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coudy
coudy
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The cut locus of a point on a compact manifold has zero measure. So this allows to use a single chart centered at the point e.g. to compute integrals (typically radial coordinates in a chart given by the exponential).

You can exemplify this with the sphere (easy, the cut locus is a single point) and tori. The tori are related to Fourier analysis, so this hopefully speaks to applied mathematicians. Then show that there are more than just "square" tori. The cut locus depend on the lattice and since this is a Riemannian invariant, you can show that different tori are not isometric, and explain why we talk about the hexagonal lattice for the lattice built on $1$ and $e^{i {\pi \over 3}}$ (hint: the cut locus is a regular hexagon).

Concerning tori, you can then go slightly further along the relationship between Riemannian geometry and harmonic analysis, the link between isometry and isospectrality, and end up with the famous "Can we hear the shape of thea drum" talk and the 16-dimensional Milnor counterexample of two isospectral but not isometric tori, if you have time.

The cut locus of a point on a compact manifold has zero measure. So this allows to use a single chart centered at the point e.g. to compute integrals (typically radial coordinates in a chart given by the exponential).

You can exemplify this with the sphere (easy, the cut locus is a single point) and tori. The tori are related to Fourier analysis, so this hopefully speaks to applied mathematicians. Then show that there are more than just "square" tori. The cut locus depend on the lattice and since this is a Riemannian invariant, you can show that different tori are not isometric, and explain why we talk about the hexagonal lattice for the lattice built on $1$ and $e^{i {\pi \over 3}}$ (hint: the cut locus is a regular hexagon).

Concerning tori, you can then go slightly further along the relationship between Riemannian geometry and harmonic analysis, the link between isometry and isospectrality, and end up with the famous "Can we hear the shape of the drum" talk and the 16-dimensional Milnor counterexample of two isospectral but not isometric tori, if you have time.

The cut locus of a point on a compact manifold has zero measure. So this allows to use a single chart centered at the point e.g. to compute integrals (typically radial coordinates in a chart given by the exponential).

You can exemplify this with the sphere (easy, the cut locus is a single point) and tori. The tori are related to Fourier analysis, so this hopefully speaks to applied mathematicians. Then show that there are more than just "square" tori. The cut locus depend on the lattice and since this is a Riemannian invariant, you can show that different tori are not isometric, and explain why we talk about the hexagonal lattice for the lattice built on $1$ and $e^{i {\pi \over 3}}$ (hint: the cut locus is a regular hexagon).

Concerning tori, you can then go slightly further along the relationship between Riemannian geometry and harmonic analysis, the link between isometry and isospectrality, and end up with the famous "Can we hear the shape of a drum" talk and the 16-dimensional Milnor counterexample of two isospectral but not isometric tori, if you have time.

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coudy
coudy
  • 18.7k
  • 5
  • 75
  • 135

The cut locus of a point on a compact manifold has zero measure. So this allows to use a single chart centered at the point e.g. to compute integrals (typically radial coordinates in a chart given by the exponential).

You can exemplify this with the sphere (easy, the cut locus is a single point) and tori. The tori are related to Fourier analysis, so this hopefully speaks to applied mathematicians. Then show that there are more than just "square" tori. The cut locus depend on the lattice and since this is a Riemannian invariant, you can show that different tori are not isometric, and explain why we talk about the hexagonal lattice for the lattice built on $1$ and $e^{i {\pi \over 3}}$ (hint: the cut locus is a regular hexagon).

Concerning tori, you can then go slightly further along the relationship between Riemannian geometry and harmonic analysis, the link between isometry and isospectrality, and end up with the famous "Can we hear the shape of the drum" talk and the 16-dimensional Milnor counterexample of two isospectral but not isometric tori, if you have time.