World Sheet Problem

If one adapts the picture that particles are not points but circles, then the world lines of particles become world sheets. The topology of this sheet describes the interaction. A torus for example could describe the emergence of a particle, which then splits into two particles, merges again to one particle then disappears. Given a 4-manifold, what is the genus of a typical surface? These are not well defined questions in the continuum, unless one decides what probability measure of surfaces one takes. Already in the case of random curves, like Brownian motion, it is not obvious how to define the Wiener measure on the set of all continuous paths and this only dealt with the problem of a single particle which is not allowed to split, merge or form loops. My level set theorem naturally leads to probability spaces of co-dimension k manifolds in q manifolds. We can therefore look at the statistical problem. Ioriiginally suggested that to Jennifer Gao in 2023 as a senior thesis project but Jennifer then wrote a beautiful thesis discussing more topological aspects of graphs.

Here is a simple case. Lets take a 2-manifold G and ask the question what is the expected number of points of a random codimension 2 surface (a collection of finite points). We have already a probability space of functions where V is the set of vertices of the 2-manifold. For every function g, we have a level set that consists of a set of finitely many points. If X(g) is the the number of points, what is the expectation E[X] ? This is the world sheet problem for 2-manifolds. For 4-manifolds, the analog is to look at X(g), the Euler characteristic of (one could also look at the number of connected components, or the surface area as a functional). It is obvious that we have an extremely rich playground to experiment. My index formula relates this question locally at least with curvature and I talk a bit about this in the presentation from yesterday.

Let me look here at the simplest possible problem of this type. Take a 1 connected 1 manifold (it must be with . Take a random function g from V to {-1,1} and look at the number X(g) of roots of this function (called nodes). These are of course the level sets of g. Technically they are the edges on which the function g changes sign. The expectation of nodes is linear in n of course (even simpler than Crofton as we are in a one dimensional situation) and we have E[X] = n/2. Now, if the circle is the unit circle in a 2-manifold and the measure on functions is changed because the number of sign changes depends on the value of the function at the center. The expectation is then E[X]=n/3. The index formula (discussed in one of the Topics in geometry