Abstract

On this Valentine’s day, let’s talk about a love triangle: smooth automorphisms, symplectic geometry, and Dehn twists – a special kind of transformation that “twist” spaces in interesting ways.

In the world of orientable surfaces, Dehn twists are infinite order generators of the mapping class group. Surprisingly, in 4 dimensions Dehn twists have finite order (just 2!). What it is more striking is a result of Seidel, who showed using Floer cohomology that these transformations, which preserve the symplectic structure, when viewed inside the symplectic mapping class group, retain their infinite order in any dimension. In a way this tells us that the symplectic generalization of Dehn twists is the natural one.

In this talk we’ll start from the beginning, introducing Dehn twists from the ground up and exploring how their behaviour splits between the smooth and symplectic worlds. By the end, it will be clear why symplectic geometers can’t stop talking about Floer theory (from which no prior knowledge is required)—it’s not just a tool, it’s a window into the soul of symplectic manifolds.