Every four years, at the International Congress of Mathematicians, the Fields Medal is awarded to two, three, or four young mathematicians. To be eligible, the awardees must be under forty years of age. For the chosen few, who came from England, France, Korea and Ukraine, the award, often described as the Nobel Prize of Mathematics, is the crowning achievement of their careers [TM235 or search for “thatsmaths” at irishtimes.com].
Clockwise from top left: Maryna Viazovska, James Maynard, June Huh and Hugo Duminil-Copin. Image Credits: Mattero Fieni, Ryan Cowan, Lance Murphy.
The congress, which ran from 6th to 14th July, was originally to take place in St Petersburg. When events made that impossible, the action shifted to Helsinki and the conference presentations were moved online. The International Mathematical Union generously allowed participants to register at no cost.
) has been demonstrated. There is essentially no doubt about its validity, but no proof has been found.
are dense in the real numbers 
. The cardinality of rational numbers in the interval 
is 
. We cannot list them in ascending order, because there is no least rational number greater than 
.
![\displaystyle \mathbb{Z}[i] \equiv \{ m + i n : m, n \in \mathbb{Z} \} \,.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cmathbb%7BZ%7D%5Bi%5D+%5Cequiv+%5C%7B+m+%2B+i+n+%3A+m%2C+n+%5Cin+%5Cmathbb%7BZ%7D+%5C%7D+%5C%2C.+&bg=ffffff&fg=000000&s=0&c=20201002)
![{\mathbb{Z}[i]}](https://s0.wp.com/latex.php?latex=%7B%5Cmathbb%7BZ%7D%5Bi%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
forms a two-dimensional lattice in the complex plane. For any element ![{g \in \mathbb{Z}[i]}](https://s0.wp.com/latex.php?latex=%7Bg+%5Cin+%5Cmathbb%7BZ%7D%5Bi%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
we consider the four numbers 
as associates. The associates of 
are known as units: 
.
is a way of writing 
. This was known to Galileo. However, with the usual ordering,
is a proper subset of 
, it must be smaller than 
-th entry in row 
(read 
can be found using derangement numbers. Recall from last week’s post that we call any permutation of the elements of a set an arrangement. A derangement is an arrangement for which every element is moved from its original position.
. By a simple change of variable, we can have them lying on the real axis. But the eigenvalues of any hermitian matrix are real. This led to the Hilbert-Polya Conjecture:
are the
, and ratios of these, the positive rational numbers 
. It came as a great shock that the diagonal of a unit square could not be expressed as a rational number.
, which include rationals, irrationals like 
and transcendental numbers like 
.
for s=1.1 (red), s=1.01 (blue) and s=1.001 (black).
. The number of atoms in the universe is estimated to be about 
. When we consider permutations of large sets, even more breadth-taking numbers emerge.
The rational numbers are countable: they can be put into one-to-one correspondence with the natural numbers. In previous articles we showed how the rationals can be presented as a list that includes each rational precisely once. One approach leads to the 
can be expanded as a continued fraction:![\displaystyle x = a_0 + \cfrac{1}{ a_1 + \cfrac{1}{ a_2 + \cfrac{1}{ a_3 + \dotsb + \cfrac{1}{a_n} } }} = [ a_0 ; a_1 , a_2 , a_3 , \dots , a_n ]](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+x+%3D+a_0+%2B+%5Ccfrac%7B1%7D%7B+a_1+%2B+%5Ccfrac%7B1%7D%7B+a_2+%2B+%5Ccfrac%7B1%7D%7B+a_3+%2B+%5Cdotsb+%2B+%5Ccfrac%7B1%7D%7Ba_n%7D+%7D+%7D%7D+%3D+%5B+a_0+%3B+a_1+%2C+a_2+%2C+a_3+%2C+%5Cdots+%2C+a_n+%5D+&bg=ffffff&fg=000000&s=0&c=20201002)
are integers, all positive except perhaps 
. If 
we add it to 
; then the expansion is unique.
Every circle has the property that the distance around it is just over three times the distance across. This has been known since the earliest times [see 
, but equality is violated for this enormous value and, intermittently, for larger values of 
is chosen randomly with probability 
of being either plus one or minus one. It follows from the Kolmogorov three-series theorem that the series is “almost surely” convergent.
ThatsMaths 