The investigators are continuing their core research programs, attacking important classic problems of combinatorics, including (1) The theory of crossing numbers of graphs with connections to discrete geometry; and (2) Searching for new insights on antichains and intersecting families of subsets, and for connections between chain decompositions of the subsets of a set and symmetric Venn diagrams. Research by the investigators with a broader impact includes their contributions to (3) Computational biology: phylogenetic tree reconstruction, clustering biomolecular sequences, (4) Theoretical computer science: graph drawing, security of statistical databases; and (5) Communications: New models of real-number vertex labellings with distance conditions to find optimal or near-optimal channel assignments for large networks of transmitters. Research in combinatorics is carried out on problems that lie at the very core of our understanding of how discrete structures work and how to use them optimally. The investigators also apply combinatorial theory to interdisciplinary areas (including computational biology, theoretical computer science, and communications), where progress often requires using or inventing deeper combinatorial results which exhibit structures that are key to applications, or which provide bounds on what can be achieved by any algorithm. Progress in these areas contributes to important real-life problems, including the development of more efficient research tools in bioinformatics; the design of better visualisation tools for computer screens; increased computer and database security; and more efficient cell phone communication. The investigators are adding to their impressive record of recruiting and training graduate students and involving them in their research. They are expanding their efforts to attract more minority and women students into Ph.D. programs in mathematics. They are continuing their extensive activities in managing research journals, organizing conferences, guiding the Canada-USA Mathcamp for high school students, and contributing to the international Mathematical Competition in Modeling for undergraduates.
