
My research is in the field of algebraic topology. When one wishes to study any kind of system quantitatively, one is usually led to consider all possible configurations of the system. A simple example is that of a classical system consisting of a single particle in 3space. For this system, a configuration would be its position (a vector in R^{3}) and its momentum (another 3vector), together giving a vector in R^{6}. If the particle could move everywhere, and at any speed (ignoring Einstein's relativity and the fact that there's lot's of stuff in space already), the collection of all possible configurations would then be the vector space R^{6}. Depending on the system, one obtains other spaces of configurations, for example the surface of a ball. This collection of configurations is called a symplectic space by mathematicians or the phase space by physicists. It has turned out to be very profitable to think of the evolution of the system as a path in this space, and its geometry carries some information which is intrinsic to the system. 
Properties of the geometric structure, such as how many "holes" there are in the space, what dimensions the holes have, and how they are linked together, can carry important information about the system. As an example, think about the differences between the surface of a doughnut (below), and the surface of a ball (opposite). They have different numbers of holes of varying dimensions, and the holes on the doughnut are linked in an interesting way. Algebraic topology encodes some of this underlying geometric structure using algebra, enabling spaces (and in particular their holes!) to be manipulated as if they were numbers. This elucidates some of their intrinsic properties, which are often hard to 'see' any other way. Rational homotopy, my specialty within algebraic topology, represents an excellent compromise between the amount of information encoded and the ease of manipulation.



I study spaces (many of which might not arise as phase spaces) using three main tools: The LusternikSchnirelmann category of a space, which is the minimum number of "simple" pieces needed to build the space, the Toral Rank of space, which is maximum dimension of the simplest type of symmetry it possesses, and the approximation of spaces by 'very finite' ones, for which two classical algebraic measures of spaces, the 'homology groups' and the 'homotopy groups', are both finite. I am also interested in the geometry of Lie (pronounced "lee") algebras, some of which also come from phase spaces. 
I will be teaching Introduction to Geometry in the winter session, 2017.
The surface on the right is known as a Klein bottle. It has no real inside or outside, and can be obtained by gluing together two Mobius bands (which only have one side: see the diagram at the top left of this page) along their boundaries. I found these images at The Geometry Centre, where you can find more neat stuff like this, or perhaps learn more about these particular penguins than you ever wanted to know. ( © Copyright 199x by The Geometry Center" .) Another interesting site is Symbolic Sculptures and Mathematics where you can find some sculpture inspired by mathematics, as well as learn a little knot theory. 
For your reference, here are the sessional dates for this academic year.
If you are teaching for the first time this year, some suggestions can be found in "Tips" on teaching.
If you will be grading papers this year, some useful suggestions can be found in my talk on grading in Science and Engineering.
If you will be proctoring for me this year, please read these guidelines.
Do you have a late class or work late on campus? Would you feel more comfortable having someone to walk with to your destination? The University of Ottawa Foot Patrol believes in "Safety in Numbers" and is available for your use. If your class ends between 5pm and 2am Monday to Friday, call the Foot Patrol at 5625800 ext. 7433 for a team to escort you to your destination. The Foot Patrol will take you anywhere within a 45 minute walking distance from the University. Arrangements can also be made to have a team meet you outside your class every week. To schedule a team or for more information call ext. 7433 or stop by the office UCU 08A."
Your comments and suggestions are welcome (My email address is Barry.Jessup at uottawa.ca )