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# Classifying geometric shapes

‘We care about geometric shapes, like curves and surfaces, but we use the tools that come from algebra to study them,’ says algebraic geometer Diletta Martinelli about her field of research. She joined the Korteweg-de Vries Institute for Mathematics at the University of Amsterdam in the fall of 2019. ‘For example, if you consider the circle, you have a geometric object that is defined by the algebraic equation x2 + y2=1. This is a simple example of the interplay between algebra and geometry.’

## Three building blocks

Martinelli specialises in a branch of algebraic geometry called “higher dimensional birational geometry”, a field that classifies geometric objects defined by polynomial equations. ‘These geometric shapes can be incredibly complicated, but are fundamental objects in many areas of mathematics and science,’ she explains.

‘The goal of an algebraic geometer is to understand their properties and to achieve a complete classification of these important objects. For the classification to be meaningful, we want to have a finite amount of classes. But since there are infinitely many objects, we need to introduce some notion of equivalence. As an algebraic geometer, the first question that you should ask yourself is: when are the properties of two objects similar enough to put both of them in the same class? There is a technical notion that is called “being birationally equivalent”, which means that one object can be obtained by a small modification of the other.

For example, if two surfaces are exactly the same, up to a bunch of curves or points, then they are birationally equivalent.’ As an illustration, she mentions a cylinder and a cone. ‘Take the cylinder, wrap a string around it and squeeze towards the centre: you obtain the cone. This is an example of two surfaces that are birationally equivalent.’

How many classes are there? Martinelli: ‘The guiding conjecture of the research field says that there are only three classes. We call them building blocks. It is possible to reconstruct the geometry of any geometric object starting from the geometry of these three building blocks. We can distinguish these building blocks using the notion of curvature: they can have positive curvature - like a sphere, they can be negatively curved - like horse saddles, or flat - like a plane.’

### Becoming an algebraic geometer

Going back to the beginning of her career, Martinelli was not sure what to study after high school. But when she attended a popular science talk by algebraic geometer Marco Andreatta, she knew it had to be mathematics. ‘He talked about non-Euclidean geometry,’ she remembers.

‘Euclid was a famous mathematician from ancient Greece, who wrote books called the Elements. They were long seen as some kind of Bible, and perceived as an universal truth. But in the 18th century people started to question his assumptions and especially his 5th postulate. In this postulate he says that the sum of the internal angles of a triangle is always 180 degrees.

Several mathematicians realised that if you contradict this statement you still end up with a completely coherent geometry describing not the three-dimensional Euclidean space we know from our daily life, but more complicated objects. This observation really prompted a revolution in geometry, opening up several research fields, from hyperbolic geometry to Einstein’s theory of general relativity.’

During this talk she had the feeling that geometers describe different realities from the one that we’re used to. ‘I was fascinated by abstraction and I liked the idea that you could describe spaces that behave in different ways than the three dimensional space we’re familiar with. It reminds me of how Ursula Le Guin, a famous science fiction writer, described herself when she said she was “a realist of a larger reality”.’

### Walking on ice

After finishing her Master’s at the University of Pavia in Italy, Martinelli obtained a PhD from Imperial College London, followed by two postdocs at the University of Edinburgh and the Mathematical Science Research Institute in Berkeley. Though she loves her field, it can sometimes be quite intimidating that her area of research has such a long history.

Martinelli clearly remembers that during the first year of her PhD a professor asked how she was doing. ‘I answered that it felt like walking on ice all the time. There is so much knowledge, that I neither had time nor capacity to read everything. The professor said that this was quite normal, since it is such an ancient subject. If you want to try to do something new, you cannot understand everything that came before you, there is simply too much. You just have to accept that.’

Nevertheless, her commitment to the science doesn’t waiver. And she has an ambition to inspire others too. She has travelled to countries like Cameroon, Kenya and South-Africa, to give masterclasses about algebraic geometry to students there.

Martinelli: ‘The African Institute for Mathematical Studies in Cameroon is a centre that runs a one-year Master’s programme for students coming from all over Africa. As a teacher, you live and work in one building together with the students. It is a very immersive and enriching experience, because you share much more with the students than lecture time. It was truly fascinating to listen to all their different life experiences.’

### Closing the gender gap

And now, she joined the UvA, thanks to the MacGillavry Fellowship programme. Martinelli is very happy she managed to obtain one of these quite exclusive fellowships. Martinelli: ‘Professionally it is a good fit. It is a famous group, my colleagues are very nice, and Amsterdam is a great city to live in.’ Besides that, she believes the fellowship helps closing the gender gap in her research community. ‘There are still very few women in my field. So I believe that these hiring schemes are very much needed.’

When she started her PhD in London, she was the only female PhD student in the large geometry group at Imperial College. Martinelli: ‘This was definitely tough at times. You keep asking yourself whether you really belong there and if you are good enough to stay.’

She recalls the first time that she was invited to speak at an international conference in China. ‘I was the youngest speaker and the only woman. I remember seeing these groups of old male professors talking together during coffee breaks. It felt hard to interact with them. This can really have a negative impact, because networking at conferences is as important as the technical talks you give.’

### Building a support network

But slowly it’s getting better, Martinelli experiences. ‘Conferences put more emphasis on having female speakers and they organise networking events to help you create your own support network. At Berkeley we had a weekly women’s lunch where we talked about different things, we shared experiences of being women in maths, but also chatted about restaurants in Berkeley and things to visit in California.’

Martinelli keeps looking out for great female scientists as source of inspiration and guidance. But at the same time she realises that she is starting to become a role model herself for younger students. ‘For example, when I was in Cameroon a few female students told me that they were so happy I had come, because I was the first female lecturer they had in the whole academic year.’

She also joined the diversity sounding board, that reflects on issues of diversity at the Faculty of Science. Martinelli: ‘We have a monthly diversity journal club in which we read and comment articles on different topics. Recently, I co-hosted a session of the journal club dedicated to cultural biases in maths.

Mathematics is often seen as culture-free. But if you start reflecting on the history of the subject, you realise that, instead, it’s deeply intertwined with history and different power dynamics in the world. I believe that we should put an effort into making the field more inclusive.

Highlighting the work of researchers coming from underrepresented communities and having more awareness on the cultural history of the subject could be a way to start. We should really change the idea that mathematics is an “exclusive boys club”. We should make sure that everybody feels welcome and believes they can be a successful mathematician.’

### Going back to Africa

For the coming years, Martinelli will continue her work on the classification of geometric objects. ‘I am currently working on several projects to better understand the properties of the three building blocks,’ she says. And she hopes to return to Africa in the near future. ‘I hope to go back as soon as the pandemic is over.’

### CV

 2008-2010: BSc, University of Ferrara, Italy 2010-2012: MSc, University of Pavia, Italy 2012-2016: PhD in Algebraic Geometry, Imperial College London, United Kingdom 2016-2018: Postdoctoral researcher, University of Edinburg, United Kingdom 2019: Postdoc, University of Berkeley, United States 2019: Postdoc, University of Glasgow, United Kingdom 2013-2021: Co-organiser of workshops in Algebraic Geometry at various universities in the UK and Kenya 2018-2020: Several grants and a Research Fellowship to give workshops in Kenya, Cameroon and South Africa 2019-present: Assistant professor and MacGillavry fellow, Korteweg-de Vries Institute, University of Amsterdam, the Netherlands