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

Office Hours: Monday, Wednesday, Friday 1000-1050 or by appointment. Meets MWF 0800-0850. Problem session F1600-1650. Deady 210. Note: No class Memorial Day Monday May 28 2014. There will be no text in Math 619. However, there are some informal notes available from 1992. There are no formal prerequisites. In particular the Math 616-617 sequence is not a prerequisite. The material of Math 515 would be helpful. Green's theorem will be derived from scratch. If in doubt about your background, send me an email gilkey@uoregon.edu Homework will be due each Mondayon the material of the subsequent week. TheFridaydiscussion hour is an opportunity for you to ask questions about the homework. The homework problems will be challenging and it is essential that you have thought about the homework before comming to the discussion hour. You should also feel free to ask questions regarding the lecture that have come up then (or during class of course). I will drop your 2 lowest homework scores in computing the homework average. This is to allow for life's little emergencies in case you have to miss turning in 1 or 2 homeworks. Late homework will not be accepted.Homework Assignments

- Home Work Assignment #1 due Monday April 6 2014.
Grade: Will be based

- 25% on the homework
- 25% on the mid term Wednesday 30 April 2014
- 50% on the Final Exam Tuesday 10 June 2014 10:15.

## Course Goals

We will discuss the basic material of complex variables. Although we will develop everything "from scratch", this is a 600 level course and a fair amount of mathematical maturity will be assumed. We will begin with a treatment of complex differentiability. We will use Green's theorem to establish the Cauchy Integral Formula and to establish the basic properties of holomorphic functions (the uniform limit of holomorphic functions is holomorphic, the composite of holomorphic functions is holomorphic, the set of holomorphic functions on an open set forms a ring, a function which has one holomorphic derivative is smooth, etc.) We will discuss the Arzela-Ascoli theorem, the winding number, theorem of removable singularity, etc. We will then use the Cauchy Integral Formula to discuss Taylor series, analytic continuation, maximum modulus, the zeros of a holomorphic are isolated, and Laurent series. We will treat essential singularities just a bit and then pass to the residue calculus, poles, and discuss various techniques of computing integrals. We will discuss the complex log function and some techniques for evaluating infinite series using the residue calculus. We will then turn our attention to infinite products and construct functions with arbitrary zeros and poles. This comprises the standard material of any course on complex variables. We will then turn to various more specialized topics. Amongst those might be the following. The Gamma function plays an important role both in the study of partial differential equations (via the Mellin transform) and in analytic number theory. We will construct the gamma function, introduce the Riemann zeta function, and show there are an infinite number of primes congruent to 1,2 mod 3 and to 1,3 mod 4. The Riemannian mapping theorem states that any proper subset of C if simply connected is biholomorphic with the open unit disk and is crucial in the study of Riemann surfaces. We shall show the Schwarz lemma and establish the Riemann mapping theorem. The theory of elliptic functions involves meromorphic functions on the torus and leads to a determination of the ring of meromorphic functions on the torus. It provides a useful review of the material of the course. One could also investigate some results in several complex variables including domains of holomorphy. The precise choice of topics will depend on the interest of the class and the time permitting.## Learning outcomes

Learning outcomes are short, concrete statements describing the skills and knowledge students should be able to demonstrate after completing a course or degree program. The term can be used interchangeably with ÒgoalsÓ or ÒobjectivesÓ as long as the abilities in question will be meaningfully evaluated using exams, papers, and other accepted means. The best students can infer expected learning outcomes from reading a course description and a list of assignments. Other students need to have them pointed out explicitly, typically in a short list of well-formulated bullet points. Active verbs (evaluate, analyze, demonstrate, etc.) concretize expectations better than vague ones (appreciate, study, learn, etc.).

- Student should be able to demonstrate they can use the basic techniques of complex variables to solve specific calculational problems (evaluate definite integrals, compute residues, find Taylor series, etc.)
- Students must be able to apply the basic theorems of complex variables so give rigorous proofs not only of results which have been established in class, but other results which follow from these theorems; exemplars will be provided in the homeworks.
- This is a 600 level graduate class. Students must demonstrate that they can handle theoretical material at a high level of abstraction and give well written, clear mathematical proofs.
## 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.

