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

CBMS会议“分析，几何和偏微分方程在低维世界”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，将讨论最近的突破性进展，涉及分析，偏微分方程和多维集的几何性质之间的联系。这些问题在物理学、材料科学和工程学的多个领域都有令人惊讶和复杂的应用，观众将有一个独特的机会直接接触到纯数学前沿的看似抽象的概念和结果可以立即影响光子器件及其背后的物理学的最先进的工程。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。Mayboroda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于初级参与者来说非常重要，并将增强他们在数学领域追求职业生涯的信心。 在过去的几年里，R^n中（n-1）维集合的解析性质、偏微分方程性质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.关于调和测度相对于Hausdorff测度的绝对连续性的Riesz定理，以及一系列关于可求长性的漂亮刻画，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。我们将快速地调查这些，并集中在一个神秘的低维集世界，当几乎没有这些特征可用时，寻找有意义的和强大的类似物揭示了一些全新的现象：一个突出的作用退化偏微分方程，新的非线性积分算子的地方，经典的奇异积分，一个新的光滑距离函数，存在，为每一个低维集，的一个特殊的“椭圆”运营商的调和措施是绝对连续的Hausdorff措施在这个集合上。有关会议的更多信息可以在网站上找到：https://cbms2020.math.fsu.eduThis奖项反映了NSF的法定使命，并被认为值得通过使用基金会的知识价值和更广泛的影响审查标准进行评估来支持。

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.