A publication of the Societe Mathematique de France.
ISBN: 978-2-85629-892-3
Asterisque, Number 403
15 October 2018; Copyright Year: 2018;
Pages: 148; Softcover;
Analysis
Differential Equations
Graduate students and research mathematicians.
The authors prove that small, semi-linear Hamiltonian perturbations of the defocusing
nonlinear Schrodinger (dNLS) equation on the circle have an abundance of invariant
tori of any size and (finite) dimension which support quasi-periodic solutions. When compared
with previous results, the novelty consists in considering perturbations which do not
satisfy any symmetry condition (they may depend on x in an arbitrary way) and need not be
analytic. The main difficulty is posed by pairs of almost resonant dNLS frequencies.
The proof is based on the integrability of the dNLS equation, in particular, the fact that the
nonlinear part of the Birkhoff coordinates is one smoothing.
The authors implement a Newton-Nash-Moser iteration scheme to construct the invariant tori.
The key point is the reduction of linearized operators, coming up in the iteration scheme, to
2 × 2 block diagonal ones with constant coefficients together with sharp asymptotic estimates
of their eigenvalues.
A publication of the Societe Mathematique de France.
ISBN: 978-2-85629-893-0
Asterisque, Number 404
Published: 15 October 2018; Copyright Year: 2018;
Pages: 110; Softcover;
Analysis
Probability and Statistics
Graduate students and research mathematicians.
The authors derive Feynman-Kac formulas for the ultra-violet renormalized
Nelson Hamiltonian with a Kato decomposable external potential and for corresponding
fiber Hamiltonians in the translation invariant case. They simultaneously treat massive and
massless bosons. Furthermore, we present a non-perturbative construction of a renormalized
Nelson Hamiltonian in a non-Fock representation defined as the generator of a corresponding
Feynman-Kac semi-group.
The authors’ novel analysis of the vacuum expectation of the Feynman-Kac integrands shows
that if the external potential and the Pauli principle are dropped, then the spectrum of the N
-particle renormalized Nelson Hamiltonian is bounded from below by some negative universal
constant times g 4 n 3 for all values of the coupling constant g . A variational argument also
yields an upper bound of the same form for large g 2 N
The authors further verify that the semi-groups generated by the ultra-violet renormalized
Nelson Hamiltonian and its non-Fock version are positivity improving with respect to a
natural self-dual cone if the Pauli principle is ignored. In another application, they discuss
continuity properties of elements in the range of the semi-group of the renormalized Nelson
Hamiltonian.
A publication of the Societe Mathematique de France.
ISBN: 978-2-85629-867-1
Documents Mathematiques, Number 16
Published: 15 October 2018; Copyright Year: 2018;
Pages: 200; Softcover;
Analysis
Graduate students and research mathematicians.
Michael R. Herman was a leading specialist of dynamical system theory. After his
sudden death, he left many handwritten notes, some of them of high quality and never published.
Jean-Christophe Yoccoz, who was not only his scientific executor but also one of his first students
and one of his most cherished mathematical companions, decided to assemble the most
important notes and make them available to the community. He gathered a team of colleagues
acquainted with Herman’s research domains to classify and investigate these notes and then to
type the selected ones.
The guideline of this collective work was to adhere as strictly as possible to the original manuscript, adding, if necessary, some helpful corrections or comments. The result is this volume of
unpublished notes in which readers will discover or rediscover some facets of the mathematical
mind of Michael R. Herman.
A publication of the Societe Mathematique de France.
ISBN: 978-2-85629-890-9
Memoires de la Societe Mathematique de France, Number 157
Published: 15 October 2018; Copyright Year: 2018;
Pages: 114; Softcover;
Algebra and Algebraic Geometry
Graduate students and research mathematicians.
The author constructs a motivic Eilenberg-Mac Lane spectrum with a highly structured
multiplication over general base schemes which represents Levine’s motivic cohomology,
defined via Bloch’s cycle complexes, over smooth schemes over Dedekind domains. The
author’s method involves gluing p -completed and rational parts along an arithmetic square.
Hereby, the finite coefficient spectra are obtained by truncated etale sheaves (relying on the
now proven Bloch-Kato conjecture) and a variant of Geisser’s version of syntomic cohomology,
and the rational spectra are the ones which represent Beilinson motivic cohomology.
As an application, the arithmetic motivic cohomology groups can be realized as Ext-groups in
a triangulated category of motives with integral coefficients. The author’s spectrum is compatible
with base change giving rise to a formalism of six functors for triangulated categories of
motivic sheaves over general base schemes, including the localization triangle.
Further applications are a generalization of the Hopkins-Morel isomorphism and a structure
result for the dual motivic Steenrod algebra in the case where the coefficient characteristic is
invertible on the base scheme.
A publication of the Societe Mathematique de France.
ISBN: 978-2-85629-891-6
Memoires de la Societe Mathematique de France,Number 158
Published: 15 October 2018; Copyright Year: 2018;
Pages: 162; Softcover;
Probability and Statistics
Graduate students and research mathematicians.
There is a natural measure on loops (time-parametrized trajectories that, in the end,
return to the origin) which one can associate to a wide class of Markov processes. The Poisson
ensembles of Markov loops are Poisson point processes with intensity proportional to these
measures. In wide generality, these Poisson ensembles of Markov loops are related, at intensity
parameter 1/2, to the Gaussian free field, and at intensity parameter 1, to the loops done by a
Markovian sample path.
Here, the author studies the specific case when the Markov process is a one-dimensional diffusion.
After a detailed description of the measure, the author studies the Poisson point processes
of loops, their occupation fields, and explains how to sample these Poisson ensembles of loops
out of diffusion sample path perturbed at their successive minima.
Finally, the author introduces a couple of interwoven determinantal point processes on the
line, which is a dual through Wilson’s algorithm of Poisson ensembles of loops, and studies
the properties of these determinantal point processes.
Paperback ISBN: 9780128178010
Imprint: Academic Press
Published Date: 1st June 2019
Page Count: 350
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