**We caught up with Amedeo, who has recently completed his PhD after successfully defending his thesis, “On some problems in extremal hypergraph theory”.**

**In this interview, we will learn about Amedeo’s experience at LSE in the Department of Mathematics, his thoughts on London, his Italian background and his inspiring outreach work.**

**Hello! To start, would you please say a bit about yourself: where did you grow up, and where did you study for your degrees?**

**Hello! To start, would you please say a bit about yourself: where did you grow up, and where did you study for your degrees?**

I grew up in a beautiful small town called Caiazzo, in Southern Italy. I then went to the University of Padova in northeastern Italy for my bachelor’s and my master’s degree in maths. During my master’s, I also spent one year at the University of Warwick as an Erasmus student.

**Did you always know that you wanted to go into mathematics?**

**Did you always know that you wanted to go into mathematics?**

I cannot say exactly when I became interested in mathematics. However, I am sure that it became apparent at an early stage whilst going to school. I also believe that much of the credit should go to the many maths teachers and professors I was lucky to meet over my educational journey. They presented mathematics as a nice puzzle to solve or a challenge to overcome, and they were always very inspirational for me. So, when I started university, I also wanted to become an aspirational model for the next generation of students. I’ve continued with this mindset until today, so I still spend lots of my free time doing outreach activities for students and the public.

**That sounds great! What type of programmes are you involved in?**

**That sounds great! What type of programmes are you involved in?**

One of them is about training students for maths competitions, as this is something that I used to do, especially when I was in Padova. I am also a maths teacher for an enrichment programme called ‘maths circle’. A maths circle is a short maths lesson on some extra-curricular topics. The idea of a maths circle was originally from Russia and Eastern Europe. However, they have now started in the UK and, for the last two years, in Padova too. It is a fantastic opportunity to meet motivated students, and it is also rewarding. This is especially true when you work with the same students for a period and witness their progress. In terms of outreach, they are also open to everyone, and we always ensure that financial restrictions are not a barrier to getting involved.

**It’s great to hear that you’re giving back to the community that you came from. And maybe this is something that more of us should consider taking up.**

**It’s great to hear that you’re giving back to the community that you came from. And maybe this is something that more of us should consider taking up.**

**We mentioned your postdoc that you are just starting. Honestly speaking, when I started university, I had no idea what a postdoc was. So, could you explain what a postdoc is and how it differs from being a PhD student or a lecturer? **

**We mentioned your postdoc that you are just starting. Honestly speaking, when I started university, I had no idea what a postdoc was. So, could you explain what a postdoc is and how it differs from being a PhD student or a lecturer?**

Generally, a postdoc is any position after a PhD and before you get a permanent academic position. In terms of the job, it is not very different from what you do in your PhD in that you work on open problems, write papers, prove theorems, attend conferences, present at seminars, and so on. But it is also a period in which you become more independent. So, you also develop your research plans, your research projects, and your research programmes. And it is also the time when you start becoming involved in projects where you are the most senior member of the team. For example, I have a project with some PhD students and I’m the only post-doc in this team, so I feel some responsibility for the junior members. On the other hand, compared to lecturers, I do not have as many administrative responsibilities so I can really focus on my research, which is very good.

**So, as a PhD student, you learn how to conduct research, and the post-doc allows you the freedom to do just that. Speaking of which, would you please describe what research you do?**

**So, as a PhD student, you learn how to conduct research, and the post-doc allows you the freedom to do just that. Speaking of which, would you please describe what research you do?**

I work in an area of mathematics which is called graph theory. Graphs are mathematical structures that encode relations between pairs of objects. These objects are called vertices, and then we have edges that connect any two objects that have a certain relationship with each other. For example, imagine the vertices are your friend, and some of your friends don’t like each other. So, let’s draw an edge between any two people that don’t like each other. Therefore, if you want to host a dinner party, it might be better not to invite two people connected on this graph.

