The theory of graph limits aims to provide analytic models representing large networks/graphs. Such analytic models have found applications in various areas of computer science and mathematics. We survey basic results on analytic models of large graphs and focus in more detail on those related to dense large graphs. We conclude with some of the future research directions related to the presented concepts.

The talk will be self-contained and no previous knowledge of the area is needed.

Daniel Král' is a professor of mathematics and computer science at the University of Warwick. He obtained his Ph.D. from Charles University in Prague in 2004, and has held positions at TU Berlin, Charles University, and the Georgia Institute of Technology. His research focuses on problems in graph theory and graph algorithms, and has been awarded numerous prizes, including the European Prize in Combinatorics (2011) and the Philip Leverhulme Prize in Mathematics and Statistics (2014).

Most of Král''s current research is related to combinatorial limits. The theory of combinatorial limits provides analytic views on large discrete structures and it responds to challenges from computer science where structures such as the graph of internet connections and graphs of social networks (e.g. Facebook, LinkedIn) are enormous. These limit theories open new links between analysis, combinatorics, ergodic theory, group theory and probability theory.