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.

### 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