Morse
Theory |

**In General** > s.a. connections; crystals;
homology types [Morse homology];
jacobi metric.

* __Idea__: A modern version of the calculus of variations, which uses
infinite-dimensional manifolds, their points being the geodesics of a given
manifold.

* __Applications__: Studying symmetries in crystals (study symmetry breaking
by minimizing the potential energy).

@ __General references__: Morse 64; Milnor 73; Rassias 92; Matsumoto 01; Harvey & Lawson AM(01)
[based on de Rham-Federer theory of currents]; Nicolaescu 07; Katz 14 [manifolds with boundary]; Knudson 15 [smooth and discrete Morse theory].

@ __Morse index theorem__: Rezende LMP(98)
[proof]; Piccione & Tausk
JMP(99),
Top(02) [semi-Riemannian geometry].

@ __In general relativity__: Woodhouse CMP(76)
[and spacetime topology]; Giannoni et al JGP(00)
[particles]; > s.a. Fermat's Principle.

@ __Related topics__: Floer BAMS(87); Ghrist et al m.DS/01 [on spaces of braids, and Lagrangian dynamics].

@ __Generalizations__: Goresky & MacPherson 88 [stratified]; Perlick JMP(95)
[infinite-dimensional]; Minian T&A(12)-a1007 [discrete, for posets].

> __Online resources__:
see MathWorld page; Wikipedia page.

**Morse Function**

* __Idea__: Once a Morse
function has been defined on a manifold, information about its topology can
be deduced from its critical elements.

$ __Def__: Given a smooth
cobordism between *M* and *N*, a Morse function
is a function *f* with nowhere vanishing gradient except at isolated
points, where the Hessian is non-degenerate.

* __In general relativity__:
It can be used as time to define a Lorentzian metric starting from a Riemannian
one; The Lorentzian metric will be degenerate where *f* has
vanishing gradient, but it will not have closed timelike curves.

@ __Discrete__: Lewiner
et al CG(03)
[2D]; Chari & Joswig DM(05)
[complex of discrete Morse functions of a fixed simplicial complex]; Ayala et
al T&A(09)
[Morse theory and topology of graphs]; Sawicki JPA(12) [for graph configuration spaces, and quantum statistics for particles on networks]; Bloch DM(13) [polyhedral representation].

**Morse Inequalities**

$ __Def__: If *f *: *M* → \(\mathbb R\),
with *M* a compact *n*-dimensional
differentiable manifold, *c*_{k} the
number of non-degenerate critical points with index *k*,
and *R*_{k}(*M*) the *k*-th
Betti number, then

*R*_{k}(*M*) – *R*_{k–1}(*M*)
+ ... ± *R*_{0}(*M*)
≤ *c*_{k} –
*c*_{k+1} + ... ± *c*_{0} ,

with equality if *k* = *n*.

* __Corollary__: For all integers *k*, *c*_{k }> *R*_{k}(*M*).

**Morse Vector Field**

$ __Def__: Given a manifold *M* with boundary consisting of two disjoint
components, ∂*M* = *M*_{0} ∪ *M*_{1},
a Morse vector field is a vector field which points inwards on *M* and outwards on *M*.

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feb 2016