Miruna-Stefana Sorea
SISSA, Trieste, Italy

We study families of polynomial dynamical systems inspired by biochemical reaction networks. These systems are known to enjoy very strong dynamical properties and, due to their remarkable algebraic and combinatorial structures, they were called toric dynamical systems by G. Craciun, A. Dickenstein, A. Shiu and B. Sturmfels. We consider the class of disguised toric dynamical systems, which contains toric dynamical systems, and to which all previous dynamical properties extend naturally. The set of parameters giving rise to (disguised) toric dynamical systems is called the (disguised) toric locus. We show that some reaction networks have an empty toric locus or a toric locus of Lebesgue measure zero in parameter space, while their disguised toric locus is of positive measure. We also propose some algorithms for detecting the disguised toric locus. Moreover, we prove that the (disguised) toric locus is preserved under invertible affine transformations of the network. Finally, we emphasize the topological structure of the toric locus, showing that it is connected. This is based on several recent joint works with L. Brustenga i Moncusí, G. Craciun, S. Haque, J. Jin, M. Satriano, P. Yu.