MATH-UA 377: Differential Geometry (Revised March 22, 2020)

Calendar (subject to change)
The rest of the semester will follow O'Neill, Chapters 4, 6, 7, 8

NYU Classes

All announcements and information about the course will be emailed to you and posted here. You are responsible for keeping track of everything, including newly added assignments, exam dates, homework due dates, etc. NO EXCUSES.

Homework: Overleaf, Gradescope

Questions?

• Work together with classmates. But be careful! If you simply copy someone else's answers without struggling through the problem yourself, you will not be ready for the exams. Get help from other students but, when preparing final version of your solutions, do it alone.
• Ask me during office hours. If you cannot make it during the scheduled times, please email me to make an appointment.
• Email me

Course Grade Factors (subject to change)

• Homework (50%)
• Midterm (20%)
• Final Exam (30%)

Textbooks

Barrett O'Neill, Elementary Differential Geometry, Revised 2nd Edition. Used and new copies are available from several sellers and in the NYU bookstore.

• John Stillwell, The Four Pillars of Geometry. Download PDF using link while connected to NYU network.
• Peter Petersen, Classical Differential Geometry. Notes from UCLA undergraduate differential geometry course.
• Sheldon Axler, Linear Algebra Done Right. Download PDF using link when connected to NYU network.

Homework (Overleaf, Gradescope)

• Homework assignments will be posted every week and due by 11:59pm a week later. You are encouraged to start working on the homework immediately, so you'll be prepared with specific questions to ask me during class or office hours, the TA during recitation, or either of us by email.
• Homework solutions must be prepared using LaTeX either Overleaf or LaTeX software installed on your computer and submitted via Gradescope.
• Using Overleaf:
  • Go to Overleaf and set up an account.
  • Each week you will get a link to a project containing the new assignment. Click on it to open the project. Under Menu in the upper left corner, click on Copy Project.
  • You enter your solutions using LaTeX in the left window, and the formatted document will appear on the right.
  • If you have never used LaTeX before, please look at the Overleaf Documentation.
  • There will often be error messages that you'll need to fix. If you cannot figure out how to fix the errors, share the project with me and I'll help you.
  • If you prefer to edit the LaTeX using your own editor, you can keep the local version synced with the Overleaf version using git.
• Using Gradescope:
  • Set up your Gradescope account by going to Gradescope and sign up for this course using the entry code MG7N48.
  • To submit an assignment, save it from Overleaf on your computer as a PDF and follow the instructions on Gradescope to upload the PDF.
• Optional: Install LaTeX on your computer.

Special Accommodations

If you are student with a disability who needs special accommodations, please contact New York University’s Moses Center for Students with Disabilities (CSD) at 212-998-4980 or mosescsd@nyu.edu. You must be registered with CSD to receive accommodations. Information about the Moses Center can be found at www.nyu.edu/csd. The Moses Center is located at 726 Broadway on the 2nd floor. (See Diagnosing A Learning Disability In College for more information. If you believe you might have a learning disability, please get tested.)

About the course

This course is an introduction to differential geometry, which can be described as studying the geometric properties of a curved space using calculus. The focus will be on the geometry of curves and surfaces in 3-space but with an eye on how to extend the concepts to abstract geometric spaces. After building the foundations of the subject, specific topics to be covered include the Gauss-Bonnet theorem, which links the differential geometric properties of a surface to its topological properties, and the geometry of hyperbolic space, which is the negatively curved analogue of the sphere.

Prerequisites

• Precalculus, single and multivariable calculus
• Linear algebra, preferably at the honors level. Key concepts include abstract vector spaces and subspaces, linearly independence, bases, linear transformations.
• Strong skills in doing rigorous calculations and proofs.
• Strong problem solving skills, including identifying and testing useful examples and counterexamples, when trying to prove or disprove a statement.
• Willingness to struggle with abstract mathematical concepts, statements, calculations, and proofs.