In June 2022 we were pleased to welcome Viresh Patel to the Combinatorics Group as a Lecturer in Optimisation. We asked him to tell us a bit about his mathematical interests and academic journey.
Tell us about your academic journey and what brought you to Queen Mary.I studied maths at Cambridge and became interested in combinatorics and discrete mathematics both because of some inspiring lecturers there and also because I enjoyed some of the computer projects I did that were offered there at the time. In my last year, with some encouragement from a friend in the year above, I decided to apply for a PhD.
I did my PhD at the LSE. I was given quite a bit of freedom to explore my interests and I ended up working on problems related partially ordered sets. After that I did postdocs in Durham, Birmingham, and at Queen Mary, and each of the postdocs helped to shape me in different ways through the guidance of my advisors. In Durham, I worked at the computer science department, where I had a chance to broaden my knowledge and research interests; in Birmingham, I gained the confidence to work on more difficult questions than I had before; and at Queen Mary, I became more independent.
At that point, I was really only looking for permanent positions; I didn't see too many opportunities in my area in the UK, but there was a very good opportunity in the Netherlands, where several permanent positions were advertised as part of a large collaboration called "Networks" at the interface of probability, discrete mathematics, and algorithms. it was a perfect fit for my profile! I had never seriously considered moving abroad, but again, after encouragement from friends and colleagues, I decided it would be a good career move and it would be interesting experience generally. It helped that I had some collaborators and friends there. So I applied, and when I was offered the position, I took it and moved to Amsterdam in 2015. I enjoyed my time in the Netherlands and have many close connections there now. I had a lot of opportunity for pursuing new research directions, designing and teaching new courses, organising events, supervising PhD students and so forth.
In June 2022, I moved back to Queen Mary, this time as a lecturer. During my time as a postdoc at Queen Mary, I remember the combinatorics group being friendly, welcoming and supportive with varied research interests, and this is part of what attracted me back to Queen Mary. The School as a whole is also very diverse and friendly. I'm also happy to be back in London, which I consider to be home.What general area is your research in?I work in the general area of discrete mathematics and have quite broad interests. Graph theory and algorithms feature prominently in my work. A graph consists of a (finite) collection of "nodes" where some pairs of nodes are connected by "edges". Graphs can be used to represent different types of networks e.g. social networks, transport networks, or computer networks. Graph theory is the study of the general properties of graphs where we don’t necessarily have particular applications in mind (here are some fun articles on graphs and networks).
A theme that runs through several of my papers is that of a Hamilton cycle. A Hamilton cycle in a graph is a cycle that passes through every vertex; it is the fundamental object of interest in the famous Travelling Salesman Problem, which you can read about here.
And could you tell us something about a specific problem you have worked on?Recently I have been working with colleagues in Amsterdam on computational counting problems. Here one is interested in finding fast algorithms for (approximately) counting certain objects of interest, often in graphs. We helped develop a technique for obtaining such algorithms that is based on showing that certain polynomials associated with graphs have no roots in certain regions of the complex plane.One of my favourite examples concerns counting graph colourings. Suppose we are given a graph and we wish to colour the nodes of the graph so that no two nodes that are connected receive the same colour. Such a colouring is called a proper colouring of the graph.
One easy way to properly colour the nodes is to give every node a different colour, but what if we are only allowed to use a limited number of colours? If our graph has the property that each node is connected to at most r other nodes, then it turns out that the graph can be properly coloured with r+1 colours (this is not too hard to show). For such a graph, we can then ask: how many “different” ways are there to properly colour the nodes with r+1 colours? A naive approach would be to try all possible qn ways of colouring the nodes and checking which are proper colourings. This is extremely slow and we would like something much faster: however we have to pay for this by asking not for the exact number of proper (r+1)-colourings but an approximate number which is say within 1% of the true number.
One of our contributions was to show that such an algorithm exists provided that the roots of certain types of polynomials called chromatic polynomials always have real part bigger than r+1. This established a connection between two seemingly different problems, and gives a new way of approaching the algorithmic problem.What do you enjoy most about your area of maths?Probably what first attracted me to combinatorics and graph theory was the very concrete nature of the problems. These are often problems one can start thinking about without too much background, and which give the feeling that there is an elegant solution just around the corner. (Of course, usually the solution is not just around the corner and one requires knowledge and experience of various techniques.)
Also the connection to computer science has always been appealing to me. Over time I've come to appreciate and enjoy how tools from other areas of mathematics, e.g. probability and algebra can be used in surprising ways to solve the concrete problems of combinatorics and also in the other direction how techniques from combinatorics have been used to solve problems in other areas of mathematics.What is your approach to teaching?During my PhD I taught many tutorials where I went over solutions to exercises with small groups of students. The most effective way I found was to ask students guiding questions that would allow them to discover the solution for themselves. This has many advantages: it's engaging, students gain experience with the thought process of solving problems, and discovering something is far more memorable than having it presented to you. Of course sometimes I would have to give a strong push in a certain direction, but even then, the hint usually highlighted the key idea. I try to replicate this in lectures as far as is possible. This means asking guiding questions (perhaps rhetorical) to give the sense that a concept or argument is being naturally discovered. Where this is no longer possible, the student hopefully sees where ingenuity and insight has entered the argument.