23 Mathematical challenges

To mark the launch of Britain’s new science agency ARIA, which aims to tackle the toughest problems, we made a list of the 23 most important mathematical challenges of our time.

In 1900, the German mathematician David Hilbert published a list of 23 important mathematical problems. It proved hugely influential, inspiring researchers throughout the 20th century. Since they appeared, 17 of Hilbert’s problems have been solved or partially resolved—one just last month, as it happens.

Now the London Institute for Mathematical Sciences has compiled a new list of 23 mathematical challenges for the 21st century. It is offering it as a roadmap to the British government’s fledgling science agency, ARIA.

Why mathematical? The answer is that the government’s Advanced Research and Inventions Agency was founded with the specific aim of creating “a paradigm shift in an area of science”. Mathematics is the language of science, and the most transformative scientific breakthroughs depend on it. Number theory, for instance, is at the root of modern cryptography.

Our list is also useful for another reason. Identifying the toughest challenges is a reminder not only to ARIA, but to the whole scientific community, to aim high. Moonshot missions can rally a team to exceed their perceived limits.

To figure out what our 23 challenges should be, we hosted a one-day symposium of physicists and mathematicians at our offices at the Royal Institution—a fitting venue, since this building in Mayfair has overseen some of the greatest breakthroughs in the history of science. Before the symposium, we also wrote to 23 leading researchers around the world to ask for their contributions. We collected over 100 challenges, which we then whittled down to 23.

We used three criteria to choose our challenges. First, we favoured problems whose investigation is likely to prove fruitful, even if it doesn’t lead to an actual solution. Second, we sought a balance between concrete problems with a definite answer and speculative challenges that are loosely posed. These tend to correspond to established fields and ones that are just emerging. Third, we tried to balance our list across the full range of the mathematical sciences.

To prime itself for these challenges, ARIA might consider David Hilbert’s epitaph. The words on the mathematician’s tomb at Göttingen are, “Wir müssen wissen. Wir werden wissen.”

This translates as, “We must know. We shall know.”

1 Theory of everything

We lack a single theory that describes the universe. Gravity, described by general relativity, is not consistent with our quantum field theory of the other three forces. Will this be resolved by string theory, loop quantum gravity, or something new? What are the testable consequences of such a theory, which is beyond the limit of human experimentation?

2 Riemann hypothesis

Attempts to settle the Riemann hypothesis have inspired whole new branches of mathematics. For example, the Riemann zeta function is the simplest kind of L-function, and these seem to play a role in modern mathematics similar to polynomials in ancient mathematics. What new concepts are needed to resolve this most important of open problems?

3 Thermodynamics of life

According to Darwin’s theory, evolution is the result of mutation, selection and inheritance. But from a physics perspective, we don’t understand how life got started in the first place. What is the thermodynamic basis for emergent self-replication and adaptation, of which biology is just one instance? Can it be used to create digital artificial life?

4 The structure of innovation

Despite advances in our understanding of evolution, what drives innovation remains elusive. Technological innovation operates in an expanding space of building blocks, in which combinations of technologies become new technologies. Can we characterise innovation in a mathematical way, so that we can predict it and influence it through interventions?

5 Physics of self-assembly

Self-assembly is how proteins fold, snowflakes form and viruses assemble. It can be used to manufacture complex and nanoscale objects at low cost. Because it is a physical embodiment of computation, it is deeply linked with decidability. Can statistical physics be combined with computability theory to build a comprehensive theory of self-assembly?

6 Cosmological constant

Only a small fraction of the observable universe is made up of known matter. The majority is conjectured to be dark matter and dark energy, for which there is no consensus in explanation. Why does the zero-point energy of the quantum vacuum not cause a large cosmological constant? What cancels it out? Is new fundamental physics needed to reformulate gravity?

7 Langlands Program

There is evidence of a grand unified theory for mathematics, called the Langlands Program. It seeks to relate automorphic forms in geometry and number theory to representation theory in algebra. Wiles’ proof of Fermat’s Last Theorem can be viewed as just one instance of it. How can we advance and extend this Program, and what fruit will it bear when we do?

8 Intelligent AI

Far from approaching artificial general intelligence, AI has not progressed beyond high-dimensional curve-fitting. What mathematical insights could lead to more intelligent AI, such as causal reasoning, functional modules or a representation of the environment? Are there fundamental limits to AI, and what might this tell us about human intelligence?

9 Repairable instead of robust

To be sure of success in the face of uncertainty, we make plans that can cope with the unexpected. One way is to be robust: able to absorb a known setback. Another is to be repairable: easily modifiable in the face of unknown setbacks. Our approaches to threats, such as war or climate change, tend to be robust. What would a theory of repairability look like?

