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

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SYMMETRY, INVARIANCE AND THE STRUCTURE OF MATTER 对称，不变性和物质的结构
Proceedings of the International Congress of Mathematicians (ICM 2018) Pub Date : 2019-05-01 DOI: 10.1142/9789813272880_0208
R. James
{"title":"SYMMETRY, INVARIANCE AND THE STRUCTURE OF MATTER","authors":"R. James","doi":"10.1142/9789813272880_0208","DOIUrl":"https://doi.org/10.1142/9789813272880_0208","url":null,"abstract":"We present a mathematical view of the structure of matter based on the invariance of the classical equations of physics.","PeriodicalId":318252,"journal":{"name":"Proceedings of the International Congress of Mathematicians (ICM 2018)","volume":"58-60 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":"123121191","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}
引用次数: 4
RESONANCES IN HYPERBOLIC DYNAMICS 双曲动力学中的共振
Proceedings of the International Congress of Mathematicians (ICM 2018) Pub Date : 2019-05-01 DOI: 10.1142/9789813272880_0147
S. Nonnenmacher
{"title":"RESONANCES IN HYPERBOLIC DYNAMICS","authors":"S. Nonnenmacher","doi":"10.1142/9789813272880_0147","DOIUrl":"https://doi.org/10.1142/9789813272880_0147","url":null,"abstract":"The study of wave propagation outside bounded obstacles uncovers the existence of resonances for the Laplace operator, which are complex-valued generalized eigen-values, relevant to estimate the long time asymptotics of the wave. In order to understand distribution of these resonances at high frequency, we employ semiclassical tools, which leads to considering the classical scattering problem, and in particular the set of trapped trajectories. We focus on \"chaotic\" situations, where this set is a hy-perbolic repeller, generally with a fractal geometry. In this context, we derive fractal Weyl upper bounds for the resonance counting; we also obtain dynamical criteria ensuring the presence of a resonance gap. We also address situations where the trapped set is a normally hyperbolic submanifold, a case which can help analyzing the long time properties of (classical) Anosov contact flows through semiclassical methods.","PeriodicalId":318252,"journal":{"name":"Proceedings of the International Congress of Mathematicians (ICM 2018)","volume":"33 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":"124893382","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}
引用次数: 1
REPRESENTATIONS OF GALOIS ALGEBRAS 伽罗瓦代数的表示
Proceedings of the International Congress of Mathematicians (ICM 2018) Pub Date : 2019-05-01 DOI: 10.1142/9789813272880_0098
V. Futorny
{"title":"REPRESENTATIONS OF GALOIS ALGEBRAS","authors":"V. Futorny","doi":"10.1142/9789813272880_0098","DOIUrl":"https://doi.org/10.1142/9789813272880_0098","url":null,"abstract":"","PeriodicalId":318252,"journal":{"name":"Proceedings of the International Congress of Mathematicians (ICM 2018)","volume":"40 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":"121948365","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}
引用次数: 4
MEAN FIELD ASYMPTOTICS IN HIGH-DIMENSIONAL STATISTICS: FROM EXACT RESULTS TO EFFICIENT ALGORITHMS 高维统计中的平均场渐近性:从精确结果到高效算法
Proceedings of the International Congress of Mathematicians (ICM 2018) Pub Date : 2019-05-01 DOI: 10.1142/9789813272880_0168
A. Montanari
{"title":"MEAN FIELD ASYMPTOTICS IN HIGH-DIMENSIONAL STATISTICS: FROM EXACT RESULTS TO EFFICIENT ALGORITHMS","authors":"A. Montanari","doi":"10.1142/9789813272880_0168","DOIUrl":"https://doi.org/10.1142/9789813272880_0168","url":null,"abstract":"Modern data analysis challenges require building complex statistical models with massive numbers of parameters. It is nowadays commonplace to learn models with millions of parameters by using iterative optimization algorithms. What are typical properties of the estimated models? In some cases, the high-dimensional limit of a statistical estimator is analogous to the thermodynamic limit of a certain (disordered) statistical mechanics system. Building on mathematical ideas from the mean-field theory of disordered systems, exact asymptotics can be computed for high-dimensional statistical learning problems. This theory suggests new practical algorithms and new procedures for statistical inference. Also, it leads to intriguing conjectures about the fundamental computational limits for statistical estimation.","PeriodicalId":318252,"journal":{"name":"Proceedings of the International Congress of Mathematicians (ICM 2018)","volume":"66 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":"122276375","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}
引用次数: 6
A SELECTIVE SURVEY OF SELECTIVE INFERENCE 选择性推论的选择性调查
