Convexity and Applications

摘要奖：DMS 1504701，首席研究员：伊丽莎白·M·沃纳这些研究项目是渐近几何分析和仿射凸几何。首席研究员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，如果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分子的正面/背面、侧面/侧面和向上/向下的位置。分子运动的方向和速度需要另外三个数字，所以要描述分子的当前状态，使人们能够根据位置和速度预测其未来的运动，总共需要六个独立的数字。如果一个人想要跟踪房间里100个不同的空气分子，那么他将需要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是研究这种高维现象。拟议项目的重要特征一方面是强调仿射不变和高维对象和现象，另一方面是与数学和数学科学的其他领域(如概率论、统计学、信息论和量子信息论)的联系。学习的主题包括：凸体和对数凹函数的熵，仿射不变量及其不等式，凸体的结构性质，与最优化有关的凸性和几何方面的概率方法，Banach空间和量子信息论。这项研究的一个重点是仿射不变量及其相关的不等式的发展。主要研究人员和她的合作者开始系统地研究与凸体和对数凹函数相关的(仿射不变)泛函及其相应的不等式。其中最重要的这类凸体泛函是仿射表面积和p-仿射表面积。最近，PI和她的合作者将这些函数扩展到对数凹函数。与仿射表面积有关的仿射等周不等式比它们的欧几里得同类更强大，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，该猜想在三维及更高维度上仍是开放的。PI最近观察到，仿射表面积和Renyi熵之间的联系允许信息论和凸仿射几何之间的相互作用。

AbstractAward: DMS 1504701, Principal Investigator: Elisabeth M. Werner These research projects are in asymptotic geometric analysis and affine convex geometry. One main emphasis of the principal investigator's 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, if one wants 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 one to predict its future motion from position and velocity one would need six separate numbers in all. If one wants to track 100 distinct molecules of the air in the room then one 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.Important features of the proposed project are, on the one hand, emphasis on affine invariant and high dimensional objects and phenomena, and, on the other hand, links with other areas of mathematics and mathematical sciences (such as probability, statistics, information theory and quantum information theory). Topics to be studied include: Entropies for convex bodies and log concave functions, affine invariants and their inequalities, structural properties of convex bodies, probabilistic methods in convexity and geometric aspects related to optimization, Banach spaces, and quantum information theory. A focus of this research program is the development of affine invariants and their related inequalities. The principal investigator and her collaborators started the systematic study of (affine invariant) functionals associated with convex bodies and log concave functions and their corresponding inequalities. Among the most important such functionals for convex bodies are affine surface area and p-affine surface area. Those were recently extended to log concave functions by the PI and her collaborators. The affine isoperimetric inequalities related to the affine surface areas 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 three and higher. Connections, observed recently by the PI, between affine surface areas and Renyi entropy allow interactions between information theory, and convex affine geometry.