Jun 15, 2018 at 10:00 Am
Title: Computer driven questions, pre-theorems and theorems in geometry
Abstract: Several numbers can be associated to a free homotopy class X of closed curves on a surface S with negative Euler characteristic. Among these,
– the self-intersection number of X
– the word length of X
– the length of the geodesic corresponding to X (given a metric with constant negative curvature
– the number of free homotopy classes of a given word length in the mapping class group orbit of the class X.
The interrelations of these numbers exhibit many patterns when explicitly determined or approximated by running a variety of algorithms on a computer.
We will discuss how these computations lead to counterexamples to existing conjectures and to the discovery of new patterns . Some of these new patterns, are so intricate and unlikely that they are certainly true (even if not proven yet), so they become “pre-theorems”. Many of these pre-theorems later became theorems. An example of such a theorem states that the distribution of the self-intersection of free homotopy classes of closed curves on a surface, appropriately normalized, sampling among given word length, approaches a Gaussian when the word length goes to infinity. An example of a counterexample (no pun untended) is that there exists pairs of length equivalent free homotopy classes of curves on a surface S that have different self-intersection number. (Two free homotopy classes X and Y are length equivalent if for every hyperbolic metric M on S, length(X) = length(Y).)