the economy and even in the nature of the bureaucracy itself and the way it's running society. But the positive aspect of this is now we have, as I said earlier on, the biggest proletariat you ever seen in history, an educated, skilled working class, an educated youth and expectations, obviously. And this system that they've created inevitably will enter into crisis because it now has the contradictions of capitalism grafted on, let's say, to the old bureaucratic system. And the answer to the question, what is China is? It's an ex-deform worker state, an ex-planned economy where the bureaucracy, in order to maintain its privileges, has adopted more and more capitalist methods to the point where the nature of the economy itself has been transformed. It's a peculiar development, but history knows all kinds of mutations, as we know. There's no pure, clean development, you know, feudalism and capitalism, etc., etc. But what we have now is a modern industrialized China, powerful proletariat, massive means of productions at that disposal, but the contradictions of capitalism which have now been brought into that economy. And as the world crisis of capitalism develops, China will be impacted and China in turn will impact on that. And the working class of China will play a key role in the future struggle for socialism. They will, in the process, rediscover their own history and they will seek out an answer to many of the questions posed by the crisis that's developing. And it's not by chance that the ideas of socialism are beginning to become popular amongst a layer of the youth in China. It's inevitable because the huge inequalities and injustices, the censorship, the youth can't tolerate that and they will fight and they will resist and they will react against it. And the crisis will produce class struggle in China on a scale that we've never seen before in history. At the moment there's a repressive regime trying to hold society down. You cannot do that forever, no matter how many cameras you put on the streets, no matter how big the internet police is, no matter how much resources you dedicate to controlling society. If you fail to guarantee growing living standards and progress in society as a whole and you slow down and you enter into crisis, inevitably that is going to unleash the class struggle. And the problem with a dictatorial regime like this is that sometimes you can see it, an eruption of class struggle can come seemingly from nowhere. And yet it's borrowing away under the surface, preparing to come to the surface at some point, and then we will see a huge transformation in society. Now we have to have the correct analysis of what China is because it determines your program. It determines your attitude towards such a system. And it's clear that the program in China has to be nationalize all the multinational corporations operating in China under workers control. Renationalize what's being privatized under democratic workers control and integrate those together with what's left of the state sector, which is still sizable, but introduce genuine democratic workers control over the whole economy and the concept of electing the officials. The right of recall, i.e. the workers who elect their delegates to a higher body have the right to recall them if they don't carry out the policies or if they're dissatisfied with them. And the concept of no bureaucrat, no official must earn more than the average wage of a skilled worker in China. That's something this bureaucracy could never ever accept, but it's a necessity to guarantee the genuine communist future of China and not all we have today. I leave it there because my time has expired. I have a lot more I would like to say, but this is the basic outline of the position that we've developed over the past decades, I would say. And there is more material written material available if companies obviously want to deepen their understanding of this process. And I leave it there.


One of the great companies that does a lot of really cool electronics, consumer electronic devices, has a name that everybody argues over how to pronounce and I've just been instructed by Daryl Mia that it is Scosh. Correct. Scosh Industries. All right. So you've got a really, it's a funny thing to be excited about that you have a cable that really excites me. Now this is an audio podcast but with some video. So describe what you have in your hand here. So this cable right here is magnetic. The whole cable is actually magnetic, it's a USB-C to USB-C and what's great is it creates a really clean environment.  Strike line cobra. Yep, because it coils up on itself and it kind of resembles a cobra or a snake. So he's not talking about the ends are magnetic. He's saying the entire length of the cable. The cable itself is actually magnetic, not the ends. Let me feel what this feels like when we, I'm going to loop it out here and hold it. And it actually feels like a noodle. I mean it's limp, it's not stiff or anything like that but as soon as I try to put it into a coil it sticks to itself, right? Yeah. That is really pleasing because I am anal and I do like my stuff neat. And as an added feature if you get the diameter just right you can use it as a kickstand on your iPhone 12 or newer. As a MagSafe kickstand. I love it. Exactly. You also said you can stick it to a metal filing cabinet. Yep, yep, yep. Very good. If you have a desk that's a metal desk you can just stick it right along your file cabinet and create a nice clean environment. Okay, I've got to get a strike line cobra. There's no question about that. So now we've got, one of the things that my family just swears by is keeping a battery pack for jump starting cars. I've got grandkids. They leave the lights on. Every car needs a power jumper. This is our 2000 PCAMP power jumper here. It'll jump start vehicles up to eight liters and eight cylinders and it has a built-in flashlight and USB-C and USB-A output. If the people listening don't already have one of these you should absolutely get one. I drive an EV but guess what's dead in my Tesla Model 3? My 12-volt battery is dead right now so I actually could use this. I don't know if that would actually work in the Tesla but this is great so you can add the little