2024-07-29 ~ 2024-08-02
2024-04-12 ~ 2024-04-14
2024-04-08 ~ 2024-04-12
2024-03-18 ~ 2024-03-23
2024-03-04 ~ 2024-03-08
Intelligent Computing for Inverse Problems and Data Science
会议编号:
M240701
时间:
2024-07-29 ~ 2024-08-02
浏览次数:
240
SPDEs and multiscale analysis
会议编号:
M240402
时间:
2024-04-12 ~ 2024-04-14
浏览次数:
330
2024年4月11日-14日,由清华大学丘成桐数学科学中心荆文甲老师,华中科技大学高婷老师,北京雁栖湖应用数学研究院张琦老师组织的随机偏微分方程与多尺度分析研讨会(SPDEs and multiscale analysis)将在清华三亚国际数学论坛如期举行,本次会议共有24位(含线上)来自国内数所院校或研究机构的数学学者参加,会议期间将探讨有关随机偏微分方程与多尺度分析等相关研究问题。共有22场相关学术报告。
Elliptic Discrete Integrable Systems
会议编号:
M240401
时间:
2024-04-08 ~ 2024-04-12
浏览次数:
757
会议摘要(Abstract)
Elliptic discrete integrable systems are among the richest of the whole class of integrable equations, both continuous as well as discrete. Their solutions in terms of special functions involve novel features, such as bi-elliptic addition formulae and elliptic generalisations of classical functions and orthogonal polynomials. A lot of progress have been made on various aspects of the theory, but some important elliptic models have received not as much attention as they deserve. One of the aims of the workshop is to repair this inbalance. Thus, the workshop will bring together experts who have worked on various aspects of integrability and elliptic function theory, and thus aim at bringing progress in dealing with these challenging but rewarding model systems.
在整个连续和离散可积方程中,椭圆离散可积系统被认为蕴含着最为丰富的信息。关于这类系统的求解,涉及到特殊函数的一些新的性质,如双-椭圆加法公式、经典的函数与正交多项式的椭圆推广等。对于椭圆离散可积系统理论的研究在许多方面都取得了不小的进展,但一些重要的椭圆可积模型尚未获得应有的关注。这也是此次研讨会的举办背景之一。研讨会将汇集可积系统和椭圆函数理论各个方面的专家学者,交流在这些具有挑战性的可积模型方面的研究进展,发展相关理论。
举办意义(Description of the aim)
Elliptic integrable systems are, in principle, exactly solvable model equations defined through ordinary or partial difference equations and that contain parameters associated with an elliptic curve. In the continuous case the outstanding examples are the Krichever-Novikov equation and the fully anisotropic Landau-Lifschitz equation, where in the former case the equation contains an arbitrary quartic polynomial in the dependent variable, to which one can associate an elliptic curve, while in the latter case the anisotropy parameters in general position are associated with the moduli of an elliptic curve. The discovery of discrete analogues of these equations and their significance within the framework of integrable systems forms the main motivation for this proposal. In addition, classes of ordinary difference equations of discrete-time many-body systems exhibit features which are radically different and richer than those of the corresponding continuous-time many-body systems. (Recently a novel example of such an elliptic many-body system arising as reduction of the BKP equation was found by Zabrodin). Furthermore, there is an entire classification of difference equations of Painlevé type (i.e. equations whose general solutions are meromorphic w.r.t. the movable singularities) at the top of which stands the elliptic Painlevé equation discovered in 1999 by H. Sakai. About a year earlier V. Adler discovered a discrete analogue of the Krichever-Novikov equation, which stands at the top of the well-known Adler-Bobenko-Suris (ABS) classification of quadrilateral lattice equations, which are four-point partial difference equations integrable in the sense of being multidimensionally consistent (i.e., consistently embeddable in a higher-dimensional lattice). Regarding the Landau-Lifschitz equations, there are three independent versions known: one proposed by Nijhoff and Papageorgiou in 1989, another constructed by Adler and Yamilov in 1996, and yet a third proposed by Adler himself in 2000. Almost nothing is known to date about these three models, not even if/how they are interrelated. Nonetheless, it would be of great interest to know their solution structures and their connection with other integrable systems. Finally, there is also a three-dimensional system of equations, namely an elliptic version of the Kadomtsev-Petviashvili equation, which was first proposed by Date, Jimbo and Miwa in 1983 and further studied recently by Fu and Nijhoff.
