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European Mathematical Society Publishing House
2024-03-29 13:06:25
9
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JEMS&vol=14&iss=6&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
14
2012
6
Higher-dimensional cluster combinatorics and representation theory
Steffen
Oppermann
Universität Köln, KÖLN, GERMANY
Hugh
Thomas
University of New Brunswick, FREDERICTON, CANADA
Higher Auslander algebras were introduced by Iyama generalizing classical concepts from representation theory of finite-dimensional algebras. Recently these higher analogues of classical representation theory have been increasingly studied. Cyclic polytopes are classical objects of study in convex geometry. In particular, their triangulations have been studied with a view towards generalizing the rich combinatorial structure of triangulations of polygons. In this paper, we demonstrate a connection between these two seemingly unrelated subjects. We study triangulations of even-dimensional cyclic polytopes and tilting modules for higher Auslander algebras of linearly oriented type $A$ which are summands of the cluster tilting module. We show that such tilting modules correspond bijectively to triangulations. Moreover mutations of tilting modules correspond to bistellar flips of triangulations. For any $d$-representation finite algebra we introduce a certain $d$-dimensional cluster category and study its cluster tilting objects. For higher Auslander algebras of linearly oriented type $A$ we obtain a similar correspondence between cluster tilting objects and triangulations of a certain cyclic polytope. Finally we study certain functions on generalized laminations in cyclic polytopes, and show that they satisfy analogues of tropical cluster exchange relations. Moreover we observe that the terms of these exchange relations are closely related to the terms occurring in the mutation of cluster tilting objects.
Commutative rings and algebras
Order, lattices, ordered algebraic structures
General
1679
1737
10.4171/JEMS/345
http://www.ems-ph.org/doi/10.4171/JEMS/345
The structure of a local embedding and Chern classes of weighted blow-ups
Anca
Mustaţă
University College Cork, CORK, IRELAND
Andrei
Mustaţă
University College Cork, CORK, IRELAND
Deligne-Mumford stack, local embedding, etale lift, Chern classes, weighted blow-ups, moduli space of stable maps
For a proper local embedding between two Deligne--Mumford stacks $Y$ and $X$, we find, under certain mild conditions, a new (possibly non-separated) Deligne--Mumford stack $X'$, with an etale, surjective and universally closed map to the target $X$, and whose fiber product with the image of the local embedding is a finite union of stacks with corresponding etale, surjective and universally closed maps to $Y$. Moreover, a natural set of weights on the substacks of $X'$ allows the construction of a universally closed push-forward, and thus a comparison between the Chow groups of $X'$ and $X$. We apply the construction above to the computation of the Chern classes of a weighted blow-up along a regular local embedding via deformation to a weighted normal cone and localization. We describe the stack $X'$ in the case when $X$ is the moduli space of stable maps with local embeddings at the boundary. We apply the methods above to find the Chern classes of the stable map spaces.
Algebraic geometry
General
1739
1794
10.4171/JEMS/346
http://www.ems-ph.org/doi/10.4171/JEMS/346
Representation of Itô integrals by Lebesgue/Bochner integrals
Qi
Lü
University of Electronic Science and Technology, CHENGDU, CHINA
Jiongmin
Yong
University of Central Florida, ORLANDO, UNITED STATES
Xu
Zhang
Chinese Academy of Sciences, BEIJING, CHINA
Itô integral, Lebesgue integral, Bochner integral, range inclusion, Riesz-type representation theorem
In [Yong 2004], it was proved that as long as the integrand has certain properties, the corresponding It\^o integral can be written as a (parameterized) Lebesgue integral (or a Bochner integral). In this paper, we show that such a question can be answered in a more positive and refined way. To do this, we need to characterize the dual of the Banach space of some vector-valued stochastic processes having different integrability with respect to the time variable and the probability measure. The later can be regarded as a variant of the classical Riesz Representation Theorem, and therefore it will be useful in studying other problems. Some remarkable consequences are presented as well, including a reasonable definition of exact controllability for stochastic differential equations and a condition which implies a Black–Scholes market to be complete.
