Proceedings of the International Congress of Mathematicians (ICM 2018)最新文献

{"title":"POTENTIAL AUTOMORPHY OF Ĝ-LOCAL SYSTEMS","authors":"J. Thorne","doi":"10.1142/9789813272880_0061","DOIUrl":"https://doi.org/10.1142/9789813272880_0061","url":null,"abstract":"","PeriodicalId":318252,"journal":{"name":"Proceedings of the International Congress of Mathematicians (ICM 2018)","volume":"104 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115698652","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A REVIEW ON HIGH ORDER WELL-BALANCED PATH-CONSERVATIVE FINITE VOLUME SCHEMES FOR GEOPHYSICAL FLOWS","authors":"M. Castro, M. Asunción, E. D. F. Nieto, J. Gallardo, J. M. G. Vida, J. Macías, T. Morales, S. Ortega, C. Parés","doi":"10.1142/9789813272880_0190","DOIUrl":"https://doi.org/10.1142/9789813272880_0190","url":null,"abstract":"","PeriodicalId":318252,"journal":{"name":"Proceedings of the International Congress of Mathematicians (ICM 2018)","volume":"105 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122662788","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"MATHEMATICAL ANALYSIS AND NUMERICAL METHODS FOR MULTISCALE KINETIC EQUATIONS WITH UNCERTAINTIES","authors":"Shi Jin","doi":"10.1142/9789813272880_0194","DOIUrl":"https://doi.org/10.1142/9789813272880_0194","url":null,"abstract":"","PeriodicalId":318252,"journal":{"name":"Proceedings of the International Congress of Mathematicians (ICM 2018)","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127517002","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"THE EFFECT OF DISORDER AND IRREGULARITIES ON SOLUTIONS TO BOUNDARY VALUE PROBLEMS AND SPECTRA OF DIFFERENTIAL OPERATORS","authors":"S. Mayboroda","doi":"10.1142/9789813272880_0113","DOIUrl":"https://doi.org/10.1142/9789813272880_0113","url":null,"abstract":"","PeriodicalId":318252,"journal":{"name":"Proceedings of the International Congress of Mathematicians (ICM 2018)","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127681978","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"KNOTS, THREE-MANIFOLDS AND INSTANTONS","authors":"P. Kronheimer, T. Mrowka","doi":"10.1142/9789813272880_0024","DOIUrl":"https://doi.org/10.1142/9789813272880_0024","url":null,"abstract":"Low-dimensional topology is the study of manifolds and cell complexes in dimensions four and below. Input from geometry and analysis has been central to progress in this field over the past four decades, and this article will focus on one aspect of these developments in particular, namely the use of Yang–Mills theory, or gauge theory. These techniques were pioneered by Simon Donaldson in his work on 4-manifolds, but the past ten years have seen new applications of gauge theory, and new interactions with more recent threads in the subject, particularly in 3-dimensional topology. This is a field where many mathematical techniques have found applications, and sometimes a theorem has two or more independent proofs, drawing on more than one of these techniques. We will focus primarily on some questions and results where gauge theory plays a special role. 1 Representations of fundamental groups 1.1 Knot groups and their representations. Knots have long fascinated mathematicians. In topology, they provide blueprints for the construction of manifolds of dimension three and four. For this exposition, a knot is a smoothly embedded circle in 3-space, and a link is a disjoint union of knots. The simplest examples, the trefoil knot and the Hopf link, are shown in Figure 1, alongside the trivial round circle, the “unknot”. Knot theory is a subject with many aspects, but one place to start is with the knot group, defined as the fundamental group of the complement of a knotK R. We will write it as (K). For the unknot, (K) is easily identified as Z. One of the basic tools of 3-dimensional topology is Dehn’s Lemma, proved by Papakyriakopoulos in 1957, which provides a converse: Theorem 1.1 (Papakyriakopoulos [1957]). If the knot group (K) is Z, then K is the unknot. It is a consequence of Alexander duality that the abelianization of (K) is Z for any knot. (This is the first homology of the complement.) So we may restate the result above as saying that the unknot is characterized by having abelian fundamental group. The work of the first author was supported by the National Science Foundation through NSF grants DMS1405652 and DMS-1707924. The work of the second author was supported by NSF grant DMS-1406348 and by a grant from the Simons Foundation, grant number 503559 TSM. MSC2010: primary 57R58; secondary 