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Limits of discrete structures

Final Report Summary - STRUCLIM (Limits of discrete structures)

As we have more and more access to large data sets, there is an increasing need for mathematical tools to analyze them. Important examples include data related to large and complex networks such as social networks or connections in the brain. This was one of the motivations for the emerging field of graph limit theory that uses methods from the mathematical field of analysis to study very large networks. Graph limit theory is a rather diverse area. Depending on the density of connections in a network, very different methods can be used. There are two separate and well established approaches for high density and very low density networks. On the other hand the situation is much less clear for intermediate density networks (most real life networks belong to this category) and there are many non-equivalent approaches. One of the major achievements of the project is a new approach, called action convergence, that unifies sparse, dense and many of the intermediate density theories. Action convergence is applicable to networks where previous methods failed to give a detailed picture and moreover it can be extended to general matrices. Illustrative applications of action convergence deal with fundamental objects called random graphs and random matrices. Substantial new results are obtained in the area of random regular graphs.

Another area of research is in the field of higher order Foureir analysis (HOF) started by W.T. Gowers around 2000. Classical Fourier analysis is one of the most powerful method to deal with ordered data. HOF is a generalization of classical Fourier analysis and it is connected to graph limit theory in many ways. It was observed that many results in HOF are similar to results on dynamical systems but the exact nature of this connection was not known. The StrucLim project revealed a deep and fruitful connection between HOF and dynamical systems using probability theory. This led to new results in both areas.