Telephone: 519-661-3638 x 86516

Fax: 519-661-3610

E-mail: jardine@uwo.ca

Algebraic topology is the study of algebraic approximations of space, and has been one of the driving forces of Mathematics since early in the twentieth century; it effectively began with the work of Poincaré in the late 1890s. The theory acquired great depth and computational power over 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 wealth of homotopy theoretic calculational methods to algebraic geometry and number theory. This enterprise continues to this day, and has always been a central theme of research in algebraic K-theory.

The modern period for this branch of homotopy theory began in the mid 1980s with the discovery, by Jardine and Joyal, of homotopy theories for wide classes of objects in algebraic geometry, and has progressed in recent years, through the work of multiple researchers, 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, but can also be used to study models for parallel processing systems and homotopy types of dynamical systems.

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. The book was
the first to appear in the subject area in more than 25 years, and
describes much of the present state of the art in the combinatorial
approach to homotopy theory.

Combinatorial homotopy theory is used in much of modern Mathematics, and is finding new applications in Science and Engineering, particularly in the study of networks, the development of models for parallel processing, and geometric analysis of large data sets. Jardine is the cofounder, with Gunnar Carlsson of Stanford University, of the "Algebraic Topological Methods in Computer Science" conference series (Stanford, 2001; Western Ontario, 2004; Paris VII, 2008; Muenster, 2010), and 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. Jardine was the Lead Organizer for the research program "Geometric Applications of Homotopy Theory" which ran at the Fields Institute in Toronto in 2007.

Jardine is the cofounder, with Dan Grayson (University of Illinois at Urbana-Champaign), of the Algebraic K-theory Preprint Archive at the University of Illinois at Urbana-Champaign. This was one of the original, and remains one of the most successful subject-area preprint servers in Mathematics.