In 1974 the mathematical Michael Freedman began to study a very common problem in topology, an extension of the geometry. It was the “Poincaré Conjecture”, which establishes that any shape, with some generic characteristics, must be equivalent to a sphere.

Poincaré, when he postulated the conjecture in 1904, was specifically thinking about three-dimensional shapes, but modern mathematicians started to consider all possible dimensions, applying the conjecture in two distinct streams: "smooth shapes", which do not bring corners or vertices, allowing you to create calculations from any point or position; and the “topological formats”, where there is the presence of vertices where calculation is impossible.


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Image shows book cover that rescues an essential foundation of mathematics
The book "The Disc Embedding Theorem” rescues the mathematical work of Michael Freedman and his solution to the Poincaré Conjecture in four-dimensional formats. Image: Oxford/Disclosure

It is such a complex and specific kind of mathematical foundation that, some say, you can only understand it if Freedman himself were with you, guiding your study. Because of this, its use in research is extremely rare and, consequently, few people actually understand it.

Some publishers of a new book, however, want to rescue this knowledge generation , passing it on through the work "The Disc Embedding Theorem” (“The Disk Embedding Theorem”, in literal translation). At approximately 500 pages, the authors want to make the subject so divisive into something a college student can learn in just one semester.

“[With the book] We leave nothing else to the imagination,” said Arunima Ray of the Max Planck Institute for Mathematics, who signs the book's authorship with Stefan Behrens (Bielefeld University), Boldizsár Kalmár (Budapest University for Technology and Economics ), Min Hoon Kim (National Chonnam University, South Korea) and Mark Powell (Durham University). “Everything is properly noted and marked”.

Other Poincaré scholars had already made progress on shapes up to five dimensions, but Freedman focused on the four-dimensional topological conjecture – possibly the most difficult in the field: it basically claims that every topological shape that consists of a four-dimensional homotopic sphere is, too , strongly equivalent to an ordinary four-dimensional sphere.

Formally speaking: two continuous functions that move from one topological space to another are called “homotopic”. The four-dimensional conjecture is said to be one of the most complex because, at the time Freedman solved it, the tools used by mathematicians did not work very well within the restricted four-dimensional environment.

To understand how Freedman solved this problem, it is important to establish some parameters: a four-dimensional homotopic sphere is not the same thing as an ordinary sphere. Here, it is characterized by how the curves drawn within it interact with each other.

In the case of the four-dimensional conjecture, these curves are two two-dimensional planes. It's easier to understand considering the image below, where the single-dimensional curves intersect within a two-dimensional space:

Graph shows a line crossing a curve at two points
Graphic that simplifies the understanding of Freedman's analysis for Poincaré's conjecture. Image: Quanta Magazine/Reproduction

In the example above, both curves bring something called “intersection algebraic number”. To find out what this number is, we just set the value of “-1” for all the crossed points where the curve is rising; and the value of “+1” for the crossed points where the curve is descending. Since we have a point for each in the image, then we know that the algebraic number is “0”.

This same number is characteristic of the homotopic sphere and also of the common sphere, with the difference that the common sphere can have its curves drawn so as not to intersect at any time. The number remains “0”, but without counting the intersection points positioned above.

Here begins the problem solved by Freedman: he needed to show that it is always possible to take pairs of curves with intersection number “0” and “push” them away from each other, without changing that number. If you have pairs of curves with intersecting numbers “0” and you can move them apart, then you have proven that the space where they are embedded must be that of an ordinary sphere.

The studies of men took seven years of almost social isolation (and you there, thinking it's bad the departures caused by pandemic), where Freedman interacted with virtually no one. Eventually, he achieved his goal, but his work nearly disappeared when he himself failed to prove that it was “presentable” at any student level – unlike other scientific fields, there is no “jury panel” that judges mathematical works like correct.

Freedman presented his findings to several strong minds in the community who helped him spread this knowledge, but to achieve a more distributed status he needed a written notice of the exam so that people who had never seen him before could read and learn by own account. He never did that.

Freedman now works on Station Q, a Microsoft project within the University of Santa Barbara, California, to develop a quantum computer with topological calculus capabilities: “I probably didn't handle the exposure of written material as carefully as I should have. ", he said.

The book is already available in the US, but it is not expected to arrive in a translated version for Brazil.

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