- Snapshots of modern mathematics
- Diderot Mathematical Forum 2013: “Mathematics of Planet Earth”
- Pierre de Fermat and Andrew Wiles in Czech Republic stamps
- Stefan Banach (March 30, 1892 – August 8, 1945)
- Diderot Mathematical Forums
- Guessing the Numbers
- What is mathematics for Ehrhard Behrends
- What is mathematics for Krzysztof Ciesielski
- The Three Ducks Trick
- What is mathematics for Franka Brueckler

# Plus Magazine Monthly Column – April 2017 – Women of Mathematics and other things

*Plus http://plus.maths.org is a free online magazine about mathematics aimed at a general audience. It is part of the Millennium Mathematics Project, based at the University of Cambridge and our aim is to open a door onto the world of maths for everyone. We run articles, videos and podcasts on all aspect of mathematics, from pure maths and theoretical physics to mathematical aspects of art, medicine, cosmology, sport and more. Plus has a news section, covering news from the world of maths as well as the maths behind the mainstream news, reviews of books, plays and films, as well as puzzles for you to sharpen your wits.*

**Women of mathematics**

There’s a famous book on the history of mathematics, written in 1937 by E.T. Bell, called *Men of Mathematics*. Viewed with modern eyes, there’s something obviously wrong with the title: where are the women? In Bell’s defence, mathematicians have been almost exclusively male throughout history (with some notable exceptions). This has changed, but gender imbalance is still a problem in mathematics. What does this mean for female mathematicians?

It’s a question we explored in a series of interviews to accompany the *Women of Mathematics * photo exhibition, which opened in Cambridge this month. The exhibition was launched in Berlin in the summer of 2016, and has been travelling ever since. It originally contained portraits of thirteen female mathematicians from throughout Europe. The Cambridge part of the exhibition includes the portraits of another six women, all working at the Centre for Mathematical Sciences in Cambridge.

As they were being photographed, we took the opportunity to interview these women about their life and work, with interesting results. Anne-Christine Davis, a Professor of Theoretical Physics at Cambridge and one of our older interviewees, told us that the old days were truly dismal for female mathematicians. After completing her PhD, Davis was told to get married, because “what else is there left to do for a woman”? When thinking of applying for a job, she was told that males would be preferred, since “a man has a family to look after, but a woman has a husband to look after her”.

Thankfully, things have changed “beyond all recognition” as Davis puts it. Attitudes like the ones she encountered are deemed wholly unacceptable today, and female mathematicians can go about their work without being constantly reminded of their gender. There are still disadvantages of course: there is a lack of role models, female mathematicians are still in a minority wherever they go, and some battle with guilt over pursuing a career alongside (or instead of) a family. But this shouldn’t deter them. The one piece of advice our interviewees would give to budding (female) mathematicians is to pursue their dreams regardless of what anyone says and to not give up. Perhaps the most important message came from Natalia Berloff, Professor of Applied Mathematics at Cambridge: “Female or male. I don’t know if it makes such a big difference. We’re defined by our profession.”

To watch, read or here the interviews with the six mathematicians, see here.

**The hidden beauty of the multiplication table**

One thing that might enthuse those budding mathematicians right from the start are the patterns that hide within a simple multiplication table. As Zoheir Barka explained in a recent article on *Plus*, you start with an ordinary multiplication table in which an entry is the multiple of the number at the top of its column and the number on the left of its row. Now colour each cell of the table a particular colour if it’s a multiple of a particular number. For example, in the table below all multiples of 2 have been coloured blue (we have added a row of 0s at the top and a column of 0s on the left, as this gives a nice frame to the pattern without disrupting the coherence of the table).

Colouring the multiples of 4 gives a slightly more interesting pattern:

Once you allow yourself to colour multiples of several numbers in different colours — for example multiples of 2 red, multiples of 3 orange and multiples of 4 yellow, there is almost no limit to the patterns you can create. Have a look at the pictures below and see if you can figure out the multiples of which numbers have been coloured by which colours.

**The maths of beer**

Multiplication patterns are mathematics pure and simple, but what if you prefer your maths to be actually useful? In that case, beer — or to be precise, Guinness — delivers a perfect example. Guinness is famous for its lovely head, which is much creamier than that on most other beers. The reason is that, while the bubbles in other beers are made of air, the bubbles on Guinness are made of nitrogen. The nitrogen needs to be added to the Guinness when it is poured. If you buy your Guinness in a pub, then this is achieved by a pipe linked to a nitrogen supply. But what if you buy your Guinness in a can?

For many years, Guinness can drinkers had to make do without the creamy head, until *widgets * were introduced: these are small nitrogen containers inside the can, which release exactly the right amount of nitrogen when the can is opened. Getting the widget right, however, isn’t easy and requires a lot of careful thought, including mathematical thought. The process has been analysed by the entire applied mathematics department at the University of Limerick and is described in a paper called *The initiation of Guinness*. The same group have also done a complete analysis of the maths of coffee making.

While Guinness has to thank maths for the widget, maths has to thank Guinness for one of the best-known tests used by statisticians. In the first part of the twentieth century Guinness was ahead of its time when it came to quality control. It would obviously have been unfeasible to check every single barrel or bottle of Guinness that was being produced. Instead, quality controllers made careful measurement of samples of Guinness and used these to assess its general quality and how much quality varied. This amounts to a difficult statistical problem. It was Guinness employee William Gossett who solved it by devising a statistical test to compare measurements. This was published in 1908, anonymously under the name of “Student”. It’s now widely known as *Student’s t-test. * To find out more about the maths of beer, and also of the maths of food, read Chris Budd’s *Plus * articles Where’s the maths in beer? and How much maths can you eat?

**Taming big data**

When we shared the above stories on the *Plus* social media feeds, the accompanying images, videos and links were widely shared and followed — though admittedly not as much as the pictures of kittens we occasionally post. Sharing information has become normal practice for many of us. We carry computers in our pockets, constantly connected and sharing information, pictures and videos as part of our daily lives. In 2016, for instance, in every single minute Google translated over 69 million words, over 400 hours of video were uploaded to YouTube and more than 200,000 photos were shared on Facebook Messenger.

We are surrounded by technology that collects, transmits, manipulates and ultimately needs to understand reams of information of an order of magnitude which is hard to comprehend.

The need to understand this *big data*, as the mass (and sometimes mess) of data that arises in the modern world is called, comes up in all sorts of different contexts: from the biomedical sciences to finance, the internet, software and hardware development and security, and image processing, to name just a few.

Mathematics has long been called the “language of the universe”, underpinning the development of modern science and technology. There is every reason to believe that mathematics, too, will provide the language of data: both the data itself (the values of the qualitative or quantitative variables) and the information (the content and meaning) it contains. To find out more about this burgeoning area of research we spoke to Carola Schönlieb, the director of a new research institute, the Cantab Capital Institute for Mathematics of Information.

“In fact, it is not the data itself that is so important, but rather the information contained within it,” says Schönlieb. “Using fundamental techniques from the mathematical sciences, it is possible to understand the limitations of what can be found from the data, and whether this information can be found in the next few seconds, minutes, hours, or if we have to run an algorithm forever without ever providing an answer. We can also use maths and stats to understand how certain or uncertain we should be about conclusions we draw from data.” We’re looking forward to hearing about the work of the institute in the future — and you can rest assured your photos of kittens serve a mathematical purpose!

Marianne Freiberger and Rachel Thomas

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