Proposal for EPSRC postdoctoral fellowship in applied probability by Dr. Matthew I. Roberts

Matthew I. Roberts 博士关于 EPSRC 应用概率博士后奖学金的提案

基本信息

批准号：
EP/K007440/1
负责人：
Matthew Roberts
金额：
$ 27.64万
依托单位：
University of Bath
依托单位国家：
英国
项目类别：
Fellowship
财政年份：
2013
资助国家：
英国
起止时间：
2013 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FK007440%2F1
关键词：
Proposal EPSRC postdoctoral fellowship applied

项目摘要

Probability has always been a field of mathematics with many applications in the real world: gambling strategies, growth of animal populations, spread of disease, or the performance of financial markets. More recently, with increases in knowledge and in computing power, new areas of interest have appeared, many of which rely on structures that resemble trees or large networks. To name three specific examples, it is now feasible to study the evolution of the DNA of species; to have efficient methods of organising the large amounts of data on our computers; and to understand the large clusters of computers that make up the internet.Evolution of DNA and branching Brownian motionBrownian motion was described by the botanist Robert Brown as he watched particles of pollen moving in water. He noticed that small changes caused by water molecules hitting the particles caused a slow, macroscopic, random movement of the pollen grains.DNA strings are extremely complex. Even the simplest organisms can contain millions of molecules. Each time a cell divides it creates two copies of every bit of this data. Inevitably mistakes occur, but most of these mistakes have a tiny effect given the extra information built into the cell. Nonetheless these small fluctuations slowly precipitate to create large-scale changes which contribute to the evolution of the species. These random movements caused by tiny mistakes make Brownian motion a good model for this process.So the evolution of the DNA of one organism can be modelled using Brownian motion; but each organism also breeds, creating copies of its DNA that then independently mutate and evolve. This description leads us to consider a model called branching Brownian motion, which is a tree-like structure in which each branch moves in space according to a Brownian motion. Probabilists have extensively studied the overall spread of this process: in biological terms, how fast a species will evolve if left to its own devices. If a species does not evolve as fast as its environment is changing then it will quickly become extinct. We can then ask how long the species will survive for, and how fast the population will grow.Data structures and sorting algorithmsAs larger and larger files are required to store the enormous amounts of data on our computers, it is important for that data to be organised in such a way that it can be easily accessed. One such method is known to computer scientists as quicksort, and has been extensively studied.The process works by sorting the data into a tree-like structure, which can then be accessed at speed by making a relatively small number of checks at the branch points of the tree. Very fine detail is now known about the height of this tree, which corresponds to how many checks must be made to find the hardest-to-reach bits of data. However, almost nothing is known on how much of the data must be stored at the highest levels of the tree, which would tell us how often we have to access the furthest (and slowest) corners of our drives.The internet and large random networksThe internet is made up of huge numbers of computers (and web pages) linked together. This creates a complicated structure that is permanently changing. The connectivity properties of the network are very important for the speed of the internet: on the local scale this boils down to whether one computer can reach another, and how many links it takes to make that connection.The same ideas can be used to examine other related structures like social networking sites. There are a large number of points, connected to each other by links. As time progresses each of these links may appear or disappear. Very small alterations can cause the large-scale behaviour of the system to change suddenly, affecting the speed at which data can be shared.From bacteria to blue whales, the BBC Micro to broadband internet, probability theory provides tools for studying all of these structures.

概率一直是数学领域，在现实世界中有许多应用：赌博策略、动物种群的增长、疾病的传播或金融市场的表现。最近，随着知识和计算能力的增加，出现了新的感兴趣领域，其中许多领域依赖于类似于树或大型网络的结构。举三个具体例子，现在研究物种DNA的进化是可行的；拥有有效的方法来组织计算机上的大量数据； DNA 的进化和分支布朗运动植物学家罗伯特·布朗在观察水中移动的花粉颗粒时描述了布朗运动。他注意到水分子撞击颗粒所引起的微小变化会导致花粉粒发生缓慢、宏观、随机的运动。DNA 串极其复杂。即使是最简单的生物体也可以包含数百万个分子。每次细胞分裂时，它都会为该数据的每一位创建两个副本。错误不可避免地会发生，但考虑到细胞中内置的额外信息，大多数错误的影响都很微小。尽管如此，这些微小的波动慢慢地促成了大规模的变化，从而促进了物种的进化。这些由微小错误引起的随机运动使布朗运动成为这一过程的良好模型。因此，一个生物体 DNA 的进化可以使用布朗运动来建模；但每个生物体也会繁殖，产生其 DNA 副本，然后独立变异和进化。这种描述使我们考虑一种称为分支布朗运动的模型，它是一种树状结构，其中每个分支根据布朗运动在空间中移动。概率学家广泛研究了这一过程的整体传播：从生物学角度来说，如果任其发展，一个物种的进化速度有多快。如果一个物种的进化速度赶不上环境变化的速度，那么它很快就会灭绝。然后，我们可以询问该物种将生存多久，以及人口增长的速度。数据结构和排序算法由于需要越来越大的文件来在我们的计算机上存储大量数据，因此将这些数据存储在文件中非常重要。以易于访问的方式组织。其中一种方法被计算机科学家称为快速排序，并且已被广泛研究。该过程的工作原理是将数据排序成树状结构，然后可以通过在分支点进行相对少量的检查来快速访问该结构树的。现在我们已经知道了关于这棵树的高度的非常详细的信息，这对应于必须进行多少次检查才能找到最难访问的数据位。然而，几乎不知道有多少数据必须存储在树的最高层，这将告诉我们必须多久访问一次驱动器最远（也是最慢）的角落。互联网和大型随机网络互联网由大量连接在一起的计算机（和网页）组成。这创造了一个永久变化的复杂结构。网络的连接属性对于互联网的速度非常重要：在本地范围内，这归结为一台计算机是否可以到达另一台计算机，以及建立该连接需要多少个链接。可以使用相同的想法来检查其他相关结构，例如社交网站。有大量的点，通过链接相互连接。随着时间的推移，这些链接中的每一个都可能出现或消失。非常小的改变就可能导致系统的大规模行为突然改变，影响数据共享的速度。从细菌到蓝鲸，从 BBC Micro 到宽带互联网，概率论为研究所有这些结构提供了工具。