In April 2023, the UK government updated the BEIS guidelines (now managed by DSIT – the Department for Science, Innovation and Technology) to formally include pure mathematics as a qualifying activity under the UK R&D Tax Credits scheme. While this is a significant and welcome shift, there’s still a lot of confusion in the industry about what this actually means in practice - especially for businesses not traditionally seen as “maths-heavy.”
That’s why we’ve broken down what qualifies asR&D in the field of pure mathematics, why it matters, and how companies in life sciences, technology, finance, and beyond can benefit - if they know what to look for.
What Qualifies as R&D in Pure Mathematics?
To qualify for R&D tax relief, work must aim to achieve an advance in science or technology - and in the case of pure mathematics, that means producing new mathematical knowledge, frameworks, or methods.
Critically, your work must involve scientific or technological uncertainty - a problem whose solution isn’t obvious, even to professionals in the field. It also needs to be part of a structured project, not just speculative exploration, and must aim to benefit the wider mathematical community, not just your internal operations.
Qualifying work must also be part of a structured, goal-driven project rather than casual exploration. The innovation should benefit the broader field of mathematics, not just solve an internal business need. Creating novel mathematical models, structures, or proofs counts; simply adapting or applying established methods typically does not. The key is demonstrating that the work contributes something meaningfully new to the discipline.
So what does this mean in lay-man terms?
Let’s say you’re working on a new algorithm to model biological systems more efficiently. If this involves developing a new class of equations or proving convergence properties that don’t currently exist, that’s potentially R&D. Or maybe your team is trying to prove a mathematical conjecture that has direct application in quantum computing or climate models - again, if you're breaking new ground, this could qualify.
However, if you’re simply applying existing statistical tools or running known calculations in a new business setting, that doesn’t count. The distinction lies in whether you're creating new knowledge or just using what’s already known.
Examples of R&D in Pure Mathematics for UK R&D Tax Credits
Here is a breakdown by industry of a few examples of R&D in the field of mathematics that can potentially qualify for R&D Tax credits in the UK
Life Sciences: Modelling Pest Population Dynamics
Developing novel mathematical models to simulate and predict pest outbreaks under varying environmental and agricultural conditions, especially when existing models are too static or linear.
Example: Creating a system of differential equations to predict the lifecycle and spread of wireworms in UK potato crops, incorporating dynamic variables like rainfall, soil pH, and temperature fluctuation to improve intervention planning and reduce crop loss.
Finance: High-Dimensional Risk Modelling
Constructing advanced probabilistic frameworks to model correlated risks in large, complex financial portfolios, where traditional models oversimplify interdependencies.
Example: Developing a copula-based stochastic model for pricing catastrophe-linked insurance derivatives, accounting for regional climate volatility and loss correlation across geographies, something existing actuarial models could not robustly quantify.
Technology: Proving Algorithmic Convergence
Establishing new mathematical proofs to demonstrate convergence or performance guarantees for machine learning algorithms, especially where current literature lacks rigour or scalability.
Example: Proving the convergence of a custom gradient descent method used in a deep learning model for autonomous drone navigation, where conventional proof techniques were insufficient due to the model's non-convex, adaptive architecture.
Cybersecurity – Designing Cryptographic Primitives
Developing new mathematical structures or assumptions to support cryptographic protocols that cannot be reduced to existing schemes or proven secure using current techniques.
Example: Creating a lattice-based key exchange protocol resistant to quantum attacks, underpinned by a novel hardness assumption and supported by a formal reduction to a newly defined class of mathematical problems, extending beyond current post-quantum cryptographic standards.
Environmental Science – Atmospheric Pollution Modelling
Building mathematical models to capture pollutant dispersion across varying terrains and climate patterns, especially where standard linear diffusion models underperform.
Example: Developing a non-linear partial differential equation (PDE) system to simulate ammonia dispersion from agricultural sites in upland areas, accounting for complex topography, wind turbulence, and local humidity, variables neglected in standard Gaussian plume models.
Oil & Gas – Reservoir Simulation under Uncertainty
Developing new mathematical techniques to model fluid flow in underground reservoirs where geological uncertainty and sparse data make existing models unreliable.
Example: Constructing a probabilistic inverse modelling framework using stochastic differential equations to estimate oil recovery rates in fractured shale formations, incorporating sparse seismic data and complex rock-fluid interactions that standard deterministic models cannot resolve.
Examples that Do Not Qualify
To help clarify, here are activities that do not meet HMRC’s threshold for R&D in pure mathematics:
- Applying Standard Methods in New Contexts
Using known statistical tests (e.g. t-tests, regression) in a new industry sector, even if the application is complex. - Parameter Tuning of Existing Algorithms
Optimising variables in a pre-built algorithm using trial-and-error, without creating new mathematical insight. - Mathematical Support for Commercial Planning
Creating financial models, business forecasts, or pricing spreadsheets, even if they are complex, f they rely on existing mathematical knowledge. - Automating Reporting Using Mathematical Tools
Streamlining analytics or data visualisation processes using off-the-shelf software (e.g. Excel, Power BI) or pre-built libraries.
Why This Matters for R&D Tax Relief Claims
The formal inclusion of pure mathematics in the DSIT guidelines opens up a new avenue for innovative companies to benefit from R&D tax credits, but only if the claims are made carefully and correctly.
Mathematics-focused work is often embedded within wider scientific or technical projects, making it easy to overlook. Even when identified, claims in this area are sometimes dismissed or poorly documented because the work doesn’t appear "experimental" in the traditional sense. That’s why it’s crucial to clearly articulate:
- The uncertainty faced
- The structured approach taken
- The mathematical advance achieved
- How the work contributes to the field beyond your business
Need Advice on Complex or Unusual R&D Claims?
At ResearchQX We specialise in helping innovative companies uncover hidden R&D in areas like mathematics, life sciences, and data science, and turn that into value through HMRC-compliant claims.
Book a free consultation today or get in touch with our team of R&D tax specialists.
