The space of real symmetric matrices whose principal minors are all non-zero is called a space of principally regular matrices. This space parametrizes the space of totally non-degenerate quadratic forms which are local models of ridgy Lagrangians introduced as a tool in the arborealization program. I will give a complete description of the space in a case of 2×2 and 3×3 matrices, discuss the general case, and formulate our main conjecture. I will also discuss a combinatorial rule satisfied by the signs of principal minors as an entry point to understand an asymptotic of a number of connected components of the space for a general n.

Refreshments will be served preceding the talk, starting at 2:45.