Now, in graph theory, we study the properties of structures that can be found in graphs. This involves a good combination of tools from algebra, probability, and analysis, and I like it when two areas of maths meet, and you realise that they are speaking slightly different languages, but they are mainly talking about the same problems. This is when tools from one area can solve problems from another and vice versa. Sometimes it is surprising, but sometimes, it is a matter of different notation, and two areas are calling the same object different things.

**Could you tell us a bit about the recent breakthrough result in Ramsey theory?**

Let me start by a small example of a Ramsey problem. Suppose you have six people, some of them are friends, and some are not. You can show that you can always find three people where either they are all friends with each other or none of them are friends with each other. More generally, in Ramsey theory, we consider a graph on *n *vertices, connect each pair of vertices by an edge, and colour each edge either red or blue. Then, we ask, “which structures in this graph can we find where all of the edges are one colour?”.

A longstanding challenge is to determine how big *n *has to be so that we can always find *k *vertices such that the edges between them are either all red or all blue. The exact answer is only known for small values of *k*, and the best general bound comes from an old result of Erdős and Szekeres from 1935, which showed that taking *n*=4* ^{k}* is enough. The result you mentioned, due to Marcelo Campos, Simon Griffiths, Robert Morris, and Julian Sahasrabudhe, shows that

*n*=3.9922

*is already enough. Although this might seem like a very small improvement, in fact, it is a pivotal one, being the first exponential improvement since 1935. Moreover, it is plausible that the constant 3.9922 can be reduced even more.*

^{k}**That sounds very interesting. **

**That sounds very interesting.**

**You have been at LSE for four years of your PhD. What advice would you give to someone that is just starting at LSE? **

**You have been at LSE for four years of your PhD. What advice would you give to someone that is just starting at LSE?**

LSE is a great place to study and do a PhD. The department is not very big, so there is also a strong sense of community. I was able to have a lot of interaction with my supervisors and everyone in the graph theory community here. I also had many opportunities to interact with everyone else in the department, which only enhanced the experience.

My first suggestion is to enjoy every moment at LSE and take advantage of all the opportunities that present themselves. Being based in London means that you are always close to many universities and people working in the areas that may interest you, so you can always socialise and network.

My second suggestion is about research. Sometimes research is difficult because there are weeks when you will be stuck on your problems. However, it is always good to remember that this is normal. I would advise you to be happy with the little steps that you make, and not pretend that you will be able to solve the problem in a couple of weeks. I often find that there is not a specific moment when I solve the problem, instead, there is a moment when I can combine all the hard work and small steps I made in advance.

There are problems that I started thinking about early on in my PhD, and either made a little progress or I was stuck, and from time to time, I think about them again. As stated before, this is completely normal. If you have never been stuck on a problem, then you are working on problems that are not interesting!

**Thank you for that advice. Some of it is probably good life advice as well. **

**Thank you for that advice. Some of it is probably good life advice as well.**

**I have two final questions for you. Firstly, what is your favourite thing about London? And secondly, I would be remiss if I did not ask an Italian for a restaurant recommendation.**

**I have two final questions for you. Firstly, what is your favourite thing about London? And secondly, I would be remiss if I did not ask an Italian for a restaurant recommendation.**

London is a huge city, and I like that it is very international. I have had the opportunity to meet people from all over the world, and this is one of the few places where you can do that. It is nice to learn more about different cultures, especially those that are geographically very far away from us.

In terms of restaurants, I’d like to recommend a fantastic pizzeria as I come from the south of Italy. My favourite one is called 50 Kalò and it is near Trafalgar Square. It’s a very famous pizzeria from Napoli, and a few years ago they opened this branch in London.

**Excellent, I will take up that suggestion. It’s been a pleasure, Amedeo! Thank you for your time. **

**Excellent, I will take up that suggestion. It’s been a pleasure, Amedeo! Thank you for your time.**