# MATH5700 Modern Differential Geometry and Topology

MATH5700 is an Honours and Postgraduate Coursework Mathematics course. See the course overview below.

Units of credit: 6

Exclusions: MATH3531, and MATH3701 (MATH5700 jointly taught with MATH3701)

Course offering: Term 3

This course provides a good understanding of basic topological properties, constructions and reasoning in three dimensional space, classical curves and surfaces, and understand the meaning of curvature for curves and surfaces, and appreciates the connections between topology and differential geometry for surfaces.

On completion you will know how to compute the fundamental group for a range of topological spaces and understand the importance of homotopy relation. You will know how to differentiate between two manifolds using algebraic
topology tools and differential geometry tools. You will gain an appreciation for the importance of quadrics to approximate surfaces at a point, and you will be able to make explicit computations for a wide variety of examples, computing Frenet frames for curves, and first and second fundamental forms for many surfaces. Algebraic surfaces and surfaces of revolution will provide a good source of examples. You will understand the idea of a developable surface and its applications.

The above outcomes are related to Research, inquiry and analytical thinking abilities, communication,
and Information literacy.

More information: This recent course handout (pdf) contains information about course objectives, assessment, course materials and the syllabus.

The Online Handbook entry contains information about the course. (The timetable is only up-to-date if the course is being offered this year.)

If you are currently enrolled in MATH5700, you can log into UNSW Moodle for this course.

#### Course Overview

Topology and differential geometry both deal with the study of shape: topology from a continuous and differential geometry from a differentiable viewpoint.

This course begins with an introduction to general topology. We then study curves in space and how they bend and twist, and the topology of curves. We then consider surfaces, studying the first and second fundamental forms introduced by Gauss, the various measures of curvature and what they mean for the external and internal appearance and properties of surfaces. We prove the important Gauss-Bonnet theorem and use it to examine topological properties of surfaces, such as the Euler Characteristic. We finish with a look at the hyperbolic plane and a look forward to general Riemannian geometry.