15 Jan – Harald Helfgott: Expander graphs in number theory: the prime divisibility graph

An expander graph may be defined in any of several ways: in terms of boundaries of vertex sets, or in terms of eigenvalues of the graph Laplacian, or by random walks… Expander graphs have become a central object of study in discrete mathematics. Besides having many applications to the study of algorithms, they appear in group theory, in combinatorics and also in number theory. After taking a look at the use of number theory to construct expanders, the talk will then focus on the use of expanders in number theory – and on one new application in particular.

Helfgott will discuss a graph that encodes the divisibility properties of integers by primes. In joint work with M. Radziwiłł, this graph is shown to have a strong local expander property almost everywhere. He will go briefly over several consequences, including stronger versions of results by Tao and Tao-Teräväinen, as well as other results beyond the so-called parity barrier.


Born in Peru in 1977, Harald Andrés Helfgott went on a scholarship to Brandeis and then to Princeton, where he took his doctorate in 2003. He was then a postdoc at Yale University and Université de Montréal, and had a first permanent position in the UK. He has been a researcher at the Centre national de la recherche scientifique (CNRS) in Paris since 2010, becoming a senior researcher in October 2014 (currently on leave). Since 2015, he has been a Humboldt Professor at the University of Göttingen. He is also an honorary professor at Universidad Nacional de San Marcos and has an honorary doctorate from Universidad Nacional de Córdoba.

Download the poster here