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 2022年亚洲世界杯预选赛 from 2:30 - 3:00. the regular talk will begin at 3:00.