Difference between revisions of "A Portal Special Presentation- Geometric Unity: A First Look"
A Portal Special Presentation- Geometric Unity: A First Look (view source)
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<p>[01:34:28] Which is just looking like the exterior algebra, $$\wedge^{*}(C)$$ on the chimeric bundle; that means that it is graded by degrees. [The] chimeric bundle has dimension-14, so there's a zero-part, a one-part, a two-part, all the way up to 14. Plus, we have forms in the manifold. And so the question is if I want to look at $$\Omega^i$$ valued in the adjoint bundle, there's going to be some element $$\Phi_{i}$$, which is pure trace. | <p>[01:34:28] Which is just looking like the exterior algebra, $$\wedge^{*}(C)$$ on the chimeric bundle; that means that it is graded by degrees. [The] chimeric bundle has dimension-14, so there's a zero-part, a one-part, a two-part, all the way up to 14. Plus, we have forms in the manifold. And so the question is if I want to look at $$\Omega^i$$ valued in the adjoint bundle, there's going to be some element $$\Phi_{i}$$, which is pure trace. | ||
<p>[01:35:12] Right? Because it's the same representations appearing where in the usually auxiliary directions as well as the geometric directions. So we get an entire suite of invariance together with trivially associated invariants that come from using the Hodge star operator on the forms. I'm just going to call them for completeness | <p>[01:35:12] Right? Because it's the same representations appearing where in the usually auxiliary directions as well as the geometric directions. So we get an entire suite of invariance together with trivially associated invariants that come from using the Hodge star operator on the forms. I'm just going to call them for completeness, $$\tilde\Phi_{i}$$, I'm not going to deal with them. | ||
<p>[01:35:41] Now, this is a tremendous amount of freedom that we've just gained. Normally we keep losing freedom, but this is the first time we actually begin to see that we have a lot of freedom and we're going to actually retain some of this freedom to the end of the talk. But the idea being that I can now start to define operators which correspond to the "Ship in the Bottle" problem. | <p>[01:35:41] Now, this is a tremendous amount of freedom that we've just gained. Normally we keep losing freedom, but this is the first time we actually begin to see that we have a lot of freedom and we're going to actually retain some of this freedom to the end of the talk. But the idea being that I can now start to define operators which correspond to the "Ship in the Bottle" problem. | ||