Mathematical analysis

Quantifying change, with a key role played by fundamental notions of continuity and approximation. This research area includes, for example, Fourier and harmonic analysis, operator theory, ordinary and partial differential equations (PDEs), probability theory, stochastic analysis and applications of analysis. Mathematical Analysis has links with all other areas of mathematics (pure and applied).

The UK is strong across all areas of Mathematical Analysis and we aim to maintain its world-leading position. This strategy recognises the research area’s many intradisciplinary links to all fields of mathematics and strong links to other disciplines, which will help it contribute strongly to the EPSRC Outcomes. It also considers how best to support the ‘people pipeline’, given the major investment in relevant Centres for Doctoral Training (CDTs) over the last Delivery Plan.

Over the course of the current Delivery Plan, we aim to have:

  • Maintained a research area of high-quality research across the breadth of Mathematical Analysis to sustain support for core areas and drive developments in key challenges
  • Encouraged novel interactions between Mathematical Analysis and application areas, particularly research relating strongly to EPSRC Outcomes. There are many opportunities to engage with research challenges related to EPSRC Ambitions, as a result of this area’s underpinning nature and links to all areas of mathematics (e.g. Algebra, Geometry, Mathematical Biology, Mathematical Physics, Continuum Mechanics, Non-Linear Systems, Numerical Modelling, Statistics and Applied Probability, and Data Science)
  • Increased support for early-career analysts and so build capacity across the breadth of the area, to respond to growing demand for broad Mathematical Analysis skills. This recognises the relatively large number of well-trained analysts in the UK delivered by investment in three CDT's. We wish to maintain this momentum and build on the greater connectedness of analysis, both in the UK and internationally

We aim to maintain this area as a proportion of the EPSRC portfolio.

Highlights:

Traditionally strong in Mathematical Analysis, the UK is at the leading edge of research in dynamical systems, ergodic theory, harmonic analysis, PDEs and their applications, operator algebras and operator theory, and stochastic analysis. Mathematical Analysis is of high importance to the mathematical sciences as a whole, linking many areas of pure mathematics to applied areas. For example, there are strong interactions with algebra, applied mathematics, geometry, materials science, physics, and statistics and applied probability. Over the last five to ten years, the UK has seen a blurring of boundaries as Mathematical Analysis finds new applications and the area is expected to keep growing. World-leading groups are located across the UK at many big centres, as well as several smaller institutions (Evidence source 1,2,3,4).

The high quality of Mathematical Analysis in this country is demonstrated, for example, by a plethora of prizes and the ability to attract international leaders in the field to the UK. The Research Excellence Framework (REF) 2014 exercise noted, since the Research Assessment Exercise (RAE) 2008, an increase in quantity and quality of research in theoretical partial differential equations and stochastic analysis (some of which was world-leading at the highest level), (Evidence source 1,4,5).

The connection of modern statistics to mathematics remains strong and is growing. For example, both theory and methodology of functional data analysis rely on ideas and results from Mathematical Analysis, as noted by the International Review of Mathematical Sciences (IRM) 2010 and by REF expert panel B. Mathematical Analysis has potential to contribute strongly to developments in data science over the Delivery Plan period. Connections with developments in data science should be encouraged (Evidence source 1-7).

As of April 2016, over half the EPSRC portfolio in Mathematical Analysis was in training, largely in the form of three CDTs. These now generate relatively large numbers of well-trained analysts. It is important that the best analysts in the UK are supported at the next stage of their career.

This research area can potentially influence all EPSRC Outcomes over the medium to long term, and contribute most strongly to the following Ambitions:

C1: Enable a competitive, data-driven economy

Harmonic analysis and PDEs are the starting point for many applications in imaging and sparse recovery. Functional analysis is fundamental to machine learning. Stochastic analysis underpins high-dimensional and nonparametric statistics. 

R5: Build new tools to adapt to and mitigate climate change

Climate modelling is based on results from dynamical systems out of equilibrium, and a wealth of analysis insights.

H3: Optimise diagnosis and treatment

Mathematical Analysis will improve/optimise information captured in image analysis and aid refinement of diagnostic tools by helping devise better models (e.g. blood flow and heart modelling is often based on PDEs of fluid dynamics). Numerical modelling of biological systems is based on the continuum PDE representation of the relevant physical phenomena.

H4: Develop future therapeutic technologies

Mathematical Analysis is expected to contribute through its role in, and development of, image analysis. Techniques from Mathematical Analysis can extract information from medical scanners.

Research area connections

This diagram shows the top 10 connections between Research Areas within the EPSRC research portfolio. The depth of the segment relates to value of grants and the width of the segment relates to the number of grants shared by those two Research Areas. Please click to see the related Research Area rationale.

Maintain

We aim to maintain this area as a proportion of the EPSRC portfolio.

Visualising our Portfolio (VoP)
Visualising our portfolio (VoP) is a tool for users to visually interact with the EPSRC portfolio and data relationships.

EPSRC support by research area in mathematical analysis (GoW)
Search EPSRC's research and training grants.

Contact Details

In the following table, contact information relevant to the page. The first column is for visual reference only. Data is in the right column.

Name: Katharine Moore
Job title: Manager
Department: Mathematical Sciences
Organisation: EPSRC
Telephone: 01793 444246