## MATH 533, Fall 2017

Course Title: Lie Theory I

Instructor: Nigel Higson

Class Meeting Times: Mondays, Wednesdays and Fridays, 2:30-3:20 in 104 Osmond

Office Hours: By appointment in 228 McAllister (contact me in class or by email)

Overview: Lie theory brings together multivariable calculus, differential equations, topology, group theory, linear algebra and more in a fascinating way. It is the theory of “continuous symmetries” such as the group of rotations of a circle or a sphere, to mention only the most elementary examples, and applications occur throughout mathematics and beyond. This is an introductory course on the subject, and I’ll begin very concretely by discussing Lie groups of matrices and the exponential map. It will soon be clear why it is natural to recast the theory in the language of manifolds, and this is what I will do next (in the reverse direction, many very interesting examples and techniques in manifold theory come from Lie theory). Once the fundamental theorems relating Lie groups and Lie algebras have been covered, I’ll turn to the basic taxonomy of Lie groups (nilpotent, solvable, semisimple), and then to one or two topics in representation theory, both finite-dimensional and infinite-dimensional.

Course and Grading Policies: Attendance at all lectures is very strongly encouraged! In addition to class attendance and participation, a variety of homework problems and assignments will be provided that I hope will help deepen everyone’s understanding of the material discussed in class. There will be no exams. Grades will be assessed on the basis of class participation (50%) and homework (50%).

Text: There is no required textbook. There are many very good introductions to Lie theory in both book form and lecture note form, quite a few of them easily available online. See this list.

Prerequisites: A good grasp of linear algebra; some familiarity with abstract algebra and smooth manifolds. If in doubt, please contact me.

Academic Integrity: Academic integrity is the pursuit of scholarly activity in an open, honest and responsible manner. Academic integrity is a basic guiding principle for all academic activity at The Pennsylvania State University, and all members of the University community are expected to act in accordance with this principle. Consistent with this expectation, the University's Code of Conduct states that all students should act with personal integrity, respect other students' dignity, rights and property, and help create and maintain an environment in which all can succeed through the fruits of their efforts.

Students with Disabilities: Penn State welcomes students with disabilities into the University's educational programs. If you have a disability-related need for reasonable academic adjustments in this course, contact Student Disability Resources at 814-863-1807 (V/TTY). For further information, please visit Student Disability Resources web site: http://equity.psu.edu/student-disability-resources/ . In order to receive consideration for accommodations, you must contact SDR and provide documentation (see the documentation guidelines at http://equity.psu.edu/student-disability-resources/). If the documentation supports your request for reasonable accommodations, SDR will provide you with an accommodation letter identifying appropriate academic adjustments. Please share this letter with me and discuss the accommodations with me as soon as possible.

Information on Available Counseling & Psychological Services: Students with an interest in obtaining counseling services may wish to contact the Penn State Counseling & Psychological Services Office. More information about the Counseling & Psychological Services Office can be found here: http://studentaffairs.psu.edu/counseling/

Instructor: Nigel Higson

Class Meeting Times: Mondays, Wednesdays and Fridays, 2:30-3:20 in 104 Osmond

Office Hours: By appointment in 228 McAllister (contact me in class or by email)

Overview: Lie theory brings together multivariable calculus, differential equations, topology, group theory, linear algebra and more in a fascinating way. It is the theory of “continuous symmetries” such as the group of rotations of a circle or a sphere, to mention only the most elementary examples, and applications occur throughout mathematics and beyond. This is an introductory course on the subject, and I’ll begin very concretely by discussing Lie groups of matrices and the exponential map. It will soon be clear why it is natural to recast the theory in the language of manifolds, and this is what I will do next (in the reverse direction, many very interesting examples and techniques in manifold theory come from Lie theory). Once the fundamental theorems relating Lie groups and Lie algebras have been covered, I’ll turn to the basic taxonomy of Lie groups (nilpotent, solvable, semisimple), and then to one or two topics in representation theory, both finite-dimensional and infinite-dimensional.

Course and Grading Policies: Attendance at all lectures is very strongly encouraged! In addition to class attendance and participation, a variety of homework problems and assignments will be provided that I hope will help deepen everyone’s understanding of the material discussed in class. There will be no exams. Grades will be assessed on the basis of class participation (50%) and homework (50%).

Text: There is no required textbook. There are many very good introductions to Lie theory in both book form and lecture note form, quite a few of them easily available online. See this list.

Prerequisites: A good grasp of linear algebra; some familiarity with abstract algebra and smooth manifolds. If in doubt, please contact me.

Academic Integrity: Academic integrity is the pursuit of scholarly activity in an open, honest and responsible manner. Academic integrity is a basic guiding principle for all academic activity at The Pennsylvania State University, and all members of the University community are expected to act in accordance with this principle. Consistent with this expectation, the University's Code of Conduct states that all students should act with personal integrity, respect other students' dignity, rights and property, and help create and maintain an environment in which all can succeed through the fruits of their efforts.

Students with Disabilities: Penn State welcomes students with disabilities into the University's educational programs. If you have a disability-related need for reasonable academic adjustments in this course, contact Student Disability Resources at 814-863-1807 (V/TTY). For further information, please visit Student Disability Resources web site: http://equity.psu.edu/student-disability-resources/ . In order to receive consideration for accommodations, you must contact SDR and provide documentation (see the documentation guidelines at http://equity.psu.edu/student-disability-resources/). If the documentation supports your request for reasonable accommodations, SDR will provide you with an accommodation letter identifying appropriate academic adjustments. Please share this letter with me and discuss the accommodations with me as soon as possible.

Information on Available Counseling & Psychological Services: Students with an interest in obtaining counseling services may wish to contact the Penn State Counseling & Psychological Services Office. More information about the Counseling & Psychological Services Office can be found here: http://studentaffairs.psu.edu/counseling/