Communications on Pure and Applied Mathematics最新文献

{"title":"Planar Coulomb Gas on a Jordan Arc at Any Temperature","authors":"K. Courteaut, K. Johansson, F. Viklund","doi":"10.1002/cpa.70065","DOIUrl":"https://doi.org/10.1002/cpa.70065","url":null,"abstract":"We study a planar Coulomb gas confined to a sufficiently smooth Jordan arc in the complex plane, at inverse temperature . Let <jats:disp-formula/> be the normalized partition function, where is the capacity of , the entropy of the logarithmic equilibrium measure on , and . We compute the free energy as the number of particles tends to infinity, including the constant term: <jats:disp-formula/> Here is an explicit Selberg integral, is half the Loewner energy of a certain Jordan curve associated to plus a term corresponding to the endpoints of the arc, and is the Fekete energy, related to the zero temperature limit of the model. We also prove an asymptotic formula for the Laplace transform of linear statistics for sufficiently regular test functions. As a consequence, the centered empirical measure converges to a Gaussian field with explicit asymptotic mean, and asymptotic variance given by the Dirichlet energy of the bounded harmonic extension of the test function outside of the arc. A key tool in our analysis is the arc‐Grunsky operator associated to , reminiscent of but different from the classical Grunsky operator. We derive several basic properties of the arc‐Grunsky operator, including an estimate analogous to the strengthened Grunsky inequality and the relation to the Dirichlet integral. In our proofs arises from the Fredholm determinant and a substantial part of the paper is devoted to relating this to the Loewner energy.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"234 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149218629","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Stability of Viscous Three-Dimensional Stratified Couette Flow via Dispersion and Mixing","authors":"Michele Coti Zelati,&nbsp;Augusto Del Zotto,&nbsp;Klaus Widmayer","doi":"10.1002/cpa.70048","DOIUrl":"10.1002/cpa.70048","url":null,"abstract":"<p>This article explores the stability of stratified Couette flow in the viscous <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>3</mn>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 <annotation>$3d$</annotation>\u0000 </semantics></math> Boussinesq equations. In this system, mixing effects arise from the shearing background, and gravity acts as a restoring force leading to dispersive internal gravity waves. These mechanisms are of fundamentally different nature and relevant in complementary dynamical regimes. Our study combines them to establish a bound for the non-linear transition threshold, which is quantitatively larger than the inverse Reynolds number <span></span><math>\u0000 <semantics>\u0000 <mi>ν</mi>\u0000 <annotation>$nu$</annotation>\u0000 </semantics></math>, and increases with stronger stratification resp. gravity.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"79 12","pages":"2435-2490"},"PeriodicalIF":3.5,"publicationDate":"2026-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.70048","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147739342","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Recursive Reduction Quadrature for the Evaluation of Laplace Layer Potentials in Three Dimensions","authors":"Shidong Jiang, Hai Zhu","doi":"10.1002/cpa.70057","DOIUrl":"https://doi.org/10.1002/cpa.70057","url":null,"abstract":"A high‐order quadrature scheme is constructed for the evaluation of Laplace single and double layer potentials and their normal derivatives on smooth surfaces in three dimensions. The construction begins with a harmonic approximation of the density on each patch, which allows for a natural harmonic polynomial extension in a volumetric neighborhood of the patch in the ambient space. Then by the general Stokes theorem, singular and nearly singular surface integrals are reduced to line integrals preserving the singularity of the kernel, instead of the standard origin‐centered 1‐forms that require expensive adaptive integration. These singularity‐preserving line integrals can be semi‐analytically evaluated using singularity‐swap quadrature. In other words, the evaluation of singular and nearly singular surface integrals is reduced to function evaluations at the vertices on the boundary of each patch. The recursive reduction quadrature largely removes adaptive integration that is needed in most existing high‐order quadratures for singular and nearly singular surface integrals, resulting in exceptional performance. The scheme achieves twelve‐digit accuracy uniformly for close evaluations and offers a speedup of five times or more in constructing the sparse quadrature‐correction matrix compared to previous state‐of‐the‐art quadrature schemes.