As Number Theory has been described as the Queen of mathematics, Class Field Theory is the crown jewel of Late 19th and early 20th Century number theory. Its key statements are theorems regarding a beautiful reciprocity between the arithmetic of a Number field, and the abelian field extensions of said number field. It has a number of links to other areas of mathematics, such as algebraic geometry, various algebraic theories such as group/Galois cohomology, and even areas of theoretical physics, and is a key special case of various modern theories such as the (in)famous Langlands programme. 

Given its complexity however, it also has a very humble background, with a natural historical progression from elementary number theory, beginning with a simple question: For a given positive whole number n, what primes can be expressed in the form x^2+ny^2? 

The aim of this course is introduce the key concepts of class field theory from a historical and hands-on point of view. It will include topics such as the theory of quadratic forms, genus theory, ramification, Class Field Theory and, if we have time, Elliptic curves and Complex Multiplication. 

Please contact me with questions regarding the course at danielfunckmaths@gmail.com

The study of modular forms traces back its roots to the late 19th and early 20th century with Gauss, Eisenstein and Ramanujan, and is a fascinating blend of analysis and algebra, and play a central part of modern number theory. They have many surprising applications to number theory, including a beautiful proof of Lagrange's four square theorem and the ground-breaking proof of Fermat's last theorem in 1995. This course aims to give an introductory understanding of this broad topic. 

The aim of this course is to explain the foundational concepts, results and methods of the classical theory of modular forms, and understand the role they play in modern number theory. 

Please contact me with questions regarding the course at danielfunckmaths@gmail.com