Rational neural networks are feedforward neural networks with a rational activation function. These networks found their applications in approximating the solutions of PDE, as they are able to learn the poles of meromorphic functions. In this talk, we are going to consider the simplest rational activation function, sigma = 1 / x, and study the geometry of family such architectures. We will show that the closure of all possible shallow (one hidden layer) networks is an algebraic variety, which called a neurovariety.
The moduli space of pointed rational curves has a natural action of the symmetric group permuting the marked points. In this talk, I will present combinatorial and recursive formulas for the induced representation on the cohomology of the moduli space. These formulas are derived from wall crossings of birational models, governed by Hassett’s theory of weighted stable curves and Choi-Kiem’s theory of delta-stability of quasimaps. These results allow us to investigate positivity and log-concavity of the representation. Based on joint works with Jinwon Choi and Young-Hoon Kiem.
A theory of heights of rational points on stacks was recently introduced by Ellenberg, Satriano and Zureick-Brown as a tool to unify and generalize various results and conjectures about arithmetic counting problems over global fields. In this talk I will present a moduli theoretic approach to heights on stacks over function fields inspired by twisted stable maps of Abramovich and Vistoli. For some well-behaved class of stacks, we obtain moduli spaces of points of fixed height whose geometry controls the number of rational points on the stack. I will outline an approach for more general stacks which is closely related to the geometry of the moduli space of vector bundles on a curve. This is based on joint work with Park and Satriano.