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"78 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148902301","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The SIR Model in a Moving Population: Propagation of Infection and Herd Immunity","authors":"Duncan Dauvergne,&nbsp;Allan Sly","doi":"10.1002/cpa.70054","DOIUrl":"10.1002/cpa.70054","url":null,"abstract":"<p>In a collection of particles performing independent random walks on <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>Z</mi>\u0000 <mi>d</mi>\u0000 </msup>\u0000 <annotation>$mathbb {Z}^d$</annotation>\u0000 </semantics></math> we study the spread of an infection with SIR dynamics. Susceptible particles become infected when they meet an infected particle. Infected particles heal and are removed at rate <span></span><math>\u0000 <semantics>\u0000 <mi>ν</mi>\u0000 <annotation>$nu$</annotation>\u0000 </semantics></math>. We show that when <span></span><math>\u0000 <semantics>\u0000 <mi>ν</mi>\u0000 <annotation>$nu$</annotation>\u0000 </semantics></math> is small, with positive probability the infection survives forever and grows linearly. Furthermore, after the infection reaches a region, it quickly passes through and leaves behind a <i>herd immunity</i> regime consisting of recovered particles, a small positive density of susceptible particles, and no infected particles. One notable feature of this model is the simultaneous existence of supercritical and subcritical phases on either side of an infection front of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>O</mi>\u0000 <mo>(</mo>\u0000 <mn>1</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$O(1)$</annotation>\u0000 </semantics></math> width.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"79 11","pages":"2375-2429"},"PeriodicalIF":3.5,"publicationDate":"2026-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.70054","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148409134","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Nonlocal Approximation of Minimal Surfaces: Optimal Estimates From Stability","authors":"Hardy Chan,Serena Dipierro,Joaquim Serra,Enrico Valdinoci","doi":"10.1002/cpa.70063","DOIUrl":"https://doi.org/10.1002/cpa.70063","url":null,"abstract":"ABSTRACT Minimal surfaces in closed 3‐manifolds are classically constructed via the Almgren‐Pitts approach. The Allen‐Cahn approximation has proved to be a powerful alternative, and Chodosh and Mantoulidis (in Ann. Math. 2020) used it to give a new proof of Yau's conjecture for generic metrics and establish the multiplicity one conjecture. The primary goal of this paper is to set the ground for a new approximation based on nonlocal minimal surfaces. More precisely, we prove that if is a stable ‐minimal surface in then: enjoys a estimate that is robust as (i.e., uniform in ); The distance between different connected components of  must be at least of order  (optimal sheet separation estimate); Interactions between multiple sheets at distances of order are described by the Dávila–del Pino–Wei system. A second important goal of the paper is to establish that hyperplanes are the only stable ‐minimal hypersurfaces in , for sufficiently close to 1. This is done by exploiting suitable modifications of the results described above. In this application, it is crucially used that our curvature and separations estimates hold without any assumption on area bounds (in contrast to the analogous estimates for Allen‐Cahn).","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"53 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148894638","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Stochastic Gradient Descent in High Dimensions for Multi-Spiked Tensor PCA","authors":"Gérard Ben Arous,&nbsp;Cédric Gerbelot,&nbsp;Vanessa Piccolo","doi":"10.1002/cpa.70056","DOIUrl":"https://doi.org/10.1002/cpa.70056","url":null,"abstract":"<p>We study the high-dimensional dynamics of online stochastic gradient descent (SGD) for the multi-spiked tensor model. This multi-index model arises from the tensor principal component analysis (PCA) problem with multiple spikes, where the goal is to estimate the unknown signal vectors within the <span></span><math>\u0000 <semantics>\u0000 <mi>N</mi>\u0000 <annotation>$N$</annotation>\u0000 </semantics></math>-dimensional unit sphere through maximum likelihood estimation from noisy observations of a <span></span><math>\u0000 <semantics>\u0000 <mi>p</mi>\u0000 <annotation>$p$</annotation>\u0000 </semantics></math>-tensor. We determine the number of samples and the conditions on the signal-to-noise ratios (SNRs) required to efficiently recover the unknown spikes from natural random initializations. We show that full recovery of all spikes is possible provided a number of sample scaling as <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>N</mi>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 <mo>−</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 </msup>\u0000 <annotation>$N^{p-2}$</annotation>\u0000 </semantics></math>, matching the algorithmic threshold identified in the rank-one case. Our results are obtained through a detailed analysis of a low-dimensional system that describes the evolution of the correlations between the estimators and the spikes, while sharply controlling the noise in the dynamics. We find the spikes are recovered sequentially in a process we term “sequential elimination”: once a correlation exceeds a critical threshold, all correlations sharing a row or column index become sufficiently small, allowing the next correlation to grow and become macroscopic. The order in which correlations become macroscopic depends on their initial values and the corresponding SNRs, leading to either exact recovery or recovery of a permutation of the spikes. In the matrix case, when <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 <mo>=</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$p=2$</annotation>\u0000 </semantics></math>, if the SNRs are sufficiently separated, we achieve exact recovery of the spikes, whereas equal SNRs lead to recovery of the subspace spanned by them.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"79 10","pages":"2291-2369"},"PeriodicalIF":3.5,"publicationDate":"2026-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.70056","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148696382","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Effective Velocities in the Toda Lattice","authors":"Amol Aggarwal","doi":"10.1002/cpa.70046","DOIUrl":"10.1002/cpa.70046","url":null,"abstract":"<div>\u0000 \u0000 <p>In this paper, we consider the Toda lattice <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>p</mi>\u0000 <mo>(</mo>\u0000 <mi>t</mi>\u0000 <mo>)</mo>\u0000 <mo>;</mo>\u0000 <mi>q</mi>\u0000 <mo>(</mo>\u0000 <mi>t</mi>\u0000 <mo>)</mo>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(bm {p}(t); bm {q}(t))$</annotation>\u0000 </semantics></math> at thermal equilibrium, meaning that its variables <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msub>\u0000 <mi>p</mi>\u0000 <mi>i</mi>\u0000 </msub>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(p_i)$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msup>\u0000 <mi>e</mi>\u0000 <mrow>\u0000 <msub>\u0000 <mi>q</mi>\u0000 <mi>i</mi>\u0000 </msub>\u0000 <mo>−</mo>\u0000 <msub>\u0000 <mi>q</mi>\u0000 <mrow>\u0000 <mi>i</mi>\u0000 <mo>+</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msub>\u0000 </mrow>\u0000 </msup>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(e^{q_i-q_{i+1}})$</annotation>\u0000 </semantics></math> are independent Gaussian and Gamma random variables, respectively. This model can be thought of a dense collection of many “quasiparticles” that act as solitons. We establish a law of large numbers for the trajectory of these quasiparticles, showing that they travel with approximately constant velocities, which are explicit. Our proof is based on a direct analysis of the asymptotic scattering relation, an equation that approximately governs the dynamics of quasiparticles locations. This makes use of a regularization argument that essentially linearizes this relation, together with concentration estimates for the Toda lattice's (random) Lax matrix.</p></div>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"79 9","pages":"2223-2288"},"PeriodicalIF":3.5,"publicationDate":"2026-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147719935","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Benjamin–Ono Equation in the Zero-Dispersion Limit for Rational Initial Data: Generation of Dispersive Shock Waves","authors":"Elliot Blackstone,&nbsp;Louise Gassot,&nbsp;Patrick Gérard,&nbsp;Peter D. Miller","doi":"10.1002/cpa.70044","DOIUrl":"10.1002/cpa.70044","url":null,"abstract":"&lt;p&gt;The leading-order asymptotic behavior of the solution of the Cauchy initial-value problem for the Benjamin–Ono equation in &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;L&lt;/mi&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$L^2(mathbb {R})$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is obtained explicitly for generic rational initial data &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;u&lt;/mi&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$u_0$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. An explicit asymptotic wave profile &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;u&lt;/mi&gt;\u0000 &lt;mi&gt;ZD&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;t&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mo&gt;;&lt;/mo&gt;\u0000 &lt;mi&gt;ε&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$u^mathrm{ZD}(t,x;epsilon)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is given, in terms of the branches of the multivalued solution of the inviscid Burgers equation with initial data &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;u&lt;/mi&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$u_0$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, such that the solution &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;u&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;t&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mo&gt;;&lt;/mo&gt;\u0000 &lt;mi&gt;ε&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$u(t,x;epsilon)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of the Benjamin–Ono equation with dispersion parameter &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;ε&lt;/mi&gt;\u0000 &lt;mo&gt;&gt;&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$epsilon &gt;0$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and initial data &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;u&lt;/mi&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$u_0$&lt;/annotation&gt;\u0000 ","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"79 9","pages":"2107-2157"},"PeriodicalIF":3.5,"publicationDate":"2026-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.70044","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147586397","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Optimal Convergence Rates in Multiscale Elliptic Homogenization","authors":"Weisheng Niu,&nbsp;Yao Xu,&nbsp;Jinping Zhuge","doi":"10.1002/cpa.70047","DOIUrl":"10.1002/cpa.70047","url":null,"abstract":"&lt;div&gt;\u0000 \u0000 &lt;p&gt;This paper is devoted to the quantitative homogenization of multiscale elliptic operator &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mo&gt;∇&lt;/mo&gt;\u0000 &lt;mo&gt;·&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;mi&gt;ε&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;∇&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$-nabla cdot A_varepsilon nabla$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, where &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;mi&gt;ε&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mo&gt;/&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;ε&lt;/mi&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mo&gt;/&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;ε&lt;/mi&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mtext&gt;…&lt;/mtext&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mo&gt;/&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;ε&lt;/mi&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$A_varepsilon (x) = A(x/varepsilon _1, x/varepsilon _2,ldots, x/varepsilon _n)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;ε&lt;/mi&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;ε&lt;/mi&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;ε&lt;/mi&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mtext&gt;…&lt;/mtext&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;ε&lt;/mi&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mo&gt;∈&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;mo&gt;]&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mi&gt;n&lt;","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"79 9","pages":"2158-2222"},"PeriodicalIF":3.5,"publicationDate":"2026-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147681955","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Fibrations by Lagrangian Tori for Maximal Calabi–Yau Degenerations and Beyond","authors":"Javier Fernández de Bobadilla, Tomasz Pełka","doi":"10.1002/cpa.70060","DOIUrl":"https://doi.org/10.1002/cpa.70060","url":null,"abstract":"We introduce a general technique to construct Lagrangian torus fibrations in degenerations of Kähler manifolds. We show that such torus fibrations naturally occur at the boundary of the <jats:italic>A'Campo space</jats:italic> . This space extends a degeneration over a punctured disc to a hybrid space involving its real oriented blowup and the dual complex of the central fiber; equipped with an exact modification of a given Kähler form which becomes fiberwise symplectic at the boundary. Therefore, it allows to move Lagrangian tori from the “radius zero” fibers at the boundary to the nearby fibers of the original degeneration using the symplectic connection. As a consequence, for <jats:italic>any</jats:italic> maximal Calabi–Yau degeneration we prove that a generic region of each nearby fiber admits a Lagrangian torus fibration with asymptotic properties expected from the (conjectural) Strominger–Yau–Zaslow fibration.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"41 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148409135","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}