Convexity and Applications

PI的研究是在两个密切相关的领域渐近几何分析和凸几何，以及它们的应用。经典凸性研究的几何凸体在欧氏空间的一个固定的维度。渐近几何分析研究有限维凸体的几何性质。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，以更好地理解凸集的结构。PI能够证明各种结果，其中凸集的这种结构方面发挥了作用：用多面体近似凸体;建立所谓的顺序统计之间的联系（这是统计学的基本对象）和Orlicz规范;确定某些（凸）的“尺寸”PI研究的另一个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。其中最重要的这样的泛函是仿射表面积和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不等式更强大，并且与其他重要的不等式相关，例如Santalo-不等式和逆Santalo-不等式。后者与Mahler猜想有关，而Mahler猜想在3维及更高维空间中仍然是开放的。一个科学或工程问题的数学描述往往需要大量的独立数，从而导致一个高维几何空间。 例如，如果您想指定房间中一个气体分子的位置，则需要使用三个数字报告分子的前/后、侧到侧和上/下位置。分子运动的方向和速度需要另外三个数字，所以要描述足够的分子当前状态，让我们能够从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值。 随着这些维度的增加，采样和计算的难度迅速增加，科学家和数学家有时称之为“维度灾难”。“然而，随着维度的增加，也会出现一些模式，这些模式在低维度中是不可见的。我们可以利用这些模式，从而利用“维度灾难”，使其成为“维度祝福”。这是一个目的，这项补助金，以研究这种高维现象。

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.