Probability theory and stochastic processes
General
1795
1823
10.4171/JEMS/347
http://www.ems-ph.org/doi/10.4171/JEMS/347
On the Lawrence–Doniach model of superconductivity: magnetic fields parallel to the axes
Stan
Alama
McMaster University, HAMILTON, ONTARIO, CANADA
Lia
Bronsard
McMaster University, HAMILTON, ONTARIO, CANADA
Etienne
Sandier
Université Paris 12 – Val de Marne, CRÉTEIL CEDEX, FRANCE
Calculus of variations, elliptic equations and systems, superconductivity, vortices
We consider periodic minimizers of the Lawrence–Doniach functional, which models highly anisotropic superconductors with layered structure, in the simultaneous limit as the layer thickness tends to zero and the Ginzburg–Landau parameter tends to infinity. In particular, we consider the properties of minimizers when the system is subjected to an external magnetic field applied either tangentially or normally to the superconducting planes. For normally applied fields, our results show that the resulting “pancake” vortices will be vertically aligned. In horizontal fields we show that there are two-parameter regimes in which minimizers exhibit very different characteristics. The low-field regime resembles the Ginzburg–Landau model, while the high-field limit gives a “transparent state” described in the physical literature. To obtain our results we derive sharp matching upper and lower bounds on the global minimizers of the energy.
Partial differential equations
Global analysis, analysis on manifolds
General
1825
1857
10.4171/JEMS/348
http://www.ems-ph.org/doi/10.4171/JEMS/348
On a new normalization for tractor covariant derivatives
Matthias
Hammerl
Universität Wien, WIEN, AUSTRIA
Petr
Somberg
Charles University, PRAHA 8, CZECH REPUBLIC
Vladimír
Souček
Charles University, PRAHA 8, CZECH REPUBLIC
Josef
Šilhan
Masaryk University, BRNO, CZECH REPUBLIC
Parabolic geometry, prolongation of invariant overdetermined PDE's, BGG sequence, tractor covariant derivatives
A regular normal parabolic geometry of type $G/P$ on a manifold $M$ gives rise to sequences $D_i$ of invariant differential operators, known as the curved version of the BGG resolution. These sequences are constructed from the normal covariant derivative $\nabla^\omega$ on the corresponding tractor bundle $V,$ where $\omega$ is the normal Cartan connection. The first operator $D_0$ in the sequence is overdetermined and it is well known that $\nabla^\omega$ yields the prolongation of this operator in the homogeneous case $M = G/P$. Our first main result is the curved version of such a prolongation. This requires a new normalization of the tractor covariant derivative on $V$. Moreover, we obtain an analogue for higher operators $D_i$. In that case one needs to modify the exterior covariant derivative $d^{\nabla^\omega}$ by differential terms. Finally we demonstrate these results on simple examples in projective, conformal and Grassmannian geometry. Our approach is based on standard techniques of the BGG machinery.
Global analysis, analysis on manifolds
Differential geometry
General
1859
1883
10.4171/JEMS/349
http://www.ems-ph.org/doi/10.4171/JEMS/349
Geometric optics and instability for NLS and Davey–Stewartson models
Rémi
Carles
Mathématiques, CC 051, MONTPELLIER CEDEX 5, FRANCE
Eric
Dumas
Université Grenoble I, SAINT MARTIN D'HERES CEDEX, FRANCE
Christof
Sparber
University of Illinois at Chicago, CHICAGO, UNITED STATES
Nonlinear Schrödinger equation, Davey–Stewartson system, geometric optics, instability
We study the interaction of (slowly modulated) high frequency waves for multi-dimensional nonlinear Schr¨odinger equations with gauge invariant power-law nonlinearities and nonlocal perturbations. The model includes the Davey–Stewartson system in its elliptic-elliptic and hyperbolic-elliptic variants. Our analysis reveals a new localization phenomenon for nonlocal perturbations in the high frequency regime and allows us to infer strong instability results on the Cauchy problem in negative order Sobolev spaces, where we prove norm inflation with infinite loss of regularity by a constructive approach.