jumper things, the red and the black clips, you can carry it to save somebody else too, right? Yeah, oh yeah. You can jump start multiple vehicles multiple times on one battery charge. The battery charge lasts about six months of these. I'm going to turn around here and go down a little bit. Very good. Why don't you walk in front of me over here and take over on this side? This is Baselink's 2.0. It's our second version of it. It is a modular charging system. You can configure it however you want. You can add different components so you can take two or three of the Qi wireless stands and put them together. Let me describe what this looks like. So he's got one block that's plugged into power and it's got an AC outlet but then it's got a 67 watt USB-C outlet. Okay, so that's just standalone but then he can connect up to that standard Qi charger with a... This is going to be Qi 2 certified. There you go. So you have a wireless charger here and then down below you can charge your AirPods or your wireless... So you've got a phone and AirPods. You can stick that one on. And then the next thing looks like something I'd stand by bicycle up in. This is a BERT stand and it's really good for phones and tablets and it has three USB-C outlets here. It's a shared 60 watts. So you could do up to 60 watts to one device or you could share the 60 watts but it's got these little bars like a bike stand so you could just slot in a bunch of phones instead of them slapping around on top. I have never seen that design. Then what's the last module? This is the Apple Watch charger. Okay, but you can mix and match these within this unit. This is just a kit that we're showing with one of everything that you can select but you can customize it. You could add on two or three of the BERT stands. You've got six kids you can do these on. Absolutely. Depending on how you want to customize it. You could set up a phone here and a phone here so that you don't grab the wrong phone. I love it. We're really known for magnets so they magnetize together and it just kind of clicks right into place. And through the magnets you run in the power? Well the power, no the power is through the right here. He's pulling it apart here. Oh okay, oh I see, I see that's a power adapter on the edge. The magnet just connects them together and holds them so they don't separate. So again, need an orderly like we intended, right? And what is this whole scheme called? This is Baselinks 2.0 Modular Charging System. Very cool, now I see like what looks like a water bottle but it also looks like a MagSafe charger. It is not a charger, it is a phone holder which is, this is the Magic Mount Flask. And so what's great about this is it has an adjustable phone holder on the front here. It also has a flip cap so you can flip it, sip it. And mount it. So if you're one of these people that carries a water bottle around all the time you can also sit and watch your phone. You can watch your videos, you can watch your phones, you can use it as a tripod.  Oh that's perfect. So you can hydrate your pet and hydrate yourself, you know, like that. And that's called the Magic Mount Flask. And this will actually be an Indiegogo campaign that starts February 13th. Very good. So if people want to find out more about the Strikeline Cowabra and all the other things that you, the Baselink 2.0, where would they find out about all of this? scosh.com. S-C-O-S-C-H-E dot com. Very good. Thank you very much. This is great.


This video is the second part of the Wienberg lecture on Wienberg's algorithm and Katz-Mudy algebras. There should be a link to the first part of the lecture in the description of the video below. So the first part of the lecture we covered Wienberg's original paper describing his algorithm and his calculation of some reflection groups. This lecture I'm going to discuss mainly Conwen Sloan's approach to Wienberg's results and in particular describe Conway's calculation of the Dinken diagram of the 26-dimensional even Lorentzian lattice. I'll start just by having a quick review of some of the things from the first lecture. First of all, we recall the lattice i n comma 1 is the lattice in Lorentzian space r n comma 1. It consists of all vectors m1 up to mn mn plus 1 with mi integers and the norm of this vector, the square of its length is given by m1 squared plus mn squared minus mn plus 1 squared. So this is a minus sign here as in special relativity. We recall that the even lattice is defined similarly except that all the mi are in z or all the mi are in z plus a half and also the sum of the mi is even. Here we take n congruent to 1 modulo 8 and this has the result that all vectors have even norm and what the first part of this lecture was doing was it was calculating the reflection groups of these lattices using Wienberg's algorithm and we saw that we could calculate the Dinken diagram and we found the Dinken diagram was finite for i n1 for n less than equal to 19. So here we have a finite Dinken diagram and the reflection group is a fundamental domain of finite volume but for i n comma 1 for n greater than or equal to 20, Wienberg showed that the Dinken diagram was infinite and similarly it's finite for the even lattices for i 91 and i i 17 1 but it's infinite for i i 25 1 so this is big jump in behavior when n goes from 19 to 20. So what we're going to do is to start off by looking at this case here. So first of all I want to describe the relation between norm zero vectors of this lattice and so we'd take norm zero vectors in this lattice i i 25 1 and nemia lattices so nemia lattices are the lattices in 24 dimensions they're even and they're unimodular that means that volume of a fundamental domain of the lattice and the translation is just one and there are 24 of these found by nemia and the most famous one is the leach lattice in which there's no norm two vectors so the norm two vectors are normally called roots because they correspond to reflections. Now if we take a nemia lattice so let's take a nemia lattice L and add on a little two-dimensional lattice with with inner product matrix that looks like this so it's Lorentzian it's it's got norm zero vectors and positive norm vectors and negative norm vectors then this is actually isomorphic to