## Final Report Summary - H13P (Hilbert's 13th Problem)

The fellow, Feng, has been offered a tenure track position in the prestigious mathematics department of Auburn University, Alabama. This has meant that the fellowship has been terminated midterm, before the work plan could be completed. However, a significant amount of work has been achieved and the scientist in charge, Good, and the fellow continue to collaborate on the research objectives outlined below.

The thirteenth problem from Hilbert’s famous list asks whether every continuous function of three variables can be written as a composition of continuous functions of two variables. It took 50 years for significant progress to be made on Hilbert’s 13th problem. In 1954, Vitushkin found a negative result along the lines that Hilbert conjectured. In fact, Vitushkin partially proved that the complexity of the functions in term of representation by compositions is characterized by the number of variables and the order of differentiability. However, Kolmogorov’s Superposition Theorem (KST), proved by Kolmogorov and Arnold, shows that every continuous real-valued function of two or more variables from [0,1] can be written as a composition of continuous functions of just one variable along with just one function of two variables, namely addition.

Recently, Feng, together with Professor Paul Gartside characterized exactly which topological spaces admit a Kolmogorov type theorem and extend the KST to deal with vector-valued maps. For possible applications, Feng also developed a constructive superposition theorem for continuous functions of any number of real variables.

The project was split into 5 key inter-related research objectives.

(RO1) The aim of Research Objective 1 was to implement Feng’s constructive version of the KST as usable computer code.

(RO2) Research Objective 2 was to investigate the smoothness of the functions arising in superposition theorems.

(RO3) This objective was to carefully analyse the combinatorial argument used by Vitushkin to extend his results.

(RO4) The fourth research objective proposed a novel approach to studying the possible superposition theorems via the topological structure of the set of critical points of a function.

(RO5) The final objective was to seek appropriate function space topologies in which continuous functions are approximated arbitrarily closely by infinitely differentiable ones.

The point of RO1 is to enable the highly theoretical KST to be applied in ‘real-world’ applications. We have coded a two-variable superposition theorem using the high level language Python. However, the code runs extremely slowly. We held a series of research meetings with Dr Martin Escardo a leading expert on real number computation from the School of Computer Science at the University of Birmingham. After carefully going through the constructive version of the KST, the conclusion of these meetings was that there is a theoretical barrier to substantial improvements in computation time for any such code. This arises from calculating the modulus of continuity of the (nowhere differentiable) outer functions arising in the KST. At the workshop we organized July 2013 in the University of Oxford, we investigated the applications of the Superposition theorem in image processing, progressive image transmission and removable watermarks (useful in, say, pay-per-view video streaming) with Dr. Pierre-Emmanuel Leni from University of Franche-Comte, France. Interestingly, Leni has shown that significant practical applications of the KST are possible without particularly accurate calculations of the outer functions. Other application may include neural networks and data reconstruction.

In research objectives (RO2)-(RO5), we use several different approaches to address questions raised by Vitushkin, Kolmogorov and Arnold about superposition of smooth functions. Using the combinatorics of superposition of finite functions (RO3), we have extended Vitushkin’s theorem to address the case of Lipschitz functions. This result is currently being prepared for publication. Our studies here seem to demonstrate that this is the best extension one could hope for using the combinatorics of superposition of finite functions: there is not quite as much flexibility in Vitushkin’s argument as appears at first sight. In the research object (4), we studied the structure of the critical sets of two variable functions that are finite compositions of continuously differentiable functions of one variable and addition. Although it is very hard to completely determine this structure, our strong conjecture is that a ‘Cantor’ type set of circles in the plane cannot be the critical set of a function that is a finite composition of continuously differentiable functions and addition. Such an example would answer a long-standing series of questions in the negative and we are reasonably confident that we will be able to confirm this conjecture. The studies suggest that the proof will require some new techniques.

In the research object (5), we proved that the analytic functions in the form of Kolmogorov’s Superposition Theorem are strongly dense in the space of continuous functions of n variables. Using this result, we can obtain a startling result: there is an algebraic PDE whose solutions are dense (in the strong topology) in the space of all continuous functions of n variables. We have one publication in preparation describing the new results. Unfortunately, the studies don’t show how to approximate an arbitrary analytic function and its first derivative simultaneously using the analytic functions in the form of Kolmogorov’s Superposition Theorem. A negative answer to this will yield a negative answer to the series questions by Vitushkin, Kolmogorov and Arnold, but a positive answer to this will open an amazing window to many applications.

In addition to the work on the research objectives, three papers are being prepared (A Constructive Superposition Theorem, Strong Analytic Approximation and Universal PDEs, and On the Complexity of Superpositions). Feng has given a number of seminars about results from the project in the UK at Oxford and Birmingham (School of Mathematics and School of Computer Science), in the US (University of Pittsburgh and Lawrence Berkley National Laboratory) and China (Capital Normal University Beijing and Shandong University). A workshop (Around Hilbert’s 13th Problem) was held at the Mathematical Institute, Oxford with 10 researchers from the UK, China, the USA, France and Croatia.