Communications on Pure and Applied Mathematics最新文献

{"title":"Equivariant toric geometry and Euler–Maclaurin formulae","authors":"Sylvain E. Cappell, Laurenţiu Maxim, Jörg Schürmann, Julius L. Shaneson","doi":"10.1002/cpa.70016","DOIUrl":"https://doi.org/10.1002/cpa.70016","url":null,"abstract":"We first investigate torus‐equivariant motivic characteristic classes of toric varieties, and then apply them via the equivariant Riemann–Roch formalism to prove very general Euler–Maclaurin‐type formulae for full‐dimensional simple lattice polytopes.We consider ‐equivariant versions and of the <jats:italic>motivic Chern</jats:italic> and, resp., <jats:italic>Hirzebruch characteristic classes</jats:italic> of a toric variety (with corresponding torus ), and extend many known results from the non‐equivariant context to the equivariant setting. For example, the equivariant motivic Chern class is computed as the sum of the equivariant Grothendieck classes of the ‐equivariant sheaves of <jats:italic>Zariski</jats:italic> <jats:italic>‐forms</jats:italic> weighted by . Using the motivic, as well as the characteristic class nature of , the corresponding generalized <jats:italic>equivariant Hirzebruch</jats:italic> <jats:italic>‐genus</jats:italic> of a ‐invariant Cartier divisor on is also calculated.Further global formulae for are obtained in the simplicial context based on the Cox construction and the <jats:italic>equivariant Lefschetz–Riemann–Roch theorem</jats:italic> of Edidin–Graham. Alternative proofs of all these results are given via <jats:italic>localization techniques</jats:italic> at the torus fixed points in ‐equivariant ‐ and, resp., homology theories of toric varieties, due to Brion–Vergne and, resp., Brylinski–Zhang. These localization results apply to any toric variety with a torus fixed point. In localized ‐equivariant ‐theory, we extend a classical <jats:italic>formula of Brion</jats:italic> for a full‐dimensional lattice polytope to a weighted version. We also generalize the <jats:italic>Molien formula</jats:italic> of Brion–Vergne for the localized class of the structure sheaf of a simplicial toric variety to the context of . Similarly, we calculate the <jats:italic>localized Hirzebruch class</jats:italic> in localized ‐equivariant homology, extending the corresponding results of Brylinski–Zhang for the <jats:italic>localized Todd class</jats:italic> (fitting with the equivariant Hirzebruch class for ).As main applications of our equivariant characteristic class formulae, we provide a geometric perspective on several <jats:italic>weighted Euler–Maclaurin‐type formulae for full‐dimensional simple lattice polytopes</jats:italic> (corresponding to simplicial toric varieties), coming from the <jats:italic>equivariant toric geometry via the equivariant Hirzebruch–Riemann–Roch</jats:italic> (for an ample torus invariant Cartier divisor). Our main results even provide generalizations to <jats:italic>arbitrary equivariant coherent sheaf coefficients</jats:italic>, including algebraic geometric proofs of (weighted versions of) the Euler–Maclaurin formulae of Cappell–Shaneson, Brion–Vergne, Guillemin, and so forth (all of which correspond to the choice of the structure sheaf), via the equivariant Hirzebruch–Riemann–Roch formalism. In particu","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"26 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145246966","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":"Dimer models and conformal structures","authors":"Kari Astala, Erik Duse, István Prause, Xiao Zhong","doi":"10.1002/cpa.70014","DOIUrl":"https://doi.org/10.1002/cpa.70014","url":null,"abstract":"Dimer models have been the focus of intense research efforts over the last years. Our paper grew out of an effort to develop new methods to study minimizers or the asymptotic <jats:italic>height functions</jats:italic> of general dimer models and the geometry of their <jats:italic>frozen boundaries</jats:italic>. We prove a <jats:italic>complete classification</jats:italic> of the regularity of minimizers and frozen boundaries for <jats:italic>all dimer models</jats:italic> for a natural class of polygonal domains, much studied in numerical simulations and elsewhere. In particular, we show that the frozen boundaries are always algebraic curves. Our classification also implies that the Pokrovsky‐Talapov law holds for all dimer models at a generic point on the frozen boundary and, in addition, shows a very strong local rigidity of dimer models, which can be interpreted as a <jats:italic>geometric universality</jats:italic> result. Indeed, we prove a converse result, showing that any geometric situation for any dimer model is, in the simply connected case, realized already by the lozenge model. To achieve these goals we develop a new study on the <jats:italic>boundary regularity</jats:italic> for a class of Monge–Ampère equations in <jats:italic>non‐strictly convex</jats:italic> domains, of independent interest, as well as a new approach to minimality for a general dimer functional. In the context of polygonal domains, we give the first general results for the existence of <jats:italic>gas domains</jats:italic> for minimizers.