Differential geometry (Math 1060)

Announcements

• None currently.

Course information

This is an undergraduate course in differential geometry. The primary objects of study are curves and surfaces in 3-space. This course builds on multivariable calculus and linear algebra and could be a precursor to the study of manifolds, topology, and Riemannian geometry.

Course objectives

• learn basic properties of curves and surfaces (lengths, area, curvature, torsion) and how to compute them;
• prove foundational results (fundamental theorem of curves, theorem egregium, Gauss-Bonnet);
• understand theoretical trends in differential geometry (local vs global properties, intrinsic vs extrinsic quantities);
• improve your understanding of linear algebra and calculus by applying it;
• investigate various applications of differential geometry (isoperimetric inequality, regular isotopy of plane curves, knots, map-making, Euler characteristic/topology);
• improve your ability to read/write proofs and critique arguments;
• expose you to things you haven’t seen and challenge you through problem solving.

Recommended prerequisites

We will use multivariable calculus and linear algebra heavily and without much review during lecture. Some experience with writing proofs may also be helpful.

Textbook

do Carmo, Differential geometry of curves and surfaces

Course expenses

Just the textbook. In practice, the text may not be absolutely necessary for the course, although you may want it as a reference.

Grading

homework 20%, midterm 20%, final project 20%, final exam 20%, participation 10%

• a grade of 92% or higher is guaranteed an A
• a grade of 84% or higher is guaranteed an B
• a grade of 76% or higher is guaranteed an C

For students taking this course S/NC, a minimum grade of 70% and a minimum participation grade of 70% is required to guarantee a grade of S.

Contact information

• Instructor: Bena Tshishiku (bena_tshishiku at brown.edu)
• TA: Nathan Smith (nathan_smith at brown.edu)

Course events

Lectures: Tu-Th 9-10:20am in Smith-Buonanno Hall 206

Office hours:

• Bena: TBD
• Nathan: TBD

Important dates:

• Midterm: TBD
• Final projects: Dec 2, 4, 9
• Final exam: Dec 13 at 9am

Homework

There will be weekly assignments posted below. The homework is designed to increase your engagement with the material, with your peers, and with me.

Collaboration: Please collaborate! Working together with your classmates can help you learn the material better. Some of the homework problems may be difficult, so it is recommended that you talk to someone when you are stuck. You are required to write your solutions alone and acknowledge the students you worked with. For any solution you submit, you should understand it well enough that you can explain it to someone else and answer questions about it. If you find yourself writing down things that you can’t explain, you should go back and think more about the problem.

LaTeX: Homework solutions must be typed in LaTeX. If you’re new to LaTeX, you can either download LaTex or use sharelatex which allows you download, edit, and compile LaTeX files online. See here for some commonly used symbols in LaTeX. Also Detexify is a useful tool for finding the commands for various symbols. The source code for the assignments should be a helpful guide. The most basic thing to know/remember is that math always goes in between dollar signs.

Late homework policy: For your homework grade, I will drop the score from your lowest assignment. View this as a one-time “get out of jail free card” in the event that you oversleep, forget, have a midterm, etc. As a general rule, late homework will not be accepted. If you have an emergency (medical, family, …), you will need to send a dean’s note before the assignment is due.

Homework assignments.

HW1 (due 9/12).

Participation

Occasionally, we may have small assignments or quizzes in class (randomly). These count toward the participation part of your grade (10%). Attending class and taking notes is a basic part of course participation. If you have to miss class, I expect that you will get notes from a classmate.

Course materials

• For asynchronous discussions (e.g. questions about homework) we will use Campuswire.
• If you need additional references (outside of lecture and do Carmo), I encourage you to find the resource that works best for you. There are other popular texts by Banchoff-Lovett and Shifrin. You can also find further discussion on Wikipedia and Youtube.

Final Project

Working in groups of 2, you’ll choose a topic and give an N-minute presentation at the end of the semester.

The topic should be something related to the course that interests you. Below are some topic ideas to get you started, but you don’t have to choose one of these.

As part of completing the final project, I will ask you to submit:

• A project proposal
• A project outline
• Final presentation slides

Some potential topics will be posted here eventually.

Tentative schedule (subject to change)

• Week 1:
  • Thurs (9/4).
• Week 2:
  • Tues (9/9).
  • Thurs (9/11).
  • Fri (9/12). HW1 due
• Week 3:
  • Tues (9/16).
  • Thurs (9/18).
  • Fri (9/19). HW2 due
• Week 4:
  • Tues (9/23).
  • Thurs (9/25).
  • Fri (9/26). HW3 due
• Week 5:
  • Tues (9/30).
  • Thurs (10/2).
  • Fri (10/3). HW4 due
• Week 6:
  • Tues (10/7).
  • Thurs (10/9).
  • Fri (10/10). HW5 due
• Week 7:
  • Tues (10/14).
  • Thurs (10/16).
  • Fri (10/17). HW6 due
• Week 8:
  • Tues (10/21).
  • Thurs (10/23).
  • Fri (10/24).
• Week 9:
  • Tues (10/28).
  • Thurs (10/30).
  • Fri (10/31). HW7 due
• Week 10:
  • Tues (11/4).
  • Thurs (11/6).
  • Fri (11/7). HW8 due
• Week 11:
  • Tues (11/11).
  • Thurs (11/13).
  • Fri (11/14). HW9 due.
• Week 12:
  • Tues (11/18).
  • Thurs (11/20).
  • Fri (11/21).
• Week 13:
  • Tues (11/25).
  • Thurs (11/30). No class (thanksgiving)
  • Fri (12/1).
• Week 14:
  • Tues (12/2).
  • Thurs (12/4).
• Week 15: (reading period)
  • Tues (12/9).
  • Thurs (12/11).