topology
A Klein bottle can't hold any liquid because it doesn't have an inside. How do you construct this strange thing and why would you want to? 

Why doing maths is like being Lewis Carroll's Red Queen and how to keep going beyond the formidable age of 84. 
Maryam Mirzakhani is being honoured for her "rare combination of superb technical ability, bold ambition, farreaching vision, and deep curiosity". 
The paths of billiard balls on a table can be long and complicated. To understand them mathematicians use a beautiful trick, turning tables into surfaces. 
How to make a hard problem easy by changing the way you look at it. 
The London Underground turns 150 today! It's probably the most famous rail network in the world and much of that fame is due to the iconic London Underground map. But what makes this map so special? 
Progress in pure mathematics has its own tempo. Major questions may remain open for decades, even centuries, and once an answer has been found, it can take a collaborative effort of many mathematicians in the field to check
that it is correct. The New Contexts for Stable Homotopy Theory programme, held at the Institute in 2002, is a prime example of how its research programmes can benefit researchers and its lead to landmark results.

Data, data, data — 21st century life provides tons of it. It's paradise for researchers, or at least it would be if we knew how to make sense of it all. This year's AAAS annual meeting in Vancouver
devoted plenty of time to the question of how to understand large amounts of data. And there's one method we
particularly liked. It's based on the kind of idea that gave us the London tube map.

Topologists famously think that a doughnut is the same as a coffee cup because one can be deformed into the other without tearing or cutting. In other words, topology doesn't care about exact measurements of quantities like lengths, angles and areas. Instead, it looks only at the overall shape of an object, considering two objects to be the same as long as you can morph one into the other without breaking it. But how do you work with such a slippery concept? One useful tool is what's called the fundamental group of a shape. 