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Manifolds, from Mr. Isomorphisms

2017-03-15 ยท Mr. Isomorphisms
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Ralph, you and I were dropping the word "manifold" on this site a year or so ago. It got me wondering if it is either necessary or helpful.

For example, what cases are we cutting out? What kind of situations are not manifolds?

  • manifolds with singularities (this is what we really meant, I think, since aberrant days are remembered by all)

  • algebraic spaces (recommended on MathOverflow as a precursor to understanding stacks = 2-sheaves)

-verdier duality (a condition which fails for non-manifolds)

In the 19th century, manifolds were always subsets of R^n (quoting bill fulton, representation theory). They might have holes or handles and they might have curvature. The curvature I think is what interests us in finance. Most term structures are curved, and order books are irregular as well.

All in all, I am still up in the air about whether this is a useful word to use in finance.