Apart from elliptic models, there are also classes of elliptic solutions of integrable lattice equations that exhibit novel features, such as a new concept of elliptic Nth root of unity, appearing in elliptic solutions of higher rank lattice equations of Gel’fand-Dickey type.
There are many outstanding problems in the theory of elliptic discrete systems, and resolving them has become urgent, as they stand at the top of the tree of integrable systems: knowing their resolution would imply the resolution of similar questions for the degenerate cases, which comprises almost all integrable equations.
In recent years, various novel mathematical techniques and methods have been developed to study these discrete systems and their solutions, bringing together ideas stemming from several branches of mathematics and physics, that are usually distinct, now come together: asymptotic analysis, algebraic geometry, representation theory, spectral/isomonodromy analysis, random matrix theory, exactly solvable models, theory of special functions and combinatorial geometry. These will form the unifying themes of the workshop, which will comprise the following topics:
--Elliptic solutions of higher-rank and multidimensional lattice equations;
--The construction of higher-rank elliptic lattice systems;
--Bi-elliptic addition formulae (as appearing in the soliton solutions of the ABS Q4 equation);
--Elliptic discrete time many-body systems;
--The solution structure of the variants of the lattice Landau-Lifschitz equations;
--Elliptic Lax pairs and isomonodromic deformation problems;
--Elliptic orthogonal polynomials and corresponding elliptic discrete integrable systems;
--Reductions to elliptic autonomous and non-autonomous ordinary difference equations, such as the QRT map and elliptic Painlevé equation;
--Algebro-geometric solutions of integrable lattice equations, and higher-genus lattice systems;
--Elliptic solutions of recurrence relations for integer (Somos) sequences;
--Connections with models of quantum theory and statistical mechanics.
椭圆离散可积系统指包含与椭圆曲线相关的参数的精确可解的常差分方程或偏差分方程。在连续情况下,典型的例子是Krichever-Novikov方程和各向异性Landau-Lifschitz方程。前者方程中包含有关于因变量的任意四次多项式,可以将椭圆曲线与之关联;而后者涉及的各向异性参数与椭圆曲线的模参数相关。这些方程的离散形式的发现及它们在可积系统框架内具有的重要意义构成申请此次研讨会的主要背景。在描述离散时间的多体系统的常差分方程中表现出与相应的连续时间多体系统截然不同且更丰富的性质。最近,Zabrodin给出了一个从BKP方程约化获得椭圆多体系统的新例子。目前关于Painlevé型 (即其通解相对于可移动奇点是亚纯函数的方程) 差分方程已有完整的分类,分类中处于顶端的方程是1999年由H. Sakai发现的椭圆离散Painlevé方程。大约在Sakai的工作的一年之前,V. Adler发现了Krichever-Novikov方程的离散形式,该方程处于后来给出的Adler-Bobenko-Suris (ABS)著名的关于多维相容的四边形方程分类的顶端 (多维相容性定义了一种离散可积性,指低维方程可以相容地嵌入到更高维空间)。关于Landau-Lifschitz方程,目前有三个已知的独立的离散版本:一个由Nijhoff和Papageorgiou于1989年提出,一个由Adler和Yamilov于1996年构建,还有一个由Adler于2000年给出。迄今为止,人们对这三个离散的Landau-Lifschitz模型知之甚少,甚至还不清楚它们之间是否存在联系或如何关联。十分期待理解它们的解结构以及它们与其他可积系统的联系。最后,还有一个与Kadomtsev-Petviashvili方程相关的椭圆三维方程组,由Date、Jimbo和Miwa于1983年提出,最近由Fu和Nijhoff对其做了进一步研究。
除了椭圆模型以外,离散可积方程还存在多种类型的椭圆函数解,并涉及到一些新的特征。例如在Gelfand-Dickey型高阶离散方程的椭圆孤子解的研究中引入了新的“椭圆单位根”的概念。
椭圆离散系统理论中存在许多尚未解决且亟待研究的问题。发展椭圆可积系统的研究方法具有重要意义。事实上,椭圆可积系统处于可积系统分类的顶端,这类系统通过退化几乎包括了所有的可积方程。
近年来,不断有新的数学技巧和方法出现,用于研究这类系统及其精确解。这些研究逐步将来自于数学和物理的若干看似不同的分支结合起来,包括:渐近分析、代数几何、表示理论、谱/等单值性分析、随机矩阵理论、精确可解模型、特殊函数理论和组合几何等。这些贯穿于此次研讨会的主题内容,具体地,我们将关注:
--高阶和高维离散方程的椭圆函数解;
--高阶椭圆离散系统的构造;
--双-椭圆加法公式(出现于ABS的Q4方程的孤子解中);
--椭圆离散多体系统;
--各种离散Landau-Lifschitz 方程的解的结构;
--椭圆Lax 对和等单值形变问题;
--椭圆正交多项式与相应的椭圆离散可积系统;
--椭圆自治与非自治常差分方程的约化,如QRT映射与椭圆Painlevé方程;
--离散可积系统的代数几何解与高亏格离散系统;
--整数(Somos)序列的递推结构的椭圆解;
--量子理论模型与统计力学方面的联系。
The interaction between geometry, quantum field theory and nonlinear analysis
会议编号:
M240302
时间:
2024-03-18 ~ 2024-03-23
浏览次数:
859
会议摘要(Abstract)
The interaction between geometry (Riemannian, algebraic, Kaehler, metric,...), quantum field theory and analysis (elliptic and parabolic PDEs, calculus of variations) has brought spectacular advances and generated deep insight in all those disciplines.