the 26th dimensional even Lorentzian lattice that's because there is only one such lattice up to isomorphism so this gives you a map from L to norm zero vectors because if we choose coordinates for this we can use coordinates lambda mn with lambda in L and mn integers and this thing has norm lambda squared plus 2 mn then we notice the vector w equals zero zero one has norm zero so w squared is zero so this is a way of going from lattices to orbits of norm zero vectors in this lattice here on the other hand if we've got a norm zero vector w then we can in this 26th dimensional even Lorentzian lattice what we can do is we take your orthogonal complements of w and this is 25 dimensional and it's not quite positive definite because it's got norm zero vector w in it but if we then quotient out by w this then becomes a nemire lattice so this gives us a bijection between me or isomorphism classes of nemire lattices and orbits of primitive norm zero vectors in this lattice here and you can use this in both directions you can either use the classification of nemire lattices to find the primitive norm zero vectors or you can run it backwards and use the classification of primitive norm zero vectors to classify the the nemire lattices so we can give an example of this for example we can ask can we find a norm zero vector in this lattice here corresponding to the leach lattice and this was solved by Conway and Sloane so we've got to find some vector m0 m1 up to m24 m25 and whose orthogonal complement modulo w itself is the leach lattice in particular that there can be no norm two vectors orthogonal to this and you notice if if two of these numbers were equal so if mn was equal to mn plus one then we would have a norm two vector orthogonal to this which just looks like zero one minus one zero zero zero and so on so mn is is not equal mn plus one for um for for any m i should say you may as well arrange these they're increasing order so we've got to find an increasing series of integers um such that um no two of them are equal up to sine and there's an obvious way of doing this which is we just take one two up to 24 now we've got to put something there and now there's another constraint which we must have omega squared equal zero um and that means we've got to find a number here whose square is equal to the sum of the squares of these numbers and by a weird coincidence there is such a number which is 70 because zero squared plus one squared so in all the work plus 24 squared is equal to 70 squared um incidentally 24 is the only non-trivial solution to this equation apart from you can have nought squared plus one squared equals one squared but apart from that um there are no other cases when the sum of the first n squares is a square um and now the vector omega per over omega is the leach lattice um it's not all that easy to show there are no other norm two vectors orthogonal to this i mean we've shown there are no obvious norm two vectors orthogonal to this but there you know how do you know there are some aren't some non-obvious ones um so um next we come on to the result by Conway Parker and Sloan which asks what is the covering radius of the leach lattice leach lattice is normally denoted by a capital lamb to stand for leach um so this means what what you do is you take spheres of radius r around lattice points um so so they just cover the whole of the the real vector space corresponding to the leach lattice so this shouldn't be confused with the packing radius and the packing radius asks what is the biggest sphere you can put around every point such that the spheres don't overlap the covering radius asks what is the smallest sphere you can put around every point so that the the insides of these spheres actually cover space so for example if i've got a um a triangular lattice in two dimensions um the the um the the packing radius you'd be looking for spheres that um don't overlap so so these would give you spheres of the packing radius the covering radius on the other hand you're looking for bigger spheres that cover the lattice so here you would have bigger spheres at this time they that they are going to be overlapping and you see that these slightly bigger spheres are actually covering r2 whereas these spheres are disjoint so this is the covering radius that we're going to be interested in and the points at maximal distance from all lattice points like this point here are going to be called deep holes they're sort of the in some sense the biggest holes you can find in the lattice that this are um as far as possible from lattice points um and um what they found is the covering radius of the leach lattice is the square root of two exactly this is a really amazing fact um their calculation of this is was rather long and difficult um in general calculating the covering radius of a lattice is rather hard you just have to um um um sort of get your hands dirty doing a lot of calculation considering all cases so most of the papers by Conway and Sloan um as i'm talking about this lecture been reproduced in the book sphere packings lattices and groups um in particular i just wanted to show you uh um what their calculation of the covering radius of the leach lattice looks like so here's their paper it sits chapter 23 of the book and if you look through it you see you're getting large numbers of case by case calculations here they're writing down vectors of the leach lattice um here they have a big table showing all the different ways of finding a certain an inside the leach lattice i explain that in a moment and it sort of goes on like this for many many pages there are there are huge numbers of cases to consider and the reason they had to consider so many cases is there are actually the answers rather complicated there are 23 orbits of deep hole so they had to actually find all these different orbits and show there are no other orbits and the amazing thing is these correspond to the 23 nemy lattices that are not the leach lattice 24 nemy lattices one of which is the leach lattice it's the 23 others that appear corresponding to the 23 holes um now how do the nemy lattices correspond to holes well um let's take a typical nemy lattice for instance we can look at the nemy lattice e8 cubed and you look