TITLE: Jean Bourgain's relatively lesser known significant contributions
QUESTION [56 upvotes]: Jean Bourgain passed away on December 22, 2018.
A great mathematician is no longer with us.
Terry Tao has blogged about Bourgain's death and mentioned some of his more recent significant contributions, such as the proof of Vinogradov's conjecture with Guth and Demeter. He was of course very prolific and his work spanned many areas. There are a few questions at mathoverflow (e.g., here and here) regarding his technical contributions.
I am sure there are some of his contributions that may not be as widely known, but deserve to be so. Perhaps focusing on results with unexpected aplications may be a good start, but any answers are welcome.
I was unable to find the list of talks at the conference in his honour at the IAS mentioned and linked to in Tao's post, but that may be just me.
Thanks to Josiah Park for pointing out the link to the IAS Bourgain conference, titled 'Analysis and Beyond'. In fact, the videos of the talks are also there.
Edit: Revisiting this question, I read again all the wonderful and informative answers. Mathoverflow is a greeat community.
Edit 2: Sept 2020: There is an upcoming collection of papers on Bourgain's work to be published by the Bulletin of the American Mathematical Society. Terry Tao has blogged about it here, as well as uploading his contribution to arXiv .

REPLY [16 votes]: One body of works which I found mind Boggling is related to the question: "Can you hear the degree of a map". I think the general theme is about the connection between Fourier analysis and algebraic topology. The beautiful videotaped lecture by Haim Brezis is about this topic. On the negative side, Bourgain and Kozma's paper One cannot hear the winding number "constructed an example of two continuous maps $f$ and $g$ of the circle to itself with the same absolute value of the Fourier transform but with different winding numbers, answering a question of Brezis." On the positive side (going back to Cauchy) under further restrictions on $f$, the degree (and other topological invariants) can be "heard" (and is  equal to $\sum_{n=-\infty}^{\infty}|\hat f(n)|^2n$), and Bourgain mainly with Brezis and other coauthors have made various important contributions.