### 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