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"33 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145241308","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":"Fast randomized least‐squares solvers can be just as accurate and stable as classical direct solvers","authors":"Ethan N. Epperly, Maike Meier, Yuji Nakatsukasa","doi":"10.1002/cpa.70013","DOIUrl":"https://doi.org/10.1002/cpa.70013","url":null,"abstract":"One of the greatest success stories of randomized algorithms in linear algebra has been the development of fast, randomized solvers for highly overdetermined linear least‐squares problems. However, none of the existing algorithms is backward stable, preventing them from being deployed as drop‐in replacements for existing QR‐based solvers. This paper introduces sketch‐and‐precondition with iterative refinement (SPIR) and FOSSILS, two <jats:italic>provably</jats:italic> backward stable randomized least‐squares solvers. SPIR and FOSSILS combine iterative refinement with a preconditioned iterative method applied to the normal equations and converge at the same rate as existing randomized least‐squares solvers. This work offers the promise of incorporating randomized least‐squares solvers into existing software libraries while maintaining the same level of accuracy and stability as classical solvers.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"8 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145195048","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":"Sharp commutator estimates of all order for Coulomb and Riesz modulated energies","authors":"Matthew Rosenzweig, Sylvia Serfaty","doi":"10.1002/cpa.70010","DOIUrl":"https://doi.org/10.1002/cpa.70010","url":null,"abstract":"We prove functional inequalities in any dimension controlling the iterated derivatives along a transport of the Coulomb or super‐Coulomb Riesz modulated energy in terms of the modulated energy itself. This modulated energy was introduced by the second author and collaborators in the study of mean‐field limits and statistical mechanics of Coulomb/Riesz gases, where control of such derivatives by the energy itself is an essential ingredient. In this paper, we extend and improve such functional inequalities, proving estimates which are now sharp in their additive error term, in their density dependence, valid at arbitrary order of differentiation, and localizable to the support of the transport. Our method relies on the observation that these iterated derivatives are the quadratic form of a commutator. Taking advantage of the Riesz nature of the interaction, we identify these commutators as solutions to a degenerate elliptic equation with a right‐hand side exhibiting a recursive structure in terms of lower‐order commutators and develop a local regularity theory for the commutators, which may be of independent interest. These estimates have applications to obtaining sharp rates of convergence for mean‐field limits, quasi‐neutral limits, and in proving central limit theorems for the fluctuations of Coulomb/Riesz gases. In particular, we show here the expected ‐rate in the modulated energy distance for the mean‐field convergence of first‐order Hamiltonian and gradient flows.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"30 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145133855","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":"Convergence properties of dynamic mode decomposition for analytic interval maps","authors":"Elliz Akindji, Julia Slipantschuk, Oscar F. Bandtlow, Wolfram Just","doi":"10.1002/cpa.70011","DOIUrl":"https://doi.org/10.1002/cpa.70011","url":null,"abstract":"Extended dynamic mode decomposition (EDMD) is a data‐driven algorithm for approximating spectral data of the Koopman operator associated to a dynamical system, combining a Galerkin method with functions and a quadrature method with quadrature nodes. Spectral convergence of this method subtly depends on an appropriate choice of the space of observables. For chaotic analytic full branch maps of the interval, we derive a constraint between and guaranteeing spectral convergence of EDMD. In particular, the computed eigenvalues converge exponentially fast (in ) to the eigenvalues of the Koopman operator, taken to act on the dual space of a certain Banach space of analytic functions.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"74 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145035521","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":"Infinite quantum signal processing for arbitrary Szegő functions","authors":"Michel Alexis, Lin Lin, Gevorg Mnatsakanyan, Christoph Thiele, Jiasu Wang","doi":"10.1002/cpa.70007","DOIUrl":"https://doi.org/10.1002/cpa.70007","url":null,"abstract":"We provide a complete solution to the problem of infinite quantum signal processing (QSP) for the class of Szegő functions, which are functions that satisfy a logarithmic integrability condition and include almost any function that allows for a QSP representation. We do so by introducing a new algorithm called the Riemann–Hilbert–Weiss algorithm, which can compute any individual phase factor independent of all other phase factors. Our algorithm is also the first provably stable numerical algorithm for computing phase factors of any arbitrary Szegő function. The proof of stability involves solving a Riemann–Hilbert factorization problem in nonlinear Fourier analysis using elements of spectral theory.