Two notes on conformal geometry

Notes on conformal invisibility devices

As a consequence of the wave nature of light, invisibility devices based on isotropic media cannot be perfect. The principal distortions of invisibility are due to reflections and time delays. Reflections can be made exponentially small for devices that are large in comparison with the wavelength of light. Time delays are unavoidable and will result in wave-front dislocations. This paper consid...

Notes on Symplectic Geometry

1. Week 1 1 1.1. The cotangent bundle 1 1.2. Geodesic flow as Hamiltonian flow 4 2. Week 2 7 2.1. Darboux’s theorem 7 3. Week 3 10 3.1. Submanifolds of symplectic manifolds 10 3.2. Contact manifolds 12 4. Week 4 15 4.1. Symplectic linear group and linear complex structures 15 4.2. Symplectic vector bundles 18 5. Week 5 21 5.1. Almost complex manifolds 21 5.2. Kähler manifolds 24 6. Week 6 26 6....

Notes on Noncommutative Geometry

Foreword This book is not just a survey of noncommutative geometry — later NCG; rather, it strives to answer some naive but vital questions: What is the purpose of NCG? What is it good for? Can NCG solve open problems of classical geometry inaccessible otherwise? In other words, why does NCG matter? What is it anyway? Good answer means good examples. A sweetheart of NCG called non-commutative t...

Notes on Axiomatic Geometry

Notes on Differential Geometry

Part 1. Geometry of Curves We assume that we are given a parametric space curve of the form (1) x(u) =   x 1 (u) x 2 (u) x 3 (u)   u 0 ≤ u ≤ u 1 and that the following derivatives exist and are continuous (2) x (u) = dx du x (u) = d 2 x du 2 1. Arc Length The total arc length of the curve from its starting point x(u 0) to some point x(u) on the curve is defined to be (3) s(u) = u u 0 √ x ·x...