There are two cases:
- (topological) manifolds
- differential manifolds
Questions: are all compact manifolds / differential manifolds homotopic / diffeomorphic to the sphere in that dimension?
- for topological manifolds: this is a generalization of the Poincaré conjecture.Original problem posed, for topological manifolds.Last to be proven, only the 4-differential manifold case missing as of 2013.Even the truth for all was proven in the 60's!Why is low dimension harder than high dimension?? Surprise!AKA: classification of compact 3-manifolds. The result turned out to be even simpler than compact 2-manifolds: there is only one, and it is equal to the 3-sphere.For dimension two, we know there are infinitely many: classification of closed surfaces
- for differential manifolds:Not true in general. First counter example is . Surprise: what is special about the number 7!?Counter examples are called exotic spheres.Totally unpredictable count table:
Dimension Smooth types 1 1 2 1 3 1 4 ? 5 1 6 1 7 28 8 2 9 8 10 6 11 992 12 1 13 3 14 2 15 16256 16 2 17 16 18 16 19 523264 20 24 is an open problem, there could even be infinitely many. Again, why are things more complicated in lower dimensions??
Mathematical consistency of quantum field theory by
Ciro Santilli 35 Updated 2025-03-28 +Created 1970-01-01
It is hard to say if this channel is good because of the awesome information, or if because of the absolute cutness of that British presenter. Maybe it is both.
Communicating at a distance, from Greek "tele" for distance!
A very cool thing about telecommunication is, besides how incredibly fast it advanced (in this sense it is no cooler than integrated circuit development), how much physics and information theory is involved in it. Applications of telecommunication implementation spill over to other fields, e.g. some proposed quantum computing approaches are remarkably related to telecommunication technology, e.g. microwaves and silicon photonics.
This understanding made Ciro Santilli wish he had opted for telecommunication engineering when he was back in school in Brazil. For some incomprehensible reason, telecommunications was the least competitive specialization in the electric engineering department at the time, behind even power electronics. This goes to show both how completely unrelated to reality university is, and how completely outdated Brazil is/was. Sad stuff.
Testing and Circuit for a Condenser microphone by RSD Academy (2018)
Source. Not very numerical, but shows a simple working breadboard circuit and an oscilloscope. He whistles with his mouth to get a pretty pure frequency.
That type of microphone requires a bias voltage. The circuit is in Ciro's ASCII art circuit diagram notation:
DC_9---R_10k--+--MICROPHONE--+--G
| |
+-------V------+
Soundwaves on an oscilloscope by Animated Science (2015)
Source. Dude speaking to microphone. Some analysis of how different sounds look like. No circuit diagram. Pinned article: ourbigbook/introduction-to-the-ourbigbook-project
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