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)模拟结果的可靠性如何？ 我将研究一类称为马尔可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的计算挑战性问题（例如，在被随机噪声模糊的数字图像中，可能的真实图像是什么？）等领域 用户必须决定运行算法的时间，以便结果不会受到初始条件的影响。 这个决定并不总是明确的，很容易被输出误导。 唯一的保证来自理论分析。为此，我的目标是证明在一些简化模型中使初始偏差的影响任意小所需时间的上限（形式上，我将马尔可夫链的收敛速度限制在均衡），我希望这可以作为实践中出现的更复杂情况的指导方针。 (2)接下来，这是纯组合学中的一个概率问题。 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）。 我们说一个排列“包含模式4231”，如果你能在排列中找到四个与4231相同相对顺序的数字（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）。 例如，在一个示例中，86425713包含模式4231，因为数字8573是按此顺序出现的。 考虑不包含模式4231的大小为N的排列的集合。 当N很大时，数学家们试图（但成功有限）确定这个集合大约有多大。 我观察到，从这个集合中随机选择的元素可能具有一些惊人的结构特性，我的研究目标是了解为什么会发生这种情况。 (3)最后，我将研究免疫系统的数学建模方面，使用包含随机性的模型，以考虑结果的不确定性。一个问题涉及突变，这是本质上随机的事件。当与流感病毒等病原体作战时，我们的免疫系统使用T细胞，T细胞旨在追捕病毒的某些可识别标志。但是如果病原体的后代在正确的位置发生突变，它们就可以逃脱T细胞。我想估算一下T细胞能够消灭所有病原体的概率，然后一些后代积累了足够的突变，能够逃避所有的T细胞。第二个问题是，我们的“先天”免疫系统是否可以迅速控制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.