From: mk_thisisit
An article by an outstanding Polish mathematician was recognized as the best article in algebraic topology in the last five years [00:00:36]. This distinction, received at an award ceremony in Beijing, was given for a paper on the TZD property for the automorphism of free groups [02:54:15]. Although the article is not directly about algebraic topology, it fell under this field for the purpose of the award, as group theory does not have its own separate category in this particular recognition [02:29:24]. The achievement also contributed to the mathematician’s promotion to a full professor at Oxford [02:48:49].
Symmetries of Symmetries
The award-winning work relates to what has been popularized in Polish media as “the symmetry of all symmetries” [02:49:52]. This concept refers to the automorphism groups of free groups [03:09:07], which are considered universal symmetries [02:59:58]. The research proved that these groups are very similar to classical groups (like special linear groups) that appear universally in mathematics [03:12:09]. These classical groups are used in various fields, including sorting algorithms, number theory, and geometry, controlling classical geometries [03:33:05].
A key finding was that these groups possess property (T), which means they can be symmetries of certain objects but never of cyclic spaces [03:44:00]. The research further demonstrated that the symmetries of all symmetries also cannot be symmetries of Euclidean spaces [03:57:13]. This theorem allows for the creation of new expander graphs with additional properties [04:20:04]. Expanders are crucial for fast data processing in fields like sorting algorithms and daily algorithmics [04:31:00]. The proof of this theorem involved a significant computational aspect [03:10:07].
Fiberization Theorem
A second major theorem discussed by the mathematician is related to fiberization [00:43:08]. Fiberization is a phenomenon in which a space can be understood as a smaller object that changes over time [04:57:04]. For example, spacetime can be seen as a two-dimensional sphere (the surface of the Earth) changing in time, creating a three-dimensional space [04:59:01]. Each point on the sphere, as it moves through time, leaves a “fiber” or streak, and the entire space is composed of these fibers [05:01:05]. This concept involves reducing a space by one dimension and considering how objects evolve through time [05:04:00].
The speaker’s theorem builds upon a theorem by Jan Ig concerning three-dimensional varieties with hyperbolic geometry [05:09:07]. While Ig’s theorem uses specific tools for three-dimensional manifolds, this new theorem generalizes the concept [05:17:09]. It shows that if an object, especially a three-dimensional manifold, possesses a certain algebraic property, it will fiberize [05:27:08]. This means that if the algebraic property is present, the space can be “separated” or broken down into fibers, regardless of its initial dimensionality [05:40:06]. Colleagues from the United States have recently found a simpler proof for this fiberization theorem [04:17:09].
The mathematician’s work often involves looking at higher dimensional situations, particularly spaces with an odd number of dimensions, such as 130 dimensions, where particles spread out rapidly [01:52:03]. The goal is to demonstrate that such a space always originates from a higher dimension (e.g., 136 dimensions) that changes over time [01:57:08]. This approach makes it easier to visualize and understand complex spaces [02:11:09].