Sieve theory and gaps between primes

Prime numbers are the subject of many conjectures - there is a huge gap between what we believe to be true and what we can prove. However, in the last few decades several breakthroughs have been achieved with the aid of sieve theory. A notable example is Yitang Zhang’s work on bounded gaps between primes, which may be viewed as a step towards resolving the twin prime conjecture. In my talk, I will first give a brief, accessible survey of sieve methods, with a particular focus on their application to twin primes. I will also talk about some recent progress in improving upper bounds on those gaps between m consecutive primes which appear infinitely often.