MAS6360 Geometry I

Both semesters, 2017/18 20 Credits
Lecturer: Dr Nick Gurski
Aims Outcomes Assessment Full Syllabus

The aim of this module is to introduce the students to the theory of differential geometry, of crucial importance in modern mathematical physics, and to give some applications involving optics and symplectic geometry.

There are no prerequisites for this module.
No other modules have this module as a prerequisite.


Outline syllabus

Semester 1
Curves in R2
  • basic notions & examples
  • curvature
Surfaces in R3
  • basic notions & examples
  • metric quantities
  • curvature
  • parallel vector fields & geodesics
  • Gauss' Theorem & the Gauss-Bonnet Formula
Semester 2
  • Symplectic matrices
  • Light rays and lenses: Gaussian optics
  • Symplectic forms
  • Linear Optics
  • Generating functions
  • Geometrical Optics (if time permits)
  • Nonlinear Phenomena (if time permits)



Aims

  • give students a foundation in differential geometry;
  • To demonstrate the utility of calculus and advanced calculus in the analysis and description of curves and surfaces
  • give students an opportunity to think about notions related to curves and surfaces, e.g., curvature and umbilics;
  • introduce the distinction between local and global properties in geometry;
  • To provide an introduction to symplectic geometry, an important part of modern pure mathematics, in parallel with optics, which is one of its most important applications;
  • To provide a knowledge of symplectic matrix theory

Learning outcomes

  • calculate and work with the curvature function for curves in the plane;
  • calculate and work with various curvature functions for curves in space;
  • understand essential concepts in differential geometry, such as curvature;
  • understand basic concepts in symplectic geometry, and its importance in optics and in mathematical physics;
  • understand the notion of Lagrangian subspace and its role in symplectic linear algebra.

40 lectures, no tutorials

Assessment

The module will be assessed by a formal, closed book, two and a half hour examination at the end of each semester.

Full syllabus

Semester 1

I. Curves in R2 & their Curvature (4 lectures). Parametrisation. Tangent vectors. Arc lengths. Unit-speed parametrisation. Smooth curves. Curvature.
II. Surfaces in R3 (2 lectures). Parametrisation. Tangent planes. Smooth surfaces. Maps of surfaces.
III. Metric Quantities of a Surface (2 lectures). First fundamental form. Local isometries. Conformal maps. Surface areas. Area-preserving maps.
IV. Curvature of a Surface (3 lectures). Second fundamental form. Weingarten map. Normal curvature. Principal curvatures & vectors. Mean & Gaussian curvature.
V. Parallel Vector Fields & Geodesics (4 lectures). Vector fields along a curve. Covariant differentiation. Parallel vector fields. Geodesic curvature. Geodesics. Shortest paths.
VI. Gauss' Theorem & the Gauss-Bonnet Formula (4 lectures). Holonomy & relation to Gaussian curvature. Gauss' Theorem. Local Gauss-Bonnet Formula. Triangulation. Euler characteristic. Global Gauss-Bonnet Formula. Applications.

Semester 2
  • Symplectic matrices : Reminder of the properties of orthogonal matrices. Symplectic matrices. Standard symplectic form on R2n. Insufficency of det = 1. (About 2 lectures)
  • Light rays and lenses. Gaussian optics : Refraction and Snell's Law. Thin lenses and small angles. The correspondence between Gaussian Optics and SL(2) = Sp(2). (About 4 lectures)
  • Symplectic forms : Symplectic forms on a vector space. Skew of a subspace; a symplectic vector space has even dimension. Lagrangian subspaces. Symplectic maps. (About 3 lectures)
  • Linear Optics : Systems in R3 without rotational symmetry and Sp(4). (About 4 lectures)
  • Generating functions : Generating functions for 2n = 2 and general 2n. Cases and optical interpretation. (About 4 lectures)
  • Nonlinear phenomena (if time permits) : Wavefronts and caustics. The normal bundle. Familes of curves and surfaces. Envelopes and discriminants. (About 3 lectures)
(The order of topics may be varied as the course progresses.)