Welcome to the People in STEM Interview Series! Our first interview is with Dr Anca Mustata and Dr Andrei Mustata, lecturers and reasearchers in the field of algebraic geometry in University College Cork.

We discuss algebraic geometry, working as a mathematician and their experience taking part in maths Olympiads in Romania. Algebraic geometry is a fascinating branch of mathematics looking at the geometric properties of solutions to polynomials.

**Polynomials** are algebraic expressions consisting of 2 or more terms, e.g. 5x² + 3x + 2, or 10x + 7y.

**Q: Could you tell us a bit about your work?**

**Andrei: **So both of us work in the same field, Algebraic Geometry. Essentially you can say that the main object of the work is to look at many polynomials with many variables, make them zero and get the solutions. The geometrical objects you get in this way are rather complicated, and the main point is to find some type of invariant.

Let’s say you have infinitely many such shapes, but for some of them, you can go from one to the other by continuous reform. If I would be able to go from one to the other, then I will put them into the same class. Otherwise you will have infinitely many objects, but if you are allowed to group them together as continuous families, then you have a better understanding of the objects.

These types of invariants are not only in algebraic geometry, they are in geometry and topology. For example, if you want to look at surfaces in a three dimensional space they can be like a sphere, a donut, or lots of donuts put together.

Basically in there the invariants are how many holes you have in them, if you have one donut then you have one hole and if you have many donuts you have the number of holes of the number of donuts. This type of invariant completely classifies the topological sets.

If you want to look at them algebraically, they are not the same. For the same topological shape you will have lots of different surfaces, and you can move from one to the other.

**Q: What is the most challenging part of the work?**

**Anca: **The most challenging part for me is to find the right balance between teaching and research and all the other activities.

In the actual research, the most challenging part is often just formulating the problem. For example, Andrei was talking about invariants for surfaces. And for this you can imagine circles, and counting holes make sense.

But if you look at higher dimensions it doesn’t work. So you ask yourself, okay, why? Should I add another number? Or maybe you might need more complicated structures like groups, you might consider it with mods, you might need rings… It’s kind of a mathematical skill constructed, as we speak.

**Andrei: **How do you get this ring structure? So basically, in sets, you have two operations, taking union and taking intersection. If you take a geometry object, you can look at all the sub-geometric objects. Again, I would only look at them up to the movements and formations. Same thing, I could take union of such objects or intersections.

You have to be a bit careful, for example, when I’m taking a plane intersecting itself it’s just a plane. But if you move it a little bit, in a 3 dimensional space, they intersect at a line. So I want to do the intersection but up to the formations, then you have to be a bit more delicate.

A union basically translated to addition. If I’m taking two planes, then taking unions is one plane plus another plane. The intersection translates to the product, so these invariants have the structure of addition and multiplication. That’s what a ring is.

**Q: Do you ever get stuck at a problem? How do you deal with it?**

**Andrei: **Yes and we’re still dealing with it.

**Anca: **You can read new findings in the literature; you can think of other ways to approach it; You can try easier problems, stop for a few years and then come back…

**Q: How did you get interested in Algebraic Geometry?**

**Anca: **I didn’t do algebra at the beginning, I did more of geometry, the differential kind; For algebra —I found it not challenging enough at the beginning and didn’t go deeper during college, and only later I was struck by its difficulty but also beauty. I picked Algebraic Geometry because Andrei was very enthusiastic and he was saying: ”Oh , you have to do that it’s so cool!”

**Q: How did you find the process of writing scientific papers?**

**Andrei:** That’s the hardest part for me. It’s kind of a torture. It’s very depressing to start writing. I’m working a lot in my head so I don’t write things down. I don’t decide to write it down until everything is finished. But then there’s a lot of small details and it’s very depressing to start because you know it’s going to be complicated. The writing part is the part I like the least.

**Anca:** I find it satisfying in the end. You realise that you need to reformulate when you write some thoughts. Again, in Olympiad, when you get to write you consider all the cases, and you would realise you are missing something.

**Andrei:** Usually when you write down you realise your mistakes.

**Q: Would you recommend mathematics research as a career to settle on at a young age?**

**Andrei:** I think it depends, I don’t think it’s something you can recommend in general. I think it’s something people should do out of passion. If they have the passion then they will automatically want to do it.

**Q: Do you prefer researching more or lecturing more?**

**Andrei: **Researching. I remember this quote from a Russian professor: ”Lecturing is like donating blood.” It’s taking part of the time and energy to prepare for this lecture, this time is what you would otherwise donate to researching.

**Anca: **I recommend it on the other hand. When you lecture, it helps you to understand something better.

**Andrei: **Lecture is very good when you are trying to understand a subject. But then… I’m not sure if it’s as good for the student. If you get too interested in the topic, then you want to go more in depth and your lectures become more challenging… but beyond a certain limit it’s very unpleasant for the student.

**Q: How did you get involved in Olympiad?**

**Anca:** In Romania, we were involved in the Olympiad when we were children, that was how I started doing maths, to a large degree. We had the first Olympiad when I was 11, later there was this final round, there was one place in my school, and a friend and I were equal. So I said okay, you go. So she went and created all this math circle and she came back with all these stories and I felt like I had missed out on something so big.

And so in fifth class I got more serious about it and I went. I think math is the first subject that you’d be involved with an Olympiad. Later there were more things like physics, chemistry, literature… but if you were caught in maths, it would be harder for the others to reach you for the long term.

**Andrei:** There was as good an atmosphere in other Olympiads as in the maths Olympiad.

**Anca: **I went to the literature olympiad, and it was a bit strange. It was very subjective.**Andrei: **The maths was really nice. The best part in my point of view is before the national, they take you for a week in the mountains for a week and they prepare you. That was amazing. We did a bit of hiking, and the atmosphere was really great. I usually didn’t do well in the national olympiad, because I didn’t care. The only thing I wanted was to do well enough so that I could go to these camps. It was something about the atmosphere that caught me, and I really wanted to go.

Thank you for reading the first instalment of the Science Angles People in Stem Interview Series. Don’t miss out on any Science Angles content – follow us on Instagram and Facebook at @scienceangles.

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