The Probabilistic Method

9970822调查员将继续研究概率方法，这是已故的保罗·鄂尔多斯留下的遗产，目前仍处于非常活跃的阶段。当人们想要证明具有某些性质的对象的存在时，最初的、仍然是基本的应用是离散数学。粗略地给出了随机对象的适当定义，并证明了随机对象以正概率具有期望的性质。这种方法论与理论计算机科学中随机性的使用有很强的交叉，相互作用是双向的。如果随机算法可以被证明有正的成功机会，那么成功的存在是肯定的。此外，随机算法的输出本身也非常有趣。随着随机物体的演化，存在着特定的临界区，称为阈值函数，在这些区域中，事件发生的概率从接近零迅速移动到接近一。使用数理逻辑的方法，研究者试图描述用给定逻辑语言表示的所有事件的可能阈值函数。随机性现在被认为在许多计算机算法中扮演着重要的角色。研究人员特别研究打包算法。如何处理一组部分重叠的请求--带宽、起飞时隙等--以满足最大数量的请求？使用随机贪婪算法，请求以随机洗牌的顺序进行处理，如果与之前的批准不冲突，则每个请求都会被批准。有时，这种自然且易于实现的算法可以给出近乎最佳的结果，尽管对它的分析被证明是特别微妙的。第二个区域虽然相关，但仍处于渗流效应中。大型系统(通常)是非线性的--它们在令人惊讶的短时间内经历了一个相变(液体到气体，低犯罪率到高犯罪率)，这种转变既是定性的，也是定量的。在适当的尺度下，调查人员应展开这一过渡，以便更好地理解这一现象。

9970822The investigator will continue his study of The Probabilistic Method, a legacy of the late Paul Erdos that remains in a very active stage. The original, and still basic, applications are to discrete mathematics when one wishes to prove the existence of an object having certain properties. Very roughly, a random object is appropriately defined and it is shown that the random object has the desired properties with positive probability. The methodology strongly intersects with the use of randomness in Theoretical Computer Science, the interaction going both ways. If a random algorithm can be proven to have positive chance of success then the existence of a success is guaranteed. Further, the output of random algorithms is very much of interest for its own sake. As the random object evolves there are certain critical regions, dubbed threshold functions, where the probability of events move rapidly from near zero to near one. Using methods from mathematical logic the investigator attempts to describe the possible threshold functions for all events expressible in a given logical language.Randomness is now recognized to play an important role in many computer algorithms. The investigators particularly study packing algorithms. How can a set of partially overlapping requests - for bandwidth, takeoff slots or whatever - be handled to satisfy the maximal number of requests? With the random greedy algorithm the requests are taken in randomly shuffled order and then each is approved if not conflicting with previous approvals. Oftimes this natural and easily implemented algorithm can be shown to give near optimal results though analysis of it has proved to be particularly subtle. A second, though related, area is in percolation effects. Large systems are (often) nonlinear - they undergo a phase transition (liquid to gas, low crime to high crime) which is qualitative as well as quantitative in a surprisingly short period of time. With the appropriate scaling the investigator shall spread out this transition so as better to understand the phenomenon.