Convexity and Applications

主要研究方向是渐近几何分析和仿射凸几何。她的研究重点之一是高维物体和现象。这导致了她的研究在物理、生物和医学、计算机科学、优化和经济学以及材料科学等领域的应用：事实上，科学或工程问题的数学描述往往需要大量独立的数字，从而产生高维几何空间。例如，指定一个房间中一个气体分子的位置、方向和速度总共有六个独立的数字。如果你想跟踪房间里100个不同的空气分子，那么你需要600个独立的数字坐标来收集所有相关的测量数据。随着这些维度的增加，采样和计算的难度迅速上升，数据科学家有时将这种现象称为“维度诅咒”。然而，也有随着维度的增加而出现的模式，这些模式在低维度中看不到。我们可以利用这些模式，从而将“维度的诅咒”转化为“维度的祝福”。该奖项的目的之一就是研究这种高维现象。该项目的重要特点是研究高维物体和现象及其与其他数学和数学科学领域的联系，如概率论、统计学和信息论。特别有趣的是高维凸体上的仿射不变泛函。其中最重要的这类泛函是仿射表面积和p-仿射表面积(由实数p参数表示的一族泛函)。它们对应的仿射等周不等式，是由Pi和合作者为所有p建立的，比它们的欧几里德同行更强，并与著名的Mahler猜想有关，该猜想在四维和更高维上仍然是开放的。主要研究人员证明了p-仿射表面积与凸体的锥测度的熵直接相关，从而在凸几何和信息论之间建立了联系。这一联系也将在对数凹函数的背景下进一步探索，对数凹函数是凸体在函数域中的自然延伸。此外，仿射表面积自然而然地出现在用多面体逼近凸体的问题中，这是进一步研究的主要课题。其目标是建立对近似中涉及的所有相关参数的最优依赖，例如近似多面体的维度和顶点数量。主要研究人员和她的合作者最近还将仿射表面积的概念推广到函数环境以及球面和双曲空间。在这些环境中建立相应的不平等是进一步研究的主题。该奖项反映了NSF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 principal investigator's research is in asymptotic geometric analysis and affine convex geometry. One main emphasis of her research is on high-dimensional objects and phenomena. This leads to applications of her research in areas as diverse as physics, biology and medicine, computer science, optimization and economics and material science: Indeed, a mathematical description of a scientific or engineering question often requires lots of independent numbers, leading to a geometric space of high dimension. For example, specifying the location, direction and speed of one gas molecule in a room six separate numbers in all. If you want to track 100 distinct molecules of the air in the room, then you will need 600 independent numerical coordinates to collect all of the relevant measurements. As these dimensions increase then the difficulty of sampling and computation go up rapidly, a phenomenon data scientists sometimes call "the curse of dimensionality." However, there are also patterns that emerge as dimension increases which are not visible in low dimensions. We can exploit those patterns, thus converting the "curse of dimensionality" into "blessing of dimensionality". It is one purpose of this award to study such high-dimensional phenomena. Important features of this project are the study of high-dimensional objects and phenomena and their links with other areas of mathematics and mathematical sciences, such as probability, statistics and information theory. Of particular interest are the affine invariant functionals on convex bodies in high dimensions. Among the most important such functionals are affine surface area and the p-affine surface area (a family of functionals parametrized by a real number p). Their corresponding affine isoperimetric inequalities, established by the PI and collaborators for all p, are stronger than their Euclidean counterparts and related to the famous Mahler conjecture which is still open in dimensions four and higher. It was shown by the principal investigator that p-affine surface areas are directly related to entropies of cone measures of convex bodies which establishes a link between convex geometry and information theory. This link will be further explored, also in the context of log concave functions which are a natural extension of convex bodies in the realm of functions. Moreover, affine surface area appears naturally in questions on approximation of convex bodies by polytopes, a further main topic of study. The goal is to establish optimal dependence on all the relevant parameters involved in the approximation, for example the dimension and the number of vertices of the approximating polytopes. The principal investigator and her collaborators also extended the notions of affine surface area recently to a functional setting and to spherical and hyperbolic space. To establish the corresponding inequalities in those settings is a further topic of study.This 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.