Geometry and Topology

Prerequisite(s)：

Fundamental Algebra and Analysis I

Course description:

This course is designed for first year students who intend to pursue studies in fundamental mathematics or applied mathematics major. It is also open to all students who are interested in modern mathematics. The teaching objective is to expose the students to various topics in geometry and topology, which are either classical topics that deserve to be known by every mathematician, or are essential to modern geometry and topology. These topics include general topology, fundamental groups, projective geometry, hyperbolic geometry, and differential geometry of surfaces in 3-dimensional spaces.

Learning objectives:

By the end of this course, students will be able to:

1) Understand the concept of topological spaces and maps between them; understand how the language of point set topology applies to analysis; understand the concept of various universal constructions in the category of topological spaces such as quotient, disjoint union, products, sequential limit/colimit (although not necessarily in the language of category theory).

2) Understand the concept of fundamental group and its functorial properties (again not necessarily in the language of category theory); understand why Euclidean spaces have trivial fundamental groups, whereas the circle has fundamental group isomorphic to the additive group of integers; understand that the above facts yield proofs of highly nontrivial results such as the fundamental theorem of algebra.

3) Understand the definition of projective plane and curves defined by homogenous polynomials; understand how projective plane formalize the otherwise vague concept of “points at infinity”; understand projective transformation cross ratio; understand the proof of Pascal’s Theorem based on projective geometry.

4) Understand the concept of tangent vectors and tangent spaces of surfaces in 3-dimensional Euclidean spaces; understand the concept of Gaussian curvature and why it is an “implicit property” of a surface; in the case of triangulated surfaces, understand the concept of Euler characteristics and the Gauss-Bonnet theorem.

5) Understand the concept of hyperbolic lines, Mobius transformations and hyperbolic metric based on the upper half-plane model;

Detailed topics covered:

1) General topology: topological spaces, maps, (quasi)-compactness, Hausdorff spaces, connected and totally disconnected spaces, (special cases of) profinite spaces deriving new topological spaces out of existing ones (products, quotients...), Tychonoff’s Theorem.

2) Fundamental groups: fundamental groups and their functorial properties, homotopy invariance, the fundamental group of the circle, fundamental theorem of algebra.

3) Projective geometry: projective spaces, projective transformations, curves defined by homogenous polynomials, Pascal’s Theorem.

4) Differential geometry: smooth surfaces in 3-dim spaces, tangent spaces; the first and second fundamental forms, Gaussian curvature, Gauss-Bonnet theorem.

5) Hyperbolic geometry: the upper half plane model, Mobius transformations, hyperbolic metric/area.