/sci/ - Science & Math

>>5434943
That is quite hard to beat ;-), I would have to opt for the non-commutative analogue or

<div class="math">\mathcal H^r(\mathcal M)=\bigoplus_{p+q=r}\mathcal H^{p,q}(\mathcal M),\;\mathcal H^{p,q}(\mathcal M)=\overline{\mathcal H^{q,p}(\mathcal M)}</div>
For complex M, pic very related.
>>5435037
Not him, but I found Higson and Roe's "Analytic K-Homology" to have a decent and modern review in terms of the algebraic K-theory. Nothing is better than the original papers, however.
>>5435058
It is the Atiyah-Singer index theorem, a beautiful result in the study of manifolds with endless applications. In general, the theorem allows us to calculate certain analytic data - the indices i.e. sums and differences of the numbers of independent solutions to differential equations of various kinds - in terms of topological data about the base manifold, e.g. the Betti numbers or the number of holes in the manifolds. The index of the Dirac operator for example is important to physicists as it determines the number of generations of leptons and quarks in the conventional compactifications of string theory with 6 internal dimensions, among numerous other things.

>>5434943
That is quite hard to beat ;-), I would have to opt for the non-commutative analogue or

<div class="math">\Omega^k(M)=\rm{im}\Delta_k\oplus\ker\Delta_k=\Delta(\Omega^k(M))\oplus\mathcal H^k;~~\forall\alpha\in\Omega^k(M),\;\exists\omega\in\Omega^k(M):\Delta\omega=\alpha\Longleftrightarrow \alpha\perp\mathcal H^k</div>
For complex M, pic very related.
>>5435037
Not him, but I found Higson and Roe's "Analytic K-Homology" to have a decent and modern review in terms the aspects of algebraic K-theory. Nothing is better than the original papers, however.
>>5435058
It is the Atiyah-Singer index theorem, a beautiful result in the study of manifolds with endless applications. In general, the theorem allows us to calculate certain analytic data - the indices i.e. sums and differences of the numbers of independent solutions to differential equations of various kinds - in terms of topological data about the base manifold, e.g. the Betti numbers or the number of holes in the manifolds. The index of the Dirac operator for example is important to physicists as it determines the number of generations of leptons and quarks in the conventional compactifications of string theory with 6 internal dimensions, among numerous other things.