Geometric topology is the topological study of geometric objects, particularly manifolds and related objects. The main goals of this course are as follows:
- Introduce the two key approaches for the modern study of geometric topology: algebraic and differential methods.
- Exhibit connections and interactions of different fields, particularly algebra, analysis and topology, which often lead us to discoveries of deep results in mathematics.
Our treatment will use ingredients from undergraduate-level algebra, calculus, and general topology. The methods and expositions will be as elementary as possible.

Students are expected to understand basic definitions and theorems in general topology, as well as possess a basic knowledge of groups and multivariable calculus. Necessary facts from algebra and calculus will be reviewed in class.

The course will be based on the lecture notes provided by the instructor. I recommend the following as useful references:
- J. Munkres, Topology (2nd Edition), Pearson, 2000.
- W. Massey, A basic course in algebraic topology, Springer-Verlag, 1991.
- M. Spivak, Calculus on manifolds, Benjamin, 1965.
- J. Milnor, Topology from the differentiable viewpoints, Princeton University Press, 1997 (revised reprint of the 1965 original).
- V. Guillemin, A. Pollack, Differential Topology, American Mathematical Society, 2010 (Reprint edition).

Topics include the following:
- Review of topological spaces and continuous maps
- Definition of manifolds with some history
- Homotopy of maps
- Fundamental groups
- Covering spaces
- Seifert van-Kampen theorem
- Applications of fundamental groups
- Knots and fundamental groups
- Alexander polynomials
- Review of multivariable calculus and inverse function theorem
- Smooth manifolds
- Regular values and transversality
- Sard's theorem
- Mod 2 degree of maps
- Orientations and degree
- Applications to vector fields

The course will consists of lectures, homework and exams/projects.