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

• 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

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，其中许多依赖于类似于树木或大型网络的结构。为了列举三个特定的例子，现在可以研究物种DNA的演变。具有有效的方法来组织计算机上的大量数据；为了理解构成互联网的大量计算机簇。植物学家罗伯特·布朗（Robert Brown）看着花粉颗粒在水中移动时，植物学家罗伯特·布朗（Robert Brown）描述了DNA和分支布朗运动的动作。他注意到，击中颗粒的水分子引起的小变化导致花粉晶粒的缓慢，宏观的随机运动。DNA弦非常复杂。即使是最简单的生物也可以包含数百万个分子。每当一个单元格划分时，它都会创建两个数据的每个副本。不可避免地会出现错误，但是鉴于单元格中内置的额外信息，这些错误中的大多数都具有很小的效果。尽管如此，这些小小的波动慢慢地沉淀出来，产生了大规模的变化，这有助于物种的演变。这些由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，可以使用布朗运动对一个生物体的DNA的演变进行建模。但是每个生物也会繁殖，创建其DNA的副本，然后独立变异和进化。该描述使我们考虑了一个称为分支布朗尼运动的模型，布朗尼运动是一种类似树状的结构，每个分支都根据布朗运动在太空中移动。概率主义者已经广泛研究了这一过程的总体传播：从生物学的角度来看，如果将物种留给自己的设备，则物种会发展的速度。如果一个物种的发展速度不如其环境变化，那么它将迅速灭绝。然后，我们可以询问该物种将生存多长时间，以及人口的生长速度。数据结构和分类算法更大，并且需要更大的文件才能将大量数据存储在我们的计算机上，对于可以轻松访问的方式来组织数据非常重要。计算机科学家已知一种这样的方法是QuickSort，并且已经进行了广泛的研究。该过程通过将数据分类为类似树状的结构来起作用，然后可以通过在树的分支点上进行相对较少的检查来以速度访问该过程。现在已经知道这棵树的高度非常细节，这对应于必须进行多少个检查才能找到最难到达的数据位。但是，几乎一无所知。这会产生一个正在永久变化的复杂结构。网络的连通性属性对于互联网的速度非常重要：在本地规模上，这归结为一台计算机是否可以到达另一台计算机，以及建立该连接所需的链接。相同的想法可用于检查其他相关结构，例如社交网站。有很多点，通过链接相互连接。随着时间的流逝，这些链接中的每一个都可能出现或消失。很小的变化会导致系统的大规模行为突然改变，从而影响数据可以共享的速度。从细菌到蓝鲸，BBC微型到宽带互联网，概率理论为研究所有这些结构提供了工具。

项目成果

Growth rates of the population in a branching Brownian motion with an inhomogeneous breeding potential

• DOI: 10.1016/j.spa.2014.12.008
• 发表时间: 2012-03
• 期刊: Stochastic Processes and their Applications
• 影响因子: 1.4
• 作者: J. Berestycki;'Eric Brunet;J. Harris;S. Harris;Matthew I. Roberts

The number of ends of critical branching random walks

• 发表时间: 2014-01
• 期刊: arXiv: Probability
• 作者: Elisabetta Candellero;Matthew I. Roberts

Mixing Time Bounds via Bottleneck Sequences

通过瓶颈序列混合时间限制

• DOI: 10.1007/s10955-017-1917-5
• 发表时间: 2017
• 期刊: Journal of Statistical Physics
• 影响因子: 1.6
• 作者: Addario-Berry L

The many-to-few lemma and multiple spines

• DOI: 10.1214/15-aihp714
• 发表时间: 2011-06
• 期刊: Annales De L Institut Henri Poincare-probabilites Et Statistiques
• 影响因子: 1.5
• 作者: S. Harris;Matthew I. Roberts

Vanishing Corrections for the Position in a Linear Model of FKPP Fronts

FKPP 锋面线性模型中位置的消失修正

• DOI: 10.1007/s00220-016-2790-9
• 发表时间: 2016
• 期刊: Communications in Mathematical Physics
• 影响因子: 2.4
• 作者: Berestycki J