## MATH 497C, Fall 2015

Course Title: Lie Groups in Two, Three and Four Dimensions

Instructor: Nigel Higson

Assistant: Qijun Tan

Meeting Times: Mondays, Wednesdays, Thursdays and Fridays, 10:10-11:00, in 113 McAllister. Generally I will lecture on Mondays, Wednesdays and Fridays, and Qijun will hold court on Thursdays, but we may switch days from time to time.

Office Hours: My office hours are Tuesdays and Wednesdays, 3:40-4:40, in 228 McAllister. Qijun’s office hours are Mondays and Fridays, 9:00-10:00, in 411 McAllister.

Overview: The main protagonists in the course will be groups like SO(3) (the 3x3 rotation matrices) or SL(2,R) (the 2x2 unimodular matrices). These are (matrix) Lie groups and they arise in many places in mathematics and physics. We shall study some of their abstract features, especially as revealed through their close relationship with the rather mysterious Lie algebras, and we shall also study them as they arise in applications.

We shall more or less follow the following book:

Wulf Rossmann, Lie groups. An introduction through linear groups

Oxford Graduate Texts in Mathematics (2002)

But in places we shall be more focused on examples and applications than Rossmann, and correspondingly less focused on the general theory.

Prerequisites: The most important requirement will be some familiarity with linear algebra (vector spaces, and subspaces, bases, linear transformations, determinants, some additional multilinear algebra), with multivariable calculus, and with basic point-set topology (open and closed sets, convergence, continuity, etc). There will be an emphasis on understanding and explanation (that is, proofs) rather than computation (although we shall do computations too). Contact me if you have any questions or concerns.

MASS Program: This course will form part of the 2015 MASS program at Penn State University. See the MASS web pages for further information about the program. Students outside of the MASS program who wish to take this course ought to contact me.

Homework: There will be weekly homework assignments, due on Mondays.

Midterm: There will be a midterm exam on Monday October 5. The exact time and format of the exam will be announced later.

Final Exams: A final oral exam will be scheduled for each student on December 11, 14 or 16. The exam will follow the standard MASS final exam format. More about that later.

Project: Each student will be required to complete a project and present it during the final exam (this is part of the standard MASS exam format). A list of possible project titles will be distributed at about the time of the midterm, along with further details about the project requirement.

Academic Integrity Statement: All Penn State policies regarding ethics and honorable behavior apply to this course (see links below for policy statements). Academic integrity is the pursuit of scholarly activity free from fraud and deception and is an educational objective of this institution. All University policies regarding academic integrity apply to this course. Academic dishonesty includes, but is not limited to, cheating, plagiarizing, fabricating of information or citations, facilitating acts of academic dishonesty by others, having unauthorized possession of examinations, submitting work of another person or work previously used without informing the instructor, or tampering with the academic work of other students. For any material or ideas obtained from other sources, such as the text or things you see on the web, in the library, etc., a source reference must be given. Direct quotes from any source must be identified as such. All exam answers must be your own, and you must not provide any assistance to other students during exams. Any instances of academic dishonesty WILL be pursued under the University and Eberly College of Science regulations concerning academic integrity.

(For a more compelling account of what honesty and integrity should mean, at least for a scientist, consider these famous words of Richard Feynman.)

Disability Statement: Penn State welcomes students with disabilities into the University's educational programs. Every Penn State campus has an office for students with disabilities. The Office for Disability Services (ODS) Web site provides contact information for every Penn State campus: http://equity.psu.edu/ods/dcl. For further information, please visit the Office for Disability Services Web site: http://equity.psu.edu/ods. In order to receive consideration for reasonable accommodations, you must contact the appropriate disability services office at the campus where you are officially enrolled, participate in an intake interview, and provide documentation: http://equity.psu.edu/ods/doc-guidelines. If the documentation supports your request for reasonable accommodations, your campus’s disability services office will provide you with an accommodation letter. Please share this letter with your instructors and discuss the accommodations with them as early in your courses as possible. You must follow this process for every semester that you request accommodations.

