Math 829F: Riemann Surfaces

Meetings:
Time: 1.00 pm - 2.15 pm
Days: Monday - Thursday
Room: Jabara (JB) 336
Office Hours: 12.00 pm - 1.00 pm, Monday - Thursday, and by appointment


Contact Info:
Instructor: Justin Ryan
Office: Jabara (JB) 362
Phone: (316) 978 - 5089
Email: ryan@math.wichita.edu, or
Email: jryan@fceia.unr.edu.ar
Slack (preferred method): riemannsurfaces.slack.com
Instructions: Just click the link above and use your @math.wichita.edu email address to sign up (it's free!). If you don't have an @math.wichita.edu email adress, that's okay. Just send me an email and I'll invite you to the group.

Basic Course Information

Prerequisites:
Departmental consent.


Course Description:
This course covers introductory material in the study of Riemann Surfaces using methods of topology, differential geometry, and complex analysis. The course can be repeated with departmental consent.


References:
The main (required) reference for this course is:
Book: J. Jost, Compact Riemann Surfaces

The following may be used as supplementary references:
Book: L. Ahlfors and L. Sario, Riemann Surfaces
Notes: W. Schlag, A concise course in complex analysis and Riemann surfaces

We will also use quite a lot of material from other areas of mathematics. A bibliography will be generated as we go along, but students are not required to buy nor read such outside sources.


Supplies:
A three-ring binder is suggested, as well as a hole-punch, which is to be "used relentlessly," as stated in the New York Times. I also recommend that you bring pens in various colors, in order to replicate what is on the board. You should bring your book every day, but the binder can be left at home. It will be used to organize the materials and hand-outs that are distributed in class, as well as your own notes.


Class Protocol:
Attendance is not required. However, if you are not present, you will not be able to complete the activities that correspond to that day's discussion and work in class. Students are asked to be on time, and to notify me if they will be absent. They are asked to observe common norms of civility in class and in interactions with me and with classmates outside of class.

Detailed Course Information

Course Content:
I have a general idea of how I think this course should run and what topics we should study, but nothing is set in stone. For the most part the course will follow Jost's book. All other references will be supplementary. The number of topics that we study, and the pace at which we study them, will depend on the experience and interest levels of the students. Regardless, this course will have two parts.

Part I: Introduction (~3-4 weeks):
We will begin the course by covering the introductory topics in Differential Topology and Complex Analysis necessary to define and study basic properties of Riemann surfaces. In the process we will construct and study numerous examples. We will then study some of the basic differential geometric properties of Riemann surfaces. This portion of the course will cover chapters 1 and 2 of Jost's book.

Part II: Further Study (~3-4 weeks):
Once we have a good foundation, we will begin a more extensive study of Riemann surfaces. Potential topics for further study include (but are not limited to): Harmonic Theory, Hodge Theory, Teichmüller Theory, Divisors and the Riemann-Roch Theorem, and the Uniformization Theorem. We will probably start this portion of the course with chapter 4 of Jost's book, then mix in topics from chapters 3 and 5 as time permits.


Assignments and Coursework:
The coursework for this class will be divided into four categories.

Recommended Exercises (0%):
I will assign about 10-15 exercises per week that will not be collected nor graded, but that you will be expected to (and strongly encouraged to!) at least attempt.

Weekly Assignments (30%):
Each week I will assign 3-4 exercises to be typed in LaTeX and handed in. These exercises are meant to cover the key ideas from the past week's lectures, and to make sure you are keeping up with the pace of the course. To avoid wasting paper, these assignments should be submitted via email or Slack each week before an agreed upon deadline. Late assignments will not be accepted.

Midterm Exam (20%):
Between parts I and II of the course, you will have a midterm exam. It will consist of important definitions, fundamental proofs, and a selection of recommended exercises.

Good Problems (50%):
Each student will complete a unique "Good Problem" to be presented during the last week of class. The Good Problems will cover material from part II of the course.


Grading:
As a Graduate "Topics Course" the numerical grade scale for this course is different than that of a traditional undergraduate course. Please familiarize yourselves with it and ask any questions well before the end of the course. Your final average will be weighted as indicated above in the Coursework section of this syllabus.

Letter Grade	Numerical Grade
A	90 - 100
A-	85 - 89
B+	80 - 84
B	75 - 79
B-	70 - 74
C	60 - 69
D	50 - 59
F	< 50

Measurable Student Learning Outcomes

General Course Outcomes:
The subject area identified as Riemann Surfaces is a very classical and extremely rich body of knowledge lying in the intersection of topology, complex analysis, and differential geometry. Accordingly, one applies results and methods from each of these areas. That is not to say that the study of Riemann Surfaces is only an application of those methods and results. On the contrary, the synthesis and interaction of the topological, geometric, and complex analytic structures on a Riemann surface make its study unique and very rich. Moreover, a Riemann Surface is a one-dimensional complex manifold, and therefore serves as the most fundamental of examples in Complex Differential Geometry. It is also a complex curve, a principal object of study in Algebraic Geometry. Upon completion of the course, students will be able to compute the fundamental geometric quantities associated to Riemann Surfaces, and apply algebraic and topological tools to investigate their structure. Students will be well-prepared for further study in related fields such as those mentioned here.

