Tag Archives: Mathematics

Research with Impact: Economics, Mathematics and Statistics

Through the REF 2021 rankings, we were pleased to see our impact results in Economics, Maths, and Statistics demonstrate a significant improvement from 2014, with 33% of our impact work ranked world-leading, 50% internationally excellent, and the remaining 17% internationally recognised. Discover our research case studies below. Visit the REF website for full details.

Demographic structure and economic trends: planning for Europe’s financial future

According to a 2019 OECD report, Fiscal Challenges and Inclusive Growth in Ageing Societies, ‘The number of people over 65 for each working-age person will at least double in most G20 countries by 2060′. This is important because age groups differ in their savings behaviour, productivity levels, labour input, contribution to innovation, and investment opportunities. Research by Professor Yunu Aksoy and Professor Ron Smith offered new insight into the specific nature of these differences as well as providing an innovative theoretical model for predicting future trends. Taken up by central banks around the world, this work has contributed to an increased focus on demographics amongst the global central banking community and influenced fiscal policy decisions in a number of countries.

Notably, a 2017 secondment to the Bank of Spain allowed Professor Aksoy to develop his research, to publish alongside Bank economist Henrique Basso, and to help answer the question of ‘how to adapt fiscal and social policies to demographic changes’, identified by the Bank’s head of research as ‘one of the most important issues’ that the Bank is currently facing. Basso went on to sit on a European Central Bank taskforce charged with investigating the future of pension schemes in the EU (a multi-billion-Euro question), which drew directly on Professor Aksoy and Professor Smith’s work to inform its research and recommendations. The work has also been invoked in German debates around immigration (where a recent policy change aimed to facilitate the entry of young, skilled workers to reinvigorate the country’s economy).

Making the right decisions for patients: competition, choice and inequality in publicly-funded healthcare

Successive UK governments have emphasised the importance of patient choice as a means for service users to voice and realise their preferences, and as a way to encourage competition in markets for publicly-funded healthcare. Dr Walter Beckert’s work on patient choice has challenged the assumption that increasing competition helps to improve care provision, showing that competition can reinforce existing inequalities between demographic groups and that patients often turn for guidance to primary healthcare practitioners who have competing priorities of their own.

Dr Beckert is an academic panel member for the Competition and Markets Authority (CMA) and his research has shaped the methodology used by the CMA to analyse the impact of hospital mergers, therefore influencing each of the eight hospital merger decisions made by that body since its first in 2013 (determining the allocation of at least £560 million in public money). More broadly, Dr Beckert’s work with the Health Foundation, a campaigning charity, has helped to change opinion within the sector as to the value of competition and has contributed to a strategic shift away from competition and towards a more collaborative approach to service delivery, as realised in the 2019 NHS Long-Term Plan. Health Foundation CEO Jennifer Dixon points out that ‘the reach of Walter’s work therefore potentially affects all the population in England – circa 55 million’.

New strategies for portfolio management: applying new estimates of equity yields and equity duration

Accurate estimates of expected rates of returns and investment time horizons are crucial for investment managers when it comes to assessing the risk and return characteristics of equity portfolios. Collaborative research originated by Birkbeck’s Dr David Schröder and continued in partnership with Florian Esterer, an asset management professional, proposed a new method for predicting returns and estimating risk on individual shares, using earnings forecasts created by analysts across the market. They also proposed for the first time the concept of equity duration, which sits alongside the well-established idea of bond duration (a measurement of sensitivity to interest rate changes).

Analysts and investment managers at asset management firms across the financial sector took up Schröder and Esterer’s work, featuring the research in internal research reports and in some cases inviting them in to deliver in-person explanations. Feedback from the Head of European Qualitative Research at one of these major international firms described Schröder and Esterer’s research as ‘crucial’ in reframing its approach ‘to measuring the duration of global stocks’, demonstrating direct impact on the billions of pounds of assets that this firm holds under its management. Esterer was also able to put the research into practice at his own workplace, which also manages tens of billions in equity holdings.