at the dink and diagram of the reflection group of this nemy lattice so we draw three copies of e8 one two three four five six seven so here's one um and here's another and here's another and so so this is the dink and diagram of e8 cubed and now i'm going to make it into an affine dink and diagram and to make it into an affine dink and diagram for e8 you just add on these three points here and now i'm going to make these correspond to a configuration of vectors in the leach lattice as follows so if we've got two points in the dink and diagram not joined i want v to a distance four from w and if they are joined v w then v minus w squared should be six so um this looks absolutely bizarre you're taking an affine dink and diagram and using that to construct a configuration of vectors in the leach lattice and these um and if you take a configuration like this these turn out to be the vertices of a deep hole the vertices of a deep hole just mean the lattice vectors nearest to the deep hole and Coyne Sloan found that they got this exact correspondence between um um the affine dink and diagrams of the knee my lattice and the deep holes of the leach of the leach lattice this this correspondence was first suggested by Richard Parker who somehow noticed um i think he knew a couple of holes in the leach lattice and sort of noticed these seem to correspond to the knee my lattices a1 to the 24 and a2 to the 12 and pointed this out to Conway and Conway and Sloan then got rather excited about this and checked this work for all the other knee my lattices i've no idea how Richard Parker came up with this idea because the knee my lattices were a pretty obscure topic at the time the leach lattice was also pretty obscure so it was somehow putting together these two um very obscure um um objects so um so how can we explain this um well we we can explain this by recalling that knee my lattices correspond to norm zero vectors in in the 26 dimensional Laurentian lattice so here we take this is going to be a knee my lattice um and if we take um one of these norm zero vectors in the 26 dimensional even Laurentian lattice we can look at the roots of the 26 dimensional even Laurentian lattice which are um orthogonal to um w and these will be just the roots of the affine knee my lattice this is because the orthogonal complement of w is just the knee my lattice plus a little zero dimensional lattice and it's not difficult to check the root system of that is it's just the chorus you just get the corresponding affine um group um and Conway discovered that the dink and diagram of um the leach lattice so the dink and diagram of ii 25 comma one is the leach lattice and this doesn't even seem to make sense um because you know the dink and diagram of this lattice is going to be some sort of um graph with various distances between points and then leach lattice is a set of points in Euclidean space so how how on earth do you mean by saying they're the same well they're actually isometric so both of these are metric spaces the dink and diagram consists of simple roots and you can talk about the distance between two simple roots and of course the leach lattice is a subset of 24-dimensional space so it's a metric space um so let's see how they are actually isometric well um what Conway did was he applied Windberg's algorithm to the 26 dimensional even Laurentian lattice and you've got to choose coordinates for this and the obvious coordinates are to choose you know integers m1 up to m25 as i had before but Conway chose another coordinate system he chose to write it as the leach lattice plus 0 1 1 0 so um the vectors are going to be the form lambda mn where lambda is in the leach lattice and m and n are integers um and um this thing has norm lambda squared plus 2 mn and there are two conventions sometimes put a minus one there instead and i again this is something i sometimes get confused about so there may be a few sign errors now for windberg's algorithm you remember you have to start by choosing a sort of special controlling vector which i mean to take to be 0 0 1 so the orthogonal complement of p divided by p is is is just the leach lattice and now we're going to find the simple roots of this 26 dimensional lattice in in order of their inner product with p so we want rp to be 0 1 2 and so on and of course we want r squared equals 0 so if rp is 0 well then you see that um that the vector lambda mn would have the property that lambda squared was 2 and the leach lattice is no vectors of norm 2 so that should be a 2 not a 0 so um so there are no um no simple roots that have inner product 0 with p now let's look at the vectors with rp equals 1 well well here we get the vectors lambda 1 um lambda squared over 2 minus 1 for any vector lambda in the leach lattice so we've got an infinite number of roots of level 1 in accordance with windberg's discovery that the the root system is infinite so this is the first batch of simple roots and the amazing thing that conway noticed is there are no more and um this depends on the fact that lambda has covering radius root 2 and let's see why this is true so um suppose we've got some simple roots so suppose lambda ab is simple now this means that it has um inner product less than or equal to naught with all simple roots of the form mu m so sorry mu 1 um mu squared over 2 minus 1 and if you work out what this condition is you just work out the inner product of these two vectors and you find its equivalent to saying that lambda minus v over a squared is greater than or equal to 2 plus 2 over a squared sorry that should be a mu minus um lambda over a squared is greater than 2 plus 2 over a squared which is greater than 2 and this is not possible because the leach lattice is covering radius root 2 so you can't find any vectors mu that have any vectors lambda over a that have distance greater than root 2 from all all vectors mu um so um now this explains um the correspondence between neomyel lattices and deep holes because the deep hole is just the set of um simple roots in the leach lattice corresponding to the dink and diagram of the neomyel lattice and conway and slone now use this to reinterpret minberg's results on the reflection group of i n comma 1 and um i n comma 1 can almost be