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European Mathematical Society Publishing House
2024-03-29 15:56:25
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https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JEMS&vol=18&iss=10&update_since=2024-03-29
Journal of the European Mathematical Society
J. Eur. Math. Soc.
JEMS
1435-9855
1435-9863
General
10.4171/JEMS
http://www.ems-ph.org/doi/10.4171/JEMS
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
18
2016
10
Symplectomorphism group relations and degenerations of Landau–Ginzburg models
Colin
Diemer
University of Miami, CORAL GABLES, UNITED STATES
Ludmil
Katzarkov
University of Miami, CORAL GABLES, UNITED STATES
Gabriel
Kerr
Kansas State University, MANHATTAN, UNITED STATES
Homological mirror symmetry, symplectomorphisms, Landau–Ginzburg models, minimal model program, toric varieties
We describe explicit relations in the symplectomorphism groups of hypersurfaces in toric stacks. To define the elements involved, we construct a proper stack of these hypersurfaces whose boundary represents stable pair degenerations. Our relations arise through the study of the one-dimensional strata of this stack. The results are then examined from the perspective of homological mirror symmetry where we view sequences of relations as maximal degenerations of Landau–Ginzburg models. We then study the $B$-model mirror to these degenerations, which gives a new mirror symmetry approach to the minimal model program.
Differential geometry
2167
2271
10.4171/JEMS/640
http://www.ems-ph.org/doi/10.4171/JEMS/640
A categorification of non-crossing partitions
Andrew
Hubery
Universität Bielefeld, BIELEFELD, GERMANY
Henning
Krause
Universität Bielefeld, BIELEFELD, GERMANY
Non-crossing partition, hereditary algebra, Grothendieck group, Weyl group, Coxeter group, exceptional sequence, symmetrisable generalised Cartan matrix, perpendicular category
We present a categorification of the non-crossing partitions given by crystallographic Coxeter groups. This involves a category of certain bilinear lattices, which are essentially determined by a symmetrisable generalised Cartan matrix together with a particular choice of a Coxeter element. Examples arise from Grothendieck groups of hereditary artin algebras.
Associative rings and algebras
Combinatorics
Group theory and generalizations
2273
2313
10.4171/JEMS/641
http://www.ems-ph.org/doi/10.4171/JEMS/641
Kirkwood gaps and diffusion along mean motion resonances in the restricted planar three-body problem
Jacques
Féjoz
IMCCE, PARIS, FRANCE
Marcel
Guàrdia
Université Paris 7 Denis Diderot, PARIS CEDEX 13, FRANCE
Vadim
Kaloshin
University of Maryland, COLLEGE PARK, UNITED STATES
Pablo
Roldán
Escola Tècnica Superior d'Enginyeria Industrial de Barcelona, BARCELONA, SPAIN
Three-body problem, instability, resonance, hyperbolicity, Mather mechanism, Arnol’d diffusion, Solar System, Asteroid Belt, Kirkwood gap
We study the dynamics of the restricted planar three-body problem near mean motion resonances, i.e. a resonance involving the Keplerian periods of the two lighter bodies revolving around the most massive one. This problem is often used to model Sun–Jupiter–asteroid systems. For the primaries (Sun and Jupiter), we pick a realistic mass ratio $\mu=10^{-3}$ and a small eccentricity $e_0>0$. The main result is a construction of a variety of non local diffusing orbits which show a drastic change of the osculating (instant) eccentricity of the asteroid, while the osculating semi major axis is kept almost constant. The proof relies on the careful analysis of the circular problem, which has a hyperbolic structure, but for which diffusion is prevented by KAM tori. In the proof we verify certain non-degeneracy conditions numerically. Based on the work of Treschev, it is natural to conjecture that the time of diffusion for this problem is $\sim \frac{-\ln (\mu e_0)}{\mu^{3/2} e_0}$. We expect our instability mechanism to apply to realistic values of $e_0$ and we give heuristic arguments in its favor. If so, the applicability of Nekhoroshev theory to the three-body problem as well as the long time stability become questionable. It is well known that, in the Asteroid Belt, located between the orbits of Mars and Jupiter, the distribution of asteroids has the so-called Kirkwood gaps exactly at mean motion resonances of low order. Our mechanism gives a possible explanation of their existence. To relate the existence of Kirkwood gaps with Arnol'd diffusion, we also state a conjecture on its existence for a typical $\epsilon$-perturbation of the product of the pendulum and the rotator. Namely, we predict that a positive conditional measure of initial conditions concentrated in the main resonance exhibits Arnol’d diffusion on time scales $\frac{- \ln \epsilon }{\epsilon^{2}}$.
Mechanics of particles and systems
Dynamical systems and ergodic theory
2315
2403
10.4171/JEMS/642
http://www.ems-ph.org/doi/10.4171/JEMS/642
Overconvergent subanalytic subsets in the framework of Berkovich spaces
Florent
Martin
Universität Regensburg, REGENSBURG, GERMANY
Berkovich spaces, semianalytic sets, subanalytic sets, overconvergent
We study the class of overconvergent subanalytic subsets of a $k$-affinoid space $X$ when $k$ is a non-archimedean field. These are the images along the projection $X \times \mathbb B^n \to X$ of subsets defined with inequalities between functions of $X \times \mathbb B^n$ which are overconvergent in the variables of $\mathbb B^n$. In particular, we study the local nature, with respect to $X$, of overconvergent subanalytic subsets. We show that they behave well with respect to the Berkovich topology, but not to the $G$-topology. This gives counter-examples to previous results on the subject, and a way to correct them. Moreover, we study the case dim$(X)=2$, for which a simpler characterisation of overconvergent subanalytic subsets is proven.
Algebraic geometry
Mathematical logic and foundations
Field theory and polynomials
Several complex variables and analytic spaces
2405
2457
10.4171/JEMS/643
http://www.ems-ph.org/doi/10.4171/JEMS/643