04/08/2009, 09:32 AM

I have been watching these excellent (3) video lectures by E.Ghys in English online:

Ghys Video lectures Osculating Curves 2008

There is a lot of difficult and interesting stuff presented clearly. Some of that I do not understand.

Ghys speaks of osculating circles being disjoint. Of osculating conics having a subtler ordering than just being disjoint. Of Taylor polynomials being analogue to the disjoint osculating curves. Of differential equations representing these osculating curves, or, vice versa, these osculating curves being geometric representation of differential equations.

Then he proceeds to show Hopf fibration of a neighboroughood of a real point using crossing of 2 complex axis and sphere S3 around the crosssing. He arrives at knots etc. This is all fantastic.He says putting complex lines instead of real works like a local microscope which is farther enhanced by using higher and higher degree osculating curves =differential equations.

Now since Hopf fibration uses a pair of complex lines could it be that tetration allows to take the pair of points or values from INSIDE the local neighboroughood of a real point as revealed by Hopf fibration etc and compute the corresponding real value and vice versa?

Since Hopf fibration is topological, the values there are related to number of knots of ratios of areas, there are no coordinates inside there.

But please have a look at lectures, I may be misinterpreting something.Ghys also puts forward unsolved problems.

There is also an excellent film and web site about this stuff, in simpler way.

Dimensions Movie Ghys

Ivars

Ghys Video lectures Osculating Curves 2008

There is a lot of difficult and interesting stuff presented clearly. Some of that I do not understand.

Ghys speaks of osculating circles being disjoint. Of osculating conics having a subtler ordering than just being disjoint. Of Taylor polynomials being analogue to the disjoint osculating curves. Of differential equations representing these osculating curves, or, vice versa, these osculating curves being geometric representation of differential equations.

Then he proceeds to show Hopf fibration of a neighboroughood of a real point using crossing of 2 complex axis and sphere S3 around the crosssing. He arrives at knots etc. This is all fantastic.He says putting complex lines instead of real works like a local microscope which is farther enhanced by using higher and higher degree osculating curves =differential equations.

Now since Hopf fibration uses a pair of complex lines could it be that tetration allows to take the pair of points or values from INSIDE the local neighboroughood of a real point as revealed by Hopf fibration etc and compute the corresponding real value and vice versa?

Since Hopf fibration is topological, the values there are related to number of knots of ratios of areas, there are no coordinates inside there.

But please have a look at lectures, I may be misinterpreting something.Ghys also puts forward unsolved problems.

There is also an excellent film and web site about this stuff, in simpler way.

Dimensions Movie Ghys

Ivars