### Math 1040: Fundamental problems of geometry, Spring 2016MWF 2-3 Wilson Hall 103

Instructor: Melody Chan
office: 311 Kassar House
course website: http://www.math.brown.edu/~mtchan/2016Spring_1040.html
office hours: Mondays 3-4, Fridays 4:05-5

Teaching Assistant: David Schwein
email: david_schwein@brown.edu
TA Office hour: Sundays at 7pm, Kassar House Common Room

Course description

This is a one-semester course in undergraduate geometry. I will follow the textbook "Mostly Surfaces" by Schwartz. Tentatively, we'll cover chapters 1-10, and additional topics selected from the book, time permitting.

The course will be (mostly) about surfaces: spaces that, near every point, "look like" the plane R^2. We'll take this as an excuse to look at some fundamental notions in geometry and topology (metric spaces, homeomorphisms, gluing, fundamental groups and covering spaces) and to explore some beautiful topics in Euclidean, spherical, and hyperbolic geometry.

The prerequisite for this class is Math 0520 or Math 0540 (linear algebra). You will also need to know some multivariable calculus. Some experience with proofs would also be helpful. I will need some notions from other parts of mathematics, like point-set topology, real analysis and group theory. However, I will develop what we need as we go, so you don't necessarily have to have prior experience with these topics. In fact, it may be helpful to see some of these topics as they relate to geometry before you take them up again in your other math classes.

I intend to make this class interesting and useful for any math major or anyone with a serious interest in and enjoyment of mathematics. I will expect a good amount of work outside class on the problem sets. If you are unsure about whether to take this course, please feel free to discuss it with me.

Topics

Metric spaces, gluing, the fundamental group, covering spaces. Euclidean, spherical, and hyperbolic geometry. Hyperbolic surfaces and other topics, time permitting.

Required textbook

Schwartz, Mostly Surfaces, AMS, first edition.
This book is available at the bookstore for \$49. You can also read it online here, or you can click here for an earlier draft of the book, provided by the author.

The list of errata is here.

Supplementary books will be announced.

A midterm study guide and running lecture notes starting March 9 are posted here.

Homework (20%), midterm (30%), final exam (40%), show and tell (10%).

Exams and schedule

The midterm will be held Wednesday March 16 in the evening, 6-8pm, in Kassar House Foxboro Auditorium. If you have a legitimate conflict with this time, you must let me know by February 15.

The final exam will be held Saturday May 14, 2pm-5pm in WILSON 302 (NOTE LOCATION CHANGE).

Class will be cancelled Friday April 8.

Reading week schedule: There will be class on Friday April 29 and Monday May 9. There will be no class during the week of May 2-6.

Homework

Homework will be assigned once a week and due on Mondays at 5pm.

Problem Set 1, due Wednesday February 10, postponed because of snow day. Solutions
Problem Set 2, due Monday February 15. Solutions
Problem Set 3, due Wednesday February 24. Solutions
Problem Set 4, due Monday February 29 Solutions
Problem Set 5, due Monday March 7 Solutions
Problem Set 6, due Monday March 14 Solutions
Problem Set 7, due Monday March 21 Solutions
Problem Set 8, due Monday April 4 Solutions
Problem Set 9, due Monday April 11 Solutions
Problem Set 10, due Monday April 18 Solutions
Problem Set 11, due Monday April 25 Solutions

Although no late homework is accepted, your lowest homework score will be dropped.

Geometry Show and Tell

There will be a Geometry Show and Tell assignment, consisting of 1 geometric object of your choice, a short written description, and a 5 minute presentation. Click here for details. You may sign up to do your show and tell during any class period. Depending on the size of the class, we may also hold an extra show and tell on Monday May 9. Please bear in mind that this assignment will be assesssed more rigorously than your average show and tell!

Collaboration

You are welcome to collaborate with other students in the class on your homework. You are required to write up your solutions separately and write the names of the students with whom you worked on the assignment. (You may only use the Internet as a general reference, at the level of generality of Wikipedia.)

Accommodations for students with disabilities

Any student with a documented disability is welcome to contact me as early in the semester as possible so that we may arrange reasonable accommodations. As part of this process, please be in touch with Student and Employee Accessibility Services by calling 401-863-9588 or online at
http://brown.edu/Student_Services/Office_of_Student_Life/seas/index.html