# Out of the norm: an introduction

A norm is a general notion of distance that applies to vectors (most people have typically met vectors in two or three dimensions at school, for instance during GCSE maths). Several areas of mathematics arise out of the idea of a norm, such as Banach spaces and Banach algebras, the two areas I used to research as a PhD student and PostDoc.

Though mathematicians are stereotypically seen as bad communicators, it isn’t universally true. While mathematicians may on average be more socially awkward than, say, hairdressers, as in any walk of life there’s plenty of extroverts. It’s not even true of those at the top of the subject. For instance, two holders of the Fields Medal—seen as the mathematical equivalent of the Nobel prize (which isn’t awarded for mathematics, but not because Alfred Nobel’s wife had an affair with a mathematician, especially since he didn’t have a wife)—Tim Gowers, who won his for solving several long-open problems in my area, and Terry Tao both keep excellent academic blogs alongside non-technical material (Terry’s most accessible or general posts now seem to go to his Buzz and Google+ feeds).

Personality discussions aside, pure mathematicians face two main obstacles to communicating: firstly, the nature of their research is abstract, technical (even to most other mathematicians), and usually not immediately applicable to the real world; and secondly, the near-universal response you receive upon mentioning that you research mathematics is “I was never any good at maths” (I last heard it today). So, sadly, the social norm for many mathematicians is not to put their head above the parapet and talk about maths (another reason for the blog’s name).

There’s no one strategy to overcoming these hurdles. Useful approaches are: appealing to analogy (ideally to something else mathematical, but simpler or more familiar) or giving concrete and toy examples, which The Princeton Companion to Mathematics, edited by Gowers, does brilliantly (everyone interested in 20th century mathematics and every PhD student should have this book); talking about how the mathematics is used in applications is not only politically useful, but may also gives some vague sense of the area; as a combination of the previous two points, choosing carefully what topics to talk about is extremely important, so if someone sincerely asks “What do you research?”, perhaps find something loosely related but more interesting than your own work; writing, drawing, calculating, and reasoning in front of an audience, using pictures, and one-on-one questioning can all be engaging; finally, liberally borrowing the best explanations available, while trying out your own.

Many people believe that maths is just harder and longer calculations like multiplication and division: if you can correct this view, you’ve probably succeeded somewhere. Or you can embrace and subvert this expectation as a popular lecture by Gowers does, where he discusses multiplying and factorising large natural numbers (eventually showing his audience the Fast Fourier Transform, and Fermat’s Little Theorem). As for the most common ‘never good at maths’ reaction above, there’s advice out there too (I briefly tried out: “Don’t worry, we do the maths so you don’t have to”);

I think (or at least hope) the general public is very slowly warming to mathematics, despite the attempts of many people’s boring schooling. We’ve always had popularisers such as the late maths and science journalist Martin Gardner and professor Ian Stewart, whose books were a great childhood inspiration to me. But now we also have projects like maths busking, serious discussion of statistics on Radio 4’s More or Less, Vi Hart’s effortlessly endearing Doodling in Math Class video blog, and more logic puzzles than ever before in newspapers.

I hope to cover some of the same ground as all the influences above, mainly aimed at a general audience, but with some discussion of my specific research areas, and hopefully bring along some of my own view of the mathematical world.