Minima of variational integrals may provide optimal solutions to geometric problems, and parabolic PDEs may deform initial geometrical structures into optimal ones, thereby solving important geometric problems. Functionals from QFT contain rich structures that can be exploited for the construction of subtle geometric invariants. In turn, such problems typically lead to very difficult analytical challenges. The resulting PDEs are not only highly non-linear, but from a variational perspective usually are not contained in the range of the Palais-Smale condition, and therefore, standard methods usually break down. This challenge gave an important impetus to the theory of non-linear PDEs. One line of research exploited convexity properties, typically arising from non-positive curvature, another one depended on an extremely careful study of the formation of singularities, which in turn had to use geometric features or algebraic properties from QFT.
During this workshop, we want to bring people together to explore various current research questions in this field, including
-- Analytic methods for studying moduli spaces in algebraic geometry
-- A general mathematical theory of the action functionals of QFT and the resulting challenges for PDE theory to construct minimizers or other critical points and to understand their regularity properties
-- Relations between QFT functionals and geometric constructions, like Kapustin-Witten and Higgs fields
-- Bernstein and Dirichlet problems for minimal submanifolds of Euclidean spaces and spheres
-- The role of PDEs in metric space geometry
-- The geometry of positive sectional curvature
-- The approximation of geometric objects by discrete ones
我们希望通过本次研讨会组织与会者一起探讨几何、量子场论和分析相互作用领域的前沿问题,包括:
-- 代数几何中模空间的分析方法;
-- 量子场论中作用泛函的一般数学理论及产生的偏微分方程理论挑战,如构造极小解或其它临界点,研究它们的正则性;
-- 量子场论中作用泛函与几何构造的关系,如Kapustin-Witten 和 Higgs 场;
-- 欧氏空间和球面中极小子流形的Bernstein 和 Dirichlet 问题;
-- 偏微分方程在度量空间几何中的作用;
-- 正曲率几何学;
-- 几何对象的离散逼近。
举办意义(Description of the aim)
几何,量子场论和非线性分析之间的相互作用为这些领域中带来了巨大的推动和深刻的洞察。变分问题的极小点为几何问题提供了最优解,抛物方程把初始几何结构形变为最优解,从而解决重要的几何问题。量子场论中的作用量泛函包含丰富的结构,可以用于构造精妙的几何不变量。反之,这些问题也常常带来困难的分析挑战,其产生的偏微分方程不仅是高度非线性的,而且从变分观点看通常不能包含在满足Palais-Smale条件的范围中,因此,标准的方法往往失效。这种挑战为我们提供了发展非线性分析理论的重要动力,一条途径是运用凸性性质,常见于非正曲率情形,另一条途径是依赖于对奇性形成的精细研究,这反过来需要利用量子场论的几何特性或代数性质。在本次研讨会期间,我们将探讨有关几何,量子场论和非线性分析之间相互作用领域的前沿研究问题。
Advanced Finite Elements Methods for Nonlinear PDEs
会议编号:
M240301
时间:
2024-03-04 ~ 2024-03-08
浏览次数:
1054
会议主题(Theme)
本次会议以“非线性偏微分方程的高效有限元方法”为主题,针对科学与工程计算中的非线性偏微分方程,研究其高精度有限元方法。会议拟邀请50位国内外相关领域中的杰出学者,开展学术报告和交流合作。此次会议将极大增进非线性问题数值解法的前沿研究和发展趋势的交流与讨论,为青年学者开拓学术视野,创造合作机遇。
The theme of this meeting is "Advanced Finite Element Methods for Nonlinear PDEs", focusing on high-precision finite element methods for nonlinear partial differential equations in scientific and engineering calculations. The conference plans to invite 50 outstanding scholars from relevant fields both domestically and internationally to conduct academic presentations, exchange and cooperation. This conference will greatly enhance the exchange and discussion of cutting-edge research and development trends in numerical solutions for nonlinear problems, broaden academic horizons for young scholars, and create opportunities for cooperation.
