Exploiting combinatorial structures for algebraic and geometric decompositions

We will develop novel tools to solve several important real-world problems: (i) Proving safety of programs (Computer Science), (ii)
Computing network reliability (Industrial Engineering), (iii) Causality (Statistics), and (iv) Geometry of particle interactions (Physics).
These problems are all traditionally modeled as polynomial systems. However, given that solving a general system is very difficult, they all lack scalable algorithms. The main idea is that our applications’ systems tend to have additional structural properties. Our vision is to exploit these specific properties to sidestep the difficulty of solving general systems, and obtain dedicated solution methods for realworld cases. The most profound impact is in program verification, focusing on proving the presence/absence of bugs or vulnerabilities in code. Given the ever-increasing role of software in safety-critical operations, e.g. avionics and healthcare, it is vital to perform software verification reliably and exactly. Our results will have a huge downstream effect on every aspect of verification, ultimately leading to more secure and trustworthy software in all sorts of applications. Another focus is Network Reliability with significant applications in economics and epidemiology. We will develop novel methods to compute the reliability. Finally, we will study the Amplituhedron: a geometric object that dramatically simplifies calculations of particle interactions in Physics