﻿ Analysis and Topology I

# Mao Sheng (盛茂)

## Course Description

Maybe everybody knows that analysis is the branch of mathematics dealing with “limits” and related theories such as “continuity”, “differentiation”, “integration” etc. But what is less well-known is that the underlying foundation on which people build the skyscraper of modern analysis is the completeness (or in other words, the continuum) of real numbers. Starting with completeness, one can prove theorems like

The intermediate value theorem (IVP). Let be a continuous function. Then for any value between and , there exists such that .

The extremal value theorem (EVP). Let be a continuous function. Then attains both a maximal value and a minimal value on .

An even less well-known fact is that it is the topology of the real line (associated with the continuum of the real numbers) that really stands behind the above two theorems: the connectedness for the IVP and the compactness for the EVP.

What is topology? Well, it is a monster with two faces: the first face is called the “analysis face” on which you can find “neighborhood”, “open set”, “continuity”, “compact” etc, and the second face is called the “geometry face” on which you can find words like “space”, “connectivity”, “deformation” etc. There are many interesting theorems, including

Brouwer’s fixed point theorem. If you stir a cup of coffee, there is some point in the cup that ends up in the same position as it started in.

Hairy ball theorem. If you have a hairy ball, you can not get the hairs to lie flat everywhere along the surface - some of the hair has got to stick up.

Some of the theorems may look “obvious” while others may look “absurd”. However, to give a rigorous proof is always challenging.

In the first half of this course we will focus on the analysis face of topology: we start with rigorous treatment on the topology of the real line, and their application to analysis of one-variable functions. Then we extend the ideas/conceptions from the real line , first to finite dimensional Euclidean spaces , and then to more general spaces, whose objects could be functions or shapes etc, and explore their applications in analysis and geometry. Then in the second half of this course we will turn to the geometry face of topology. In particular we will explain how ideas from analysis (including multi-variable analysis and other facts from the analysis face of topology) can be applied to prove Brouwer’s fixed point theorem and the hairy ball theorem.