Lie algebras, Lie groups, and their representation theories are instrumental in our description of symmetries in physics and elsewhere. They also occupy a central place in pure mathematics where they often provide a bridge between...

# Algebra and Number Theory

**Algebra** is the study of abstract mathematical structures that generalise well-known number systems such as the integers or the reals, and their arithmetic operations such as addition and multiplication. Some of the most important algebraic structures are rings (such as the integers), fields (such as the reals) and groups (such as the symmetries of the cube). The study of algebra is of fundamental importance to our understanding of the natural world. The electromagnetic and nuclear forces, for example, are best understood in terms of the so-called three unitary groups SU(3), SU(2) and U(1).

Traditionally, **Number Theory** referred to the study of the integers. However, in modern mathematics number theorists are interested in the properties of much broader classes of numbers. Whilst considered an area of pure mathematics, Number Theory is pervasive in every-day life, with prime numbers forming the key to secure e-shopping and internet banking. Famous unsolved problems in number theory are the Goldbach conjecture—the claim that every even integer greater than 2 can be written as the sum of two prime numbers—and the Riemann hypothesis, one of the million dollar millenium problems.