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European Mathematical Society Publishing House
2024-03-29 01:16:36
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https://www.ems-ph.org/meta/jmeta-stream.php?jrn=CMH&vol=87&iss=1&update_since=2024-03-29
Commentarii Mathematici Helvetici
Comment. Math. Helv.
CMH
0010-2571
1420-8946
General
10.4171/CMH
http://www.ems-ph.org/doi/10.4171/CMH
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European Mathematical Society Publishing House
Zuerich, Switzerland
© Swiss Mathematical Society
87
2012
1
Strict sub-solutions and Mañé potential in discrete weak KAM theory
Maxime
Zavidovique
Université de Lyon, LYON CEDEX 07, FRANCE
Discrete weak KAM theory, Aubry–Mather theory, continuous and discontinuous critical sub-solutions
In this paper, we explain some facts on the discrete case of weak KAM theory. In that setting, the Lagrangian is replaced by a cost $c\colon X\times X \rightarrow \mathbb{R}$, on a “reasonable” space $X$. This covers for example the case of periodic time-dependent Lagrangians. As is well known, it is possible in that case to adapt most of weak KAM theory. A major difference is that critical sub-solutions are not necessarily continuous. We will show how to define a Mañé potential. In contrast to the Lagrangian case, this potential is not continuous. We will recover the Aubry set from the set of continuity points of the Mañé potential, and also from critical sub-solutions.
Dynamical systems and ergodic theory
Partial differential equations
General
1
39
10.4171/CMH/247
http://www.ems-ph.org/doi/10.4171/CMH/247
On the topology of fillings of contact manifolds and applications
Alexandru
Oancea
Université Pierre et Marie Paris 06, PARIS, FRANCE
Claude
Viterbo
Ecole Normale Superieure, PARIS CEDEX 05, FRANCE
Topology of symplectic fillings of contact manifolds, obstructions to contact embeddings
The aim of this paper is to address the following question: given a contact manifold $(\Sigma, \xi)$, what can be said about the symplectically aspherical manifolds $(W, \omega)$ bounded by $(\Sigma, \xi)$? We first extend a theorem of Eliashberg, Floer and McDuff to prove that, under suitable assumptions, the map from $H_{*}(\Sigma)$ to $H_{*}(W)$ induced by inclusion is surjective. We apply this method in the case of contact manifolds admitting a contact embedding in $\mathbb{R}^{2n}$ or in a subcritical Stein manifold. We prove in many cases that the homology of the fillings is uniquely determined. Finally, we use more recent methods of symplectic topology to prove that, if a contact hypersurface has a subcritical Stein filling, then all its SAWC fillings have the same homology. A number of applications are given, from obstructions to the existence of Lagrangian or contact embeddings, to the exotic nature of some contact structures. We refer to the table in Section~\ref{table} for a summary of our results.
Differential geometry
Several complex variables and analytic spaces
Manifolds and cell complexes
General
41
69
10.4171/CMH/248
http://www.ems-ph.org/doi/10.4171/CMH/248
Height pairings, exceptional zeros and Rubin’s formula: the multiplicative group
Kâzim
Büyükboduk
Koç University, ISTANBUL, TURKEY
Zeros, cyclotomic units, height pairings, $p$-adic $L$-functions
In this paper we prove a formula, much in the spirit of one due to Rubin, which expresses the leading coefficients of various $p$-adic $L$-functions in the presence of an exceptional zero in terms of Nekovář’s $p$-adic height pairings on his extended Selmer groups. In a particular case, the Rubin-style formula we prove recovers a $p$-adic Kronecker limit formula. In a disjoint case, we observe that our computations with Nekovář’s heights agree with the Ferrero–Greenberg formula (more generally, Gross’ conjectural formula) for the leading coefficient of the Kubota–Leopoldt $p$-adic $L$-function (resp., the Deligne–Ribet $p$-adic $L$-function) at $s=0$.
Number theory
General
71
111
10.4171/CMH/249
http://www.ems-ph.org/doi/10.4171/CMH/249
Pure states, nonnegative polynomials and sums of squares
Sabine
Burgdorf
Universität Konstanz, KONSTANZ, GERMANY
Claus
Scheiderer
Universität Konstanz, KONSTANZ, GERMANY
Markus
Schweighofer
Universität Konstanz, KONSTANZ, GERMANY
Pure states, extremal homomorphisms, order units, nonnegative polynomials, sums of squares, convex cones, quadratic modules, preorderings, semirings
In recent years, much work has been devoted to a systematic study of polynomial identities certifying strict or non-strict positivity of a polynomial $f$ on a basic closed set $K\subset\mathbb{R}^n$. The interest in such identities originates not least from their importance in polynomial optimization. The majority of the important results requires the archimedean condition, which implies that $K$ has to be compact. This paper introduces the technique of pure states into commutative algebra. We show that this technique allows an approach to most of the recent archimedean Stellensätze that is considerably easier and more conceptual than the previous proofs. In particular, we reprove and strengthen some of the most important results from the last years. In addition, we establish several such results which are entirely new. They are the first that allow $f$ to have arbitrary, not necessarily discrete, zeros in $K$.
Order, lattices, ordered algebraic structures
Number theory
Commutative rings and algebras
Real functions
113
140
10.4171/CMH/250
http://www.ems-ph.org/doi/10.4171/CMH/250
On the uniform perfectness of the groups of diffeomorphisms of even-dimensional manifolds
Takashi
Tsuboi
University of Tokyo, TOKYO, JAPAN
Diffeomorphism group, uniformly perfect, commutator subgroup
We show that the identity component Diff$^r(M^{2m})_0$ of the group of $C^r$ diffeomorphisms of a compact $(2m)$-dimensional manifold $M^{2m}$ ($1\leq r\leq \infty$, $r\neq 2m+1$) is uniformly perfect for $2m\geq 6$, i.e., any element of Diff$^r(M^{2m})_0$ can be written as a product of a bounded number of commutators. It is also shown that for a compact connected manifold $M^{2m}$ ($2m\geq 6$), the identity component Diff$^r(M^{2m})_0$ of the group of $C^r$ diffeomorphisms of $M^{2m}$ ($1\leq r\leq \infty$, $r\neq 2m+1$) is uniformly simple, i.e., for elements $f$ and $g$ of Diff$^r(M^{2m})_0\setminus \{$id$\}$, $f$ can be written as a product of a bounded number of conjugates of $g$ or $g^{-1}$.
Manifolds and cell complexes
Dynamical systems and ergodic theory
General
141
185
10.4171/CMH/251
http://www.ems-ph.org/doi/10.4171/CMH/251
The Dolgachev surface
Selman
Akbulut
Michigan State University, EAST LANSING, UNITED STATES
Dolgachev surface, handlebody decomposition
We prove that the Dolgachev surface $E(1)_{2,3}$ admits a handlebody decomposition without 1- and 3-handles, and we draw the explicit picture of this handlebody. We also locate a “cork” inside of $E(1)_{2,3}$, so that $E(1)_{2,3}$ is obtained from $E(1)$ by twisting along this cork.
Global analysis, analysis on manifolds
Manifolds and cell complexes
General
187
241
10.4171/CMH/252
http://www.ems-ph.org/doi/10.4171/CMH/252