Partial differential equations
General
1885
1921
10.4171/JEMS/350
http://www.ems-ph.org/doi/10.4171/JEMS/350
Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation
Monica
Musso
Ponificia Universidad Catolica de Chile, SANTIAGO, CHILE
Frank
Pacard
École Polytechnique, PALAISEAU, FRANCE
Juncheng
Wei
University of British Columbia, VANCOUVER, CANADA
Nonradial bound states, nonlinear Schrödinger equations, balancing condition, Lyapunov–Schmidt reduction method
We address the problem of the existence of finite energy solitary waves for nonlinear Klein-Gordon or Schrödinger type equations $\Delta u - u + f(u) =0$ in $\mathbb R^N$, $u \in H^1 (\mathbb R^N)$, where $N\geq 2$. Under natural conditions on the nonlinearity $f$, we prove the existence of infinitely many nonradial solutions in any dimension $N \geq 2$. Our result complements earlier works of Bartsch and Willem ($N=4$ or $N \geq 6$) and Lorca-Ubilla ($N=5$) where solutions invariant under the action of $O(2) \times O(N-2)$ are constructed. In contrast, the solutions we construct are invariant under the action of $D_k \times O(N-2)$ where $D_k \subset O(2)$ denotes the dihedral group of rotations and reflexions leaving a regular planar polygon with $k$ sides invariant, for some integer $k\geq 7$, but they are not invariant under the action of $O(2) \times O(N-2)$.
Partial differential equations
General
1923
1953
10.4171/JEMS/351
http://www.ems-ph.org/doi/10.4171/JEMS/351
$\mathcal L$-invariants and Darmon cycles attached to modular forms
Victor
Rotger
Universitat Politècnica de Catalunya, BARCELONA, SPAIN
Marco Adamo
Seveso
Università degli Studi di Milano, MILANO, ITALY
Darmon point, $\mathcal L$-invariant, Shimura curves, quaternion algebra, $p$-adic integration
Let $f$ be a modular eigenform of even weight $k\geq 2$ and new at a prime $p$ dividing exactly the level with respect to an indefinite quaternion algebra. The theory of Fontaine-Mazur allows to attach to $f$ a monodromy module $\mathbf{D}^{FM}_f$ and an $\mathcal{L}$-invariant $\mathcal{L}^{FM}_f$. The first goal of this paper is building a suitable $p$-adic integration theory that allows us to construct a new monodromy module $\mathbf{D}_f$ and ${\mathcal{L}}$-invariant ${\mathcal{L}}_f$, in the spirit of Darmon. The two monodromy modules are isomorphic, and in particular the two ${\mathcal{L}}$-invariants are equal. Let $K$ be a real quadratic field and assume the sign of the functional equation of the $L$-series of $f$ over $K$ is $-1$. The Bloch-Beilinson conjectures suggest that there should be a supply of elements in the Selmer group of the motive attached to $f$ over the tower of narrow ring class fields of $K$. Generalizing work of Darmon for $k=2$, we give a construction of local cohomology classes which we expect to arise from global classes and satisfy an explicit reciprocity law, accounting for the above prediction.
Algebraic geometry
General
1955
1999
10.4171/JEMS/352
http://www.ems-ph.org/doi/10.4171/JEMS/352
The Kähler Ricci flow on Fano manifolds (I)
Xiuxiong
Chen
University of Wisconsin-Madison, MADISON CITY, UNITED STATES
Bing
Wang
Princeton University, PRINCETON, UNITED STATES
We study the evolution of pluri-anticanonical line bundles $K_M^{-\nu}$ along the Kähler Ricci flow on a Fano manifold $M$. Under some special conditions, we show that the convergence of this flow is determined by the properties of the pluri-anticanonical divisors of $M$. For example, the Kähler Ricci flow on $M$ converges when $M$ is a Fano surface satisfying $c_1^2(M)=1$ or $c_1^2(M)=3$. Combined with the works in [CW1] and [CW2], this gives a Ricci flow proof of the Calabi conjecture on Fano surfaces with reductive automorphism groups. The original proof of this conjecture is due to Gang Tian in [Tian90].
Manifolds and cell complexes
General
2001
2038
10.4171/JEMS/353
http://www.ems-ph.org/doi/10.4171/JEMS/353