10 The operating system of life

Networks of gene regulation govern morphogenesis and determine cell identity. The concision of viruses suggests that this genetic software uses subroutines, like digital software. What are the laws governing genetic information processing? Can they shed light on the operating system of life, setting the stage for a biological analogue of the silicon revolution?

11 The mathematical universe

Wigner noted the unreasonable effectiveness of mathematics in physics. Today, we are seeing the reverse: attempts to advance physics, such as string theory, are driving mathematics. Is there a convergence between these two disciplines, and should this inform how much we fund and advance mathematics? Can Tegmark’s mathematical universe be made rigorous?

12 Describing network structure

Network science, which extracts meaning from real-world networks, is popular but unsophisticated. To realize its potential, it must build on more rigorous concepts from graph theory and beyond. Can we formalise notions of network geometry and topology that are compatible with their continuum analogues, and a thermodynamics to describe deviations away from them?

13 Theory of free will

The existence of free will has no grounding in the laws of physics that are known. Some attempts have been made to link it to quantum phenomena, but a theory of free will remains elusive. Is it a phantom, a consequence of life, or a more general attribute of the present moment? What new physics is required to understand this seemingly vital concept?

14 Collective creativity

Collective creativity started 300 years ago when the scientific paper sped up research by enabling fast marginal advances over slow major ones. Now, anonymous collaboration platforms such as Wikipedia suggest we can speed up much more. But why and when does collective creativity work? Can platforms like Gowers’ Polymath transform the process of discovery?

15 Programmable matter

We can cause surfaces and volumes to change shape on command using polymers that respond to temperature and current. What are the scope and limits of such programmable matter? Can we use differential geometry, recent advances in algorithmic origami and other mathematical tools to provide a language for reverse engineering useful shapes and mechanisms?

16 Foundation of QFT

Can quantum field theory, which describes all elementary particles and interactions, be made rigorous? An open problem is to prove that for any compact gauge group, a Yang-Mills theory exists in four dimensions and predicts a lightest particle with positive mass. This will likely require new kinds of mathematics and offer a new perspective on physics.

17 Mathematical dualities

Dualities play a key role in how we form insights in physics and mathematics. Examples include the Geometric Langlands correspondence, dualities across quantum field and string theories, and the ADE classification. Are dualities an artefact of how we decipher new theories, or do they have a more fundamental cause? Can we systematise them to discover more?

18 Engineerable AI

Evolution and innovation both make use of interoperable functional modules to increase the odds of successful tinkering. But deep learning algorithms, by contrast, are globally wired. This makes them difficult to build up in a hierarchical way, as well as hard for humans to understand. Can we formulate a framework for AI that is engineerable?

19 Theory of simplicity

In an increasingly complex world, we seek simplicity in how we organise technology and society. But we don’t know how to describe simplicity, much less construct it. Physicists have developed various models of complexity, but what would a theory of simplicity look like? Is it connected to being able to readily reconfigure to new environments?

20 AI-assisted conjectures

Good conjectures can inspire new branches of mathematics. They come from spotting patterns and applying instinct. Because mathematics is exact and there are no equivalence coincidences, automated pattern detection is immune from the bias normally found in high dimensional search. Can machines help identify candidate conjectures and speed up theoretical research?

21 Mathematics of causality

Causality is fundamental to how we make predictions and structure society. Yet our mathematics for describing it is poor. Can a more sophisticated theory of causality help unlock challenges such as intelligent AI, the operating system of life and even how we build physical theories? How can we move from a microscopic to a macroscopic notion of causality?

22 Emergence of virtue

The fully-informed rational-agent basis of economics is inadequate for describing real-world behaviour, especially virtuous activity. Can insights from the microscopic view of behavioural science and the macroscopic view of thermodynamics form the basis of a cooperative game theory that accounts for the emergence of virtue in individuals and organisations?

23 Theory of immortality

Ageing is ascribed to the accumulation of errors—an inevitable consequence of the increase of disorder. But mounting experimental evidence suggests that ageing is not a fact of life. Is it a thermodynamic necessity, or is it instead favoured by natural selection? Can we mathematically describe the pros and cons of ageing? Is it possible to slow or even stop it?

Further reading

  • B. Yandell, The Honors Class: Hilbert's Problems and Their Solvers, CRC Press (2001).

    J. F. Nash Jr. and M. Rassias (Editors), Open Problems in Mathematics, Springer (2016).

    S. Smale, Mathematical Problems for the Next Century, Mathematical Intelligencer, 20, 7 (1998).

    23 Mathematical Challenges, Defense Advanced Research Projects Agency (2008).