embedded in the leach lattice it can't quite be embedded because this has um vectors of odd norm and all vectors in in the 26 dimensional larynxian lattice of even norm so let's take the dink and diagram d n looking something like this and um we can embed this inside the leach lattice considered as the root system so it is as the dink and diagram of the 26 dimensional even larynxian lattice as i said conway showed that the leach lattice is this and then if we look at the orthogonal complement of d n this is going to be the even sub lattice of i 25 minus n comma 1 and um this consists of all vectors m 1 up to um m 25 minus minus n um m something such that the mi of our integral and the sum of the mi is even now this lattice usually has the same reflection group as i 25 minus n 1 usually does that there are a few cases when it doesn't but we will we will see later um so um all we have to do is to work out what is the dink and diagram of this lattice here and conway and slown showed that the dink and diagram of um the orthogonal complement of d n can be obtained as follows take all points um p of lambda so that p union the d n is a spherical dink and diagram now the reason for this is that this is the condition that um the projection of p in the lattice d and perp has norm greater than zero and if you want this projection to be a root it certainly has to have norm greater than zero there's also actually another condition that it has to satisfy that i'll be discussing a little bit later but for the moment um we're not going to worry about it um so let's see what happens as we go through the various possibilities for d n so let's take d n but n greater than or equal to eight um i'm just going to write out the case for n equals eight so we get one two three four five six seven eight so here's the dink and diagram d eight and there are just two ways we can add a point to this to get um a spherical dink and diagram first of all we can add a point here and then we get d n a one so we just add an a one and this turns out to give you a norm two root in um i uh 25 minus n one the other thing we can do is we can add on a new root here this gives us a d n plus one dink and diagram and this gives us a norm one root so this uh um explains where we're getting norm two norm one roots from there from the two ways of extending d n to a bigger spherical dink and diagram so now let's look at d seven where something a little uh um little extra happens so let's take d seven one two three four five six seven and now and there are three things we can do because as well as we can add a point there or we can add a point here just as before and these give us norm two and norm one roots but there's a third thing we can do we can add on a root here and we get an e eight dink and diagram and these give us norm one roots well um this also gives us norm one roots so what's the difference between these sorts of norm one roots well the difference is that these norm one roots are parity vectors so you remember a parity vector is a root r such that the inner product of r with v is congruent to v v mod two for all vectors v um so you remember um this corresponds to the case of um i um 18 one and you remember i commented in the previous lecture that in the case of i 18 one vinberg and kaplan sky i found that they actually got three sorts of roots and the reason for the third sort of root is coming from this e eight dink and diagram and it's it's giving you this this funny extra parity vector um you can't do this for i 17 one because that would be an e nine dink and diagram and the e nine dink and diagram doesn't give you a a spherical or positive definite root system and we can do the same thing for d six so so this is the d seven case d six is kind of similar one two three four five six and again we can add roots there or we can add a root here and this again gives us norm one um this gives us norm two and this time we can add in a root here and this gives us an e seven and this gives us now gives us norm two roots and again that these norm two roots are parity vectors so we we we get a second batch of norm two roots for the case of i 19 one um so um this accounts for vinberg's diagrams um also it makes it rather easier to calculate the dink and diagrams because you just have to search for certain vectors in the leach lattice which turns out to be rather easier than running vinberg's algorithm um so what happens for the case d five so let's take a look so so here just as before um the possibilities are we could add a point then get a norm two root or we could add a vector here and get a norm one root or we could add a vector i think i'm using purple for these we'd add a vector here and this time we get an e six root well except it's not a root because this gives us norm three vectors and these are not roots um and this accounts for why um why if you take the lattice i um 21 the reflection the dink and diagram is actually infinite i mean roughly speaking this was a root then then then we'd find a finite fundamental domain for the reflection root but because this gives us something isn't a root we we only get an infinite fundamental domain the question is why is this not a root um so if you take the vector here we get a perfectly good positive norm vector in the lattice i 21 and it turns out there's an extra condition we need to satisfy for this to be a root um dink and diagrams have something called an opposition involution um and what this does is um it's an auto it you if you if you take the root system contained in um euclidean space rn then um rn has an automorphism minus one you can just multiply all vectors by minus one and this can be written as a product of reflections times an automorphism of the dink and diagram um and this automorphism turns out to be of order one or two and we can see what it is in various cases if you take a n then the opposition involution just does the obvious thing so it's non trivial unless n equals one if you look at d n it's a bit subtle it depends on whether n is odd or even so if for d n n even it does nothing for d n n odd it flips these two vectors here and for e six it flips um the two endpoints and for e seven it does nothing it can't do anything because e seven has a trivial