Hadamard products, Bergman fans and Rigidity: Using dumb multiplication in a clever way

The Hadamard product can be considered the "naive" product of linear spaces in the same ambient space: just multiply every pair of vectors together coordinate-wise. At first this seems a rather bad idea, but this simple concept has many different applications throughout mathematics. In my talk I will discuss a new method for counting the number of points contained in a generic fibre of the Hadamard product of linear spaces that utilises Bergman fans – tropical linear representations of matroids that exist even when no actual linear representation does. I will show that the stable intersection of a Bergman fan and the flip of a Bergman fan (which we have cunningly named the flip product of matroids) exactly describes the number of points of a generic Hadamard product fibre. I will further describe how this can then be utilised in the field of rigidity theory. This talk involves joint work with Oliver Clarke, Matteo Gallet, Georg Grasegger, Daniel Green Tripp and Ben Smith