Professor, Department of Mathematics
Ph.D., University of British Columbia (1981)
Homotopy Theory, Algebraic K-theory, Algebraic Geometry, Number Theory
Current research interests
Homotopy theory and its applications
Algebraic Topology is the study of algebraic approximations of space.
The subject area has been one of the driving forces for research in Mathematics for more than a century, beginning with seminal work of Poincaré in the late 1890s. The theory acquired great depth and computational power through the years, and achieved a precise level of axiomatic simplicity with Quillen's introduction of closed model structures in the 1960s. At the same time, the Grothendieck school in Paris began a grand project to apply the extant wealth of homotopy theoretic calculational methods in Algebraic Geometry and Number Theory. This enterprise continues to this day: the fusion of geometric concepts and homotopical technique has been a defining characteristic of algebraic K-theory since the 1970s, and applications have spread to other fields.
The modern period for this branch of homotopy theory began in the mid 1980s with the discoveries, by Jardine and Joyal, of homotopy theories for wide classes of objects in Algebraic Geometry. It has progressed through the work of many researchers since that time, with the introduction of motivic homotopy theory, derived algebraic geometry, and topological modular forms. The homotopy theories which arise from Algebraic Geometry are widely applicable: they engulf all cohomology theories, they form the basis for the modern geometric theory of symmetries and higher symmetries, and they give information about the homotopy groups of spheres. The overall theory is the subject of Jardine's book Local Homotopy Theory, which was published by Springer-Verlag in 2015.
Jardine is also the coauthor, with Paul Goerss (Northwestern University), of the book Simplicial Homotopy Theory, which was published by Birkhäuser in 1999, and then republished in 2009 and 2015. This book describes the combinatorial approach to homotopy theory, and is a basic reference in the subject area. Combinatorial homotopy theory is the study of set theoretic representations of spaces and homotopy. It is used in much of modern Mathematics, and is finding new applications in various disciplines related to the mathematical sciences: examples include quantum gravity theories, models for parallel processing systems, topological data analysis, and homotopy type theory in Logic.