This volume consolidates selected articles from the 2016 Apprenticeship Program at the Fields Institute, part of the larger program on Combinatorial Algebraic Geometry that ran from July through December of 2016. Written primarily by junior mathematicians, the articles cover a range of topics in combinatorial algebraic geometry including curves, surfaces, Grassmannians, convexity, abelian varieties, and moduli spaces. This book bridges the gap between graduate courses and cutting-edge research by connecting historical sources, computation, explicit examples, and new results.

Fitness, Apprenticeship, and Polynomials (B. Sturmfels).- From Curves to Tropical Jacobians and Back (B. Bolognese, M. Brandt, L. Chua).- Tritangent Planes to Space Sextics: the Algebraic and Tropical Stories (C. Harris, Y. Len).- Theta Characteristics of Tropical K4-Curves (M. Chan, P.t Jiradilok).- Secants, Bitangents, and Their Congruences (K. Kohn, B. Ivar Utstøl Nødland, P. Tripoli).- Equations of M0,n (L. Monin and J. Rana).- Minkowski Sums and Hadamard Products of Algebraic Varieties (N. Friedenberg, A. Oneto, and R.L. Williams).- paKhovanskii Bases of CoxNagata Rings and Tropical Geometry (M.B. Guillen, D. Corey, M. Donten-Bury, N. Fujita, G. Merz).- Equations and Tropicalization of Enriques Surfaces (B. Bolognese, C. Harris, J. Jelisiejew).- Specht Polytopes and Specht Matroids (J.D. Wiltshire-Gordon, A. Woo, and M. Zajaczkowska).- The Degree of SO(n,C) (M. Brandt, J. Bruce, T. Brysiewicz, R. Krone, E. Robeva).- Computing Toric Degenerations of Flag Varieties (L. Bossinger,S. Lamboglia, K. Mincheva, and F. Mohammadi).- The Multidegree of the Multi-Image Variety (L. Escobar, A. Knutson).- The Convex Hull of Two Circles in R3 (E.D. Nash, A. Firat Pir, F. Sottile, L. Ying).- The Hilbert Scheme of 11 Points in A3 is Irreducible (T. Douvropoulos, J. Jelisiejew, B. Ivar Utstøl Nødland, Z. Teitler).- Towards a Tropical Hodge Bundle (Bo Lin and Martin Ulirsch).- Cellular Sheaf Cohomology in polymake (L. Kastner, K. Shaw, A.-L. Winz).- Index.