Workshop on Asymptotic Geometry in Paris

奖项：dms -0535305首席研究员：Elisabeth werner该奖项资助美国参与者参加2006年7月在法国巴黎Henri Poincare研究所举行的“渐近分析与应用研讨会”。该研讨会是“大维度现象”特别研究项目的一个组成部分，关注有限维物体在维度无界增长时的性质。适合这一议程的特定主题是测量传输技术，测量集中，凸体的定量测量，等周不等式和大偏差不等式。在这些研究中出现的一些重要特征是阈值或相变效应，类似于在统计物理学或渐近组合学中看到的那些，而且这项工作似乎也与计算复杂性的问题有关。科学或工程问题的数学描述往往需要大量独立的数字，从而导致高维几何空间。例如，如果你想指定一个气体分子在房间里的位置，那么你需要报告分子的正面/背面，侧面和上下位置，使用三个数字。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值。随着这些维度的增加，采样和计算的难度也迅速上升，科学家和数学家有时将这种现象称为“维度诅咒”。然而，随着维度的增加，也有一些模式出现，上述资助美国前往国际研讨会的资助将研究最近发现的一些模式。

AbstractAward: DMS-0535305Principal Investigator: Elisabeth WernerThis award supports travel of US participants to a "Workshop onAsymptotic Analysis and Applications" in July 2006 at theInstitut Henri Poincare in Paris, France. The workshop is acomponent of the special trimester research program there on"Phenomena in Large Dimensions" and concerns properties offamilies of finite-dimensional objects as the dimension growswithout bound. Particular topics that fit this agenda aremeasure transport techniques, concentration of measure,quantitative measures of convex bodies, isoperimetricinequalities, and large deviation inequalities. Some of theimportant features emerging in these studies are threshold orphase transition effects, similar to those seen in statisticalphysics or asymptotic combinatorics, and this work also seems tohave connections to problems in computational complexity.A mathematical description of a scientific or engineeringquestion often requires lots of independent numbers, leading to ageometric space of high dimension. For example, if you want tospecify the location of one gas molecule in a room then you needto report the front/back, side-to-side, and up/down locations ofthe molecule, using three numbers. The direction and speed of themolecule's motion takes another three numbers, and so to describeenough of the molecule's current state to allow us to predict itsfuture motion from position and velocity we would need sixseparate numbers in all. If you want to track 100 distinctmolecules of the air in the room then you will need 600independent numerical coordinates to collect all of the relevantmeasurements. As these dimensions increase then the difficultyof sampling and computation go up rapidly, a phenomenonscientists and mathematicians sometimes call "the curse ofdimensionality." However, there are also patterns that emerge asdimension increases, and this grant for support of US travel tothe international workshop described above will study some ofthese patterns that are recent discoveries.