## MATH 597E, Spring 2015

Course Title: Loop Groups

Instructor: Nigel Higson

Meeting Times: Tuesdays and Thursdays, 11:15-12:30 in 104 Osmond.

Office Hours: By appointment, or just try your luck and stop by my office, 228 McAllister.

Overview: We shall study the geometry and representation theory of the group of smooth maps from the circle into a compact Lie group. As a warm up we shall start by studying the same topics in the context of a compact group. At the end, if time permits, we shall examine the representation theory of loop groups from an algebraic perspective and study the Dirac operator.

Texts: There will be no official textbooks. But I’ll follow the book Loop Groups by Pressley and Segal quite closely, at least at the beginning. I’ll provide other references as we go along.

Prerequisites: The main requirements will be a basic familiarity with Hilbert space theory, a basic familiarity with manifolds and differential forms, and a passing familiarity with Lie groups and Lie algebras.

Homework: I’ll hand out occasional homework problems for later discussion (rather than written submission of solution). Class participants are encouraged to work together on these.

Exams: There will be none.

Academic Integrity: 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 (or a mathematician), 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

Meeting Times: Tuesdays and Thursdays, 11:15-12:30 in 104 Osmond.

Office Hours: By appointment, or just try your luck and stop by my office, 228 McAllister.

Overview: We shall study the geometry and representation theory of the group of smooth maps from the circle into a compact Lie group. As a warm up we shall start by studying the same topics in the context of a compact group. At the end, if time permits, we shall examine the representation theory of loop groups from an algebraic perspective and study the Dirac operator.

Texts: There will be no official textbooks. But I’ll follow the book Loop Groups by Pressley and Segal quite closely, at least at the beginning. I’ll provide other references as we go along.

Prerequisites: The main requirements will be a basic familiarity with Hilbert space theory, a basic familiarity with manifolds and differential forms, and a passing familiarity with Lie groups and Lie algebras.

Homework: I’ll hand out occasional homework problems for later discussion (rather than written submission of solution). Class participants are encouraged to work together on these.

Exams: There will be none.

Academic Integrity: 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 (or a mathematician), 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.