举办意义(Description of the aim)
非线性偏微分方程的数值求解是一个既有广泛工程应用背景,又具有挑战性的困难课题。连续力学中变分形式的强非线性问题,协调有限元离散可以把原问题归结为极小值的数值计算问题,而对于一般的非变分形式的强非线性偏微分方程,如 HJB 方程,离散过程并不是那么简单。本次会议主旨是把变分计算专家和非变分形式微分方程数值解学者聚集起来,针对非线性问题的数值求解展开合作研究和讨论。本次会议的议题涉及非线性问题数值解法的前沿研究,具有重要的实际意义。
The numerical solution of Nonlinear partial differential equation is a difficult subject with both extensive engineering applications and challenges. For the strongly nonlinear problems in the variational form of continuous mechanics, the original problem can be reduced to the numerical calculation problem of the minimum by the coordinated finite element discretization, while for the strongly nonlinear partial differential equations in the general non variational form, such as the HJB equation, the discretization process is not so simple. The main purpose of this meeting is to gather experts in variational computation and numerical solvers of non variational form differential equations to conduct collaborative research and discussion on numerical solutions for nonlinear problems. The topic of this meeting involves cutting-edge research on numerical solutions for nonlinear problems, which has important practical significance.
2024年4月11日-14日,由清华大学丘成桐数学科学中心荆文甲老师,华中科技大学高婷老师,北京雁栖湖应用数学研究院张琦老师组织的随机偏微分方程与多尺度分析研讨会(SPDEs and multiscale analysis)将在清华三亚国际数学论坛如期举行,本次会议共有24位(含线上)来自国内数所院校或研究机构的数学学者参加,会议期间将探讨有关随机偏微分方程与多尺度分析等相关研究问题。共有22场相关学术报告。
2024年1月8日-1月12日,三亚波国际前沿论坛(Sanya Waves)学术会议将在清华三亚国际数学论坛如期举行,本次会议共有62位来自国内数所院校或研究机构的数学学者参加,会议期间就非线性波动方程、几何分析和广义相对论等基本问题展开研讨,共有19场相关学术报告。
2024年1月7日-1月13日,由新加坡国立大学力学工程系助理教授、中国科学技术大学近代力学系刘难生教授、香港城市大学数学系胡先鹏教授组织的粘弹性流体的动力学:从理论到机理(Viscoelastic Flow Dynamics: from Theory to Mechanisms)学术会议将在清华三亚国际数学论坛如期举行,本次会议共有49位来自国内数所院校或研究机构的数学学者参加,会议期间就粘弹性流体相关的数学研究及其物理机理展开研讨,共有35场相关学术报告。
2023年7月22日-7月29日,由清华大学郑建华、南昌大学曹廷彬、华南师范大学黄志波老师组织的“复动力系统与复方程研讨会(Workshop on Complex Dynamics and Complex Equations)”学术会议在清华三亚国际数学论坛如期举行,本次会议共有42位来自国内数所院校或研究机构的数学学者参加。
2023年3月31日-4月4日,由华中师范大学凡石磊、中国科学院刘劲松、清华大学吴云辉和浙江大学叶和溪老师组织的2023“动力系统,Teichmuller理论及相关主题”学术会议(Dynamics, Teichmuller theory and their related topics)将在清华三亚国际数学论坛如期举行,本次会议共有76位来自国内数所院校或研究机构的数学学者参加,会议期间就动力系统,Teichmuller理论及相关主题展开研讨,届时将举行28场相关学术报告。
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