Instructor: Nigel Higson

Assistant: Qijun Tan

Meeting Times: Mondays, Wednesdays, Thursdays and Fridays, 10:10-11:00, in 113 McAllister. Generally I will lecture on Mondays, Wednesdays and Fridays, and Qijun will hold court on Thursdays, but we may switch days from time to time.

Office Hours: My office hours are Tuesdays and Wednesdays, 3:40-4:40, in 228 McAllister. Qijun’s office hours are Mondays and Fridays, 9:00-10:00, in 411 McAllister.

Overview: The main protagonists in the course will be groups like SO(3) (the 3x3 rotation matrices) or SL(2,R) (the 2x2 unimodular matrices). These are (matrix) Lie groups and they arise in many places in mathematics and physics. We shall study some of their abstract features, especially as revealed through their close relationship with the rather mysterious Lie algebras, and we shall also study them as they arise in applications.

We shall more or less follow the following book:

Wulf Rossmann, Lie groups. An introduction through linear groups

Oxford Graduate Texts in Mathematics (2002)

But in places we shall be more focused on examples and applications than Rossmann, and correspondingly less focused on the general theory.

Prerequisites: The most important requirement will be some familiarity with linear algebra (vector spaces, and subspaces, bases, linear transformations, determinants, some additional multilinear algebra), with multivariable calculus, and with basic point-set topology (open and closed sets, convergence, continuity, etc). There will be an emphasis on understanding and explanation (that is, proofs) rather than computation (although we shall do computations too). Contact me if you have any questions or concerns.

MASS Program: This course will form part of the 2015 MASS program at Penn State University. See the MASS web pages for further information about the program. Students outside of the MASS program who wish to take this course ought to contact me.

Homework: There will be weekly homework assignments, due on Mondays.

Midterm: There will be a midterm exam on Monday October 5. The exact time and format of the exam will be announced later.

Final Exams: A final oral exam will be scheduled for each student on December 11, 14 or 16. The exam will follow the standard MASS final exam format. More about that later.

Project: Each student will be required to complete a project and present it during the final exam (this is part of the standard MASS exam format). A list of possible project titles will be distributed at about the time of the midterm, along with further details about the project requirement.

Academic Integrity Statement: All Penn State policies regarding ethics and honorable behavior apply to this course (see links below for policy statements). Academic integrity is the pursuit of scholarly activity free from fraud and deception and is an educational objective of this institution. All University policies regarding academic integrity apply to this course. Academic dishonesty includes, but is not limited to, cheating, plagiarizing, fabricating of information or citations, facilitating acts of academic dishonesty by others, having unauthorized possession of examinations, submitting work of another person or work previously used without informing the instructor, or tampering with the academic work of other students. For any material or ideas obtained from other sources, such as the text or things you see on the web, in the library, etc., a source reference must be given. Direct quotes from any source must be identified as such. All exam answers must be your own, and you must not provide any assistance to other students during exams. Any instances of academic dishonesty WILL be pursued under the University and Eberly College of Science regulations concerning academic integrity.

(For a more compelling account of what honesty and integrity should mean, at least for a scientist, consider these famous words of Richard Feynman.)

Disability Statement: Penn State welcomes students with disabilities into the University's educational programs. Every Penn State campus has an office for students with disabilities. The Office for Disability Services (ODS) Web site provides contact information for every Penn State campus: http://equity.psu.edu/ods/dcl. For further information, please visit the Office for Disability Services Web site: http://equity.psu.edu/ods. In order to receive consideration for reasonable accommodations, you must contact the appropriate disability services office at the campus where you are officially enrolled, participate in an intake interview, and provide documentation: http://equity.psu.edu/ods/doc-guidelines. If the documentation supports your request for reasonable accommodations, your campus’s disability services office will provide you with an accommodation letter. Please share this letter with your instructors and discuss the accommodations with them as early in your courses as possible. You must follow this process for every semester that you request accommodations.