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 2 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 $\mathbb{P}^{3}$.

Fitness, Apprenticeship, and Polynomials

We present and solve a new computational geometry optimization problem in which a set of circles with given radii is to be arranged in unspecified area such that the length of the boundary, i.e., the perimeter, of the convex hull enclosing the non-overlapping circles is minimized. The convex hull boundary is established by line segments and circular arcs. To tackle the problem, we derive a non-convex mixed-integer non-linear programming formulation for this circle arrangement or packing problem. Moreover, we present some theoretical insights presenting a relaxed objective function for circles with equal radius leading to the same circle arrangement as for the original objective function. If we minimize only the sum of lengths of the line segments, for selected cases of up to 10 circles we obtain gaps smaller than $$10^{-4}$$
 using BARON or LINDO embedded in GAMS, while for up to 75 circles we are able to approximate the optimal solution with a gap of at most $$14\%$$
 .

Packing circles into perimeter-minimizing convex hulls

Moment maps, strict linear precision, and maximum likelihood degree one

The multi-image variety is a subvariety of $\mathop{\mathrm{Gr}}\nolimits (1, \mathbb{P}^{3})^{n}$ that parametrizes all of the possible images that can be taken by n fixed cameras. We compute its cohomology class in the cohomology ring of $\mathop{\mathrm{Gr}}\nolimits (1, \mathbb{P}^{3})^{n}$ and its multidegree as a subvariety of $(\mathbb{P}^{5})^{n}$ under the Plucker embedding.

The Multidegree of the Multi-Image Variety

We give an explicit formula for the reciprocal maximum likelihood degree of Brownian motion tree models. To achieve this, we connect them to certain toric (or log-linear) models, and express the Brownian motion tree model of an arbitrary tree as a toric fiber product of star tree models.

Reciprocal Maximum Likelihood Degrees of Brownian Motion Tree Models

The Maximum Likelihood Degree of Toric Varieties

We describe convex hulls of the simplest compact space curves, reducible quartics consisting of two circles. When the circles do not meet in complex projective space, their algebraic boundary contains an irrational ruled surface of degree eight whose ruling forms a genus one curve. We classify which curves arise, classify the face lattices of the convex hulls, and determine which are spectrahedra. We also discuss an approach to these convex hulls using projective duality.

The Convex Hull of Two Circles in $\mathbb{R}^{3}$

We describe convex hulls of the simplest compact space curves, reducible
quartics consisting of two circles. When the circles do not meet in complex
projective space, their algebraic boundary contains an irrational ruled surface
of degree eight whose ruling forms a genus one curve. We classify which curves
arise, classify the face lattices of the convex hulls, and determine which are
spectrahedra. We also discuss an approach to these convex hulls using
projective duality.

Convex Hull of Two Circles in R^3

The first steps in defining tropicalization for spherical varieties have been taken in the last few years. There are two parts to this theory: tropicalizing subvarieties of homogeneous spaces and tropicalizing their closures in spherical embeddings. In this paper, we obtain a new description of spherical tropicalization that is equivalent to the other theories. This works by embedding in a toric variety, tropicalizing there, and then applying a particular piecewise projection map. We use this theory to prove that taking closures commutes with the spherical tropicalization operation.

Evan D. Nash

Papers

The Maximum Likelihood Degree of Toric Varieties

The Convex Hull of Two Circles in \(\mathbb{R}^{3}\)

Convex Hull of Two Circles in R^3

Global spherical tropicalization via toric embeddings

Global Spherical Tropicalization via Toric Embeddings