Workshop on Asymptotic Geometry in Paris

AbstractAward：DMS-0535305首席研究员：Elisabeth Werner该奖项支持美国参与者参加2006年7月在法国巴黎亨利庞加莱研究所举行的“渐近分析和应用研讨会”。 该研讨会是一个组成部分的特殊学期研究计划有“现象在大尺寸”和关注的性质families有限维物体的尺寸增长没有约束。 适合这一议程的特定主题是测量运输技术，浓度的措施，凸体的定量措施，isoperimetricinequalities和大偏差不平等。 在这些研究中出现的一些重要特征是阈值或相变效应，类似于在物理学或渐近组合学中看到的那些，并且这项工作似乎也与计算复杂性问题有关。科学或工程问题的数学描述通常需要大量的独立数，导致高维的几何空间。 例如，如果你想指定一个气体分子在房间里的位置，那么你需要用三个数字来报告分子的前/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，使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，采样和计算的难度也会迅速增加，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，而这项支持美国前往上述国际研讨会的赠款将研究其中一些最近发现的模式。

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.