Increasing Canada's Contribution to Mathematical Scholarship
Over the last 20 years, mathematician Stevo Todorcevic has solved critical problems in several fields of mathematics using a novel approach that blends some of the ideas of discrete mathematics, including combinatorics-a branch of mathematics that studies the enumeration, combination, and permutation of sets of elements-and especially the Ramsey theory.
The Ramsey theory is an area of discrete mathematics that concentrates on finding the amounts of organization that must be present in any mathematical structure of a given type. Dr. Todorcevic concentrates on finding lists of critical objects in a given mathematical structure that must be small enough to be useful but also be complete so that the answer one obtains by testing a given problem on members of the list remains unchanged when testing all other objects from the category.
Real-life examples where the ideas of the Ramsey theory have been used include the designs of computer-generated decision procedures, which are based on similar ideas of finding lists of critical objects. Also the complexity theory of Boolean functions ("switch boards") frequently uses the Ramsey theory.
Dr. Todorcevic applies the ideas of discrete mathematics to areas of mathematics that are typically concerned with study of continuous objects. He has solved various problems from topology, an area of mathematics that includes the study of properties of objects that are unchanging through continuous deformations such as twisting and stretching. As the Canada Research Chair in Mathematics, Dr. Todorcevic brings an impressive research program to the University of Toronto. His work with the Ramsey theory carries on, as he continues to discover remarkable connections between the theory and a variety of other mathematical disciplines including analysis, topological dynamics, and algebra.