57M27.","PeriodicalId":318252,"journal":{"name":"Proceedings of the International Congress of Mathematicians (ICM 2018)","volume":"56 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124703184","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"MATHEMATICS OF MACHINE LEARNING: AN INTRODUCTION","authors":"Sanjeev Arora","doi":"10.1142/9789813272880_0017","DOIUrl":"https://doi.org/10.1142/9789813272880_0017","url":null,"abstract":"Machine learning is the subfield of computer science concerned with creating machines that can improve from experience and interaction. It relies upon mathematical optimization, statistics, and algorithm design. Rapid empirical success in this field currently outstrips mathematical understanding. This elementary article sketches the basic framework of machine learning and hints at the open mathematical problems in it. An updated version of this article and related articles can be found on the author’s webpage.","PeriodicalId":318252,"journal":{"name":"Proceedings of the International Congress of Mathematicians (ICM 2018)","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120978225","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"ON EXPLICIT ASPECT OF PLURICANONICAL MAPS OF PROJECTIVE VARIETIES","authors":"Jungkai A. Chen, Meng Chen","doi":"10.1142/9789813272880_0072","DOIUrl":"https://doi.org/10.1142/9789813272880_0072","url":null,"abstract":"In this survey article, we introduce the development of birational geometry associated to pluricanonical maps. Especially, we explain various aspects of explicit studies of threefolds including the key idea of theory of baskets and other applications.","PeriodicalId":318252,"journal":{"name":"Proceedings of the International Congress of Mathematicians (ICM 2018)","volume":"97 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121736032","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"DYNAMICAL SYSTEMS EVOLVING","authors":"L. Young","doi":"10.1142/9789813272880_0035","DOIUrl":"https://doi.org/10.1142/9789813272880_0035","url":null,"abstract":"This is an expanded version of a presentation given at ICM2018. It discusses a number of results taken from a cross-section of the author’s work in Dynamical Systems. The topics include relation between entropy, Lyapunov exponents and fractal dimension, statistical properties of chaotic dynamical systems, physically relevant invariant measures, strange attractors arising from shearinduced chaos, random maps and random attractors. The last section contains two applications of Dynamical Systems ideas to Biology, one to epidemics control and the other to computational neuroscience.","PeriodicalId":318252,"journal":{"name":"Proceedings of the International Congress of Mathematicians (ICM 2018)","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123794294","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"FRONT MATTER","authors":"B. Sirakov, Paulo Ney de Souza, M. Viana","doi":"10.1142/9789813272880_fmatter04","DOIUrl":"https://doi.org/10.1142/9789813272880_fmatter04","url":null,"abstract":"","PeriodicalId":318252,"journal":{"name":"Proceedings of the International Congress of Mathematicians (ICM 2018)","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126009022","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"THE ORR MECHANISM: STABILITY/INSTABILITY OF THE COUETTE FLOW FOR THE 2D EULER DYNAMIC","authors":"J. Bedrossian, Yu Deng, N. Masmoudi","doi":"10.1142/9789813272880_0133","DOIUrl":"https://doi.org/10.1142/9789813272880_0133","url":null,"abstract":"We review our works on the nonlinear asymptotic stability and instability of the Couette flow for the 2D incompressible Euler dynamic. In the fits part of the work we prove that perturbations to the Couette flow which are small in Gevrey spaces Gs of class 1/s with s > 1/2 converge strongly in L2 to a shear flow which is close to the Couette flow. Moreover in a well chosen coordinate system, the solution converges in the same Gevrey space to some limit profile. In a later work, we proved the existence of small perturbations inGs with s < 1/2 such that the solution becomes large in Sobolev regularity and hence yields instability. In this note we discuss the most important physical and mathematical aspects of these two results and the key ideas of the proofs.","PeriodicalId":318252,"journal":{"name":"Proceedings of the International Congress of Mathematicians (ICM 2018)","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130550385","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}