### Math 1040: Fundamental Problems of Geometry

#### Professor Richard Kenyon Tel. 863-6406 rkenyon -at- math.brown.edu office: Kassar 304 Office hours: Mondays 12:30-1:30 or by appointment

Text: Mostly Surfaces, by Richard Schwartz
This text is available here(newer version) or from the bookstore.

Course outline: This is a one-semester undergraduate course in geometry. Geometry is a vast area of mathematics generally based on the notion of "metric spaces", which are sets with a distance function, or notion of length. We'll study roughly the first 10 or so chapters of the book, with possible side topics as well.

There will be homeworks collected weekly and one midterm. The final grade will be weighted as follows: Homework 35%, midterm 20%, final project 45%.

## Final Project

Write ten pages on a subject of your choice related to geometry of some sort.
Give a 15 minute presentation on it at the end of the semester.
Here are some suggested topics, or feel free to choose your own:
Minimal surfaces
Circle packing theorem
Uniformization theorem
Rigidity and flexibility of polyhedra
Knots
Tiling problems
Dehn invariant

### Homework:

Late homework will not be accepted. The lowest homework grade will be dropped.
It is expected that your homework should involve up to 10 hours of work per week.

Homework 1: Due Tuesday Feb 5 in class
Do exercises 1-5 from the text. For exercise 5, try to find an example which is fundamentally different from the one given in class.

Homework 2: Due Tuesday Feb 11 in class
Chapter 2: exercises 7, 10, 11
Chapter 3: exercises 1, 2, 3, 4

Homework 3: Due Thursday Feb 20 in class
Chapter 3: 7, 8, 9, 11

Homework 4: Due Tuesday Feb 25 in class
Chapter 4: 2, 4, 5, 6, 7

Homework 5: Due Tuesday March 5 in class
Chapter 4: 9
Chapter 5: 1, 4, 5, 6

Homework 6: Due Tuesday March 19 in class
Chapter 8: 1, 2, 3.
Chapter 9. State and prove the analog of Girault's Theorem for spherical n-gons.

Homework 7: Due Tuesday April 2 in class
Chapter 9: 6.
Chapter 10: 1,2,
What LFTs of the Riemann sphere preserve the unit circle as a set? (describe conditions on a,b,c,d).

Homework 8: Due Tuesday April 9 in class
Chapter 10: 6, 9, 10
Compute the circumference and area of a hyperbolic ball of radius r.

Homework 9: Due Tuesday April 16 in class
Chapter 11: 3, 4, 6 part 1, 7, 8

Homework 10: Due Tuesday April 23 in class
Chapter 12: 5, 6
Chapter 17: 2, 3