Further Information


Exploring sum-free sets

This post was contributed by Professor Sarah Hart of Birkbeck’s Department of Economics, Mathematics and Statistics and Head of the Pure Mathematics Research Group. Here, Professor Hart offers an insight into a major area of her research: sum-free sets

The set S = {1, 3, 5} is “sum-free”: if you add two numbers in S, the answer is outside S. So 1 + 3 = 4, and 4 is not in S.

There are many questions we could ask about sum-free sets. How big can they get? Are there infinite sum-free sets? The answer is yes – the set of all odd numbers is sum-free because the sum of any two odd numbers is even, and so lies outside the set. In fact this is also locally maximal (we cannot add any further elements to it while keeping it sum-free), because any even integer (see glossary below) is the sum of two odd integers so we cannot add any even integers to the set.

Having found this large sum-free set we might ask if we can divide up (partition) the set of integers into a small collection of sum-free sets. (Actually, the fact that 0 + 0 = 0 means we only look at the set of nonzero integers.) It turns out there is no way to partition the set of nonzero integers into a finite collection of sum-free sets, although there are partitions into an infinite number of sum-free sets.


Click the image to read Theorem of the Day's article on Professor Sarah Hart's work

Read Theorem of the Day’s article on Professor Sarah Hart’s work

The interesting thing about this set-up is that it can be generalised. The set of integers is just one example of a “group”, which is a set along with an operation which can combine two elements a, b of the set to produce a third element a*b.

For the integers, the operation is addition: two integers added together produce another integer, so a * b is defined as a + b. There are three rules that the operation must satisfy. One is “associativity”, which for the integers translates as the property that for any integers a, b and c, we have (a + b) + c = a + (b+c).

There are countless examples of groups, for example the set of symmetries of any shape. The operation is composition – so the combination of a rotation integer (see glossary below) and a reflection is the symmetry obtained by doing the rotation followed by the reflection. Unlike the group of integers, many groups are finite.

The notion of a sum-free set

The notion of a sum-free set can be generalised to any group, but we now talk about product-free sets because the notation is multiplicative. If G is a group, and the operation is *, then a subset S of G is product-free when a * b is not in S, for all a, b in S.

In the group of symmetries of the cube, for example, the set of reflections is product-free because the product of two reflections is a rotation. We can ask all the same questions. What is the biggest possible size of a product-free set? Or the smallest locally maximal product-free set? How can we partition the group into product-free sets?

My research – filled groups

I first started looking at these ideas ten years ago, in [1], but have returned to the subject recently with my PhD student Chimere Anabanti [2] – in particular we have been trying to find out more about small locally maximal product free sets. Along the way we have been able to answer a question [2] that has been open since the 1970s about so-called “filled groups” – ones whose locally maximal product free sets have a particular form.

Next steps – Solution-free sets

There is another generalisation of all this, and that is the direction I hope to move in next: we can think of product-free sets as sets having no solution to the equation a * b = c. So we can pick another equation and look for sets that don’t satisfy that equation. The general term for these types of sets is “solution-free sets”.

Professor Sarah Hart

Professor Sarah Hart

We can ask the same kinds of questions about them as for product-free sets. Examples include Sidon sets – the definition for the integers is a set where all differences are distinct; in other words there are no solutions to a – b = c – d when at least three of a, b, c and d are different.

It’s fun to ponder these ideas in their own right of course, but there are surprising and interesting links to other areas of mathematics. Product-free sets in certain groups give rise to a particularly nice type of error-correcting code. There are also applications to graph theory and to finite geometry. Sidon sets are used in research about the efficient design of sensor arrays.

Find out more

[1] Giudici and Hart “Small maximal sum-free sets”

[2] Anabanti and Hart “Locally maximal product-free sets of size 3” Preprint 10 on this page

[3] Anabanti and Hart “On a conjecture of Street and Whitehead on Locally maximal product-free sets”


  • Integer: A whole number (not a fractional number) that can be positive, negative, or zero e.g. -5, 1, 5, 8, 97, 3,043
  • Rotational symmetry: When an image is rotated (around a central point) so that it appears two or more times.
  • Maximal: A maximal element of a subset S of some partially ordered set (poset) is an element of S that is not smaller than any other element in S