Étienne Ghys: A guided tour of the seventh dimension

Abstract:
One of the most amazing discoveries of John Milnor is an exotic sphere in dimension 7. For the layman, a sphere of dimension 7 may not only look exotic but even esoteric... It took a long time for mathematicians to gradually accept the existence of geometries in dimensions higher than 3. One may wonder how topologists can develop some intuition about these geometries. How can they work "in" these abstract worlds? In this talk, I'll describe some historical developments of higher dimensional geometries, starting of course with the fourth dimension. Then, I'll try to convey some intuition about high dimension. Finally, I want to present Milnor's 7 dimensional
jewel, hoping to demystify it. I'll do my best to avoid any kind of sophisticated mathematical background.
This lecture was held at The University of Oslo, May 25, 2011 and was part of the Abel Prize Lectures in connection with the Abel Prize Week celebrations.
Program for the Abel Lectures 2011
1. "Spheres" by Abel Laureate John Milnor, Institute for Mathematical Sciences Stony Brook University, New York
2. "Manifolds, topology and dynamic" by Professor Curtis McMullen
3. "Bernoulli numbers, homotopy groups, and Milnor" by professor Michael Hopkins
4. "A guided tour of the seventh dimension", a science lecture by professor Etienne Ghys

[@Achrononmaster]

@8:30 Poincaré was so phreakin' onto it man. However, note just how good he was. He was not contradicting Newton and Laplace.

[@Achrononmaster]

@37:00 better to think in terms of real geometry. The Riemann sphere is an object over scalar and bivector dimensions. The bivector component ei∧ej is a Grassmann outer product of two basis vectors. This way in Euclidean or Lorentzian space/spacetime you can have several "complex" structures" but they represent real geometry (generators of rotations) and form an even subalgebra of the full Clifford algebra over the reals. So a "point" on a Riemann surface is some subset of instructions to dilate and rotate.

[@cypriensaito4276]

Great Etienne ! High dimension study at IHES by himself, did you know ?

[@Eloupixel]

ok, pour l'instant je comprend rien que ça soit de l'anglais ou des concepts mathématiques qu'il y a derriere mais tkt pas qu'un jour je mène une conférence avec ce mec !

[@andreemcaldas]

At 40:08... the three sphere is a what?

[@realbartonjames]

Did he say... "the roundest?" I wondered the same!

[@andreemcaldas]

@@realbartonjames Thank you, Bart! I think you got it... the roundest, because the group of symmetry is bigger.