1. Applications and longterm goals

Clemens conjecture
Conjecture 1.05.
A smooth quintic threefold has only finitely many rational curves of each degree. 
Nagata’s conjecture on curves
Conjecture 1.1.
Any curve passing through $n > 9$ general points in $\mathbb{P}^2$ with multiplicities $m_1, \ldots, m_n$ must have degree greater than \[ \frac{1}{\sqrt{n}} \sum_{i=1}^n m_i. \] 
Hartshorne’s conjecture on vector bundles
Conjecture 1.15.
Every rank $2$ vector bundle on $\mathbb{P}^n$, $n \geq 7$, splits as a sum of line bundles. 
Moduli space of metric graphs
The rational cohomology of the moduli space of metric graphs is zero above its “virtual cohomological dimension,” and is also zero in a stable range. Together, this means this module space is a formal space in the sense of rational homotopy theory.Problem 1.2.
[J. Giansiracusa] Does tropical geometry give us a reason why the moduli space of metric graphs should be a formal space? Motivation: Complex Kähler manifolds are formal.
 [Ranganathan & Payne] The nonarchimedean PoincaréLelong formula may be useful here. Is there a way to apply it?

Realization problem for tropical curves
Problem 1.25.
[Ranganathan] Can tropical scheme structure distinguish between realizable and unrealizable tropical curves (in $\mathbb{R}^3$, e.g.). 
Hilbert schemes of points
Problem 1.3.
Construct and compare $\text{Trop}(\text{Hilb}^n(X))$ and $\text{Hilb}^n(\text{Trop}(X))$. 
Problem 1.35.
[Bejleri] Given 2 monomial ideals, when is there a connected smooth curve $(\mathbb{P}^1 \hookrightarrow \text{Hilb}^n(\mathbb{C}^2))$ joining them? 
Conjecture 1.4.
[Payne] $\text{Hilb}^n(\mathbb{C}^3)$ is generically reduced. Compute the schemetheoretic tropicalization of $\text{Hilb}^n(\mathbb{C}^3)$.
 Show that the Behrend function is constant. The Behrend function can be calculated by motivic integration, which can be done tropically.

Problem 1.45.
[Payne] More generally, when are Hilbert schemes of points on threefolds reducible? 
BrillNoether theory
Problem 1.5.
[Chan] Develop a tropical version of limit linear series. 
Tropical Euler characteristic
Problem 1.55.
For a tropical scheme $Y$, define $\chi(\mathcal{O}_Y)$. 
Abelian varieties
Problem 1.6.
[Maclagan] What is a tropical abelian variety? How do we tropicalize subschemes of abelian varieties? 
Problem 1.65.
[Ranganathan] Extend the results of GiansiracusaGiansiracusa’s first paper to maximally degenerate abelian varieties. 
Combinatorics
Problem 1.7.
[Abramovich] Which questions in combinatorics are amenable to tropical scheme theory? 
Problem 1.75.
[Lorscheid & J. Giansiracusa] Use tropical scheme theory to approach MacPherson’s conjecture that the “space of oriented matroids” is homotopy equivalent to the real oriented Grassmannian. 
Cohomology and intersection theory
Problem 1.8.
Develop a theory of cohomology for tropical schemes. 
Problem 1.85.
Does tropical cohomology satisfy Hard Lefschetz? Can we use it to approach problems in combinatorial algebraic geometry? [Maclagan] Can the recent proof of the RotaHeronWelsh conjecture (by AdiprasitoHuhKatz) and its machinery be understood in the context of tropical schemes?
 [Ranganathan] Use tropical schemes to approach the topheavy conjecture of DowlingWilson in the nonrealizable case (recently proved for realizable matroids by HuhWang).

Problem 1.9.
[Maclagan] Develop schemetheoretic tropical intersection theory. Intersection theory for tropical varieties has been quite successful.
 [J. Giansiracusa] Is there a tropical version of Serre’s Torformula for intersection multiplicities?

Problem 1.95.
[Lorscheid] Prove a tropical RiemannRoch theorem by cohomological methods.
Cite this as: AimPL: Foundations of tropical schemes, available at http://aimpl.org/tropschemes.