9,412 research outputs found
Percival Lagrangian approach to Aubry-Mather theory
We present some streamlined proofs of some of the basic results in
Aubry-Mather theory (existence of quasi-periodic minimizers, multiplicity
results when there are gaps among minimizers) based on the study of hull
functions. We present results in arbitrary number of dimensions
We also compare the proofs and results with those obtained in other
formalisms
Stable Flags and the Riemann-Hilbert Problem
We tackle the Riemann-Hilbert problem on the Riemann sphere as stalk-wise
logarithmic modifications of the classical R\"ohrl-Deligne vector bundle. We
show that the solutions of the Riemann-Hilbert problem are in bijection with
some families of local filtrations which are stable under the prescribed
monodromy maps. We introduce the notion of Birkhoff-Grothendieck
trivialisation, and show that its computation corresponds to geodesic paths in
some local affine Bruhat-Tits building. We use this to compute how the type of
a bundle changes under stalk modifications, and give several corresponding
algorithmic procedures.Comment: 39 page
- …