What is the importance of ??? = ??? + 1? How to explain it on the fingers?

The author of the answer to Quora is Michael Griffin, postdoc on mathematics

Senia Scheidwasser gave very good, simple answer to this question, I recommend reading this brief version. But there is a much more surprising story of the Monstrous Moonshine hypothesis mixed with McKay's equation: from Jack Daniel's whiskey to black holes and quantum gravity.

In this story, symmetries and mathematical "groups" are often mentioned, so let's start with what is meant by the group in mathematics. A group can be represented as a way to reorder a set of objects, preserving a certain structure. The operations in the group must follow certain rules, for example, there should always be the possibility to cancel the operation, and if you perform one operation and then another, you get the third operation * in the group * .

The image source is

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If you like to represent figures, then a simple example of the group is the symmetry of the square. It can be rotated in three ways: 90 ° to the right (clockwise), 180 ° and 90 ° to the left (counterclockwise); there are four symmetries: along the vertical, horizontal and two diagonal axes); and there is one symmetry of the identity , when nothing changes. If you rotate the square 90 ° to the right, and then reflect on the vertical axis, you get another symmetry. In particular, the result will be the same as if you immediately reflected on the diagonal axis from the top left to the lower right corner. This is a kind of multiplication table for the elements of the group. In fact, we can write a multiplication table for a better understanding of the structure of the group. I did it right here. The symbol "i" in the table is the symmetry of identity, when nothing changes. "R" and "L" - rotate 90 ° to the right and to the left, respectively. "F" is a rotation of 180 °, and each line is a reflection along the axis in the direction of this line.

Some groups can be divided into smaller parts. For example, if you have two squares, there may be two copies of the same symmetry operations, each of which acts on one square independently of the other. Simple groups can not be divided into smaller independent groups, so they are like prime numbers in group theory. But finite simple groups are slightly more difficult to classify than prime numbers. During the second half of the last century, considerable progress was made in attempts to completely classify all finite simple groups. Most simple groups fit into neatly organized families. For example, one family contains all the symmetries of regular N-gons (such as an equilateral triangle, a square, a regular pentagon, etc.). But not all groups fit into some kind of normal family. There are exactly 26 "sporadic" groups that are orphans. They are usually a little more difficult to determine, but many of them can be constructed from lattice symmetries in several dimensions. The largest of the simple sporadic groups is * Monster * .

In 197? Fisher and Griss first (independently) found evidence that a very large simple group can exist if it satisfies certain properties. But only after a decade it was possible to prove that these properties are stable, and the group does exist. Griss called this elusive hypothetical group the Friendly Giant, the initials of F. G. for Fisher-Griss. But Conway, the more famous mathematician, called her the Monster - and this name was fixed. By the way, this Conway plays an important role in our history, but most likely you've heard of it before. This is the same Conway, who invented the game "Life" and proved a theorem on free will. If you do not remember, go read it!

In 197? two mathematicians, Ogg and Tits met at a conference in Paris. Tits calculated that if the Monster exists, its size will be:

** 2 ^ 46 · 3 ^ 20 · 5 ^ 9 · 7 ^ 6 · 11 ^ 2 · 13 ^ 3 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71 ≈ 8 × 10 ^ 53 **

This is a very large number. Very, very, very big. This is the approximate number of atoms in Saturn and Jupiter combined. But Ogg's attention was not drawn to size, but to prime factorization.

Ogg at that time was studying pieces called modular curves. If N is a positive integer, then there is a surface, we call it X (N), which captures some important arithmetic information about the number N (if you remember complex numbers from the school, then such a surface can be obtained by "rolling" or "folding" the complex plane by means of a series of symmetries, depending on the number N). Ogg asked about this question: if N is a prime number, then in which case will this surface (or modular curve) look like a ball, and not a donut with one or more handles (i.e., "holes" in a donut)? He found that only if N belongs to the set

**{? ? ? ? 1? 1? 1? 1? 2? 2? 3? 4? 4? 5? 71}**

These are the same primes that are used in calculating Tits for the size of a Monster! But between these two calculations there is absolutely no obvious connection. Ogg was so stunned by this apparent coincidence that he offered Jack Daniel's whiskey bottle to anyone who could explain it.

For obvious reasons, compiling a multiplication table will not help to study the Monster. If we write down the table of multiplication by hydrogen atoms, it does not fit in our galaxy. Instead, mathematicians managed to compose

*Monster's symbol table*. Yes, it sounds like a guide to the game Dungeons & Dragons, and maybe this is not the worst way to present the table. This is a kind of Necronomicon for the Monster; a table of numbers 194 × 19? giving mathematicians some deep insight into the astronomically huge Monster. The first column lists the "dimensions of irreducible representations" of the Monster. These are bizarre words, but the essence of our story is that the first two values in the first column are the numbers

**1**and

**???r3r3227. . This is where the McKay equation appears.**

Mackay famously pointed out to Conway that

[b] 196884 = 1 + 196883

Mackay famously pointed out to Conway that

[b] 196884 = 1 + 196883

Conway found McKay's hypothesis so ridiculous that he called it a fantasy (Moonshine). In this equation

**196884**is The image source is

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In addition, the J-function is the most basic modular function for the simplest modular curve X (1). This is the most "basic" function in the sense that any other modular function for X (1) can be written as a polynomial or the ratio of polynomials in the J-function. For some other modular curves, such as X (2), another basic modular function. Let's call it J_2. In fact, X (N) has a basic modular function J_N of this kind precisely when the form X (N) is a ball (without "handles" or "holes"), exactly as studied by Ogg.