automorphism group and for e eight it does nothing and now the problem is that is the fact that the this opposition involution for e six um has norm is is non trivial and it turns out that um for in order to get a root what you would need is if you take the d five wave that we were looking at and extend it to an e six we notice the opposition involution of this e six acts like this so the opposition involution of e six does not preserve um the d five sub diagram and this is what is causing the problem if you look at all the other cases we had we see that the opposition involution of the bigger dink and diagram you know preserves the smaller dink and diagram i mean you know that if we take d n inside d n plus one and the opposition involution of d n plus one is always preserving d n so this is this is what is causing um the dink and diagram of the Lorentzian lattice to become infinite and that we can now see why i twenty one has an infinite dink and diagram and the reason is that twenty is equal to the dimension of i i twenty five one minus the dimension of e six um now vinberg had a different reason for why the dink and diagram becomes infinite so vinberg pointed out there were two lattices a eleven d seven and e six cubed as i mentioned in the previous video such that the rank of the root system is only 18 whereas the lattice is dimension 19 so we seem to have two completely different explanations for why things break down one says that e six has this funny extra um opposition involution whereas vinberg's explanation says you get these two extra lattices and i mentioned 19 with not enough roots well it turns out these two explanations are really more or less the same the question is how do you get these two lattices well there are two nemy lattices a eleven d seven e six and e six four or at least that's their root systems and you can get these two lattices by taking the orthogonal complement of a d five which is contained in some e six which is contained in inside some nemy lattice so we're taking the nemy lattices that contain an e six component taking the d five inside them and then taking the orthogonal complement of that and that gives you the even sub lattice of these unimodular lattices so in both cases the problem is caused by this d five being embedded in e six in a funny way such that the orthogonal complement of the d five is not a root um so that sort of explains exactly why vinberg's algorithm breaks down at um at a particular dimension well we can go further um so um let's look at the case of um d four so this is going to correspond to the lattice i 21 comma one and dink improved the reflection group again has infinite dink and diagram and let's take a look at the possibilities well first of all we can have normed two roots and then we can add in a root here so this has norm one um and this is giving us a d five dink and diagram and then there are two ways of extending this d four to an e five we can either add a point here or we can add a point here so what's e five well well e five looks like this and you see it's actually the same as d five so um these two ways of adding extra roots are really the same as this way of adding a root um and in fact if you count how many ways of doing this there are there are 56 roots like this and 56 like this and 56 like this and for that matter there are 42 times that um and if you look at the opposition involution of d five you see it's actually fixing this d four i mean the opposition involution of this d five flips these two vertices but that's okay that maps d four to itself so this seems to be prediction that we should get a finite dink and diagram um with um 56 plus 56 plus 42 um um points comes out to 210 so um um we we we we see that first sight of a contradiction so so vinberg's algorithm is saying that the dink and diagram of this lattice is infinite but um here we see in an argument saying that the dink and diagram with a complement of this d four is finite well um the the complements of this d four is the even sub lattice of i 21 1 and the even sub lattice usually have the same automorphism group but in this particular case these things have different automorphism groups in fact the automorphism group of this lattice is three times as large as the automorphism group of that lattice in some sense well i mean they're both infinite groups one one has index three in the other um and um in order to understand what's going on it's easier to look inside a small dimension so this is in 22 dimensions which is a bit hard to figure out but we can see the same thing going on with d four so d four is contained in the lattice i four so i four is just all vectors m one not m four with m i in z and d four actually can is contained in three different copies of i four um and we can see this as follows so d four is the set of all vectors like this with the sum of the mi even and we can um extend d four in three different ways so we can either add the points m one up to m four with m i odd and this just gives us the lattice i four or we can add the points um m one up to m four with mi in z plus half and the sum even or we can add the points m one up to m four with mi in z plus half and the sum is odd um and if we do this we notice the vector a half a half a half a half actually has norm one and in fact we can find enough other vectors of norm one that we're actually getting that this lattice is actually isomorphic to i four so we've actually got three different copies of i four um you notice this is something funny that happens whenever the dimension is congruent to four modulo eight except in general we won't get norm one vectors but in dimension four we happen to get norm vectors um and the same sort of thing is happening for um if we take i 21 one and take its even sub lattice is contained in three copies of i 21 one um so um what's going on is that um um the even sub lattice and this lattice here have different automorphism groups because this has many more roots of norm four um so these aren't roots of norm four of this unimodular lattice but they are roots of norm four if it's even sub lattice so this is how how the um how the how the um dink and diagram of the even sub lattice of this lattice can actually be finite even though the dink