Real algebraic geometry and combinatorics

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Enumeration is the study of counting mathematical objects. Some of the most important enumeration problems have their orgins in algebraic geometry. These "enumerative algebraic geometry" problems have a dual nature: they have a combinatorial (discrete) side and geometric (continuous) side. In some cases, algebra can provide a bridge between these two sides. However, there many counting problems that are fundamental for a number of areas of mathematics, where we do not have a complete picture like this; in fact, there are relatively few instances where we know how all the pieces fit together. Broadly, this aim of this research is to address these knowledge gaps.******One of the main tools in this research is the Mukhin-Tarasov-Varchenko Theorem, a seemingly innocuous result about solutions to certain equations over the real numbers, which turns out to have profound consequences in enumerative geometry. The Mukhin-Tarasov-Varchenko theorem provides a framework for explaining many of the strange phenomena that arise in enumerative combinatorics. The main objective of this proposal is to find new connections linking this important theorem with other parts of real algebraic geometry and algebraic combinatorics.******This research concerns mathematical objects that are central to many areas of pure mathematics (geometry, topology, combinatorics, algebra, complex analysis), and also has applications to physics (statistical mechanics) and engineering (control theory).