Professor: Eleny Ionel, office 383L, ionel "at" math.stanford.edu
Course Assistant: Haotian Xue, office 380H, HaotianXue "at" stanford.edu
Course Description: This course is an introduction to differential geometry, studying the geometry of curves and surfaces. Some of the key notions discussed include curvature (measuring how much they bend), geodesics (the shortest path between two nearby points) and parallel transport, surfaces of constant curvature and minimal surfaces (surfaces minimizing area in a certain sense, eg soap bubbles). Some gems along the way are the Gauss-Bonnet theorem (a beautiful theorem relating the geometry and topology of a surface) and Stokes' theorem (a central ingredient in many applications, which generalizes the Divergence Theorem, Green’s theorem and Stokes’ theorem from multivariable calculus).
This is an upper division class in mathematics, so there will be a reasonable number of proofs. However, the proofs should merely be a formalization of your visual thinking and intuition. While our main focus will be on concepts and understanding the connections between them, there might be more hands-on computations than in a typical upper division math class, and computer software is recommended for visualization as needed.
Prerequisites: Math 52 or equivalent, basically a good understanding of multivariable calculus (and basic linear algebra) which we will use to study the geometry of curves and surfaces.
Course Textbook and Resources: The course does not have a textbook per say, but there are two recommended readings: Neil Donaldson, Introduction to Differential Geometry lecture notes, and Ted Shifrin, Differential Geometry: A First Course in Curves and Surfaces which provides a complementary perspective. (Both are available online and in the Files/Course Materials section of Canvas.)
The course will follow these references, but rather loosely. For that reason, attendance, and taking notes in classes, is strongly encouraged.
A more classical reference you might find useful for some of the topics is Manfredo do Carmo, Differential Geometry of Curves and Surfaces.
Course logistics: Course announcements, homework and other course materials will be posted on Canvas. Weekly homework will be due and graded on Gradescope.
Grading Policy: The course grade is based on the following components:
Homework: There will be weekly homework assignments, given out on Thursday and due the following Thursday before 11:59 pm PDT on Gradescope. No late submissions will be accepted. The lowest homework score will be dropped. You are encouraged to discuss and work together with your classmates on the homework, but you must write up your own solutions, and cite all your collaborators. You may lose points for unclear work. Keep in mind that simply copying solutions from another student or other sources is a waste of everybody's time, as well as a violation of the Stanford Honor Code.
Usually only a portion of each week's assigned problems are graded (and the selection of problems chosen to be graded will not be announced in advance).
Exams: There will be one in class midterm and one final exam. The exam dates are given below; it is your responsibility to verify right now that you can attend these exams. If you have an academic or a competition-related conflict with the scheduled exam time, please contact us as soon as possible, but no later than two weeks before the exam. If an emergency occurs and you need to miss an exam, contact us as soon as possible.
Important dates:
Access and Accommodations: Stanford is committed to providing equal educational opportunities for disabled students. Disabled students are a valued and essential part of the Stanford community. We welcome you to our class. If you experience disability, please register with the Office of Accessible Education (OAE). Professional staff will evaluate your needs, support appropriate and reasonable accommodations, and prepare an Academic Accommodation Letter for faculty. To get started, or to re-initiate services, please visit http://oae.stanford.edu/. If you already have an Academic Accommodation Letter, please use this googleform to upload it and detail the specific accommodations you will need in this course. Letters are preferred by the end of week 2, and at least two weeks in advance of any exam, so we may partner with you and OAE to identify any barriers to access and inclusion that might be encountered in your experience of this course. New accommodation letters, or revised letters, are welcome throughout the quarter; please note that there may be constraints in fulfilling last-minute requests.
Please check for exam conflicts right away. Students with academic or competition-related exam conflicts must fill out the same googleform to make arrangements for an alternate exam sitting on the same day.
Academic Integrity: The Honor Code articulates Stanford University's expectations of students and faculty in establishing and maintaining the highest standards in academic work. Examples of conduct that have been regarded as being in violation of the Honor Code (and are most relevant for this course) include copying from another student's work or allowing another student to copy from your own work; plagiarism; revising and resubmitting your work for regrading without the instructor's knowledge and consent; representing as one's own work the work of another. See http://communitystandards.stanford.edu/ for more information on the Honor Code.