Telephone: 519-661-3638 x 86516

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 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 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 its applications have spread to other fields.

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
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, give information about classical objects such as homotopy groups of
spheres, and have recently been used to study networks and homotopy types of
dynamical systems.
The overall theory is the subject of Jardine's new book *Local
Homotopy Theory*, which will be published by Springer-Verlag in
early 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. This book was
the first to appear in the 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 Science and Engineering,
for example in Mathematical Physics, models for
parallel processing systems, 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). 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.

Jardine was the cofounder, with Dan Grayson (University of Illinois at Urbana-Champaign), of the Algebraic K-theory Preprint Archive (1992-2012) at the University of Illinois at Urbana-Champaign. This was an early, successful subject-area preprint server, and was one of the original proofs of concept for online publishing in Mathematics.

**Recent Postdoctoral Fellows:**

- K. Szumilo (2014-16), PhD: University of Bonn, 2014
- C. Okay (2014-16), PhD: University of British Columbia, 2014
- H. Bacard (2012-15), PhD: Universit ́e de Nice Sophia Antipolis 2012
- M. Misamore (2013-14), PhD: University of Western Ontario, 2009. Present Position: Software Developer, Desire2Learn, Guelph
- D. Schaeppi (2013-14), PhD: University of Chicago, 2013. Present Position: Swiss National Science Foundation Fellow, University of Sheffield
- M. Robertson (2010-12, 2013-14), PhD: University of Illinois/Chicago 2010. Present Position: Assistant Professor, UCLA
- P. Lowrey (2010-13), PhD: University of Texas/Austin 2010. Present Position: Van Vleck Postdoctoral Fellow, University of Wisconsin
- J. Harper (2010-12), PhD: University of Notre Dame 2008. Present Position: Assistant Professor, Ohio State University at Newark

**Current Students**

- N. Meadows, MSc 2014, PhD 2014-2018 (expected)