1. **Stating the problem:** We want to understand what simple temporal networks (STNs) are and the underlying mathematics used to optimize them. 2. **Definition of STNs:** A simple temporal network is a mathematical model used to represent temporal constraints between events. It consists of nodes representing time points and edges representing constraints on the time differences between these points. 3. **Mathematical formulation:** Each constraint can be written as an inequality of the form $$l_{ij} \leq t_j - t_i \leq u_{ij}$$ where $t_i$ and $t_j$ are time points, and $l_{ij}$ and $u_{ij}$ are lower and upper bounds on the difference. 4. **Optimization goal:** The goal is to find assignments to all $t_i$ that satisfy all constraints simultaneously. 5. **Underlying math:** This problem is often solved using graph theory and linear programming. The STN can be represented as a weighted directed graph where edges correspond to constraints. 6. **Key algorithm:** The Bellman-Ford algorithm or shortest path algorithms are used to detect negative cycles (which indicate inconsistency) and to find feasible time assignments. 7. **Summary:** STNs use inequalities to model temporal constraints and graph algorithms to optimize and check consistency. Final answer: Simple temporal networks are models of temporal constraints represented by inequalities between time points, optimized using graph theory and shortest path algorithms like Bellman-Ford to find consistent time assignments.