Convexity and Applications

项目编号：DMS 1207917，项目负责人：Elisabeth M. werner，主要研究方向为渐近几何分析和仿射凸几何。她的研究重点是仿射不变和高维对象和现象，以及与其他数学和数学科学领域（概率论、统计学、优化、信息论和量子信息论）的联系。近年来凸几何的许多研究都是针对凸体的仿射和高维方面的研究。PI和她的合作者对凸体上一些最重要的仿射不变泛函（p-仿射表面积）进行了系统的研究。他们得到了这些不变量对所有凸体和所有p的扩展，并建立了相应的仿射等周不等式（比欧几里得不等式更强）。这些作品有许多应用。我们只提到PI的工作，与Reisner和Schuett，对马勒的猜想。虽然没有提供解决方案，但他们的结果是迄今为止最强烈的迹象，表明确实达到了多面体的最小值。最近的发展开辟了全新的方向。首先，PI发现了新的仿射不变量，并证明了这些和p-仿射表面积是信息论中的某些熵。在另一个方向上，PI、Artstein、Klartag和Schuett证明了对数凹函数的仿射等周不等式对应于熵的逆对数Sobolev不等式。这些方向将进一步探讨。她的贡献还在于将量子信息理论与高维凸几何联系起来，最终在Aubrun， Szarek和她通过渐近几何分析对“量子通道的可加性猜想”的分析中达到顶峰。PI将继续利用渐近几何分析的独特视角来解决量子信息理论中的问题。科学或工程问题的数学描述往往需要大量独立的数字，从而导致高维的几何空间。例如，如果您想指定房间中一个气体分子的位置，那么您需要使用三个数字报告该分子的前/后、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，因此，为了充分描述分子的当前状态，使我们能够从位置和速度来预测它的未来运动，我们总共需要六个单独的数字。如果你想跟踪房间里100个不同的空气分子，那么你将需要600个独立的数值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，科学家和数学家有时将这种现象称为“维度的诅咒”。然而，也有一些模式随着维度的增加而出现，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“维度的诅咒”，使其成为“维度的祝福”。研究这种高维现象，是这项资助的目的之一。

AbstractAward: DMS 1207917, Principal Investigator: Elisabeth M. WernerThese research projects are in asymptotic geometric analysis and affine convex geometry. A main emphasis of her research is on affine invariant and high dimensional objects and phenomena and on links with other areas of mathematics and mathematical sciences (probability, statistics, optimization, information theory and quantum information theory). Much of the research in convex geometry in recent years has been directed to the study of affine and high dimensional aspects of convex bodies. The PI and her collaborators carried out a systematic study of some of the most important affine invariant functionals on convex bodies, the p-affine surface areas. They obtained extensions of those invariants to all convex bodies and for all p and established their corresponding affine isoperimetric inequalities (which are stronger than their Euclidean counterparts). There are numerous applications of these works. We only mention the PI's work, with Reisner and Schuett, on Mahler's conjecture. While not providing the solution (yet), their result is the strongest indication to date that the minimum is indeed attained for polytopes. Very recent developments open totally new directions. For one, the PI found new affine invariants and proved that those and the p-affine surface areas are certain entropies from information theory. In another direction, the PI, Artstein, Klartag and Schuett showed that affine isoperimetric inequality for log concave functions corresponds to a reverse log Sobolev inequality for entropy. These directions will be explored further. Contributions are also her work linking quantum information theory and high dimensional convex geometry, culminating in Aubrun's, Szarek's and her analysis of the "Additivity conjecture for Quantum channels" via asymptotic geometric analysis. The PI will continue to exploit the unique perspective given by asymptotic geometric analysis for problems in quantum information theory.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, 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". To study such high dimensional phenomena, is one purpose of this grant.