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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...