Proceedings of the International Congress of Mathematicians (ICM 2018) Pub Date : 2019-05-01 DOI: 10.1142/9789813272880_0170
Jonathan E. Taylor
{"title":"A SELECTIVE SURVEY OF SELECTIVE INFERENCE","authors":"Jonathan E. Taylor","doi":"10.1142/9789813272880_0170","DOIUrl":"https://doi.org/10.1142/9789813272880_0170","url":null,"abstract":"It is not difficult to find stories of a crisis in modern science, either in the popular press or in the scientific literature. There are likely multiple sources for this crisis. It is also well documented that one source of this crisis is the misuse of statistical methods in science, with the P -value receiving its fair share of criticism. It could be argued that this misuse of statistical methods is caused by a shift in how data is used in 21st century science compared to its use in the mid-20th century which presumed scientists had formal statistical hypotheses before collecting data. With the advent of sophisticated statistical software available to anybody this paradigm has been shifted to one in which scientists collect data first and ask questions later. 1 The new (?) scientific paradigm Figure 1: A simplified version of the scientific method. We are all familiar with a paradigm that does allow scientists to collect data first and ask questions later: the classical scientific method illustrated in Figure 1. A scientist collects data D, generates questions of interest Q(D), then collects fresh data D for confirmation and perhaps to discover additional questions of interest. The problem with this new paradigm is that it seeks to use D to answer these questions and may not have access to D. We pause here to note that Tukey used the term question rather than the more precise term hypothesis which statisticians might reasonably impute to be a statistical hypothesis. Given the computing capabilities of modern MSC2010: primary 62-02; secondary 62J15, 62J05, 62-07.","PeriodicalId":318252,"journal":{"name":"Proceedings of the International Congress of Mathematicians (ICM 2018)","volume":"1 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":"121154773","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}
引用次数: 1
INTERFACE DYNAMICS FOR INCOMPRESSIBLE FLUIDS: SPLASH AND SPLAT SINGULARITIES 不可压缩流体的界面动力学:飞溅和飞溅奇点
Proceedings of the International Congress of Mathematicians (ICM 2018) Pub Date : 2019-05-01 DOI: 10.1142/9789813272880_0135
D. Córdoba
{"title":"INTERFACE DYNAMICS FOR INCOMPRESSIBLE FLUIDS: SPLASH AND SPLAT SINGULARITIES","authors":"D. Córdoba","doi":"10.1142/9789813272880_0135","DOIUrl":"https://doi.org/10.1142/9789813272880_0135","url":null,"abstract":"In this survey I report on recent progress in the study of the dynamics of the interface in between two incompressible fluids with different characteristics. In particular I focus on the formation of Splash and Splat singularities in two different settings: Euler equations and Darcy’s law.","PeriodicalId":318252,"journal":{"name":"Proceedings of the International Congress of Mathematicians (ICM 2018)","volume":"103 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":"115753561","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}
引用次数: 0
ON EFFECTIVE NUMERICAL METHODS FOR PHASE-FIELD MODELS 相场模型的有效数值方法
Proceedings of the International Congress of Mathematicians (ICM 2018) Pub Date : 2019-05-01 DOI: 10.1142/9789813272880_0196
T. Tang
{"title":"ON EFFECTIVE NUMERICAL METHODS FOR PHASE-FIELD MODELS","authors":"T. Tang","doi":"10.1142/9789813272880_0196","DOIUrl":"https://doi.org/10.1142/9789813272880_0196","url":null,"abstract":"In this article, we overview recent developments of modern computational methods for the approximate solution of phase-field problems. The main difficulty for developing a numerical method for phase field equations is a severe stability restriction on the time step due to nonlinearity and high order differential terms. It is known that the phase field models satisfy a nonlinear stability relationship called gradient stability, usually expressed as a time-decreasing free-energy functional. This property has been used recently to derive numerical schemes that inherit the gradient stability. The first part of the article will discuss implicit-explicit time discretizations which satisfy the energy stability. The second part is to discuss time-adaptive strategies for solving the phase-field problems, which is motivated by the observation that the energy functionals decay with time smoothly except at a few critical time levels. The classical operator-splitting method is a useful tool in time discrtization. In the final part, we will provide some preliminary results using operator-splitting approach.","PeriodicalId":318252,"journal":{"name":"Proceedings of the International Congress of Mathematicians (ICM 2018)","volume":"46 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":"114611973","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}
