Thom's theory mathematically studies changes in the topological structure of curves or vector fields. It's beautiful as it is dynamic.
Thom applied this theory to dynamical systems, which describe the time dependence of a point in a geometric space. Imagine the swinging of a clock pendulum or the movement of fish on your monitor when your screen saver kicks in. These movements can be modeled mathematically, as can any movement of energie.
Essentially, the rise and fall of a joke could easily be measured topologically. Check out the Lorenz attractor as an example of a non-linear dynamical system. Studying this system helped give rise to Chaos theory. This model (below) also depicts a crude joke (two concepts connected by repulsion).
Small changes causes equilibria or energie to appear or disappear, or to change from attracting to repelling and visa versa, leading to large and sudden changes in the behavior of the system.
Visually speaking, cusp shape (below the joke) looks a little like the topographical energie inherent in a classic one-two punch joke.
With only a few hours to live, Ole, lying in bed, smells something. Cake. Chocolate cake, his favorite! He crawls out of bed and drags himself to the kitchen.
When Lena walks in, there is Ole, sitting at the kitchen table, eating cake. She hollers at him, "Ole! What are you doing in here? You're sick! You should be in bed! You shouldn't be out here eating cake. That's for the funeral!"
The two different concepts: 1. Ole shouldn't be out of bed because he's gravely ill vs. 2. Ole not eating his favorite chocolate cake, one last pleasure before his death, because it's being saved for the funeral. Between these two different concepts lies ambiguity.
At first, it seems as if we're hearing one joke, when suddenly and abruptly, it switches and the ambiguity results in a break of energie pattern (or logic), which can be interpreted as humorous.
John Allen Paulos relates Thom's Catastrophe Theory Model to jokes and humor in the 5th chapter of his Mathematics and Humor book. Paulos' work is well-thought out and equally intuitive, which is a beautiful byproduct of theoretical mathematics.
Measuring a joke for its discontinuities (jumps, switches, reversals) allows us to visually map the logical fallacy presented in the joke. Eventually, it would be nice to depict data and information in topological lines of energie movement (revealing instantly their categories), but that day is probably well into the future (and could also result in ruining the punchline of a joke if the pattern is revealed prior to the joke being told). For now, we still read words.
I'll end this post with a soft pitchfork bifurcation joke...