Convexity and Applications

PI的研究方向是经典凸性理论和凸几何分析。一个主要目标是更好地理解凸体的结构。为了做到这一点，她使用的技术从不同领域的数学：分析，微分几何，凸性理论，概率论。她研究等周不等式， 等周不等式这为凸集的特征化和分类提供了有力的工具。通过她的调查仿射表面积-原来是一个概念的仿射微分几何和发生在仿射等周不等式-她最近导致了广泛的研究问题的近似凸机构的多面体。仿射表面积在此上下文中自然出现，因为它与凸体的边界结构有关。PI已经研究并仍在研究多面体近似凸体的不同方面。例如，在一篇论文中，她和她的合作者一起证明了一个令人惊讶的结果，即多面体的随机近似（在物体的边界上随机选择近似多面体的顶点）与最佳近似一样好。除了凸性工具，概率工具，如浓度的措施，已被证明是非常有效的凸性。PI继续她的调查，这样的概率结果，推进她的研究，在凸性的结构结果及其应用到本地Banach空间理论和量子信息theory.Past经验导致PI相信，纯理论的概念也是有用的应用。她经历了这些领域的方法和结果在其他数学领域和应用领域中找到了应用：几何层析成像，一种起源于经典凸性理论的工具，给出了一种从其截面或投影中恢复凸形的方法。这用于计算机视觉和图像分析，在生物学和医学中，凸形（器官）自然出现。几何算法在计算机科学中有应用。更明确地说，科学或工程问题的数学描述通常需要大量的独立数字，导致高维几何空间。 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，这项资助将研究其中一些最近发现的模式。

The PI's research is in classical convexity theory and convex geometric analysis. A primary goal is to get a better understanding of the structure of convex bodies. To do so she uses techniques from different areas of mathematics: analysis, differential geometry, convexity theory, probability theory. She investigates isoperimetric inequalities and affine isoperimetric inequalities. These provide powerful tools in characterizing and classifying convex sets. Through her investigation of the affine surface area -originally a concept of affine differential geometry and occurring in the affine isoperimetric inequality- she was lately led to an extensive study of questions of approximation of convex bodies by polytopes . The affine surface area appears naturally in this context as it is related to the boundary structure of a convex body. The PI has investigated and still is investigating different aspects of approximation of convex bodies by polytopes. In one paper, for instance, she -together with her collaborator- proved the surprising result that random approximation by polytopes (choosing the vertices of the approximating polytope randomly on the boundary of the body) is as good as best approximation. Besides convexity tools, probabilistic tools, like concentration of measure, have proved to be very efficient in convexity. The PI continues her investigation of such probabilistic results for advancing her research in structural results in convexity and its applications to local Banach space theory and quantum information theory.Past experience has led the PI to believe that purely theoretical concepts are also useful in applications. She has experienced that the methods and results from these areas find applications in other fields of mathematics and in applied areas: Geometric tomography, a tool having its origins in classical convexity theory, gives a method to recover convex shapes from its sections or projections. This is used in computer vision and image analysis, in biology and medicine where convex shapes (organs) occur naturally. Geometric algorithms find applications in computer science. To be more explicit, 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, and this grant will study some of these patterns that are recent discoveries.