SPECIAL MATHEMATICS COLLOQUIUM
Speaker: Tyler Foster
Abstract. There are several closely related conjectures that describe the distribution of primes inside rings of integers. These conjectures attempt to provide an asymptotic count of the number of primes inside "short intervals" - intervals whose widths grow at a certain rate as they move off to infinity. Two important examples of conjectures of this type are the Prime Number Conjecture for Short Intervals and the Hardy-Littlewood k-Tuple Conjecture for Short Intervals. We currently only have proofs of these conjectures in special cases. In the present talk, I will provide an introduction to recent and ongoing work with Efrat Bank in which we define short intervals of functions on algebraic curves over finite fields. Using tools that exist in the algebro-geometric setting, tools that are not available for number fields, we are able to prove function field analogues of the Prime Number Conjecture for Short Intervals and the Hardy-Littlewood k-Tuple Conjecture for Short Intervals. The talk will be a gentle introduction to our work. If time permits, I will explain some of the techniques we use in our proofs.