Part I Outcomes:
Chapter 1 - Topological Foundations:
The student verifies that standard examples are differentiable manifolds, calculates the fundamental group of familiar examples, and uses the notion of covering spaces to enlarge their set of familiar reference examples.

Chapter 2 - Differential Geometry of Riemann Surfaces:
The student examines explicit examples and determines whether these concrete examples satisfy the defining criteria of a Riemann Surface. The student incorporates previous knowledge of inner products into this context, in order to extend notions of geometry to the setting of a Riemann Surface. The student explores further topological properties and classifies compact Riemann surfaces. The student investigates further the interaction between the complex analytic, topological, and geometric structures of a Riemann Surface, through the major theorems of Gauss-Bonnet, Riemann-Hurwitz, and Schwartz. The student applies this to classify the complex structures of a torus.

Part II Outcomes:
Chapter 3 - Harmonic Maps:
The student combines energy techniques with explicit computations to obtain Euler-Lagrange equations that describe minimizers for familiar geometric problems. The student extends and generalizes to the concept of a harmonic map between two manifolds.

Chapter 4 - Teichmüller Spaces:
The student now synthesizes and assembles the foregoing material to describe and classify conformal structures on Riemann Surfaces. The student applies the concrete tools of Teichmüller spaces to further illuminate the space of all conformal structures on a given Riemann surface.

Chapter 5 - Geometric Structures on Riemann Surfaces:
Time permitting, the student will finish the course with a study of differential forms, divisors, and line bundles on Riemann surfaces. The student will be able to prove the Riemann-Roch theorem and apply the result to numerous applications.

University Policies and Procedures

Academic Honesty:
Students are responsible for knowing and following the Student Code of Conduct and the Student Academic Honesty policy.


Definition of a Credit Hour:
Success in this three-credit-hour course is based on the expectation that students will spend, for each unit of credit, a minimum of forty five hours over the length of the course (normally three hours per unit per week, with one of the hours used for lecture) for instruction and preparation/studying or course related activities for a total of 135 hours. Read this to learn about the policy and examples of different types of courses and credit hour offerings.


Important Academic Dates:
For the Summer semester of 2016, classes begin on June sixth and end on July twenty-ninth. The last date to drop the class with nothing on your record is June eighth, and the last date to drop the class with a W (meaning "withdrawn") instead of an F (a failing grade) is July twelfth. There are no classes on July Fourth. The final exam period is the last week of class. (Good Problem presentations should be regarded as the final exam for this course.)


Disabilities:
If you have a physical, psychiatric/emotional, or learning disability that may impact on your ability to carry out assigned course work, I encourage you to contact the Office of Disability Services (DS). The office is located in Grace Wilkie Annex, Room 150, 316-978-3309 (voice/tty) and 316-854-3032 (videophone). DS will review your concerns and determine, with you, what academic accommodations are necessary and appropriate for you. All information and documentation of your disability is confidential and will not be released by DS without your written permission.


Counseling and Testing:
The Wichita State University Counseling and Testing Center provides professional counseling services to students, faculty, and staff; administers tests and offers test preparation workshops; and presents programs on topics promoting personal and professional growth. Services are low cost and confidential. They are located in Room 320 of Grace Wilkie Hall, and their phone number is 316-978-3440. The Counseling and Testing Center is open on all days that the university is officially open. If you have a mental health emergency during the times that the Couseling and Testing Center is not open, please call COMCARE Crisis Services at 316-660-7500.


Diversity and Inclusion:
Wichita State University is committed to being an inclusive campus that reflects the evolving diversity of society. To further this goal, WSU does not discriminate in its programs and activities on the basis of race, religion, color, national origin, gender, age, sexual orientation, gender identity, gender expression, marital status, political affiliation, status as veteran, genetic information or disability. The following person has been designated to handle inquiries regarding non-discrimination policies: Executive Director, Office of Equal Employment Opportunity, Wichita State University, 1845 Fairmount, Wichita, KS, 67260-0138; telephone 316-978-3186.


Intellectual Property:
Wichita State University students are subject to Board of Regents and University policies regarding intellectual property rights. Any questions regarding these rights and any disputes that arise under these policies will be resolved by the President of the University, or the President's designee, and such decision will constitute the final decision.


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