Great advance in mathematical evaluation: resolve Kakeya’s conjecture in three dimensions | Coffee and theorems | Science | EUROtoday

A latest results of the measure concept, included into the Math Arxiv portal – even pending friends – is receiving a deserved media affect. This is the demonstration that, in three dimensions, the Kakeya units, though of quantity equal to zero, have, nevertheless, the fractal dimension of the three -dimensional atmosphere. It responds to a primary and intriguing situation raised greater than 50 years in the past within the harmonic evaluation.

The historical past of this achievement started earlier than, in 1917, with a picturesque drawback raised by the Japanese mathematician Soichi Kakeya, who requested concerning the minimal space of ​​the minimal space inside which a needle can transfer – mathematically, an oriented phase of size one – till investing the respective positions of the attention and the tip. For instance, it’s potential to do it in a circle of radius ½, whose floor is PI/4, and in addition in an equilateral triangle of top 1, of considerably decrease space. But each are removed from the proper reply that was given, round 1919, by the Ukrainian mathematician Abram Besicovitch, one of many dad and mom of the world referred to as the geometric concept of the measure.

Interestingly, Besicovitch then ignored Kakeya’s query and his motivation got here from a elementary situation: can coordinate axes be turned in order that the iterated integral is the same as the double integral, in response to Riemann’s methodology? To give it a unfavourable response, it constructed a peculiar set within the airplane whose space is zero, but it surely incorporates a phase of line of size equal to at least one – a needle – in any course, and which has subsequently been referred to as a Kakeya set, though it might have been fairer to name it from Besicovitch. He additionally did it with a building that additionally serves for higher dimensions. From it it’s simple to reply Kakeya’s query: there’s not that minimal space set, however, given an quantity of space, nevertheless small, there’s all the time a set of smaller space the place it’s potential to maneuver the needle to speculate its place.

In 1970 Charles Fefferman – Medalla Fields in 1978 – took a twist to this story. He used the Kakeya as a complete to throw a big conjecture of the harmonic evaluation, relative to the convergence of the spherical sums of Fourier collection and integrals in a number of variables. From Fefferman’s work, the necessity to know higher the properties of Kakeya units and their relationship with spherical summation operators arose. And that query went from being a mere curiosity to changing into a major situation of mathematical evaluation. From this, the query about what was the fractal dimension of the units created by Besicovitch that has now been resolved within the three -dimensional area arose.

The fractal dimension is a generalization of the idea of dimension for geometric objects strangerscomparable to fractals or Kakeya as a complete. It measures how the size, space or quantity of a set grows once we enhance it with the closest factors – these which are at a distance from the set lower than a sure worth, which we are likely to zero. In the case of a line phase, one is price one; two for a flat rectangle and three for a cylinder. However, there are units for which this dimension is fractional – for instance, the Sierpinski carpet has a fractal dimension log (8)/log (3) -, and therefore the fractal identify.

Despite having zero space, in two dimensions, Kakeya’s units have two fractal dimension, as Roy Davies demonstrated in 1971. This can be a consequence of the primary results of my thesis, which I did underneath the course of Fefferman on the University of Chicago: it’s a exact dimension of an object that launched into that work, the maximal perform of Kakeya. In two dimensions, it’s an operator that assigns it to every level, at every level, the higher finish of the averages of the perform on rectangles of eccentricity – that’s, quotient between large and lengthy – most popular and arbitrary course, which comprise that time. These rectangles, very lengthy and slender, in bigger dimensions, change into parallelepiped, cylinders or tubes and change the needles of the preliminary strategy. The Maximal perform additionally turned out to have attention-grabbing purposes to the Fourier collection and integrals.

Then I did not function the tactic in additional dimensions, as a result of in different higher the management of the overlap of the needles (parallelepípedos, cylinders or tubes) is rather more difficult. In subsequent years, different researchers comparable to Jean Bourgain – Medalla Fields in 1994 – Thomas Wolff, or, extra just lately Nets Katz and Terence Tao – additionally additionally a Fields medal in 2006 – had been within the topic – and partial progress was achieved, now crown up with the lead to three dimensions achieved by the Mathematics Hong Wang (Courant Institute of the University of New York) and the Mathematician Joshua Zahl (University of British Columbia).

His proof, which had the deference of sending me a number of weeks earlier than importing it to the Open Repository Arxiv, permits us to affirm that within the three -dimensional area the Kakeya units have the fractal dimension of the ambient area, that’s, three. To do that, they use – within the wake of my thesis – difficult calculations of the parallelepiped overlap in area, based mostly on the classical geometry of Euclid, however of such a combinatorial complexity, that its improvement wants greater than 120 pages of intricate reasoning. It is an instance of what I prefer to name as “suprematism in harmonic analysis” – for using rectangles and tubes, just like these noticed in works of the Russian pictorial motion -, however, the place acceptable, it’s a baroque suprematism, if oxymoron is allowed.

At the second, it solely works in dimension three. In addition, the estimate of the maximal perform, which is the article of higher analytical significance, remains to be to be performed. Even so, and though it should spend time till the specialists certify the article by Wang and Zahl and that is revealed in a very good journal, for the solidity that their earlier work on the topic offers them, evidently we face a jewel of the harmonic evaluation.