Another mathematician Thompson realized that McKay's observation could be developed. He noted that the following several coefficients of the original J-function can also be written as sums of values from the first column of the Monster symbol table. Moreover, you can write several coefficients of other functions J_N in the form of sums of other values from the table. At that time Thompson was still working with an incomplete table of symbols. Only in 197? Fisher, Livingstone and Thorne completed the calculation of the symbol table, and later in the same year Conway and Norton turned Thompson's observations into an exact hypothesis. They argued that there is a way to write any coefficient of a J-function as a sum of the dimensions of irreducible Monster representations (ie, records from the first column of the Monster symbol table). Moreover, it can be done in such a way that if we swap entries from the first column with entries from another column of the symbol table, we get the coefficients of one of the other functions J_N! For example, here are the first three coefficients of the original J-function (on the left side of the equations):

**196884 = 1 + 19688?**

21493760 = 1 + 196883 + 2129687? and

864299970 = 2 x 1 + 2 x 196883 + 21296876 + 84260932?

21493760 = 1 + 196883 + 2129687? and

864299970 = 2 x 1 + 2 x 196883 + 21296876 + 84260932?

where

**1**,

**196883**,

**21296876**, and

**842609326**- the first four values in the first column of the Monster symbol table. And here are the first three coefficients of the function J_2 (again, on the left side of the equations):

**4372 = 1 + 4371**

96256 = 1 + 4371 + 91884 and

1240002 = 2 × 1 + 2 × 4371 + 91884 + 113937?

96256 = 1 + 4371 + 91884 and

1240002 = 2 × 1 + 2 × 4371 + 91884 + 113937?

where

**1**,

**4371**,

**91884**and

**1139374**- the first four values in

*second*column of the Monster's symbol table. And so on: each column of the symbol table gives the coefficients of the basic modular function for some modular curves. Conway and Norton called their hypothesis

*Monstrous Fantasy*(Monstrous Moonshine).

About a year ago I had a chance to talk with Conway about how this hypothesis came about. He said that he was viewing the fresh values in the Monster's symbol table, which required so much effort to calculate, and then went down to the mathematical library and opened a book written decades earlier with tables of coefficients of modular functions. And he described this feeling of deep horror when from the pages of the old book he was looked at by the same numbers or their obvious combinations.

In 198? Griss finally showed how to build a Monster. For the first time, mathematicians were able to get rid of the "if the Monster exists" clause. Ten years later, Borchers, a former student of Conway, proved Monstruous Fantasy, using the theory of "vertex operator algebras", which he created specifically for this purpose. This theory was created on the basis of the old physical theory of the 1960s. Borchers received the Fields Medal in 1998 in many ways for this proof. This is a kind of Nobel Prize in mathematics, except that for some inexplicable reason to get it you need to be younger than 40 years old. As I heard, Ogga satisfied Borcherds' answer to his question, but Borchers does not drink, so the bottle of "Jack Daniels" remains unclaimed. On the other hand, although Conway is very pleased with the work of Borchers, but he still sees in it only a check, but not an explanation. Yes, now we know that the coefficients of the modular functions are the sums of the values of the Monster symbols, but Conway believes that we still do not have a clear picture, HOW CAN IT BE EXPECTED?

This is not the end of the story. In 200? Witten worked to resolve conflicts in quantum gravity. Quantum mechanics and general relativity are not very compatible. Witten worked on a simplified question, throwing out of the theory of relativity everything except gravity. He found reason to believe that the VOA of Monstrous Fantasy is the key to the theory of gravity in this simplified design. In this theory, the J-function is transformed into a partition function that counts different energy states. Here there are various Monster symbols that correspond to the states of a black hole. Witten asked whether some of these states of the black hole are more common than others? Going back to Monstrous Fantasy, it basically comes down to the question, how many

**units**we expect to see when we break this coefficient of the J-function? Or how many times will

**???r3r3227. ? Are the [b] units**rare? Or there is basically

**units**with a few interesting meanings scattered here and there? I think many people have such a question when they first encounter Fantasy. If only everything had been reduced to

**units**, then this would make the theory much less interesting. But this does not need to worry. Despite the fact that from the very beginning we meet

**units**, they become very rare when we move to larger coefficients, and larger characters start to take over. After the 200th coefficient, the symbols generally appear in proportion to the size of their measurement. The ratio

**1**with all other symbols about 1 to 5.8 × 10 ^ 27. This is approximately the ratio of the weight of the clip and the mass of the Earth. The second largest symbol is found in

**196883**times more often, the third - in

**21296876**times more often, etc. Returning to the Witten configuration, this means that the larger energy states for the black hole are more common, while the trivial vacuum state (3r3r???r3r3227) practically does not exist.

There are still many studies on Fantasy. We (mathematicians) observed (and in some cases proved) the phenomenon of Fantasy for other groups outside the Monster. Specialists in string theory continue to peek at our work, hoping to turn these new Fantasies into new theories of gravity.

For more technically savvy readers, who are interested in details, I recommend the book "Fantasy outside the Monster" Terry Gannon or this scientific article (in the public domain).

It may be interesting

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Author**7-09-2018, 22:15**

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