My coordinates

Otto van Koert
Seoul National University
Department of Mathematical Sciences, Seoul National University
San56-1 Shillim-dong Gwanak-gu ‎신림동 관악구
Seoul 08826
South Korea

Email: okoert'put here the at sign'
Office: Building 27, 402

A single sphere in the Seifert-Weber hyperbolic 3-manifold

Brieskorn manifolds

Homology and contact homology of Brieskorn manifolds

Here is a rather primitive program, brieskorn_homology3.txt, that computes the homology and the contact homology of Brieskorn manifolds. The source is rather ugly, but it works. Here are the steps
1 Download the file,
2 Rename it into brieskorn_homology3.cpp
3 Compile it
4 Run it in a terminal (also called dos-box).
With the gcc-compiler, run "g++ -o prog brieskorn_homology3.cpp"
A point of warning, this program just assumes that contact homology exists, and then computes it using the algorithm from Contact homology of Brieskorn manifolds In other words, the Maslov indices of the orbit spaces are computed by an explicit formula, and the paper of Milnor-Orlik is used to compute the relevant homology groups.
Grading follows SFT conventions which differs from the Conley-Zehnder index by a shift depending on the dimension. Keep this in mind for the signs of the mean Euler characteristic.
As a final remark, the mean Euler characteristic can be proved to be an invariant of a Brieskorn manifold using equivariant symplectic homology.

Spheres with Sasaki-Einstein metrics

Here are some excel files with lists of spheres with Sasaki-Einstein metrics. The lists include basic invariants, including the mean Euler characteristic (with SFT-grading conventions). Here are the list with Sasaki-Einstein manifolds of Brieskorn type. dimension 5 dimension 7 dimension 9 In these files SH stands for the plus part of equivariant symplectic homology.

Contact manifolds without invariants to distinguish them

The Brieskorn manifolds BP(2,3,7,27) and BP(3,4,4,5) (and all others in this file with the adjective not distinguished) have the property that they have pairwise isomorphic plus-part of the equivariant symplectic homology (or linearized contact homology). Furthermore, the lower bound on the indices is not good enough to guarantee that the positive part of symplectic homology is an invariant. How can one distinguish these contact manifolds? Note that this is just the tip of the iceberg: there are many more of this type in higher dimensions.

Pictures and remarks on the (restricted) three-body problem

Download the video and play with the vlc-player (other players work, too, but I haven't tried)

Return maps for global surfaces of section

Here is a link to a page with pictures. Note that these pictures are huge, so the page may take a while to load: Pictures of disk maps arising in the restricted three-body problem

A global annular surface of section in the planar, circular, restricted three-body problem (after Levi-Civita regularization and stereographic projection)

Restricted three-body in an inertial frame

AN1.mp4 (video in mp4 format)

Restricted three-body going to a rotating frame

AN2.mp4 (video in mp4 format)

Movement in the restricted three-body problem without much energy

AN3.mp4 (video in mp4 format)

Movement in the restricted three-body problem with more energy

AN4.mp4 (video in mp4 format)

Slices of the energy hypersurfaces

AN5.mp4 (video in mp4 format)

A spatial orbit: computed via Moser regularization

Orbit near light primary (mu=0.1 and c=1.3) (video in mp4 format)

A spatial orbit near the rotating Kepler problem: these were studied by Belbruno: here computed via Moser regularization

Orbit near the heavy primary (mu=0.99999 and c=0.1) (video in mp4 format)