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17-21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合的几何性质之间的联系的最新开创性进展。这些问题在物理、材料科学和工程的多个领域有着令人惊讶和错综复杂的应用，观众将有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可以立即影响最先进的光子设备工程及其背后的物理学的方式。在宣传大会时，我们将特别努力鼓励代表不足的团体参加。主讲人梅博罗达博士是一位备受尊敬的专家，与这样一位有成就的女性研究人员互动对初级参与者来说很重要，将增强他们在数学领域追求职业的信心。最近几年，关于R^n中(n-1)维集的解析性质、偏微分方程和几何性质之间的联系，取得了突破性的进展。这包括1916年F.和M.Riesz关于调和测度关于Hausdorff测度绝对连续性的第一个逆定理，以及一系列美丽的可正性刻画，从L2上Riesz变换的有界性(David-Semmes猜想的一个深思熟虑的解决方案)到Carleson测度估计和调和函数的类似性质。我们将快速考察这些，并集中在一个神秘的低维集合世界，当这些刻画几乎都不可用时，搜索有意义和强大的类似项揭示了一些全新的现象：退化偏微分方程组的突出作用，取代经典奇异积分的新的非线性积分算子，新的光滑距离函数，对于每个低维集合，存在一个特殊的“椭圆”算子，其调和测度关于这个集合上的Hausdorff测度是绝对连续的。有关会议的更多信息可在网站上找到：https://cbms2020.math.fsu.eduThis奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.