Convexity and Applications

摘要奖：DMS 1504701，主要研究者：Elisabeth M.沃纳这些研究项目是在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。这导致她的研究在物理学，生物学和医学，计算机科学，优化和经济学以及材料科学等不同领域的应用：事实上，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，如果想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，因此要描述足够的分子当前状态，以便根据位置和速度预测其未来运动，总共需要六个独立的数字。 如果你想追踪房间里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结果。 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”。 这项拨款的目的之一，是研究这些高维现象。建议项目的重要特色，一方面是强调仿射不变量和高维物体和现象，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）相联系。将研究的主题包括：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，凸性和几何方面的概率方法与优化，Banach空间和量子信息理论。该研究项目的重点是仿射不变量及其相关不等式的发展。 主要研究者和她的合作者开始系统地研究与凸体和对数凹函数及其相应的不等式相关的（仿射不变）泛函。其中最重要的这样的泛函凸体是仿射表面积和p-仿射表面积。 PI和她的合作者最近将这些扩展到对数凹函数。 与仿射表面积相关的仿射等周不等式比它们的欧几里得亲戚更强大，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）。后者与马勒猜想有关，马勒猜想在三维和更高的维度上仍然是开放的。 PI最近观察到的仿射表面积和Renyi熵之间的连接允许信息理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.