MthSc 481 (Topics in Geometry and Topology), Spring 2012
# MthSc 481 (Topics in Geometry and Topology), Spring 2012

"Mathematics, rightly viewed, possesses not only truth, but supreme
beauty." --Bertrand Russell

### About the class

This course will be an introduction to selected mainstream topics in
geometry and topology. Special focus will be given to understanding
hyperbolic geometry and its symmetries. Hyperbolic geometry is similar
to standard high-school geometry, but with a few subtle differences
(notably as the absence of the "parallel postulate") that lead to
deep consequences and beauty. At first, it may seem exotic, but it is
just as prevalent and important to mathematics and physics as its
familiar cousin, Euclidean geometry. We will begin with a quick review
of basic Euclidean geometry, and then move into few advanced aspects
including its isometries, conformal maps, and circle inversions. These
will provide a foundation for an introduction to hyperbolic
geometry. After defining it formally, we will examine some of its
basic constituents such as its isometries and trigonometry. This will
lead us to the general linear group GL(n,R), the projective linear
group PGL(n,R), and Möbius transformations. We will study the
isometries of the hyperbolic plane in more detail, and classify them
by type: elliptic, parabolic, and hyperbolic. Next, we will turn our
attention to both orientable and non-orientable surfaces, such as
spheres, tori, and projective planes. This will be done in the setting
of simplicial complexes, and in particular, quotient maps of
side-paired polygons. Finally, we will learn about the invariance of
the Euler characteristic, and use this to give the classification of
two-dimensional surfaces. If time permits, we will learn about the
celebrated Gauss-Bonnet theorem which provides a remarkable link
between the geometry and the topology of a surface.

### Resources

### Homework

Homework 1: pdf |
tex. Due
Tuesday, January 24th at 4pm.

Homework 2: pdf |
tex. Due
Tuesday, January 31st at 4pm.

Homework 3: pdf |
tex. Due
Tuesday, February 7th at 4pm.

Homework 4: pdf |
tex. Due
Tuesday, February 21st at 4pm.

Homework 5: pdf |
tex. Due
Tuesday, March 6th at 4pm.

Homework 6: pdf |
tex. Due
Thursday, March 15th at 4pm.

Homework 7: pdf |
tex. Due
Thursday, March 29th at 4pm.

Homework 8: pdf |
tex. Due
Thursday, April 5th at 4pm.

Homework 9: pdf |
tex. Due
Tuesday, April 17th at 4pm.

Homework 10: pdf |
tex. Due
Thursday, April 26th at 4pm.

### Class lecture notes

Section 1:
What is a Geometry? *10 pages. Last updated January 27, 2012*

Section 2:
Circle Inversion. *5 pages. Last updated January 27, 2012*

Section 3:
The Hyperbolic Plane. *6 pages. Last updated January 30, 2012*

Section 4:
Euclidean vs. Hyperbolic Geometry. *10 pages. Last updated
February 4, 2012*

Section 5:
"Highschool" Hyperbolic Geometry. *6 pages. Last updated
February 9, 2012*

Section 6:
Hyperbolic Triangles. *12 pages. Last updated February 15,
2012*

Section 7:
The Euclidean Isometry Group. *7 pages. Last updated March 2,
2012*

Section 8:
Möbius Inversions and the Hyperbolic Isometry Group. *13
pages. Last updated: March 5, 2012*

Section 9:
The Boundary of the Hyperbolic Plane. *6 pages. Last updated:
March 15, 2012*

Section 10:
Conjugacy Classes of the Möbius Group. *10 pages. Last updated:
March 16, 2012*

Section 11:
Some Basic Combinatorial Topology. *9 pages. Last updated:
April 12, 2012*

Section 12:
Simplicial Complexes. *12 pages. Last updated:
April 24, 2012*

Section 13:
Classification of Surfaces. *15 pages. Last updated:
April 25, 2012*

Section 14:
Curvature and the Gauss-Bonnet Theorem. *7 pages. Last updated:
April 30, 2012*

__Section 15: Minkowski Space, and other Models of the Hyperboic Plane__