The central concept of Functional Analysis is the Banach space, which provides a unifying framework for many important results in mathematics, and about which there is a well-developed theory. Loosely speaking, a Banach space is a vector space together with a notion of a distance. The distance allows one to define concepts like continuity and convergence in a more general setting. In other words this course is, as its name suggests, analysis. The basic results of Banach space theory will be presented, as well as some abstract analysis.
Analytic Number Theory applies the methods of Classical Analysis to the integers and, in particular, to the properties of prime numbers. We prove the famous Prime Number Theorem about the distribution of primes as well as Dirichlet's theorem about primes in arithmetic progression. The role of the Riemann-Zeta function is discussed and its analytic continuation proved by means of its functional equation.
2801 is an optional course provided by the UCL mathematics department for students who wish to attend. Although some material will be taught in the course, the emphasis of the course is placed heavily on the discussion of interesting problems. No credit is awarded for participation, although consistently attending students will have the opportunity to take an accredited exam in their second year. Small awards, varying previously from chocolate to beer, have been awarded for the solution of previous questions. On the loose basis of once a fortnight, attending students will receive a list of questions whose solution may require concepts encountered before or during the first year analysis and algebra courses. Additional concepts will be introduced and explained when necessary. We aim to introduce students to interesting and stimulating problems, previous problem topics have included (among many others); combinatorics, geometry, topology, logic, number theory, analysis, algebra and set theory. Any student who has a keen enjoyment of puzzles, problems and elegant mathematics is welcome to attend.
No answers are or will be provided by the lecturers, we encourage students to present their own solutions to their peers.
The three main themes are: