Department of Mathematics
Ph.D., University of British Columbia (1981)
Telephone: 519-661-3638 x 86516
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 first 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 during the intervening decades, 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 classical objects of Algebraic Topology such as 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 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 was the first to appear in its subject area in more than 25 years, and still describes much of the present state of the art in the combinatorial approach to homotopy theory. Combinatorial homotopy theory is the study of set theoretic representations of spaces and homotopies. 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, sensor network analysis, geometric analysis of large data sets, and homotopy type theory in Logic.
Jardine is the cofounder, with Gunnar Carlsson of Stanford University, of the conference series "Algebraic Topology - Methods, Computation and Science" (Stanford, 2001; Western, 2004; Paris VII, 2008; Muenster, 2010; Edinburgh, 2012; PIMS, 2014; Torino, 2016). He was a co-organizer for the research program "Computational Applications of Algebraic Topology" which was held at the Mathematical Science Research Institute in Berkeley in 2006, and was the Lead Organizer for the research program "Geometric Applications of Homotopy Theory" which ran at the Fields Institute in Toronto during the first half of 2007.
He was the cofounder, with Dan Grayson (University of Illinois at Urbana-Champaign), of the Algebraic K-theory Preprint Archive (1992-2012). This was an early subject-area preprint server, and was one of the first successful venues for online distribution of research results in Mathematics.
Current Postdoctoral Fellows: