This talk is part of the following reading seminar: Seiberg-Witten theory (Fall 2025).

Abstract

The Weinstein conjecture in 3d states that every contact form on a compact 3-manifold admits a periodic Reeb orbit. We will first explain what these words mean. Then, we will look at Taubes' proof of this conjecture through the lense of a didactically refined article of Hutchings. The whole picture is that embedded contact homology (ECH, a homology built from Reeb orbits and pseudoholomorphic curves) is isomorphic to the Seiberg--Witten complex (SW, a homology built from solutions to three-and four dimensional Seiberg--Witten equations), which is much stronger than the conjecture. In the presentation (as in Taubes' original proof), however, we will not go all the way; we will outline how to use non-triviality of SW to prove existence of a periodic Reeb orbit, which is sufficient for the conjecture. We will not assume prior knowledge of contact geometry or any Floer theory.