Fitness, Apprenticeship, and Polynomials

TL;DR

This paper explores the role of combinatorial algebraic geometry, especially polynomial computations, in training young scholars through the Apprenticeship Program at the Fields Institute, emphasizing connections to historical roots and complex geometric structures.

Contribution

It introduces a structured educational approach integrating combinatorics, polynomial computations, and geometric themes to enhance learning in algebraic geometry.

Findings

Young scholars gain practical skills in polynomial computations.

The program connects modern algebraic geometry with its historical foundations.

Application to complex geometric structures like blow-ups in projective space.

Abstract

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 $\mathbb{P}^3$.