My name is Daniel Král'. I moved to the United Kingdom at the end of September 2012 after accepting a professor position at the University of Warwick. I work in graph theory and related fields in mathematics and computer science. Most of my research is now focused on topics related to my ERC Consolidator grant LADIST, which builds on the ERC Starting grant CCOSA. I am also still involved in organizing the Czech national olympiad in informatics.
Most of my work belongs to discrete mathematics. This area of mathematics deals with mathematical structures and problems that are "discrete" in nature, as opposed to those that exhibit continuous properties. Since information in computer systems is stored as sequences of zeroes and ones, discrete mathematics finds many applications in computer science.
My PhD is in structural and algorithmic graph theory. A graph is a mathematical object that consists of elements called vertices and links between them called edges. For example, the road map in a GPS device in a car is discretized and represented by a graph, i.e. by a list of crossings (vertices) with mutual links (edges). A substantial amount of my work during and after my PhD is related to graph colorings, i.e. decompositions of vertices and edges of graphs subject to certain constraints. One of my recent results in this area has been featured in an article in the popular science magazine Pour la Science. In relation to computer science applications, I am interested in using graph decompositions and logic methods in algorithm design.
Most of my 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. The theory opened new links between analysis, combinatorics, ergodic theory, group theory and probability theory. For example, one of the major problems on sparse graph limits, the conjecture of Aldous and Lyons, is essentially equivalent to Gromov's question whether all countable discrete groups are sofic. My work includes applications of analytic methods in extremal combinatorics and it also led to new insights in structural properties of graph limits.