NSF/CBMS Research Conference in the Mathematical Sciences - "Ergodic Methods in the Theory of Fractals" - "6/18/11 - 06/23/11"

2011年6月18-23日，NSF/CBMS数学科学区域会议，分形理论中的遍历方法研究欧几里德空间的封闭紧致子集。通常的整数维度的概念被Hausdorff维度所取代。分形学在复杂动力学、数学物理、丢番图逼近等许多数学领域都有着重要的作用。遍历理论研究动力系统(具有同胚的紧致度量空间)在被允许长时间运行时的行为。在一定条件下(遍历性)，函数的时间平均收敛于它的空间平均。遍历理论被广泛应用于组合数论、数学物理、李群等数学领域。这次CBMS会议的主要焦点是发展遍历方法来解决几何、解析、概率和组合问题，唤起本科生和研究生对一些分析家和几何学家共同感兴趣的话题的关注，并提供这些领域的基本指导，重点是所涉及的基本遍历思想。将特别注意遍历理论在分形几何中的应用(涉及分形集的行为、具有均匀标度景物的度量、CP过程、Ramsey类型结果等问题)。对这门学科的兴趣是基于它的重要性，以便在解决与分形几何和加性组合学相关的不同问题方面取得进展。希勒尔·弗斯滕伯格因将概率和遍历方法应用于数学的其他领域而广为人知，包括数论和李群。他的演讲是真正的杰作，绝对独立，任何人-从专家到研究生和本科生-都可以接触到。

NSF/CBMS Regional Conference in the Mathematical Sciences, Ergodic Methods in the Theory of Fractals, June 18-23, 2011Fractal geometry studies closed compact subsets of a Euclidean space. The usual notion of a dimension which is an integer number is replaced by Hausdorff dimension. Fractals play an important role in complex dynamics, mathematical physics, Diophantine approximation and many other areas of mathematics. Ergodic theory studies the behavior of a dynamical system (a compact metric space with a homeomorphism) when it is allowed to run for a long time. Under certain conditions (ergodicity), the time average of a function converges to its space average. Ergodic theory is widely used in many areas of mathematics such as combinatorial number theory, mathematical physics, Lie groups and others.The main focus of this CBMS conference is to develop Ergodic methods to solve geometric, analytic, probabilistic and combinatorial problems, to call attention of undergraduate and graduate students to a number of topics of common interest to analysts and geometers, and to provide basic instruction in these areas with an emphasis on the fundamental ergodic ideas involved. Particular attention will be drawn to applications of Ergodic Theory to Fractal Geometry (questions involving behavior of fractal sets, measures with uniform scaling scenery, CP processes, Ramsey type results). The interest in the subject is based upon its importance in order to achieve progress in solving different problems related to the Fractal geometry and additive combinatorics.Hillel Furstenberg is widely known for his application of probabilistic and ergodic methods to other areas of mathematics, including number theory and Lie groups. His talks are true masterpieces that are absolutely self-contained and accessible to anyone - from specialists to graduate and undergraduate students.