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Welcome... ...to my personal website, which I currently employ solely to make some visualizations regarding my PhD research results accessible to the interested audience. For any other concerns, including any topics related to my subsequent career in business, please visit Looking forward to hearing from you! Ingo Wieck |
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Background
In my PhD thesis 
Explicit symplectic packings: Symplectic tunnelling and new maximal constructions, I have introduced the "sypmlectic tunnelling" method and applied it to construct a range of new explicit maximal packings for certain symplectic manifolds. The main results, the 7- and 8-packings of the 4-dimensional ball, are fully visualized in the publication, as they can be represented via static images (for reference, I include them once more below).
Additionally, I have constructed some ranges of explicit maximal packings of ruled symplectic manifolds, which were published by giving parametrized coordinates. Regarding visualization however, the static images provided in print only capture selected key states of these ranges of packings. To admit an easier, intuitive approach to these solutions, below I provide vizualizations of the full parameter range in the form of time-series parametrized videos.
Note that the packing obstruction appears in black on this webpage. For any details regarding the interpretation of these 2-dimensional representations, the method of symplectic tunnelling and credits to other authors on whose work my research has built, please refer to the publication.
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Packings of the 4-dimensional ball B4
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A maximal 7-packing |
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A maximal 8-packing |
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Packings of S2 x S2
The following videos give maximal packings for the family of ruled symplectic manifolds S2(a) x S2(b), parametrized by the ratio a/b over time, which is sufficient due to the scaling invariance of the packing construction.
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Maximal (3- and) 4-packings The 3- and 4-packings have identical maximal radius, so the 3-packings are given by removing one ball from the 4-packings. These packings do not yet require symplectic tunnelling, but are given for completeness. |
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Maximal 5-packings These packings do not yet require symplectic tunnelling, but are given for completeness. |
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Maximal 6-packings This series employs symplectic tunnelling. Previously, semi-explicit constructions via Blow-Ups in CP2 had been found by F. Schlenk. |
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Maximal 7-packings These packings employ symplectic tunnelling. For the range 5/3 < a/b < 4, previously, semi-explicit constructions via Blow-Ups in CP2 had been found by F. Schlenk. For the range 1 <= a/b < 8/7, no explicit packings had been found yet at the time of publication. I believe that this last gap can also be closed via symplectic tunneling, which opens a vast space of potential constructions. It is work in progress to search this space for globally valid solutions with an intelligent algorithmic approach. |
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Packings of S2 x∼ S2
The following videos now give maximal packings for the family of ruled symplectic manifolds S2 x∼ S2 (the non-trivial bundle), parametrized as in the publication by α and β. The videos are parametrized over time by β ∈ (0,1), which is sufficient since (α + β) can be normalized to 1 due to the scaling invariance of the packing construction.
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Maximal 3-packings These packings do not yet require symplectic tunnelling, but are given for completeness. |
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Maximal 4-packings These packings do not yet require symplectic tunnelling, but are given for completeness. |
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Maximal 5-packings These packings do not yet require symplectic tunnelling, but are given for completeness. |
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Maximal 6-packings This series employs symplectic tunnelling. |
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Maximal 7-packings This series employs symplectic tunnelling. Note that it contains both an explicit maximal 7- and 8-packing of B4: • The 7-packing at the final position • An indirect 8-packing at playtime position 0:38 (β = 6/17 ≈ 0.35), when the truncated top has the same size as the 7 packed balls. Since the truncation results from constructing the parametrized family S2 x∼ S2 by blowing down balls of different radii from CP2, the truncation can be interpreted as an 8th ball of equal radius at this stage. |
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Thank you for your interest - please do not hesitate to contact me via Xing or LinkedIn in case of any questions!