Newton-Okounkov Bodies and Khovanskii Bases for Applications

Newton-Okounkov bodies were introduced by Kaveh-Khovanskii 
and Lazarsfeld-Musta\c{t}\u{a} to extend the theory of Newton 
polytopes to functions more general than Laurent polynomials. 
This theory has at least two implications for applications. 
First is that Newton-Okounkov bodies provide an approach to 
counting the number of solutions to systems of equations that 
arise in applications. Another is that when the Newton-Okounkov 
body is an integer polytope (there is a Khovanskii basis), there 
is a degeneration to a toric variety which in principal should 
give a numerical homotopy algorithm for computing the solutions. 
This talk will sketch both applications.