"The point of noncommutative geometry is NOT to extend classical concepts...it's more complicated than that. The point is there will be totally NEW phenomenon in the noncommutative case which will have NO commutative counterpart." Alain Connes
Non commutative algebra …. It’s a bit foggy to have a clear understanding of what this consists of a priori before one encounters the first ‘race’ of objects that express this motif : The (de rham) form - cohomology groups . That has the beautiful effect of ‘global -originally’ defining a way of calculating … Practical quantities of engineering by skipping over the discipline of canonical pullback algebra (integral calculus) and deriving these things with a logical substrate that’s reflective of perhaps of the ‘global nature’ of the end quantities: the integration (over forms) … There is also a local pullback algebra: and as it were , is noncommutative , only in the sense that it is symplectic: a cross b = (-1) ^ … * b cross a This , in comparison to the quotients and subspaces - as he mentions in the beginning - over the tensor products of a linear space V, Could be called the ‘ local’ (non-commutative) algebra of forms reflective of the discipline of integral calculus…. This is the most primary example (and discipline) of non-commutative algebra : and this promise that the so called de rham cohomology - that it globally achieves integral calculus (local, though the local algebra of forms may be) with the groups H ^p : that to me are the ‘origination’ of non-commutativity. The groups are a ‘sweet spot’ between the smooth analysis’s requirement of the general concept of cohomology: the algebraic quotient of kerT by imT , that actually; there is a sequence of them …. This is the use of algebra to get at space! In total (for me) there’s two instances of noncommutativity … and there’s an imposed (incorrect sense of) causality if one assumes commutative algebra to more natural: that one perturbs from that to arrive at noncommutativity. As a student of this algebra ‘to speak of space’ , I can’t visualize myself as having derived exactly these features of noncommutativity…. He seems to be starting from bare roots , and inducing a sense of non commutativity that is not necessarily textbook, which is interesting.
Thank you very much for this upload! I just started getting into the field of non-commutative geometry as a part of my master's thesis and I was wondering if there are more scuh talk from A. Connes about it somewhere else?
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"The point of noncommutative geometry is NOT to extend classical concepts...it's more complicated than that. The point is there will be totally NEW phenomenon in the noncommutative case which will have NO commutative counterpart." Alain Connes
Non commutative algebra ….
It’s a bit foggy to have a clear understanding of what this consists of a priori before one encounters the first ‘race’ of objects that express this motif :
The (de rham) form - cohomology groups . That has the beautiful effect of ‘global -originally’ defining a way of calculating …
Practical quantities of engineering by skipping over the discipline of canonical pullback algebra (integral calculus) and deriving these things with a logical substrate that’s reflective of perhaps of the ‘global nature’ of the end quantities: the integration (over forms) …
There is also a local pullback algebra: and as it were , is noncommutative , only in the sense that it is symplectic:
a cross b = (-1) ^ … * b cross a
This , in comparison to the quotients and subspaces - as he mentions in the beginning - over the tensor products of a linear space V,
Could be called the ‘ local’ (non-commutative) algebra of forms reflective of the discipline of integral calculus….
This is the most primary example (and discipline) of non-commutative algebra : and this promise that the so called de rham cohomology - that it globally achieves integral calculus (local, though the local algebra of forms may be) with the groups H ^p : that to me are the ‘origination’ of non-commutativity.
The groups are a ‘sweet spot’ between the smooth analysis’s requirement of the general concept of cohomology: the algebraic quotient of kerT by imT , that actually; there is a sequence of them …. This is the use of algebra to get at space!
In total (for me) there’s two instances of noncommutativity … and there’s an imposed (incorrect sense of) causality if one assumes commutative algebra to more natural: that one perturbs from that to arrive at noncommutativity.
As a student of this algebra ‘to speak of space’ , I can’t visualize myself as having derived exactly these features of noncommutativity….
He seems to be starting from bare roots , and inducing a sense of non commutativity that is not necessarily textbook, which is interesting.
Thank you very much for this upload! I just started getting into the field of non-commutative geometry as a part of my master's thesis and I was wondering if there are more scuh talk from A. Connes about it somewhere else?
Amazing video!