## Archive for July, 2022

### Poincare’s Square and Unbounded Gomoku

Poincare’s hyperbolic disk model.

Henri Poincar’e was masterful in presenting scientific concepts and ideas in an accessible way. To explain that the Universe might be bounded and yet infinite, he imagined that the temperature of space decreased from the centre to the periphery in such a way that everything contracted with the distance from the centre. As travellers moved outward from the centre, everything got smaller in such a way that it would take an infinite time to reach the boundary.

### Fields Medals presented at IMC 2022

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.

### Goldbach’s Conjecture and Goldbach’s Variation

Goldbach’s Conjecture is one of the great unresolved problems of number theory. It simply states that every even natural number greater than two is the sum of two prime numbers. It is easily confirmed for even numbers of small magnitude.

The conjecture first appeared in a letter dated 1742 from German mathematician Christian Goldbach to Leonhard Euler. The truth of the conjecture for all even numbers up to four million million million (${4\times 10^{18}}$) has been demonstrated. There is essentially no doubt about its validity, but no proof has been found.

### The Size of Sets and the Length of Sets

Schematic diagram of ${\omega^2}$. Each line corresponds to an ordinal ${\omega\cdot m + n}$ where ${m}$ and ${n}$ are natural numbers [image Wikimedia Commons].

Cardinals and Ordinals

The cardinal number of a set is an indicator of the size of the set. It depends only on the elements of the set. Sets with the same cardinal number — or cardinality — are said to be equinumerate or (with unfortunate terminology) to be the same size. For finite sets there are no problems. Two sets, each having the same number ${n}$ of elements, both have cardinality ${n}$. But an infinite set has the definitive property that it can be put in one-to-one correspondence with a proper subset of itself.