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My favorite example concerns Novikov conjecture, on the homotopy invariance of higher signatures for closed manifolds with fundamental group $G$: see http://en.wikipedia.org/wiki/Novikov_conjecture (note that this Wikipedia entry rather stupidly says that it has been proved for finitely generated abelian groups: that's correct, but it was proved for MANY more groups, e.g. hyperbolic groups, countable subgroups of $GL_n(\mathbb{C})$, etc...). I think we agree that this is a conjecture in topology.

Now, look at this remarkable result by Guoliang Yu ( http://www.kryakin.com/files/Invent_mat_(2_8)/139/139_21.pdf)http://www.kryakin.com/files/Invent_mat_(2_8)/139/139_21.pdf )

"If the group $G$ admits a coarse embedding into Hilbert space, then it satisfies the Novikov conjecture".

A coarse embedding is a map $f:G\rightarrow L^2$ for which there exists control functions $\rho_{\pm}:\mathbb{R}^+\rightarrow\mathbb{R}$, with $\lim_{t\rightarrow\infty}\rho_\pm(t)=\infty$, which ``control'' $f$ in the sense hat, for every $x,y\in G$:

$$\rho_-(|x^{-1}y|_S)\leq\|f(x)-f(y)\|_2\leq \rho_+(|x^{-1}y|_S),$$ where $|.|_S$ denotes word length with respect to some finite generating subset $S$ in $G$. The existence of a coarse embedding is a weak metric condition (actually we know of basically just one class of groups which do not admit such an embedding, the ``Gromov monsters''). And this weak metric condition, quite surprisingly, implies a strong consequence in topology.

Now my point is that the two known proofs of Yu's result (the original one, and the one by Skandalis-Tu-Yu, see http://www.math.univ-metz.fr/~tu/publi/coarse.pdf) both appeal in a fundamental way to the tools of non-commutative geometry: $C^*$-algebras, $K$-theory, groupoids, Kasparov's $KK$-theory (to be precise: ``equivariant $KK$-theory for groupoids'').

Now to answer your first question: how to motivate a graduate student? Well, the subject mixes classical geometry, algebraic topology, non-commutative algebra, functional analysis, so it is one of those subjects that give you a feeling of the unity of mathematics...
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My favorite example concerns Novikov conjecture, on the homotopy invariance of higher signatures for closed manifolds with fundamental group $G$: see http://en.wikipedia.org/wiki/Novikov_conjecture (note that this Wikipedia entry rather stupidly says that it has been proved for finitely generated abelian groups: that's correct, but it was proved for MANY more groups, e.g. hyperbolic groups, countable subgroups of $GL_n(\mathbb{C})$, etc...). I think we agree that this is a conjecture in topology.

Now, look at this remarkable result by Guoliang Yu (http://www.kryakin.com/files/Invent_mat_(2_8)/139/139_21.pdf)

"If the group $G$ admits a coarse embedding into Hilbert space, then it satisfies the Novikov conjecture".

A coarse embedding is a map $f:G\rightarrow L^2$ for which there exists control functions $\rho_{\pm}:\mathbb{R}^+\rightarrow\mathbb{R}$, with $\lim_{t\rightarrow\infty}\rho_\pm(t)=\infty$, which ``control'' $f$ in the sense hat, for every $x,y\in G$:

$$\rho_-(|x^{-1}y|_S)\leq\|f(x)-f(y)\|_2\leq \rho_+(|x^{-1}y|_S),$$ where $|.|_S$ denotes word length with respect to some finite generating subset $S$ in $G$. The existence of a coarse embedding is a weak metric condition (actually we know of basically just one class of groups which do not admit such an embedding, the ``Gromov monsters''). And this weak metric condition, quite surprisingly, implies a strong consequence in topology.

Now my point is that the two known proofs of Yu's result (the original one, and the one by Skandalis-Tu-Yu, see http://www.math.univ-metz.fr/~tu/publi/coarse.pdf) both appeal in a fundamental way to the tools of non-commutative geometry: $C^*$-algebras, $K$-theory, groupoids, Kasparov's $KK$-theory (to be precise: ``equivariant $KK$-theory for groupoids'').

Now to answer your first question: how to motivate a graduate student? Well, the subject mixes classical geometry, algebraic topology, non-commutative algebra, functional analysis, so it is one of those subjects that give you a feeling of the unity of mathematics...