CBMS Conference: Smooth and Non-Smooth Harmonic Analysis

这一国家科学基金奖为将于2018年6月4日至8日在爱荷华州立大学举行的数学科学会议《调和分析：平滑与非平滑》的会议理事会提供支持。近年来，人们对理解非光滑几何的调和分析越来越感兴趣，这些几何不同于熟悉的光滑欧几里德几何，例如当邻近的点不在局部相互连接时。这些类型的几何体出现在现实世界中的例子包括大型计算机网络、数据集中的关系和分形结构，例如在晶体物质、光散射和其他存在动力系统的自然现象中发现的那些。拟议的会议将由来自爱荷华大学的Palle Jorgensen教授连续十次演讲组成，他是平滑和非光滑调和分析领域的领导者，他将在会议过程中展示这两个领域之间令人惊讶的联系。这次会议的目的是将有经验的和新的研究人员聚集在一起，以促进在这一及时的主题上的合作。*会议的更广泛影响包括促进数学领域代表性不足的少数群体的发展，以及发展具有全球竞争力的STEM劳动力。大约30名参与者将得到支持，其中许多人将是目前的研究生。这次会议将有助于这些研究生的教育和职业发展，从而为国家的下一代研究人员从事这一日益重要的学科领域做好准备。Jorgensen教授的一个引人注目且经常被引用的结果是，存在一个Cantor类集合，该集合上支持的一致度量是谱的，这意味着存在一系列频率序列，对应的指数函数形成关于该度量的平方可积函数的Hilbert空间中的正交基。受到这一惊人结果启发的研究包括：分形傅立叶分析、Ruelle算子的谱理论、Cuntz代数的表示理论、小波理论中级联算法的收敛、再生核及其边界表示、Bernoulli卷积和马尔可夫过程。这一系列题材的显著特点是，他们既跨越了流畅的设置，也跨越了非流畅的设置。约根森教授的演讲将把这些影响深远的研究领域统一在光滑和非光滑调和分析的界面上，相应的专著对当前和未来的研究人员具有重要的价值，作为这些话题联系的巩固参考。有关这次会议的信息，请访问以下网址：http://www.public.iastate.edu/~esweber/cbms2018/

This National Science Foundation award provides support for the Conference Board of the Mathematical Sciences Conference "Harmonic Analysis: Smooth and Non-smooth" to be held at Iowa State University on June 4-8, 2018.  In recent years there has been an increasing interest in understanding the harmonic analysis of non-smooth geometries that are unlike the familiar smooth Euclidean geometry, such as when nearby points are not locally connected to each other.  Real-world examples in which these types of geometry appear include large computer networks, relationships in datasets, and fractal structures such as those found in crystalline substances, light scattering, and other natural phenomena where dynamical systems are present. The proposed conference will consist of a series of ten lectures by Professor Palle Jorgensen from the University of Iowa, a leader in the fields of smooth and non-smooth harmonic analysis, who will in the course of the conference demonstrate the surprising connections between the two domains. The conference aims to bring both experienced and new researchers together to stimulate collaboration on this timely topic.  The broader impacts of the conference include the advancement of underrepresented minorities within mathematics and the development of a globally competitive STEM workforce.  Approximately thirty participants will be supported, many of whom will be current graduate students.  The conference will contribute to those graduate students' educational and professional development and hence prepare the nation's next generation of researchers to engage this increasingly important subject area.A remarkable and oft-cited result of Professor Jorgensen is that there exists a Cantor-like set with the property that the uniform measure supported on that set is spectral, meaning that there exists a sequence of frequencies for which the corresponding exponential functions form an orthonormal basis in the Hilbert space of square-integrable functions with respect to that measure.  Research that has been inspired by this stunning result includes: fractal Fourier analyses, spectral theory of Ruelle operators, representation theory of Cuntz algebras, convergence of the cascade algorithm in wavelet theory, reproducing kernels and their boundary representations, Bernoulli convolutions, and Markov processes.  The remarkable feature of this array of subjects is that they straddle both the smooth and non-smooth settings.  The lectures presented by Professor Jorgensen will unify these far-reaching research areas at the interface of smooth and non-smooth harmonic analysis, with the corresponding monograph being of great value to current and future researchers as a consolidated reference for these topical connections. Information regarding the conference is at the following URL:  http://www.public.iastate.edu/~esweber/cbms2018/