and diagram of this lattice is infinite we now have a sort of summary table as follows so let's do a summary so let's take the lattices d eight perp d seven perp d six perp d five perp d four perp d three perp d two perp so um you notice d three is just the same as a three and d two is the same as a one squared so i'm going to take these are all going to be sub lattices of rent in lattice um in 26 dimensions so here we're getting the lattice i 17 one rather we're getting the even sub lattices of these lattices here octa i um 23 one um and um we can get three different sorts of roots we can get the roots corresponding to a one d n which are all norm two roots or we can get the roots corresponding to d n plus one which are all norm one or we can get the roots corresponding to e something e n so so um um here that there aren't any um so this should be e n plus one because e nine isn't a spherical dink and diagram but here the roots of norm one um two three four five and six and there's another special property that these roots have which that they are parity vectors so um let's see what happens uh well here the extra roots of norm one or two so they're they're just roots and this gives bin bergen kaplan skyer's example when the um when when when when the um reflection group is still infinite well here um a norm three vector is not a root so we get the dividing line between the cases when this lattice is as finite dink and diagram the cases when it's an infinite dink and diagram but this case is a bit funny because these vectors are still roots of the even sub lattice the point is they have norm four but they're also parity vectors and um this means if you take the roots and take the inner product with v um this is always even and this means that um true r v over r r is now always an integer because this is four and can and we've got a factor of two here and a factor of two here so so in this particular case we still get roots and in these cases it's still we still get knock um they're still not roots so there's this funny um extra thing going on in in in in 22 dimensions where we unexpectedly get some roots um well we can also um make a closer look at the um case of d five so the case of d five um perp was the even sublattice of i 21 um and this still has um infinite dink and diagram um but we can ask can we describe the dink and diagram even though it's infinite and the answer is we can sort of and the point is the fundamental domain of the of the reflection group is tree like so what do i mean by this well let's draw um hybolic space as the unit disk and here i i'm really talking about hybolic space in 20 dimensions but i'm drawing it as if it were two-dimensional and if we draw the fundamental domain then you can think of it as being something like this what's happening is um we get a lot of roots um and these are kind of dividing up the um fundamental domain and the fundamental domain is is um looks a bit like this we get a sort of tree like structure and the reason we get a tree like structure is um we can add in the hypo planes of norms of the norm three vectors and these are not roots but they still divide up the fundamental domain into pieces of finite volume they were roots then then we would get a finite volume fundamental domain and what what we end up with is we get a sort of tree here um we can check that we're getting a sort of nice structure looking like this except there aren't three things coming from each node there are quite a lot of things coming from each node but um and um if you've got a tree you can quite often describe its automorphism group as an amalgamated product um uh seh has a nice book on this called trees and there are a few pictures of this so here's a picture of a tree and here's a picture of the upper half plane model of um hypovolic space where he's drawn a sort of tree here and this tree is acted on by sl2z acting on the upper half plane in the usual way and seh points out the fact that sl2z is acting on this tree means you can write sl2z as an amalgamated product of two groups in fact it's an amalgamated product of a cyclic group of order four and six amalgamated over a cyclic group of order two and since this is a bit hard to read and the same thing happens here but because the fundamental domain has a sort of tree-like structure we can actually describe the automorphism group of the fundamental domain as an amalgamated product so the automorphism group of the dinking diagram can be written as follows it's g e6 amalgamated over g d5 containing e6 with g d5 so what does this mean well this means that the sub the subgroup of the group of automorphisms of each lattice that preserves this e6 dinking diagram and this means the sub subgroup that preserves both the d5 and the e6 contained in it and similarly for this one and we can work out the orders of these groups this is a little group of order 72 this group has ordered 36 and this group has ordered 1440 so this gives us a description of the automorphism group of the dinking diagram it's just an amalgamated product of these three groups here and you notice that these groups have indexed two and this is kind of because this reflection is sort of trying to be a this sorry this vector it's trying to be a root giving you a reflection and if it was a reflection then this group would be this group times a cyclic group of order two and it would have indexed two and this still has indexed two but it's not quite a product um it's I mean it turns out what you're getting is not a reflection but a kind of reflection twiddled by another non-trivial automorphism so um in some sense if if if we if if the vector here was a root then we would sort of delete this bit of the dinking diagram and the automorphism group of the dinking diagram would just be this but but we have this extra bit because the vectors we're getting here are not roots um so um um what what um should meant them that um in in the case of the even sub lattice of 21 comma 1 which had a finite dinking diagram um Dolgachev and Kondo found that the this group would say the even vectors of this actually turns up naturally and algebraic geometry um in fact there's a super singular A3 with a picard group of it so sorry super singular K3 and characteristic