引用次数: 5
ON THEORIES IN MATHEMATICS EDUCATION AND THEIR CONCEPTUAL DIFFERENCES 论数学教育理论及其概念差异
Proceedings of the International Congress of Mathematicians (ICM 2018) Pub Date : 2019-05-01 DOI: 10.1142/9789813272880_0211
L. Radford
{"title":"ON THEORIES IN MATHEMATICS EDUCATION AND THEIR CONCEPTUAL DIFFERENCES","authors":"L. Radford","doi":"10.1142/9789813272880_0211","DOIUrl":"https://doi.org/10.1142/9789813272880_0211","url":null,"abstract":"In this article I discuss some theories in mathematics education research. My goal is to highlight some of their differences. How will I proceed? I could proceed by giving a definition, T, of the term theory and by choosing some differentiating criteria such as c1, c2, etc. Theories, then, could be distinguished in terms of whether or not they include the criteria c1, c2, etc. However, in this article I will take a different path. In the first part I will focus on a few well-known theories in Mathematics Education and discuss their differences in terms of their theoretical stances. In the last part of the article, I will comment on a sociocultural emergent trend.","PeriodicalId":318252,"journal":{"name":"Proceedings of the International Congress of Mathematicians (ICM 2018)","volume":"178 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":"129411879","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}
引用次数: 7
EXTREMAL THEORY OF ORDERED GRAPHS 有序图的极值理论
Proceedings of the International Congress of Mathematicians (ICM 2018) Pub Date : 2019-05-01 DOI: 10.1142/9789813272880_0179
G. Tardos
{"title":"EXTREMAL THEORY OF ORDERED GRAPHS","authors":"G. Tardos","doi":"10.1142/9789813272880_0179","DOIUrl":"https://doi.org/10.1142/9789813272880_0179","url":null,"abstract":"We call simple graphs with a linear order on the vertices ordered graphs. Turántype extremal graph theory naturally extends to ordered graphs. This is a survey on the ongoing research in the extremal theory of ordered graphs with an emphasis on open problems.","PeriodicalId":318252,"journal":{"name":"Proceedings of the International Congress of Mathematicians (ICM 2018)","volume":"1 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":"129575552","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}
引用次数: 18
CONVECTION-DIFFUSION-REACTION AND TRANSPORT-FLOW PROBLEMS MOTIVATED BY MODELS OF SEDIMENTATION: SOME RECENT ADVANCES 沉降模式引发的对流-扩散-反应和输运-流动问题:一些最新进展
Proceedings of the International Congress of Mathematicians (ICM 2018) Pub Date : 2019-05-01 DOI: 10.1142/9789813272880_0189
R. Bürger, J. Careaga, S. Diehl, Camilo Mejías, Ricardo Ruiz Baier
{"title":"CONVECTION-DIFFUSION-REACTION AND TRANSPORT-FLOW PROBLEMS MOTIVATED BY MODELS OF SEDIMENTATION: SOME RECENT ADVANCES","authors":"R. Bürger, J. Careaga, S. Diehl, Camilo Mejías, Ricardo Ruiz Baier","doi":"10.1142/9789813272880_0189","DOIUrl":"https://doi.org/10.1142/9789813272880_0189","url":null,"abstract":"The sedimentation of a suspension is a unit operation widely used in mineral processing, chemical engineering,wastewater treatment, and other industrial applications. Mathematical models that describe these processes and may be employed for simulation, design and control are usually given as nonlinear, time-dependent partial differential equations that in one space dimension include strongly degenerate convection-diffusion-reaction equations with discontinuous coefficients, and in two or more dimensions, coupled flowtransport problems. These models incorporate non-standardproperties that have motivated original research in applied mathematics and numerical analysis. This contribution summarizes recent advances, and presents original numerical results, for three different topics of research: a novel method of fluxidentification for a scalar conservation law from observation of curved shock trajectories that can be observed in sedimentation in a cone; a new description of continuous sedimentation with reactions including transport and reactions of biological components; and the numerical solution of a multi-dimensional sedimentation-consolidation system by an augmented mixed-primal method, including an a posteriori error estimation. (Less)","PeriodicalId":318252,"journal":{"name":"Proceedings of the International Congress of Mathematicians (ICM 2018)","volume":"5 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":"130678601","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}
引用次数: 2
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