MATH-UA 377: Differential Geometry Spring 2022

Important web sites

• Gradescope
• Overleaf
• Piazza

Calendar (subject to change)

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.

If you believe you might have a learning disability, please get tested. See Diagnosing A Learning Disability In College for more information.

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.
• All math questions should be posted to Piazza , so all students can see the question and answers. If, however, you have any personal questions or issues that you need to discuss with me, please email me.

Prerequisites

• Crucial skills
  • Doing rigorous calculations and proofs
  • Problem solving, including identifying and testing useful examples and counterexamples, when trying to prove or disprove a statement.
  • Struggling with abstract mathematical concepts, statements, calculations, and proofs.
• Crucial knowledge
  • Precalculus, single and multivariable calculus
  • Fundamentals of deductive logic
  • Linear algebra, preferably at the honors level. Key concepts include abstract vector spaces and subspaces, linear independence, bases, linear transformations. See, for example, the first 3 chapters of Axler, Linear Algebra Done Right (PDF available free to NYU students)

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. The approach taken will be more abstract than found in existing textbooks, and therefore we will rely mostly on notes I have written.

The homework and exams will focus on applying the concepts to both concrete and abstract examples using both calculations and logical reasoning. The latter is often called "doing proofs".

Mandatory tools you will use in this course

• Gradescope:
  • Please email me if you need a Gradescope account.
  • Homework: There will be a weekly homework assignment that will be posted on Brightspace as a link to the Overleaf project (see below). Your solutions must be written using LaTeX and saved as a PDF. You then follow the instructions on Gradescope to upload the PDF.
• LaTeX
  • Homework solutions must be prepared using LaTeX either Overleaf or LaTeX software installed on your computer.
  • 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 install and use LaTeX on your own computer, check out Getting LaTeX or Getting Started with LaTeX.
• Piazza:
  • All questions about course material and logistics should be asked here.
  • You are also encouraged to try to answer questions posted.
  • You may post anonymously.
  • Use LaTeX for mathematical formulas and equations. You must always enclose the LaTeX with $$...$$. Using $...$ does not work.
• Brightspace:
  • Will not be used much

Textbooks

Experimental Notes on Elementary Differential Geometry (Updated regularly during semester)

Peter Petersen, Classical Differential Geometry. (Notes from UCLA undergraduate differential geometry course)

Course Grade Factors (subject to change)

• Weekly Homework (50%)
• Midterm (25%)
• Final Exam (25%)
• Tweaks