EPSRC-Warwick Symposium on Geometry, Topology and Dynamics in Low Dimensions

在过去的五十年里，EPSRC-Warwick研讨会每年都会召开一次会议，以推进英国的数学状况。每次专题讨论会集中讨论一个特定主题；2017-2018年的主题是“低维的几何、拓扑和动力学”。我们将举办七场研讨会，来自英国和世界各地的专家将聚集在华威大学讨论他们的研究并提出新的研究方向。除此之外，我们还将接待一些杰出的国际长期访问者，他们也将访问英国其他大学。我们亦会举办多场面向公众的讲座。基本的几何概念，如长度、角度、面积和体积，大多数高中数学学生都很熟悉。在现代数学研究中，这些概念有了很大的发展，大大扩大了几何可以帮助解决的问题的范围。越来越奇特的几何结构出现了；例如，空间具有某种内在的“曲率”而不是完全平坦的想法是爱因斯坦宇宙观的关键，无限“自相似性”的概念产生了壮观的分形图像。奇异的几何结构通常出现在意想不到的情况下，这些情况与几何无关，它们反映了更深层次的底层结构的存在。拓扑学是二十世纪发展起来的一门学科。它是几何学的一个分支，但提供了不同的视角和不同的工具。它研究的基本对象是曲线、曲面和高维空间，关注的问题是何时一种可以连续变形成另一种。例如，在不切断绳子的情况下，绳子闭环中的结能否解开的问题是一个拓扑学问题。本次研讨会汇集了具有不同背景的顶尖研究人员，提供他们对最新发展的看法，推进他们的研究，并巩固英国作为数学研究中心在这个重要的和国际活跃领域。动力系统领域是研究随时间变化的系统的数学模型的长期行为，如物理系统或生物现象。有些数学模型很简单，例如描述台球桌上台球运动的模型，有些则极其复杂，例如太阳系卫星的运动或天气模式。几何、拓扑学和动力学之间存在着许多深刻而不明显的联系。数学家庞加莱（Poincare）在20世纪初首次提出了这些问题，此后一直受到人们的密切研究。本次研讨会将以这些联系为基础并进一步发展这些联系，以便在所有三个领域取得进展。研讨会将集中讨论低维现象，特别是发生在一、二、三维的现象。在过去十年中，在理解这些现象方面取得了惊人的进展。例如，在数学中，庞加莱提出的一个关于三维物体拓扑结构的重要问题直到最近才得到解决。在物理学中，弦理论是一种理解物理基本定律的新方法，在弦理论的背景下，人们提出了关于表面（和其他空间）的迷人的新数学猜想。本次研讨会汇集了具有不同背景的顶尖研究人员，提供他们对最新发展的看法，推进他们的研究，并巩固英国作为数学研究中心在这个重要的和国际活跃领域。

The EPSRC-Warwick Symposium has met every year for the past fifty years to advance the state of mathematics in the UK. Each symposium concentrates on a particular theme; in 2017-2018 the topic will be "Geometry, topology and dynamics in low dimensions". We will hold seven workshops where experts from the UK and around the world will gather at the University of Warwick to discuss their researches and to propose new research directions. In addition to this, we will host a number of distinguished international long-term visitors who will also visit other UK universities. We will also organise a number of lectures aimed at the public.Basic geometric concepts such as length, angle, area and volume are familiar to most from high school mathematics. In modern research mathematics these notions have evolved a great deal, considerably widening the range of problems which geometry can help to solve. Increasingly exotic geometric structures have appeared; for example, the idea that space has some sort of intrinsic "curvature" instead of being totally flat is key to Einstein's view of the universe, and the notion of infinite "self-similarity" gives rise to spectacular fractal images. Exotic geometric structures often arise in unexpected situations which do not immediately appear to be related to geometry, and they reflect the presence of deeper underlying structures.Topology is a subject developed in the twentieth century. It is an offshoot of geometry, but offers different perspectives and different tools. The basic objects it studies are curves, surfaces and higher dimensional spaces focusing on the question of when one such can be continuously deformed into another. For example, the question of whether a knot in a closed loop of string can be untied without cutting the string is a topological question.This symposium brings together top researchers with diverse backgrounds to offer their perspectives on recent developments, advance their research, and consolidate the UK as a hub for mathematical research in this important and internationally active field.The field of dynamical systems is the study of the long-term behaviour of mathematical models of systems which change over time, such as physical systems or biological phenomena. Some of the mathematical models are simple, such as that which describes the motion of a billiard ball on a billiard table, and some are extremely complex, such as the motion of satellites in the solar system or the weather patterns.There are many deep, non-obvious links between the fields of geometry, topology and dynamics. These were first brought to light by the mathematician Poincare in the early twentieth century and have been studied intensely ever since. This symposium will build on and further develop these connections in order to make advances in all three areas. The symposium will focus on low-dimensional phenomena, especially those which occur in dimension one, two and three. There has been spectacular progress in the last decade on understanding such phenomena. For example, in mathematics, an important question raised by Poincare about the topology of 3 dimensional objects has only recently been resolved. In physics fascinating new mathematical conjectures have been raised about surfaces (and other spaces) in the context of string theory which is a new approach to understanding basic laws of physics.This symposium brings together top researchers with diverse backgrounds to offer their perspectives on recent developments, advance their research, and consolidate the UK as a hub for mathematical research in this important and internationally active field.