Applications of Topology to Analysis

The first work on topology was due to L. Euler in 1736, when he published the solution to the Konigsberg bridge problem.

In that paper Euler was aware that he was dealing with a different type of geometry where distance was not relevant.

However, it is widely regarded that the birth of "modern topology"
(as a separate field of research) was due to H.Poincare in 1895 when he published "Analysis Situs".

Nowadays topology is considered as two almost distinct fields:

• (i) combinatorial topology or algebraic topology and
• (ii) point-set topology or general topology, (whose father is probably F. Hausdorff in 1914).

My interest is in applying general topology to problems in analysis and functional analysis. This is natural since these two areas played fundamental roles in the birth of general topology. In fact one of the motivations for general topology came through ideas of convergence which go back to 1817, when B. Bolzano generalised the notion of convergence.

Later, in 1906, M. Frechet, building on the earlier work of G. Cantor (1872), K. Weierstrass (1877) and D. Hilbert (1902) introduced the notion of a metric/linear metric space as a general way of considering convergence. Finally, M. Riesz and F. Hausdorff (1914) defined the notion of an abstract topological space. General topology also finds its origins in functional analysis, via calculus of variations and Fourier analysis where precise notions of compactness and convergence are required.

The connection between functional analysis and general topology was cemented in 1932 when S. Banach established the importance of completeness (and Baire category arguments) in analysis.

Since the early part of the 20th century both topology and analysis have fed off each other and this has resulted in some very beautiful theorems that lie at the interface between these two disciplines. Perhaps the best known example of these is Brouwer's fixed point theorem (later generalised by J. Schauder to more general spaces) which has had countless applications in applied mathematics, economics and analysis itself.

What I study is the application of topology to problems in analysis. In particular, I study the "Geometry of Banach Spaces". This has applications to the solutions of partial differential equations, because even before one starts trying to solve a partial differential equation one must first decide what one means by a "solution".

This leads one to consider classes of functions/generalised functions (of potential solutions). Usually, these form a locally convex topological vector space. However, in many important cases they form a Banach space. Some approaches to finding solutions of partial differential equations use sequences of approximate solutions that "converge" in some sense to a solution.

This notion of convergence within a Banach space is one of the motivations behind the study of the geometry of Banach spaces. By using the "geometry" of a Banach space one can often infer that a sequence of approximate
solutions that converge, in some weak sense, to a solution may in fact converge in a much stronger sense (eg. with respect to the norm topology). This interplay between the different topologies on a Banach space is precisely what I explore. In particular, one of the problems that I investigate is the following: "When is a sequence of functions (defined on a compact convex set K) that converge with respect to the topology of pointwise convergence on a boundary of K convergent with respect to the much stronger topology of pointwise convergence on all of K".

The solution to this problem will have many applications.

Recall that a subset B of a compact convex set K is called a boundary if each continuous real-valued affine function defined on K attains its maximum value at some point of B.