Abstract:
Additive Number Theory, also known as Additive Combinatorics, is a relatively young area of Mathematics and is part of Combinatorial Number Theory. The subject can best be described as the study of the additive structure of sets and, as such, often focuses on sumsets, that is A+B := {a+b | a in A, b in B}. Classical problems in Number Theory can often be stated in terms of sumsets. For example, Lagrange’s Four Square Theorem that every nonnegative integer can be written as a sum of four squares can be expressed in sumsets as N = S+S+S+S where N is the set of nonnegative integers and S is the collection of the squares of integers. The Goldbach Conjecture also has a sumset interpretation. As the focus is Combinatorical, we often concern our study with counting and surprisingly there are times knowing something about the size of A+B forces a structure on A and B (as in the classic Cauchy-Davenport Theorem). Density questions can be addressed and interestingly there are even applications of zero-free sumsets in computer security.
After an introduction, this talk will focus on the structural result stemming from knowing the size of a sumset, that is the aforementioned Cauchy-Davenport Theorem, as well as a related conjecture of Paul Erdős and Hans Heilbronn from the early sixties. Both problems were originally concerned with A and B as subsets of the integers modulo a prime, but we will discuss their extension to arbitrary finite groups as well as recent generalizations of the problems. Technical proofs will be avoided (the talk will be very accessible to a general audience and grad students should come), but attendees will be treated to wonderful results using tools from Combinatorics and Group Theory including the Polynomial Method and the famous Feit-Thompson Theorem. Stories of Erdős and humorous anecdotes of dropped names will most likely occur.
704 Thackeray Hall