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];
Kirwan & Penington a1906 [without non-degeneracy assumptions].

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

@ __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].

@ __And physics__: Woodhouse CMP(76) [and spacetime topology];
Giannoni et al JGP(00) [particles];
> s.a. Fermat's Principle;
topological field theories.

> __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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send feedback and suggestions to bombelli at olemiss.edu – modified 4 aug 2019