Date: 20 and 27 March 2025 13:40-15:10  

Location: Corvinus University of Budapest, building C, C427 

Language of the event: Hungarian 

 
From Hilbert’s 13th Problem to Neural Networks I (20 March 2025 13:40-15:10) 

Hilbert’s 13th problem is about the solution of a certain (general) 7th degree polynomial in terms of its three coefficients. We know that the solution function is a continuous function of these coefficients but Hilbert wanted to know whether it is possible to express it as a composition of functions only of two variable. The problem was solved by Kolmogorov and his student Arnold in 1956-57. In this first talk I’ll state the problem precisely and then I’ll show a solution based on Baire’s category theory following a paper by Kahane which was published in 1975.

From Hilbert’s 13th Problem to Neural Networks II (27 March 2025 13:40-15:10) 

Hilbert’s 13th problem is about a special way in which the solution of a certain (general) 7th degree polynomial in terms of its three coefficients can or cannot be expressed. The problem was solved by Kolmogorov and his student Arnold in 1956-57. In this second talk I’ll present — in some cases, along with their proofs — some of the theorems that lead us to the representation of continuous functions by neural networks. In the second part of the talk I’ll move from the question of representation to that of approximation.  

This research was supported by the National Research, Development and Innovation Office (NKFIH) under grant number 2024-1.2.3-HU-RIZONT-2024-00030.