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

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.

The EPSRC-Warwick Symposia have a continuous 50-year tradition of bringing together internationally leading researchers around a particular theme in order to enhance the ability of UK universities to generate top quality research in the mathematical sciences. The vital importance of mathematical research to the UK economy is clearly demonstrated in the Deloitte report commissioned by EPSRC in 2012. This concluded that, in 2010, 10% of UK jobs and 16% of UK General Value Added was directly attributable to mathematical science research, and a much greater proportion was indirectly attributable to or induced by such research. The symposia therefore play an important role in maintaining the UK at the forefront of key technological developments.

Since our symposium is based around a theoretical subject, the primary impact will be to researchers in mathematics. With history as our guide, we additionally anticipate that it will have broad long-term impact in other disciplines.

The 2017-18 EPSRC-Warwick Symposium has chosen the intradisciplinary theme of "Geometry, topology and dynamics in low dimensions''. As explained elsewhere in the proposal, we believe this theme has great potential to bring together a number of related topics, each currently very active, and thereby generate new research and open up new avenues for exploration. We have chosen seven interrelated workshop topics to maximise this potential.

The symposium will impact directly on the mathematical community in various ways. It will:

(1) Strengthen the UK knowledge base in these areas.

(2) Directly generate new research in and around these topics.

(3) Bring this activity to the attention of workers in related areas of mathematics, thereby stimulating broader interactions.

(4) Assist early-career UK researchers in acquiring the knowledge and skills to ensure that these strengths will maintained and enhanced well into the future.

(5) Bring this research to the attention of a general public, and help to encourage enthusiastic young people into the mathematical sciences.