10 largest mathematical achievements of recent years

Aperiodic mosaic of Sokolar-Taylor

Recently, I have been working on my book "Mathematics 1001", making additions for the next edition, which will be published abroad. Therefore, I have been tracking mathematical advances since about 2009. And I decided to present you the top ten most important events on this topic since that time, in the order of a subjective increase in importance.

10. Shinichi Mochizuki stated that he is proving the abc hypothesis. The event hit the bottom of the list, because so far its proof is not supported by a large circle of mathematicians. Otherwise it would occupy the first place. In the meantime, to the disappointment of interested parties, it is in limb.
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9. The goldbach ternary problem . "Starting from 7, any odd number is the sum of three primes." Since 1937, this statement is true for fairly large odd numbers, but in 2013, Peruvian mathematician Harald Gelfgott checked this statement on a computer for numbers up to 10 ³⁰ . Independently, David Plath did it.

8. Vietnamese mathematician Ngo Bao Tau with proof of the fundamental lemma that forms part of the Langlands program. Awful technical, but very important event of the program.

7. 17 tips sudoku . In 2012, McGuire, Tiugeman, and Chivario proved that the minimum number of hints that uniquely identify a task in Sudoku is 17. Although not every set of 17 hints leads to a unique solution, the theorem says that it’s impossible to build a valid 16 and tips.

6. Homotopic theory of types / axiom of univalence. A new approach to the fundamentals of mathematics under the leadership of Vladimir Voevodsky attracts close attention. In addition to mathematical interest, it promises to modify the language of higher mathematics in such a way as to make it more suitable for computerized processing.

5. Nontriangulable varieties. In the sixth place on the list is the amazing discovery of Ciprian Manolescu about non-triangulable manifolds in dimensions 5 and higher.

4. Mosaic of Sokolar-Taylor . The famous Penrose mosaic is a set of tiles that can be used to tile a plane, but only aperiodically. For many years there was a question - is it possible to do this with just one tile? Joan Taylor and Joshua Sokolar discovered such a tile.

3. The end of the project "Flyspek". In 1998, Thomas Hales announced that he had received evidence of Kepler’s hypothesis about the most effective way of packaging cannonballs. Unfortunately, his proof was too long and included a large number of computational inserts, and therefore the people checking him could not complete the test. Therefore, Heiles and the team took it upon themselves, calling for help the auxiliary computer programs Isabelle and HOL Light. The result of the work has become a significant milestone not only in discrete geometry, but also in systems for automatically obtaining evidence.

2. Splitting numbers . How many ways can you write a positive integer as a sum of smaller numbers? In 2011, Ken Ono and Ian Bruignier offered an answer to this old question.

1. Intervals between prime numbers. It is not surprising that this achievement came first. This remarkable result was obtained by Zhang Ethan in 2013. He proved that there are infinitely many consecutive primes with a difference of no more than 70 million. The hype that ensued led to the fact that James Maynard and the Polymath project, organized by Terence Tao, reduced this number to 246.

But! But?..

Where is Heirer's work on the KPZ equation ( Kardar – Parisi – Zhang )? What about new examples of incompleteness Friedman? What can I say - we are just having fun here. If you think I'm wrong, make your own list.

As a bonus - progress in computational evidence.

• The generic continuous fraction for the number π is calculated to 15 billion first members.
• The decimal representation of the number π counted up to 13.3 trillion digits
• In search of a perfect cuboid (an integral brick), this is a rectangular parallelepiped, in which all seven basic quantities (three edges, three front diagonals and a spatial diagonal) are integers. While it is clear that if it exists, then the length of one of its sides will be no less than 3 trillion.
• The Goldbach problem (the statement that any even number, starting with 4, can be represented as the sum of two prime numbers) is verified up to the number 4 * 10 ¹⁸ .
• The largest known pair of twin twin numbers are numbers on either side of the number 3756801695685 × 2 ⁶⁶⁶⁶⁶⁹ .
• The largest of the known prime numbers and the 48th of the famous numbers of Mersenne - 2 ^57885161 -1.
• In the encyclopedia of the centers of triangles already 7,719 entries.
• The longest of the known optimal lines of Golomb now has an order of 27: (0, 3, 15, 41, 66, 95, 97, 106, 142, 152, 220, 221, 225, 242, 295, 330, 338, 354, 382, 388, 402, 415, 486, 504, 523, 546, 553)
• The most impressive achievement in factorization of integers (factoring) using classic computers is the number of 232 RSA-768 digits :
  
  1230186684530117755130494958384962720772853569595334792197322452 151726400507263655187452059978646938564749477406345452525255 5733630345373152525012010101217016011000100010001751751751751751751751751751;
  
  decomposed into two prime numbers of 116 digits:
  
  3347807169895689878604416984821269081770479498371376856891243138 8982883793878002287614711652531743087737814467999489
  
  and
  
  3674604366679959042824463379962795263227915816434308764267603228 3815739666511279233373417143396810270092798736308917
• And with the help of a quantum computer - so far only 56153 = 233 * 241
• Collatz Hypothesis tested for numbers up to 2 * 10 ²¹