Random complexes and the Cohen-Lenstra heuristics

ongoing

Graph limit theory is a rapidly developing field of mathematics. Mészáros proposes to use ideas from graph limit theory to answer questions about random simplicial complexes. The motivation for investigating the cokernels of random integral matrices comes from the number theoretic conjectures of Cohen and Lenstra. Mészáros plans to make use of the fact that the questions arising studying the homology of random simplicial complexes are very similar to questions about the cokernels of random integral matrices. Thus, ideas can be transferred from one area to the other. Mészáros also proposes to investigate the expansion properties of random simplicial complexes. Many researchers at the Rényi Institute (the host institute) played a key role in developing graph limit theory and finding applications of this theory in several areas including ergodic theory, group theory, probability theory, statistical physics and computer science. Large part of Mészáros’s past and proposed projects relies on these techniques. Building on the research that he started during his three years at the University of Toronto as a postdoctoral fellow, Mészáros will bring new questions and areas where these techniques can be applied like stochastic topology. Mészáros’s work will not only provide new areas of applications, but hopefully will open new directions in graph limit theory. The arising new topics in graph limit theory can be of great interest to many members of the host institute. The fellowship would provide Mészáros an excellent opportunity for many fruitful collaborations to explore these topics further. He is also looking forward to improving his organization, management and communication skills to become a more well rounded researcher.