- A+ Truly outstanding work
- A Good Ph.D. or M.S./M.A. level work
- A- Clearly Ph.D. level work, but below average. Good at M.S./M.A. level
- B+ Work which is at the lower margin of acceptable Ph.D. level work, but quite satisfactory at the M.S./M.A. level
- B Substandard at the Ph.D. level but satisfactory at the M.S./M.A. level
- B- Barely passing at the graduate level
- C+ or below. Unsatisfactory at the graduate level 5
## 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 students' obligation 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.## 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.## Other information

The university is committed to providing an environment for work and learning that is free from unlawful discrimination, including sexual harassment (which includes sexual assault, intimate partner or dating violence, and gender-based stalking or bullying). The safety of persons who participate in university programs and activities is critical. In support of these priorities and consistent with legal obligations, university employees have a responsibility to report instances of certain inappropriate conduct as outlined in this letter. The intent of this letter is to communicate these reporting responsibilities that apply to all UO employees and to make clear how to report and to whom.Discrimination and Discriminatory Harassment: Oregon law requires that all university employees with credible evidence that any form of prohibited discrimination by or against students, faculty members, staff members, or visitors to our campus is occurring have a duty to report that information. ÒProhibited discriminationÓ includes: Discrimination on the basis of age, disability, national origin, race, marital status, religion, gender, gender identity, gender expression or sexual orientation; and Discriminatory harassment, including all forms of sexual harassment. Reports are to be made to the employeeÕs supervisor or to the Office of Affirmative Action and Equal Opportunity (OAAEO) at 541-346-3123; or via email to the Office of Affirmative Action and Equal Opportunity. Any UO supervisor who has been notified of credible evidence that prohibited discrimination is occurring has a duty to report that to the OAAEO. Penelope Daugherty, Director of OAAEO and Title IX Coordinator, 541-346-2971, penny@uoregon.edu, is the contact person for questions about the duty to report discrimination and discriminatory harassment. Child Abuse Under the Oregon Child Abuse Reporting Statutes, all UO employees have a duty to make a report to the Oregon Department of Human Services or a law enforcement agency when they have reasonable cause to believe any child with whom the employee comes in contact has suffered abuse or that any person with whom the employee comes in contact has abused a child. For instances that relate to UO authorized activities, UO employee are to report to the University of Oregon Police Department. For purposes of this reporting responsibility, a ÒchildÓ is any Òunmarried person who is under 18 years of ageÓ and ÒabuseÓ includes, but is not limited to: assault of a child; physical injury to a child caused by other than accidental means; any mental injury to a child caused by cruelty to a child; rape of a child; sexual abuse; sexual exploitation; á negligent treatment or maltreatment of a child; threat of harm to a child; buying or selling of a child; allowing a child on the premises where methamphetamine is being manufactured; and unlawful exposure to a controlled substance that subject a child to risk of harm. The duty of employees of public universities to report incidents of child abuse applies at all times, not just to those incidents occurring during working hours or on campus. For this purpose, university employees include all faculty and staff members, student workers, graduate teaching fellows, and temporary employees. Under the law, reports must be made to the local office of the Department of Human Services or to a law enforcement agency in the county where the employee making the report is located at the time of the contact. Failure to report when required to do so is a Class A violation. Persons who make reports in good faith are immune from liability for making the report. For instances that relate to UO-authorized activities, UO employees are expected to make the report immediately to the UO Police Department at 541-346-2919. Karen Logvin, Director of Work/Life Resources in Human Resources, 541-346-2962, klogvin@uoregon.edu, is the initial point of contact for further questions related to the reporting of child abuse. In addition, you will find additional information and resources regarding mandatory reporting of child abuse and neglect at "http://hr.uoregon.edu/policies-leaves/general-information/mandatory-reporting-child-abuse-and-neglect

Teaching Associate: Ekaterina Puffini. Academic Calendar

Web page updated on 31 March 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 |