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"35 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145035523","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":"Randomly sparsified Richardson iteration: A dimension‐independent sparse linear solver","authors":"Jonathan Weare, Robert J. Webber","doi":"10.1002/cpa.70012","DOIUrl":"https://doi.org/10.1002/cpa.70012","url":null,"abstract":"Recently, a class of algorithms combining classical fixed‐point iterations with repeated random sparsification of approximate solution vectors has been successfully applied to eigenproblems with matrices as large as . So far, a complete mathematical explanation for this success has proven elusive.The family of methods has not yet been extended to the important case of linear system solves. In this paper, we propose a new scheme based on repeated random sparsification that is capable of solving sparse linear systems in arbitrarily high dimensions. We provide a complete mathematical analysis of this new algorithm. Our analysis establishes a faster‐than‐Monte Carlo convergence rate and justifies use of the scheme even when the solution is too large to store as a dense vector.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"148 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145017451","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":"A priori estimates and large population limits for some nonsymmetric Nash systems with semimonotonicity","authors":"Marco Cirant, Davide Francesco Redaelli","doi":"10.1002/cpa.70009","DOIUrl":"https://doi.org/10.1002/cpa.70009","url":null,"abstract":"We address the problem of regularity of solutions to a family of semilinear parabolic systems of equations, which describe closed‐loop equilibria of some ‐player differential games with Lagrangian having quadratic behaviour in the velocity variable, running costs and final costs . By global (semi)monotonicity assumptions on the data and , and assuming that derivatives of in directions are of order for , we prove that derivatives of enjoy the same property. The estimates are uniform in the number of players . Such a behaviour of the derivatives of arise in the theory of Mean Field Games, though here we do not make any symmetry assumption on the data. Then, by the estimates obtained we address the convergence problem in a ‘heterogeneous’ Mean Field framework, where players all observe the empirical measure of the whole population, but may react differently from one another. We also discuss some results on the joint and vanishing viscosity limit.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"21 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144987320","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":"Regularity of minimal surfaces with capillary boundary conditions","authors":"Luigi De Masi,&nbsp;Nick Edelen,&nbsp;Carlo Gasparetto,&nbsp;Chao Li","doi":"10.1002/cpa.70008","DOIUrl":"10.1002/cpa.70008","url":null,"abstract":"<p>We prove <span></span><math>\u0000 <semantics>\u0000 <mi>ε</mi>\u0000 <annotation>$varepsilon$</annotation>\u0000 </semantics></math>-regularity theorems for varifolds with capillary boundary condition in a Riemannian manifold. These varifolds were first introduced by Kagaya–Tonegawa. We establish a uniform first variation control for all such varifolds (and free-boundary varifolds generally) satisfying a sharp density bound and prove that if a capillary varifold has bounded mean curvature and is close to a capillary half-plane with angle not equal to <span></span><math>\u0000 <semantics>\u0000 <mstyle>\u0000 <mfrac>\u0000 <mi>π</mi>\u0000 <mn>2</mn>\u0000 </mfrac>\u0000 </mstyle>\u0000 <annotation>$tfrac{pi }{2}$</annotation>\u0000 </semantics></math>, then it coincides with a <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>C</mi>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>,</mo>\u0000 <mi>α</mi>\u0000 </mrow>\u0000 </msup>\u0000 <annotation>$C^{1,alpha }$</annotation>\u0000 </semantics></math> properly embedded hypersurface. We apply our theorem to deduce regularity at a generic point along the boundary in the region where the density is strictly less than 1.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"78 12","pages":"2436-2502"},"PeriodicalIF":2.7,"publicationDate":"2025-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144987321","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":"Issue Information ‐ TOC","authors":"","doi":"10.1002/cpa.22217","DOIUrl":"https://doi.org/10.1002/cpa.22217","url":null,"abstract":"","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"164 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144792241","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}