MTH 675/679: Topics in Geometry
Winter 2023
Lie Groups and Lie Algebras
Hotlist:
Announcements,
Books,
Homework,
Schedule
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Tevian Dray
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Office hours:
WF 3:15–3:45 PM in Kidder 298A
(and by appt via Zoom)
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Phone: 737-5159
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Email:
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Class Meetings:
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WF 4‐5:20 PM in Kidd 238
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Texts:
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Website:
https://oregonstate.edu/~tevian/onid/MTH675
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Check the announcements page frequently.
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Course Evaluation:
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Based on homework and class participation.
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Course Description:
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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.
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Prerequisites:
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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.
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Students who have previously taken MTH 675 should register for
this course as MTH 679.
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Undergraduates may wish to consider registering for this course as MTH 405.
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Students with Disabilities:
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Accommodations for students with disabilities are determined and approved
by Disability Access Services (DAS). If you, as a student, believe you are
eligible for accommodations but have not obtained approval please contact
DAS immediately. DAS notifies
students and faculty members of approved academic accommodations and
coordinates implementation of those accommodations. While not required,
students and faculty members are encouraged to discuss details of the
implementation of individual accommodations.
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Student Conduct:
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Student conduct is governed by
OSU's Code of Student Conduct.
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Academic Calendar:
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All students are subject to the registration and refund deadlines as
stated in the
Academic Calendar.
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Reach Out for Success:
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University students encounter setbacks from time to time. If you encounter
difficulties and need assistance, it's important to reach out. Consider
discussing the situation with an instructor or academic advisor. Learn
about resources that assist with wellness and academic success at the
Reach Out for Success
website. If you are in immediate crisis, please contact the Crisis Text
Line by texting OREGON to 741-741 or call the National Suicide Prevention
Lifeline at 1-800-273-TALK (8255).
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