Sparse Discrete Structures
This project will focus on extremal and probabilistic combinatorics and on several related areas. Professor Balog and his coauthors have developed the so-called container method, and contributed to the theory of applications of flag algebras. These new methods are considered the most exciting recent breakthroughs of the field with many unexplored directions. The project’s central focus is further developing the container method which connects several interrelated problems on the borders of combinatorics and probability. The questions raised are connected to other areas, such as classical asymptotic enumeration, geometry, extremal problems for graphs and set systems, Ramsey theory, combinatorial geometry, random graphs, and probabilistic methods. One of the most important aims of the project is the effort to understand the “critical probabilities” at which various properties of interest appear in large discrete random systems. This is a classical area that originated in the 1960s when theory random graphs and percolation was born. But it is also an area that has seen significant developments in recent years, and one that has close ties with several other disciplines, including statistical physics and theoretical computer science. Professor Balog will work on additional applications arising from theoretical computer science and combinatorial number theory.
During his CAS appointment, Professor Balog will travel to Cambridge University and ETH Zurich in order to collaborate with the leading experts of the field.