## MAS441 Optics and Symplectic Geometry

 Semester 2, 2017/18 10 Credits Lecturer: Dr Kirill Mackenzie Home page Timetable Reading List Aims Outcomes Teaching Methods Assessment Full Syllabus

Symplectic geometry is the most active area of modern differential geometry. This course is an introduction to some of the key ideas, for smooth submanifolds of Rk. Certain spaces, such as the cotangent bundles of smooth manifolds and coadjoint orbits of matrix groups, are naturally equipped with symplectic structures and the course focuses on these classes of examples. The origins of symplectic methods lie in optics and the basics of this theory are included (no prior knowledge of optics is needed).

Prerequisites: MAS211 (Advanced Calculus and Linear Algebra)
No other modules have this module as a prerequisite.

## Outline syllabus

• Examples of symplectic manifolds
• Vector spaces, duality and annihilators
• Symplectic vector spaces
• Light rays and lenses: Gaussian optics
• Smooth submanifolds of Rk
• Symplectic manifolds and Poisson brackets
• Lagrangian submanifolds, Leray index

## Aims

• To provide an introduction to symplectic geometry, in the context of submanifolds of Rk, motivated in part by ray optics.
• To provide a knowledge of symplectic linear algebra, emphasizing differences and similarities with orthogonal matrices.
• To demonstrate the value of physical phenomena in understanding abstract theory in mathematics.

## Learning outcomes

• be able to identify and work with symplectic structures on vector spaces,
• understand the concept of a Lagrangian subspace and its role in symplectic geometry,
• be able to work with certain systems of lenses and the passage of lightrays through them,
• understand and be able to work with the concept of a symplectic manifold (in Rk) and the particular cases of cotangent bundles and coadjoint orbits of specific matrix groups,
• be able to work with the Leray index and understand its importance in the study of Lagrangian subspaces.

## Teaching methods

Lectures, problem solving

20 lectures, no tutorials

## Assessment

One formal 2.5 hour written examination. All questions compulsory.

## Full syllabus

• Examples of symplectic manifolds: Specific cases of cotangent bundles, coadjoint orbits of matrix groups.
• Vector spaces, duality and annihilators: Vector spaces in Rk, dual spaces, annihilators, rank-nullity theorem.
• Symplectic vector spaces: Symplectic forms on a vector space. Standard form on R2n. A symplectic vector space has even dimension; symplectic bases. Symplectic perp of a subspace. Symplectic matrices. Determinant is +1.
• Light rays and lenses. Linear optics: Refraction and Snell's Law. Thin lenses and small angles in R3 without rotational symmetry. The correspondence between linear optics and Sp(4).
• Smooth submanifolds of Rk: Local Diffeomorphism Theorem and Inverse Function Theorem (without proofs). Submanifolds defined as zero-sets and by parametrization. Tangent spaces and cotangent spaces. Introduction to vector fields.
• Symplectic manifolds and Poisson brackets: Examples revisited. The bracket of functions; the manifold is a symplectic manifold if the Poisson bracket obeys the Jacobi equation.
• Lagrangian submanifolds, Leray index: Lagrangian subspaces of a symplectic vector space. Leray index. Stratification of det B = 0. Application to symplectic manifolds.