Looking back: 2006

Oxford, 2006

In 2006, I had the privilege of attending two conferences in England (the 11^th International Command & Control Research & Technology Symposium in Cambridge and the Complex Adaptive Systems and Interacting Agents Workshop in Oxford).

This was the year that NASA launched the New Horizons spaceprobe towards Pluto (it was to arrive in 2015). Ironically, later in 2006, the International Astronomical Union somewhat controversially downgraded the status of Pluto to that of a “dwarf planet.”

Grigori Perelman’s proof of the Poincaré conjecture was declared the “Breakthrough of the Year” by the journal Science. A variety of books, such as this one, have tried to explain what the conjecture (now theorem) is about. So far, this is the only one of the seven Millennium Prize Problems to be solved.

Perelman was offered, but refused, the prestigious Fields Medal (in interviews, he raised some ethical concerns regarding the mathematical community).

Books of 2006 included the intriguing World War Z (later made into a mediocre film). Movies included Pan’s Labyrinth, Children of Men, Apocalypto, Black Book, Pirates of the Caribbean II, Cars, and The Nativity Story.

And in music, Carrie Underwood took the world by storm, singing about Jesus and about smashing up motor vehicles with baseball bats.

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Mathematical Meme Madness Month #2

The next one:

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The Poincaré Conjecture: a book review

The Poincaré Conjecture by Donal O’Shea (2007)

I recently read The Poincaré Conjecture: In Search of the Shape of the Universe by Donal O’Shea. This 2007 book was one of the flood that greeted the proof, by Grigori Perelman, of the Poincaré conjecture. It is probably the best of them.

The Poincaré conjecture relates to the 3-sphere, the 3-dimensional object sitting in 4-dimensional space defined by x² + y² + z² + t² = R². As the diagram below illustrates, this can be seen as the union of an infinite set of hollow balloons, strung along the t axis, each defined by x² + y² + z² = R² − t²

The 3-sphere x² + y² + z² + t² = R² (horizontal t scale exaggerated 10 times, and only showing 21 of the infinitely many hollow balloons).

These hollow balloons can be nested inside each other to make two solid balls (one sequence from the left to the centre, and one sequence from the right to the centre). A 3-sphere is therefore equivalent to a pair of solid balls, where the two surfaces of the balls are imagined to coincide. That is, any path “leaving” one ball immediately “enters” the other at the corresponding point. As a consequence, the 3-sphere has no boundary.

As an example, consider a path from the centre of the left sphere, moving upwards. On reaching the north pole of the left sphere, the path immediately shifts to the north pole of the right sphere, but moving downwards. Then, on reaching the south pole of the right sphere, the path immediately shifts to the south pole of the left sphere, now moving upwards again. In this way, the path eventually reaches its starting point.

The 3-sphere x² + y² + z² + t² = R² as a pair of solid balls, where the two surfaces of the balls are imagined to coincide (thus leaving the 3-sphere without a boundary).

The Poincaré conjecture – now proved by Perelman – concerns potential universes that are connected, and in small regions look like Euclidean 3-space, but are also finite in size, and lacking a boundary. Henri Poincaré conjectured that the only such potential universe with the additional property that each loop in the space could be continuously tightened to a point was the 3-sphere.

Both the road to that conjecture, and the road from the conjecture to the proof, make for fascinating reading. So, if you haven’t already read this book, look for a copy.

As a side issue, O’Shea also touches on the interesting suggestion that the universe as described in Dante’s Paradiso (see image above) is a 3-sphere. It is still a possibility that our universe has that shape too.

The Poincaré Conjecture by Donal O’Shea: 4 stars