Stochastic Systems: Theory and Models

我的研究方向是概率论及其应用。我将研究三个不同的主题，从纯粹的到应用的。******(1)仿真结果的可靠性如何？我将研究一类被称为马尔科夫链蒙特卡罗方法的蒙特卡罗模拟算法的效率。这些方法已被用于解决统计分析中具有计算挑战性的问题（例如，在被随机噪声模糊的数字图像中，什么是可能的真实图像？）和其他领域。用户必须决定运行算法的时间，这样结果才不会受到初始条件的影响。这个决定并不总是明确的，而且很容易被输出误导。唯一的保证来自理论分析。为此，我的目标是在一些简化模型中证明使初始偏差的影响任意小所需时间的上界（形式上，我将马尔可夫链的收敛速度限制在平衡状态），我希望这可以作为实践中出现的更复杂情况的指导原则。*******(2)接下来，这里有一个纯组合学的问题，从概率的角度来看。大小为N的排列是从1到N的数字的排列（例如86425713是大小为8的排列）。如果你能在这个排列中找到四个与4231的相对顺序相同的数字（即第一个是最大的，第二个是第二小的，第三个是第三小的，第四个是最小的），我们就说这个排列“包含模式4231”。例如，86425713包含模式4231，因为数字8573以这个顺序出现。考虑一组大小为N的排列，它们不包含模式4231。当N很大时，数学家们尝试（成功有限）确定这个集合的大致大小。我观察到，这个集合中随机选择的元素可能具有一些惊人的结构属性，我的研究目标是理解为什么会发生这种情况。最后，我将研究免疫系统数学建模的各个方面，使用包含随机性的模型来解释结果的不确定性。其中一个问题与突变有关，突变本质上是随机事件。当与流感病毒等病原体作斗争时，我们的免疫系统使用T细胞，它们被设计用来寻找病毒的某些可识别的迹象。但是，如果病原体的后代在适当的地方发生突变，它们可以逃离T细胞。我想估算一下T细胞在后代积累足够的突变从而能够避开所有T细胞之前消灭所有病原体的概率。第二个问题是，我们的“先天”免疫系统是否有可能迅速控制由蚊子叮咬引起的西尼罗河病毒感染，从而不需要T细胞

My research program is focused on probability theory and its applications.  I will work on three different topics, ranging from the pure to the applied.******(1) How reliable are simulation results?  I shall study the efficiency of a class of Monte Carlo simulation algorithms known as Markov chain Monte Carlo methods.  These methods have been used to tackle computationally challenging problems in statistical analysis (for example, in a digital picture that has been blurred by random noise, what is the likely true image?) and other fields.  The user must decide how long to run the algorithm so that the results are not biased by the initial conditions.  This decision is not always clear, and it is easy to get misled by the output.  The only guarantees come with theoretical analysis. To that end, I aim to prove upper bounds on the time required to make the effect of the initial bias arbitrarily small (formally, I bound the rates of convergence of the Markov chains to equilibrium) in some simplified models, which I hope can serve as guidelines for the more complex situations that arise in practice.*******(2) Next, here is problem in pure combinatorics with a probabilistic viewpoint.  A permutation of size N is an arrangement of the numbers from 1 to N (e.g. 86425713 is a permutation of size 8).  We say that a permutation "contains the pattern 4231" if you can find four numbers in the permutation that occur in the same relative order as 4231 (i.e. the first one is largest, the second one is second smallest, the third one is third smallest, and the fourth one is smallest).   E.g., 86425713 contains the pattern 4231 because the numbers 8573 appear in this order.  Consider the set of permutations of size N that do not contain the pattern 4231.  When N is large, mathematicians have tried (with limited success) to determine approximately how big this set is.  I have observed that randomly chosen elements of this set are likely to have some striking structural properties, and a goal of my research is to understand why this happens.***(3) Finally, I will work on aspects of mathematical modelling of the immune system using models that incorporate randomness, to account for uncertainties in outcomes. One question concerns mutations, which are intrinsically random events. When fighting a pathogen such as a flu virus, our immune system uses T cells, which are designed to hunt down certain identifiable signs of the virus. But the pathogen's offspring can escape a T cell if they have a mutation in the right place. I would like to estimate the probability that the T cells can destroy all of the pathogen before some offspring accumulates enough mutations to be able to evade all of the T cells. A second question asks for the probability that our "innate" immune system can control an infection of West Nile virus from a mosquito bite rapidly enough that the T cells are not needed.**