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授课人:王立威

课程:数学所讲座2018,2018

关键词:表示论;深度学习;...

发布机构:[中国科学院数学与系统科学研究院]

使用许可:CC BY-NC-ND

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深度学习是近年来人工智能、机器学习领域的重大突破,在包括无人驾驶、围棋(AlphaGo)等应用取得成功。但深度学习目前尚缺乏基础理论。本次报告中,我将讨论深度神经网络的表示理论、深度学习的泛化理论,重点介绍其中的数学问题。

2 复动力系统 [开放课件]

授课人:王跃飞

课程:数学所讲座2018,2018

关键词:复动力系统

发布机构:[中国科学院数学与系统科学研究院]

使用许可:CC BY-NC-ND

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复动力系统主要讨论复流形上全纯映射生成的动力系统。从上世纪八十年代以来,复动力系统研究在深度和广度上都得到了蓬勃发展。本报告将简要介绍该方向的概况、发展与现状。

授课人:Sandra Cerrai

课程:PIMS-CRM Summer School in Probability 2017,2017

关键词:Scientific;Mathematics;...

发布机构:[Pacific Institute fo...

使用许可:CC BY-NC-ND

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I will introduce a new class of SPDEs defined on graphs, obtained as the limit of suitable SPDEs, defined on two-dimensional domains and depending on some parameters. I will do this presenting two examples. The first example is given by some SPDEs defined on narrow channels with wings. As the width of the channel goes to zero the solutions converge to the solution of a suitable SPDE defined on the graph that can be obtained by identifying all points on the same cross section of the tubular domain. The second example is given by the analysis of the fast advection asymptotics for some stochastic reaction-diffusion-advection equations defined on the plane. To describe the asymptotics, I will consider a suitable class of SPDEs defined on a graph, corresponding to the stream function of the underlying incompressible flow.

 

授课人:Marek Biskup

课程:PIMS-CRM Summer School in Probability 2017,2017

关键词:Scientific;Mathematics;...

发布机构:[Pacific Institute fo...

使用许可:CC BY-NC-ND

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The Gaussian free field (GFF) is a fundamental model for random fluctuations of a surface. The GFF is closely related to local times of random walks via relations that originated in the study of spin systems. The continuous GFF appears as the limit law of height functions of dimer covers, uniform spanning trees and other models without apparent Gaussian correlation structure. The GFF is also a simple example of a quantum field theory. Intriguing connections to SLE, the Brownian map and other recently studied problems exist. The GFF has recently become subject of focused interest by probabilists. Through Kahane's theory of multiplicative chaos, the GFF naturally enters into models of Liouville quantum gravity. Multiplicative chaos is also central to the description of level sets where the GFF takes values proportional to its maximum, or values order-unity away from the absolute maximum. Random walks in random environments given as exponentials of the GFF show intriguing subdiffusive behavior. Universality of these conclusions for other models such as gradient systems and/or local times of random walks are within reach.

 

授课人:Marek Biskup

课程:PIMS-CRM Summer School in Probability 2017,2017

关键词:Scientific;Mathematics;...

发布机构:[Pacific Institute fo...

使用许可:CC BY-NC-ND

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The Gaussian free field (GFF) is a fundamental model for random fluctuations of a surface. The GFF is closely related to local times of random walks via relations that originated in the study of spin systems. The continuous GFF appears as the limit law of height functions of dimer covers, uniform spanning trees and other models without apparent Gaussian correlation structure. The GFF is also a simple example of a quantum field theory. Intriguing connections to SLE, the Brownian map and other recently studied problems exist. The GFF has recently become subject of focused interest by probabilists. Through Kahane's theory of multiplicative chaos, the GFF naturally enters into models of Liouville quantum gravity. Multiplicative chaos is also central to the description of level sets where the GFF takes values proportional to its maximum, or values order-unity away from the absolute maximum. Random walks in random environments given as exponentials of the GFF show intriguing subdiffusive behavior. Universality of these conclusions for other models such as gradient systems and/or local times of random walks are within reach.

 

授课人:Hugo Duminil-Copin

课程:PIMS-CRM Summer School in Probability 2017,2017

关键词:Scientific;Mathematics;...

发布机构:[Pacific Institute fo...

使用许可:CC BY-NC-ND

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Phase transitions are a central theme of statistical mechanics, and of probability more generally. Lattice spin models represent a general paradigm for phase transitions in finite dimensions, describing ferromagnets and even some fluids (lattice gases). It has been understood since the 1980s that random geometric representations, such as the random walk and random current representations, are powerful tools to understand spin models. In addition to techniques intrinsic to spin models, such representations provide access to rich ideas from percolation theory. In recent years, for two-dimensional spin models, these ideas have been further combined with ideas from discrete complex analysis. Spectacular results obtained through these connections include the proof that interfaces of the two-dimensional Ising model have conformally invariant scaling limits given by SLE curves, that the connective constant of the self-avoiding walk on the hexagonal lattice is given by ? 2 + ? 2 , and that the magnetisation of the three-dimensional Ising model vanishes at the critical point.