Symmetries are a powerful tool in science. They relate the behaviour of different experiments within the same physical system. For example, if a ball is dropped in Durham or in Zurich it will fall towards the ground with the same constant acceleration. The rotational symmetry responsible can be seen by simply sketching a picture of the Earth. However, in many interesting physical systems the symmetries are hidden. Uncovering them and their implications is a vibrant and thriving area of research.

String theory is well known as a promising candidate for a theory of quantum gravity, unifying general relativity and quantum mechanics. Gravitons, the quanta of the gravitational force, are oscillating closed loops of string. The development of string theory over the last 50 years has led to many deep insights into the fundamental nature of physical theories. Nevertheless, many interesting string models remain difficult to study because of the presence of background fields. Much like an external magnetic field causes an electron to follow a curved path, these background fields affect the dynamics of the strings.

Symmetries can come to the rescue in the guise of **integrability**.
This powerful mathematical property is a manifestation of a large hidden symmetry that can be used to conjecture and derive exact analytic results.

**Deformations** of a physical model can expose hidden structures and help us to understand its underlying properties.
A classic example is the Zeeman effect – the splitting of spectral lines in the presence of a weak external magnetic field.
The external magnetic field breaks the rotational symmetry of the atom and exposes the hidden structure.
Similar effects can be found throughout physics, including in string theory.
One example can be seen in the integrable model of strings moving on a sphere.
Deforming the shape of the sphere breaks the rotational symmetry.
However, if this is done in a particular way then the symmetry is not broken; rather it is hidden, preserving the integrability of the model.
As well as allowing us to better understand the original model, these deformations can also be used to construct new solvable string models.
Moreover, they also appear in other areas of theoretical and mathematical physics, including quantum groups and statistical mechanics.

**Duality** is one of the most exciting concepts in modern theoretical physics.
Two theories, which may look very different, are said to be dual if they are different realisations of the same underlying physics.
Therefore, one theory can provide access to inaccessible regimes of the other.
An important example is T-duality, which relates strings moving on different spaces.
T-duality and its generalisations preserve integrability; hence, can be used to construct new integrable models.
Another striking example is the AdS/CFT correspondence, a remarkable duality between strings in 10 dimensions and quantum field theory in four.
The power of this duality is that the weak-coupling regime of string theory can be used to explore the physics of strongly-coupled quantum field theory.
In the most famous case, the strings are moving on a symmetric space and the model is integrable.
The dual quantum field theory is a supersymmetric gauge theory in four dimensions with important ties to quantum chromodynamics: the Yang-Mills gauge theory that explains how protons and neutrons are held together in the nuclei of atoms.

The goal of the project is to use the tools of **integrability**, **deformations** and **dualities** to explore different areas of theoretical and mathematical physics, developing our understanding of these topics and how they are linked by hidden symmetries.

**I. String Theory.**
Integrability methods have been successfully applied to some of the most symmetric string models.
However, this becomes more challenging when the symmetries are hidden.
Can we classify those string models for which we can develop and apply these methods, and find new models with less manifest symmetries and physical applications?

**II. Gauge/Gravity Duality.**
Gauge/gravity dualities have significantly improved our understanding of strongly-coupled physics.
However, they can be difficult to construct and analyse rigorously.
Can we use symmetries to determine the gauge theory duals of integrable string models and their deformations, for example, a q-deformed AdS/CFT correspondence?

**III. Quantum Groups.**
The mathematics underlying hidden symmetries goes by the name of quantum groups and is undeniably elegant.
While it is understood at an abstract level, often the implications for physics are not so clear.
Can we develop a solid mathematical framework underpinning the fundamental algebraic structures of integrable string models and related theories?

**IV. Statistical Mechanics.**
Many of the integrable models that appear in string theory also play a role in other areas of physics, including, for example, formal aspects of statistical mechanics.
Can we transfer our new knowledge of integrability, deformations and dualities from string theory to other disciplines?