Peter B Gilkey
202 Deady Hall,1-541-346-4717 (office phone) 1-541-346-0987 (fax) email: gilkey@uoregon.edu
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

    Grades in Graduate Courses

    The faculty has reached basic agreement on the meaning of grades for graduate students in the 500- and 600-level courses: Faculty teaching 600-level courses shall have the option to use different (but functionally equivalent) assessment procedures to grade students who have been admitted to the Ph.D. program compared to students in the Master's/Pre-Ph.D. stage of the of the program.

    Academic dishonesty

    Academic Misconduct: The University Student Conduct Code (available at conduct.uoregon.edu) defines academic misconduct. Students are prohibited from committing or attempting to commit any act that constitutes academic misconduct. By way of example, students should not give or receive (or attempt to give or receive) unauthorized help on assignments or examinations without express permission from the instructor. Students should properly acknowledge and document all sources of information (e.g. quotations, paraphrases, ideas) and use only the sources and resources authorized by the instructor. If there is any question about whether an act constitutes academic misconduct, it is the obligation of the students to clarify the question with the instructor before committing or attempting to commit the act. Additional information about a common form of academic misconduct, plagiarism, is available at http://library.uoregon.edu/guides/plagiarism/students/index.html see also http://uodos.uoregon.edu/StudentConductandCommunityStandards/AcademicMisconduct/tabid/248/Default.aspx

    Title IX

    Under Title IX, I have a duty to report relevant information. The UO is committed to providing an environment free of all forms of prohibited discrimination and sexual harassment, including sexual assault, domestic and dating violence and gender-based stalking. Any UO employee who becomes aware that such behavior is occurring has a duty to report that information to their supervisor or the Office of Affirmative Action and Equal Opportunity. The University Health Center and University Counseling and Testing Center can provide assistance and have a greater ability to work confidentially with students. Note: UO employees also have a duty to report child abuse. For those classes and/or processes in which students have historically reported information regarding child abuse, the language can be expanded to provide that notice as well by adding the following statement: All UO employees are required to report to appropriate authorities when they have reasonable cause to believe that any child with whom they come in contact has suffered abuse or any person with whom they come in contact has abused a child.

    Ethical Standards

    From the President's Office 2 May 2014: The University of Oregon is a community of scholars dedicated to the highest standards of academic inquiry, learning, and service. We are also committed to the highest standards of ethics as we work to fulfill our mission. We all share responsibility for ensuring that we conduct our transactions in ways that are ethical, honest, and reflect sound fiduciary practices. To accomplish this, it is important that all UO employees review, understand, and consistently practice the standards included in the following laws, rules, and policies including:
    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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    5. The UO will continue a similar focus on these important issues under our new governance structure. We will communicate any changes to reporting protocols after July 1. We are all responsible for understanding and complying with ORS 244, applicable government regulations and policies. We also have a responsibility to raise compliance and ethics concerns through established channels. I appreciate your commitment to integrity and honesty, as it is an essential element in maintaining an ethical and secure UO workplace environment for everyone.

    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 CouncilŐs 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 srviceprovost@uoregon.edu.


  • 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 11 October 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 Email:peter.gilkey.cc.67@aya.yale.edu of Deady Spider Enterprises