MTH 679: Topics in Geometry
Spring 2016
The Geometry of the Exceptional Lie Groups

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Tevian Dray

Office hours: WF 1:30–3:30 PM in Kidder 298A (and by appt)

Phone: 737-5159

Email:

Class Meetings:

WF 4–5:20 PM in Wngr 275

Texts:

Lie Groups: A Problem-Oriented Introduction via Matrix Groups,
by Harriet Pollatsek
(available through OSU to read online here (use this link from off campus); errata available here)
The Geometry of the Octonions,
by Tevian Dray and Corinne A. Manogue
(available through OSU to read online here; prepublication version available to read online here)

Web Site: http://math.oregonstate.edu/~tevian/onid/MTH679

Check the announcements page frequently.

Course Evaluation:

Based on homework and class participation.

Course Description:

This course will discuss the geometry of both Lie groups and Lie algebras, with an emphasis on the exceptional cases and their description in terms of the octonions. Lie groups are groups of continuous symmetries, generalizing the familiar notion of rotation groups; Lie algebras are their infinitesimal versions. Symmetry groups describe many physical phenomena, and Lie groups are widely used in physics, notably in the description of fundamental particles.
In the late 1800s, Killing and Cartan classified the simple Lie algebras into 4 infinite classical families and 5 exceptional cases. The classical Lie algebras correspond to matrix groups over the reals, the complexes, and Hamilton's quaternions. In the 1960s, a unified description of the exceptional Lie algebras was given by Freudenthal and Tits in terms of the octonions, the largest of the four division algebras.
The goal of this course is to describe the structure of the corresponding exceptional Lie groups, utilizing tools from both geometry and algebra. We will discuss the general structure of Lie algebras and Lie groups, the classification theorem, and the Freudenthal–Tits magic square of Lie algebras, culminating in a treatment of the largest of the exceptional Lie groups, E₈, in terms of the octonions.

Prerequisites:

MTH 674 or the equivalent is strongly recommended as a prerequisite, as is a first (undergraduate-level) course in abstract algebra, but the course will be largely self-contained. Students with little or no prior knowledge of differential geometry who wish to take the class are encouraged to contact the instructor to discuss their background.

Students with Disabilities:

Accommodations are collaborative efforts between students, faculty and Disability Access Services (DAS). Students with accommodations approved through DAS should contact me prior to or during the first week of the term to discuss accommodations. Students who believe they are eligible for accommodations but who have not yet obtained approval through DAS should contact DAS immediately. Further information is available on the DAS website.