Peter B Gilkey
202 Deady Hall,1-541-346-4717 (office phone) 1-541-346-0987 (fax) email:
Mathematics Department, University of Oregon, Eugene Oregon 97403 USA

Math 607 Spring 2015 The Atyiah Singer Index Theorem

  • Office Hours:  Monday, Wednesday, Friday 1000-1050 or by appointment.
  • Meets Monday, Wednesday, Friday 1200-1250
  • During the course I will outline results leading to the Atiyah-Singer index theorem
    1. Characteristic classes from the point of view of curvature (Euler form, Pontrjagin Classes, Chern Classes, Hirzebruch Polynomial, Todd Genus, Chern Character).
    2. The classic elliptic complexes (de Rham, Signature, Spin, Dolbeault).
    3. Heat equation asymptotics.
    4. The Gauss-Bonnet theorem, the Hirzebruch Signature Theorem, the A-roof genus, the Riemann-Roch theorem.
    5. The Atiyah-Singer theorem in general.
    6. We will then turn to the analytic underpinnings. Fourier Transform, Sobolev Spaces, Rellich Lemma, Sobolev inequality.
    7. Brief introduction to the pseudo-differential calculus and the heat equation.
  • The final grade will be based on an oral exam at the end of the course.

    Learning outcomes

    Learning outcomes are brief statements identifying the major skills, abilities, and concepts a student is expected to acquire from your course. The word “outcomes” can be used interchangeably with “goals” or “objectives” as long as the abilities in question are meaningfully evaluated using exams, papers, and other accepted means. The point is to make your expectations more transparent by articulating what may be only implicit in your course description, lesson topics, and assignments. Three to six short sentences or bullet points will suffice. Active verbs (evaluate, analyze, demonstrate, etc.) concretize expectations better than vague ones (appreciate, study, learn, etc.). The final grade will be based on an oral exam at the end of the course which will test the major skills, abilities, and concepts a student is expected to acquire from the course.

    Useful information

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    Academic dishonesty

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    Title IX

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    Ethical Standards

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    1. ORS Chapter 244, which codifies ethics and conflict of interest policies that you are required to follow as you conduct University of Oregon business. See the guide for public officials here.
    2. The Oregon University System (OUS) has a responsibility to prevent and detect fraud, waste, and abuse and to hold accountable those who engage in it. The OUS Fraud, Waste, and Abuse policy sets forth guidelines for reporting known or suspected fraud, waste, or abuse within any OUS institution.
    3. If you are aware of fraud, waste, or abuse occurring at the UO or within the OUS, matters can be reported to campus management, OUS Internal Audit Division, or OUS Financial Concerns Hotline. Additional information is also available on the UO Business Affairs and UO vice president for finance and administration webpages.
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    Statement on Final Exams

  • 1. In the week preceding final examination during fall, winter, and spring terms: No examination worth more than 20% of the final grade will be given, with the exception of make-up examinations. No final examinations will be given under any guise. No work that will be evaluated for grades/credit will be due unless it has been clearly specified on the class syllabus within the first two weeks of the term.
  • 2. Take-home examinations will be due no earlier than the day of the formally assigned final examination for the class in question.

    This action clarifies and extends earlier faculty legislation (1911 Faculty Assembly archives) prohibiting the giving of final examinations earlier than officially scheduled.

    In addition, you should be aware of the Faculty Advisory Council’s statement on students with multiple exams:

    Examination schedules are listed each term in the Time Schedule. Students who are scheduled to take more than three examinations within one calendar day may take the additional examination(s) as makeup examination(s) later in the examination week. The instructor(s) of record for the course(s) beyond the third examination, counting in the order the examination(s) are scheduled, will arrange for (a) makeup examination(s).

    The following procedures were approved by the Undergraduate Council to address rare circumstances of competing exam times. Students with examination conflicts may contact the Office of Academic Advising for assistance.

    In the case of two examinations scheduled at the same time, the course with the largest enrollment must provide an alternate examination. For conflicts between regular courses and combined examinations, the combined examination course must provide the alternate examination. For combined examinations with conflicts, the largest combined enrollment course must provide the alternative examination.

    Questions and concerns regarding this policy should be directed first to the relevant instructor, then the department head, and finally the dean if necessary. You may also find reference to the policy on the Academic Affairs website. If additional input is needed, please contact

  • To rest on the blue of the day, like an eagle rests on the wind, over the cold range, confident on its wings and its breadth.

    Web page updated on 18 September 2014 by Peter B Gilkey 202 Deady Hall, Department of Mathematics at the University of Oregon, Eugene OR 97403-1222, U.S.A. Phone 1-541-346-4717 of Deady Spider Enterprises