387 research outputs found
Classical 6j-symbols and the tetrahedron
A classical 6j-symbol is a real number which can be associated to a labelling
of the six edges of a tetrahedron by irreducible representations of SU(2). This
abstract association is traditionally used simply to express the symmetry of
the 6j-symbol, which is a purely algebraic object; however, it has a deeper
geometric significance. Ponzano and Regge, expanding on work of Wigner, gave a
striking (but unproved) asymptotic formula relating the value of the 6j-symbol,
when the dimensions of the representations are large, to the volume of an
honest Euclidean tetrahedron whose edge lengths are these dimensions. The goal
of this paper is to prove and explain this formula by using geometric
quantization. A surprising spin-off is that a generic Euclidean tetrahedron
gives rise to a family of twelve scissors-congruent but non-congruent
tetrahedra.Comment: 46 pages. Published copy, also available at
http://www.maths.warwick.ac.uk/gt/GTVol3/paper2.abs.htm
Nodal count of graph eigenfunctions via magnetic perturbation
We establish a connection between the stability of an eigenvalue under a
magnetic perturbation and the number of zeros of the corresponding
eigenfunction. Namely, we consider an eigenfunction of discrete Laplacian on a
graph and count the number of edges where the eigenfunction changes sign (has a
"zero"). It is known that the -th eigenfunction has such zeros,
where the "nodal surplus" is an integer between 0 and the number of cycles
on the graph.
We then perturb the Laplacian by a weak magnetic field and view the -th
eigenvalue as a function of the perturbation. It is shown that this function
has a critical point at the zero field and that the Morse index of the critical
point is equal to the nodal surplus of the -th eigenfunction of the
unperturbed graph.Comment: 18 pages, 4 figure
Morse theory, closed geodesics, and the homology of free loop spaces
This is a survey paper on Morse theory and the existence problem for closed
geodesics. The free loop space plays a central role, since closed geodesics are
critical points of the energy functional. As such, they can be analyzed through
variational methods. The topics that we discuss include: Riemannian background,
the Lyusternik-Fet theorem, the Lyusternik-Schnirelmann principle of
subordinated classes, the Gromoll-Meyer theorem, Bott's iteration of the index
formulas, homological computations using Morse theory, - vs.
-symmetries, Katok's examples and Finsler metrics, relations to
symplectic geometry, and a guide to the literature.
The Appendix written by Umberto Hryniewicz gives an account of the problem of
the existence of infinitely many closed geodesics on the -sphere.Comment: 45 pages, 5 figures. Appendix by Umberto Hryniewic
- …