CBMS Conference: Analysis, Geometry, and Partial Differential Equations in a Lower-Dimensional World

CBMS会议“低维世界中的分析、几何和偏微分方程”将于2020年8月17日至21日在佛罗里达州立大学举行。在会议期间，将讨论有关解析、偏微分方程和多维集的几何性质之间的联系的最新突破性进展。这些问题在物理学、材料科学和工程学的几个领域都有令人惊讶和复杂的应用，听众将有一个独特的机会直接接触到纯数学前沿看似抽象的概念和结果如何立即影响光子器件的最先进工程及其背后的物理学。在宣传会议时，我们将特别努力鼓励代表性不足的群体参加。主讲人Mayboroda博士是一位受人尊敬的专家，与这样一位有成就的女性研究员交流对年轻的参与者来说是很重要的，这将增强他们在数学事业上的信心。最近几年，在R^n中的（n-1）维集合的解析、偏微分方程和几何性质之间的联系方面取得了突破性进展。这包括1916年关于调和测度相对于Hausdorff测度的绝对连续性的F.和M. Riesz定理的第一个逆命题，以及一系列关于可纠偏性的美丽刻画，从L2上Riesz变换的有界性（David-Semmes猜想的一个经过深思的解）到Carleson测度估计和调和函数的类似性质。我们将快速调查这些，并专注于一个神秘的低维集合世界，当几乎没有这些特征可用时，寻找有意义和强大的类似物揭示了一些全新的现象：简并偏微分方程的突出作用，新的非线性积分算子代替了经典的奇异积分，一个新的光滑距离函数，对于每一个低维集合都存在一个特殊的“椭圆”算子，其调和测度相对于该集合上的Hausdorff测度是绝对连续的。关于会议的更多信息可以在网站上找到：https://cbms2020.math.fsu.eduThis该奖项反映了美国国家科学基金会的法定使命，并通过基金会的智力价值和更广泛的影响审查标准进行评估，认为值得支持。

The CBMS conference "Analysis, geometry, and PDEs in a lower-dimensional world" will take place in Florida State University on 17-21 of August 2020. During the conference, there will be discussed recent groundbreaking advances pertaining to connections between analytic, partial differential equations, and geometric properties of multi-dimensional sets. These problems have surprising and intricate applications across several areas of physics, materials science, and engineering, and the audience will have a unique chance to have a direct exposure to the ways in which seemingly abstract concepts and results at the cutting edge of pure mathematics can immediately influence state-of-the-art engineering of photonic devices and the physics behind them. When advertising the conference, we will make special efforts to encourage underrepresented groups to attend. The main speaker Dr. Mayboroda is a well-respected expert, and interacting with such an accomplished female researcher is important for junior participants and will boost their confidence in pursuing careers in mathematics. The last few years have marked groundbreaking advances pertaining to connections between analytic, PDE, and geometric properties of (n-1)-dimensional sets in R^n. This includes the first converse to the 1916 F. and M. Riesz theorem regarding absolute continuity of harmonic measure with respect to Hausdorff measure and an array of beautiful characterizations of rectifiability, from boundedness of the Riesz transform on L2 (a much thought-after solution of the David-Semmes conjecture) to Carleson measure estimates and similar properties of harmonic functions. We will quickly survey those and concentrate on a mysterious world of lower-dimensional sets when virtually none of these characterizations is available and search for meaningful and powerful analogues revealed some completely new phenomena: a prominent role of degenerate PDEs, new non-linear integration operators in place of classical singular integrals, a new smooth distance function, an existence, for every lower-dimensional set, of a special "elliptic" operator whose harmonic measure is absolutely continuous with respect to the Hausdorff measure on this set. More information about the conference can be found on the website: https://cbms2020.math.fsu.eduThis award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.