A Portal Special Presentation- Geometric Unity: A First Look: Difference between revisions

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<p>[01:36:44] So in this case, if I have a $$\Phi$$, which is one of these invariants in the form piece, I can either take a contraction or I can take a wedge product. In the Lie algebra piece, I can either take a Lie product or because I'm looking at the unitary group, there's a second possibility, which is I can multiply everything by $$i$$ and go from [[Skew-Hermitian]] to [[Hermitian]] and take a [[Jordan product]] using anti-commutators rather than commutators.
<p>[01:36:44] So in this case, if I have a $$\Phi$$, which is one of these invariants in the form piece, I can either take a contraction or I can take a wedge product. In the Lie algebra piece, I can either take a Lie product or because I'm looking at the unitary group, there's a second possibility, which is I can multiply everything by $$i$$ and go from [[Skew-Hermitian]] to [[Hermitian]] and take a [[Jordan product]] using anti-commutators rather than commutators.


<p>[01:37:15] So I actually have a fair amount of freedom and I'm going to use a magic bracket notation, which in whatever situation I'm looking for, knows what it wants to be is does it want to do contraction? Does want to do wedge product, Lie product, Jordan product? But the point is, I now have a suite of ways of moving forms around.
<p>[01:37:15] So I actually have a fair amount of freedom and I'm going to use a "magic bracket" notation, which in whatever situation I'm looking for, [the operator] knows what it wants to be, is does it. Want to do contraction? Does want to do wedge product, Lie product, Jordan product? But the point is, I now have a suite of ways of moving forms around.


<p>[01:37:34] So for example, I can define a Shiab ("Ship in a bottle") operator. Okay. That takes, I forms valued in the adjoint bundle.
<p>[01:37:34] So for example, I can define a Shiab ("Ship in a bottle") operator. Okay. That takes, [$$\Omega_{i}$$] forms valued in the adjoint bundle.


<p>[01:37:59] To much higher degree forms valued in the adjoint bundle. So for this, in this case, for example, it would take a two form two a D minus three plus two or a D minus one form. So curvature is an add value two form. And if I had such a Cieve? operator, it would take add value tow form to add value. D minus one forms, which is exactly the right space to be an alpha coming from the derivative of an action.
<p>[01:37:59] To much higher-degree forms valued in the adjoint bundle. So for in this case, for example, it would take a two-form to a d-minus-three-plus-two or a d-minus one-form. So curvature is an add-value two-form. And if I had such a Shiab operator, it would take add-value two-forms to add-value d-minus one-forms, which is exactly the right space to be an $$\alpha$$ coming from the derivative of an action.


<p>[01:38:38] This is exactly what Einstein was doing. He took the curvature, which was large, and he bent it back and he sheared off the vial curvature and they took that part and he pushed it back along the space of metrics to give us something, which we nowadays call Ricci. Flow and ability for the curvature to direct us to the next structure.
<p>[01:38:38] This is exactly what Einstein was doing. He took the curvature, which was large, and he bent it back and he sheared off the [[Weyl curvature]] and they took that part and he pushed it back along the space of metrics to give us something, which we nowadays call [[Ricci Flow]] and ability for the curvature to direct us to the next structure.


<p>[01:38:59] Well, we're doing the same thing here. We're taking the curvature and we can now push it back onto the space of connections.
<p>[01:38:59] Well, we're doing the same thing here. We're taking the curvature and we can now push it back onto the space of connections.
<p>[01:39:17] Okay.


<p>[01:39:28] Can you just clarify what the index of the is? So you take all your forms to D minus three plus I,
<p>[01:39:28] Can you just clarify what the index of the is? So you take all your forms to D minus three plus I,
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