We present an algebraic method to study four-dimensional toric varieties by lifting matrix equations from the special linear group $SL(2,\mathbb{Z})$ to its preimage in the universal cover of $SL(2,\mathbb{R})$. With this method we recover the classification of two dimensional toric fans, and obtain a description of their semitoric analogue. As an application to symplectic geometry of Hamiltonian systems, we give a concise proof of the connectivity of the moduli space of toric integrable systems in dimension four, recovering a known result, and extend it to the case of semitoric integrable systems with a fixed number of focus-focus points. (joint work with Daniel Kane and Alvaro Pelayo) Please note: There will be a pre-talk for graduate students from 2:30 - 3:00. The regular talk will begin at 3:00.