Prime numbers are often called the building blocks of hole numbers because every hole number is made up of prime numbers multiply together in a unique way. And this means whenever you have a problem about hole numbers, which involves multiplication, like most modern cryptographic systems, you can try and understand this problem from the point of view of prime numbers. And therefore, if you were trying to break a cryptographic code or wanted to understand how safe or secure cryptographic code is, often you can study this using prime numbers and properties about the distribution of prime numbers will give important insights into whether it's feasible to crack a code or it's infeasible to and the cryptographic coding is safe. There are several examples of simple patterns within prime numbers that mathematicians really focus on as a proof of concept result for trying to develop a general understanding of the distribution of prime numbers. So one famous example is the twin prime conjecture, which is asking for the functions n and n plus two to be simultaneously prime. Another one that's important or relevant to cryptography is asking can n and two n plus one be simultaneously prime? Again, these are basic questions that fascinate mathematicians and number theists have studied for hundreds of years, but they turn out to be relevant for cryptography and they're viewed as important proof of concept results for developing a general theory about the distribution of prime numbers. Mathematicians had been thinking for hundreds of years about questions to do with the distribution of primes, patterns in prime numbers and the gap stream prime numbers. And I was very lucky in my research to come upon some discoveries that had given us partial understanding for these gaps and made progress in some of these very longstanding and notorious open problems in mathematics. One particular question that's fascinated mathematicians is the question about gaps between prime numbers and how close together prime numbers can be. Because prime numbers apart from two are all odd numbers two and three are the only pair of prime numbers that differ by exactly one, but it's believed, and this is the famous twin prime conjecture, that there should be infinitely many pairs of primes that differ by exactly two. Along with my collaborators, we were able to show that there's infinitely many pairs of primes that differ by no more than 246. And although this isn't quite the twin prime conjecture, we were able to overcome some of the important difficulties and make important partial progress towards the twin plan conjecture. This is one of the bits of work that I'm most proud of. One of the specific areas that I've worked on a lot and I'm very passionate about is an area of mathematics known as SIV methods. So these are flexible set of techniques for trying to understand the distribution of primes. And it's there where I've really developed new methods that I give us new insights in how the primes are structured and the overall distribution of, if I was given advice to a new starting researcher who wanted to get into mathematics, I think it's always very, very important to try and distill a complicated, difficult problem into a very simple model problem. Normally at the heart of any of these big complex problems, there's some very simple problem that we still don't know how to answer, and I found it very helpful to concentrate on the simple problem to try and solve that problem. And then once I've come up with a solution for the simple problem, try and work it back up into a general solution for the big problem.
