Convexity and Applications

PI的研究是在渐近几何分析和凸几何及其应用这两个密切相关的领域。经典凸性研究固定维欧氏空间中凸体的几何。渐近几何分析研究的是有限维凸体在尺寸增长到无穷大时的几何性质。在她的研究中，PI使用了这两个领域的方法，以及概率工具和集中现象，以更好地了解凸集的结构。PI能够证明凸集的这种结构方面发挥作用的各种结果：用多面体逼近凸体；在所谓的顺序统计量(这是统计学中的基本对象)和Orlicz范数之间建立联系；确定在量子信息论中自然出现的某些(凸集)的“大小”。PI的另一个研究重点是仿射不变量的发展。PI和她的合作者开始系统地研究与凸体相关的(仿射不变)泛函及其相应的不等式。其中最重要的泛函有仿射表面积和p-仿射表面积。与它们相关的仿射等周不等式比它们的欧几里得亲属更强大，并且与其他重要的不等式有关，例如Santalo-和逆Santalo-不等式。后者与马勒猜想有关，马勒猜想在3维或更高的维度上仍然是开放的。一个科学或工程问题的数学描述通常需要大量的独立数字，从而产生一个高维几何空间。例如，如果您要指定一个气体分子在房间中的位置，则需要使用三个数字报告该分子的正面/背面、左右和上下位置。分子运动的方向和速度需要另外三个数字，所以为了描述分子的当前状态，使我们能够根据位置和速度预测它的未来运动，我们总共需要六个独立的数字。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，采样和计算的难度也迅速增加，科学家和数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，从而利用“维度的诅咒”，使其成为“维度的祝福”。这笔拨款的目的之一就是研究这种高维现象。

The PI's research is in the two closely related areas of asymptotic geometric analysis and convex geometry, and their applications.Classical convexity studies the geometry of convex bodies in Euclidean space of a fixed dimension. Asymptotic geometric analysis deals with geometric properties of finite dimensional convex bodies as the dimension grows to infinity. In her research the PI uses methods from both areas, as well as probabilistic tools and concentration phenomena, to get a better understanding of the structure of convex sets. The PI was able to prove a variety of results where such structural aspects of convex sets play a role: in approximation of convex bodies by polytopes; to establish a link between the so called order statistics (which are fundamental objects in statistics) and Orlicz norms; to determine the ``sizes" of certain (convex) sets that appear naturally in quantum information theory.A further focus of the PI's research is the development of affine invariants. The PI and her collaborators started the systematic study of (affine invariant) functionals associated with convex bodies and their corresponding inequalities. Among the most important such functionals are affine surface area and p-affine surface area . The affine isoperimetric inequalities related to them are more powerful than their Euclidean relatives and related to other important inequalities, e.g. the Santalo- and Inverse Santalo- inequalities. The latter is related to Mahler's conjecture which is still open in dimension 3 and higher.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, if you want to specify the location of one gas molecule in a room then you need to report the front/back, side-to-side, and up/down locations of the molecule, using three numbers. The direction and speed of the molecule's motion takes another three numbers, and so to describe enough of the molecule's current state to allow us to predict its future motion from position and velocity we would need 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 scientists and mathematicians 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 patters, thus taking advantage of the "curse of dimensionality" to make it the "blessing of dimensionality". It is one purpose of this grant to study such high dimensional phenomena.