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SUMMARY:Santiago Barbieri (Universitat de Barcelona)\, Semi-algebraic geom
etry and generic hamiltonian stability
DTSTART:20240116T100000Z
DTEND:20240116T110000Z
DTSTAMP:20240222T022700Z
UID:indico-event-10624@indico.math.cnrs.fr
DESCRIPTION:As it is well-known\, the steepness property is a local geomet
ric transversality condition on the gradient of C2functions which proves f
undamental in order to ensure the stability over long timespans of integra
ble Hamiltonian systems that undergo a small perturbation. Though steepnes
s is generic - both in measure and in topological sense - among functions
of high enough regularity\, the original definition of this property is no
t constructive and\, up to very recent times\, the few existing criteria t
o check steepness were non-generically verified and applied only to functi
ons depending on a low number of variables. By combining Yomdin's Lemma on
the analytic reparametrization of semi-algebraic sets together with non-t
rivial estimates on the codimension of suitable real-algebraic varieties\,
in this talk I will state explicit algebraic criteria for steepness which
are generically verified and apply to functions depending on any number o
f variables. This constitutes a very important result for applications\, e
.g. in celestial mechanics. The criteria can be constructed recursively an
d are based on algebraic equalities involving the derivatives of the studi
ed function up to any given order and external real parameters\, some of w
hich belong to compact sets and some others to non-compact sets. Moreover\
, it can be shown that\, generically\, the non-compact external parameters
can be eliminated from the equalities with the help of a linear quantifie
r elimination algorithm: this represents a crucial improvement for numeric
al implementations of the criteria.References:1) S. Barbieri\, "Semi-algeb
raic Geometry and generic Hamiltonian stability"\, preprint. https://hal.s
cience/hal-04213250/2) S. Barbieri\, "On the algebraic properties of expon
entially stable integrable hamiltonian systems"\, Ann. Fac. Sci. Toulouse\
, 31(6): 1365-1390\, 20223) N. N. Nekhoroshev\, "Stable lower estimates fo
r smooth mappings and for gradients of smooth functions"\, Math USSR Sb.\,
19(3):425–467\, 19734) G. Schirinzi\, M. Guzzo\, "On the formulation of
new explicit conditions for steepness from a former result of N.N. Nekhor
oshev"\, J. Math. Phys\, 54\, 2013\n\nhttps://indico.math.cnrs.fr/event/10
624/
LOCATION:salle de conférence (Valrose)
URL:https://indico.math.cnrs.fr/event/10624/
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