Today analysis is a central ingredient in most areas of mathematics and its applications, as well as a fascinating area of study in its own right. Whether you are planing to focus on probability, dynamical systems, geometry, partial differential equations or even number theory in your 3rd and 4th year, analysis will be most helpful if not essential to master these subjects.
 
While its roots go back to Newton, Leibniz and beyond, modern analysis plays a crucial role in much of modern mathematics. For example, the analysis of partial differential equations was central to Perelman's recent resolution of the Poincare conjecture, a conjecture about the topology of three dimensional manifolds. Although analysis has an infi nitesimal nature, it also plays a key role in the study of discrete structures such as the distribution of primes among integers (via the Riemann zeta function). Finally, analysis has been successfully employed recently in the study of random processes (or use percolations) in statistical mechanics such as the Ising model.
 
To be a bit more concrete, in the first two years you have studied:
 
  • the basic principles of multivariable calculus
  • some elementary topology and the theory of metric spaces
  • some introductory complex analysis
The goal of years 3 and 4 is not only to learn "new" analysis and look at the above topics in a systematic approach towards a more general theory from a more general point of view but also to unravel some of the deep interconnections between the various branches of analysis. You can learn about infi nite dimensional normed vectorspaces whose elements are functions and maps between such spaces (Functional Analysis) - this is at the heart of finding solutions to partial differential equations. The question "what is the size of a set?" will be revisited (Measure Theory and Integration) and eventually lead to the Lebesgue theory of integration, which replaces the Riemann integral. Both the most interesting function spaces mentioned above but even more so Probability Theory require the language of measure theory. The course Geometric Complex Analysis looks at the calculus of functions of a complex variable from a geometric viewpoint, for instance, you will learn about how geometric quantities are transformed under holomorphic mappings. In Fourier Analysis you will learn how any function can be studied by splitting it up into its component frequencies.
 
The Taught Course Centre (TCC) is a collaboration between the Mathematics Departments at the Universities of Bath, Bristol, Imperial, Oxford and Warwick. The lectures are open to all postgraduate students and are taking place in the room 6M42 (Huxley Building). The link to TCC can be found here.

More detailed course descriptions