János Pach, research professor at the Rényi Institute and one of the world's leading and recognized researchers in combinatorial and algorithmic geometry, was recently a guest on the radio program "Closer to Mathematics." The interview, which can be heard on Trend FM and Radiocafé, was prompted by the fact that this year Maryna Viazovska, the Ukrainian mathematician who became world-famous for solving the 8- and 24-dimensional sphere packing problem, received the László Fejes Tóth Award and its accompanying medal. While János Pach spoke about Maryna Viazovska's achievements and career, as well as about the internationally renowned Hungarian mathematician László Fejes Tóth, he also introduced listeners to an important chapter in the history of mathematics.
The László Fejes Tóth Award was established by the Rényi Institute in 2023 with the purpose of annually recognizing the work of a mathematician who has achieved internationally significant results in the field of geometry - broadly interpreted. Regarding why the Ukrainian mathematician received the award this year, János Pach explained: Maryna Viazovska is very young. She received the Fields Medal in 2022, and this award is only given to researchers under 40 years old. She was an ideal candidate for the Fejes Tóth Award, as she is a brilliant geometer who, moreover, achieved a decisive breakthrough in connection with a classical geometrical question, the sphere packing problem, while still a postdoc, as an extension of her PhD studies. The question is not difficult to summarize briefly. Take a very large container in three-dimensional space and try to pack as many unit-sized, identical glass balls into it as possible. How many can we fit in and how should we arrange them? It became clear relatively quickly - this is usually called the Kepler conjecture - that the best approach is to place the glass balls the way apples or oranges are usually arranged in the market. We place a layer on the bottom of the container so that each ball has six neighbors, then place a similar layer on top so that the bottoms of these balls fit exactly into the depressions between the balls below them, and so on. Everyone was certain that the Kepler conjecture was true, but no one managed to prove it until 1998. Then Thomas Hales, an American mathematician, found a complex solution using computer methods. It took years to fully verify his proof. Hales' solution was based on principles that László Fejes Tóth outlined in a book published in 1953. This solved the original question, but we didn't get an answer to what the situation might be in higher-dimensional spaces. To this day, only two dimensions are known for which conjectures have been formulated at all. In 8- and 24-dimensional space, extremely symmetrical, beautiful sphere arrangements were discovered more than 50 years ago. It was suspected that these were optimal. Maryna Viazovska's world-renowned achievement was that she managed to completely solve the sphere packing problem in 8-dimensional space alone, and a few weeks later - using the same method - in 24-dimensional space with co-authors. János Pach also spoke about how the solution connects to other areas of mathematics.
Regarding why winning the László Fejes Tóth Award might be significant for the Ukrainian researcher, János Pach put it this way: With every award, the question arises whether it's important for the one giving it or the one receiving it. For anyone, it's a great recognition to receive an award named after an excellent mathematician, especially in that mathematician's homeland. At the same time, it's also important for the Rényi Institute to have honored such great scientists as László Lovász, Endre Szemerédi, and Maryna Viazovska, as this enhances the reputation of both the award and the Institute. In the "Closer to Mathematics" program, research professor János Pach also praised László Fejes Tóth's mathematical work. He shared personal memories about him and also spoke about how the influence of this exceptionally talented, school-forming geometer can still be felt in Hungarian mathematics today.