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This article discusses the design of the Apprenticeship Program at the Fields Institute, held 21 August - 3 September 2016. Six themes from combinatorial algebraic geometry were selected for the two weeks: curves, surfaces, Grassmannians, convexity, abelian combinatorics, parameters and moduli. The activities were structured into fitness, research and scholarship. Combinatorics and concrete computations with polynomials (and theta functions) empowers young scholars in algebraic geometry, and it helps them to connect with the historic roots of their field. We illustrate our perspective for the threefold obtained by blowing up six points in $mathbb{P}^3$.
We study standard monomial bases for Richardson varieties inside the flag variety. In general, writing down a standard monomial basis for a Richardson variety can be challenging, as it involves computing so-called defining chains or key tableaux. How
Hessenberg varieties are subvarieties of the flag variety parametrized by a linear operator $X$ and a nondecreasing function $h$. The family of Hessenberg varieties for regular $X$ is particularly important: they are used in quantum cohomology, in co
We propose an algorithm to compute the GIT-fan for torus actions on affine varieties with symmetries. The algorithm combines computational techniques from commutative algebra, convex geometry and group theory. We have implemented our algorithm in the
We give a notion of combinatorial proximity among strongly stable ideals in a given polynomial ring with a fixed Hilbert polynomial. We show that this notion guarantees geometric proximity of the corresponding points in the Hilbert scheme. We define
In this paper, we study the class of free hyperplane arrangements. Specifically, we investigate the relations between freeness over a field of finite characteristic and freeness over $mathbb{Q}$.