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Publish Date: 27.12.2021
Category: Outstanding research achievements, Our contribution to sustainable development goals
Sustainable development goals: 4 Quality education (Indicators)
Urban Jezernik, PhD, of the Faculty of Mathematics and Physics, University of Ljubljana and Sean Eberhard, PhD, of the University of Cambridge have made a significant contribution to solving Babai's conjecture for high-rank classical groups using random generators.
The best-selling mechanical puzzle in the world is the Rubik's Cube. Its popularity is based on the fact that each cube can be put into about 1019 different states, and it can be led from each of these states to an orderly state using at most 20 manipulations of its faces. Therefore, there is an enormous number of possibilities, but the solution is always short, at least in theory.
The background of such puzzles relies on the mathematical model for the symmetry of objects, i.e. groups. From each such group, several different puzzles can be composed. The level of difficulty of a puzzle is measured using the diameter of an appropriate Cayley graph. Each group can be broken up into finite simple groups and these simple puzzles are then solved separately. Babai's conjecture assumes that these simple pieces should act like a Rubik's Cube: the solution to the puzzle should be short. According to many deep mathematics results, this conjecture is known in the case of simple group symmetries of vector spaces having limited dimensions over enormous finite fields.
Asst. Prof. Urban Jezernik, PhD, of the Faculty of Mathematics and Physics, University of Ljubljana, and Sean Eberhard, PhD, of the University of Cambridge have proved that in the open case of simple group symmetries of vector spaces having large dimensions over limited finite fields, Babai's conjecture applies to almost all puzzles. This method is based on an in-depth study of the expected values of characters of simple groups, and on the surprising result that simple pieces act expansively on spatial vectors.
This article represents a breakthrough in solving the famous Babai conjecture, which in recent years has been one of the central areas of group theory. The result was published in Inventiones Mathematicae, one of only three mathematical journals which play a role in the Shanghai University Ranking.
Presentation of six random puzzles, whereby each is based on a simple group of symmetries of vector space having dimension 2 over power field 5. Author: Urban Jezernik
Source: Eberhard, Sean., Jezernik, Urban. Babai’s conjecture for high-rank classical groups with random generators. Invent. math. (2021). DOI: https://doi.org/10.1007/s00222-021-01065-x