# Lecture 01

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## 18th Century

### Analysis

"Analysis" essentially means "calculus-related". Its formation took place from 1650–1700, and Newton and Leibniz were its progenitors.

Two men were responsible for the expansion and consolidation of analysis in the 18th century:

• Leonhard Euler, born 1707 in Basel, Switzerland.
• Lagrange, who lived from 1736–1813. He was an Italian that moved first to Germany, then France (and who is often mistaken for French as a result of this move in conjunction with his characteristically French-sounding name).

One of the characteristic topics of study in analysis are infinite series:

$\sum_{n=1}^\infty \frac{1}{n^2} = 1 + \frac{1}{4} + \frac{1}{9}+ \frac{1}{16} + \cdots$

A famous mathematical problem, known as the "Basel Problem" after the birthplace of Leonhard Euler, was solved by him using several different methods. A proof developed by Euler is the subject of Dunham's essay in the course reader.

Aside: the term "physics" is a modern term that was once decomposed into "mathematics" and "natural philosophy". Lagrange might be considered a physicist in the modern sense.

### Parts of Mathematics

• Differentiation
• Integration
• Ordinary Differential Equations
• Partials Differential Equations
• Calculus of Variations

• Delambre
• Méchain

## 19th Century

### Algebra and Number Systems

• Hamilton
• Galois
• Dedekind

#### Hamilton

1. Hamilton-Jacobi Theory
2. Quaternion Algebra
1. arose from an attempt to generalize complex numbers to $\Re^3$
2. quaternion multiplication is not commutative, i.e. $\bar u \cdot \bar v \neq \bar v \cdot \bar u$

### New Algebraic Structures

 TO DO: link to relevant topic pages at Mathworld, Wikipedia, etc.

• Groups
• Rings
• Fields
• Vector Spaces

### Geometry

These branches of geometry, developed during the 19th century, display an increasing level of abstraction when compared with classical Euclidean geometry.

• Non-Euclidean Geometry
• Lobachevsky
• Differential Geometry
• Riemann
• Projective Geometry
• Abstract (Higher-Dimensional) Geometry

## 20th Century

At the end of the previous century it was still possible for a mathematician to be a generalist, e.g. Poincaré, Hamilton. These sorts of polymaths disappear during the 20th century as mathematics becomes increasingly specialized. Mathematics becomes the domain of specialists working in a single, or perhaps two, specific areas.

The last great polymath was John von Neumann, a Hungarian ex-patriate who worked in the United States. He developed the eponymously-named stored program architecture that enabled the realization of practical computer hardware. This architecture is now ubiquitous.

Another influential computer scientist and mathematician from this era was Alan Turing.

### Computer Science

Computers progressed from being computational devices in their early iterations, to being information devices in more modern times.

These discplines were brought together by the events of the Second World War in order to create machines that could calculate ballistics tables, and later to perform atomic modelling for the development of the atom bomb:

• Electrical Engineering
• Numerical Analysis
• Mathematical Logic

### Graph Theory

• 4-colour Theorem
• proved in the 1970s using computers
• never proven without the aid of computers

The fact that the 4-colour Theorem could not be proved without the assistance of computers caused mathematicians to expand their notion of proof to include the idea of Computer-Assisted Proof.