two with a picard group that corresponds to this lattice here and um as a result you get the this um this dinking diagram with finite reflection group turning up naturally and algebraic geometry um um you can ask what happens if you go to um higher dimensions so we've done the case of ii 25 1 well then you can ask what happens if you go to the next case which is ii 33 1 and here you seem to get a horrendous mess there seems to be no analog of the leach lattice there's nothing with covering radius root 2 the dinking diagram is absolutely huge um you can see roughly how complicated it is by by looking at the number of orbits of norm zero vectors in the following case so for ii 9 1 there's one orbit of norm primitive norm zero vector for ii 17 1 there are two orbits here we get 24 orbits found by Niemeyer um this one we get more than a billion orbits this is an estimate due to oliver king who used a refined form of the minkowski zeagle mass formula to find a um a very large lower bound for the number of orbits of norm zero vectors here so these correspond to 32-dimensional even unimodern lattices so they're at least a billion of those and classifying them probably would be within reach of a big computer but i mean the the main obstruction is rather hard to think of any point in classifying them i mean there are huge numbers of lattices they all look very similar and classifying them it's a bit like trying to classify all grains of sand on a beach i mean the grains of sand are all different but who cares um so what seems to be happening for vinberg's algorithm is that for small dimension less than or equal to about 25 things are kind of tame and we can quite often classify them for dimension greater than about 25 the reflection groups all seem to become rather wild that they're very large and rather difficult to describe and they all have infinite dinkin diagrams and so on and the thing that seems to be causing this difference is the leech lattice so the leech lattice is sort of on the boundary the leech lattice is the dink and diagonal of the 26-dimensional Lorentzian lattice and what seems to be going on is that um you can use the 26 dimensional lattice to control things in dimension less than 26 and and many of the reflection groups are reasonably nice at least if they're small discriminant but once you go beyond this things just go crazy um people have of course um um done a lot of work on classifying lattices with finite reflection groups in dimension up to 25 um the classification is a bit complicated because there are all sorts of different things you can look at so for instance the fundamental domain can be compact or it can be finite volume but not compact or it can be infinite volume and the infinite volume case is still interesting for example um Conway's example of the 26-dimensional even Lorentzian lattice is perhaps the most interesting of all um reflection groups and that has a fundamental domain of infinite volume so you you can't even restrict a finite volume you can also ask um should the um should you take a lattice or or you can ask for um could be a lattice over z or over um some sort of number field um or it could be um a non arithmetic um group and lattices over z are what we've been discussing um there are some quite striking examples over number fields in particular the biggest known examples um over um um the biggest known examples with compact fundamental domain were found by Boogayenko and he was working over a number field um generated by a square root of five um his examples are still very mysterious um they're in about eight dimensions and I don't know how to relate them to um this group here although I suspect there might be some relation um also um we can talk about whether um the reflection groups are maximal or non-maximal and I just summarise a few of um the examples um um for more about this there's a very nice survey article um written by um Benelopetsky and I'll try and put a link to that in the video description below um so the most striking result of all is by Wienberg which says there are no finite volume in large dimensions and I find this result really amazing because um for Euclidean and spherical groups there are finite volume or compact examples in all dimensions and if you look at hyperbolic groups in small dimensions there seem to be vastly more hyperbolic reflection groups than than for Euclidean and spherical reflection groups and and you'd expect this to get even worse as dimension increases and it doesn't first but when you start hitting 20 dimensions the number suddenly plummets down to almost zero um so that the largest known example is in um hyperbolic space of dimension 21 um so Esselman showed that if you look at reflection groups of lattices then this is indeed the biggest possible dimension but whether you can find examples over other number fields or non-arithmetic example feels seems to be a bit unclear um um nickel in has classified many of the examples in dimensions in in small dimensions and there are really hundreds of them you there are just too many to do by hand and you you need to get a computer running Vinberg's algorithm in order to find them um um and um there are some finiteness examples for instance there are only a finite number of maximal reflection groups in each dimension so altogether there are only a and as Vinberg bounded the possible dimensions this shows that there are a finite number of maximal hyperbolic reflection groups Daniel Allcott showed that um up to dimension 19 there are actually an infinite number of non-maximal reflection groups so so if you want to find out this you you need to restrict the maximal ones um well um what else can you do with these reflection groups well there's one very obvious thing you can do with them these reflection groups are all giving you dink and diagrams now for the finite and euclidean dink and diagrams um these are all dink and diagrams of lee algebras you get the finite dimensional semi-simple lee algebras and you get the affine cat's moody algebras so an obvious question is do you get interesting lee algebras corresponding to the hyperbolic reflection groups um so this will be the subject of the third part of this talk which will appear in the next video