Abstract:
Let E be a plane in an algebraic torus over an algebraically closed field. Given a balanced 1-dimensional fan C in the tropicalization of E, i.e. in the Bergman fan of the corresponding matroid, we give a complete algorithmic answer to the question whether or not C can be realized as the tropicalization of an algebraic curve contained in E. Moreover, in the case of realizability the algorithm also determines the dimension of the moduli space of all algebraic curves in E tropicalizing to C, a concrete simple example of such a curve, and whether C can also be realized by an irreducible algebraic curve in E. In the first important case when E is a general plane in a 3-dimensional torus we also use our algorithm to prove some general criteria for C that imply its realizability resp. non-realizability. They include and generalize the main known obstructions by Brugalle-Shaw and Bogart-Katz coming from tropical intersection theory.

Abstract:
Let X be a plane in a torus over an algebraically closed field K, with tropicalization the matroidal fan Sigma. In this paper we present an algorithm which completely solves the question whether a given one-dimensional balanced polyhedral complex in Sigma is relatively realizable, i.e. whether it is the tropicalization of an algebraic curve, over the field of Puiseux series over K, in X. The algorithm implies that the space of all such relatively realizable curves of fixed degree is an abstract polyhedral set. In the case when X is a general plane in 3-space, we use the idea of this algorithm to prove some necessary and some sufficient conditions for relative realizability. For 1-dimensional polyhedral complexes in Sigma that have exactly one bounded edge, passing through the origin, these necessary and sufficient conditions coincide, so that they give a complete non-algorithmic solution of the relative realizability problem.

Abstract:
We construct immersions of trivalent abstract tropical curves in the Euclidean plane and embeddings of all abstract tropical curves in higher dimensional Euclidean space. Since not all curves have an embedding in the plane, we define the tropical crossing number of an abstract tropical curve to be the minimum number of self-intersections, counted with multiplicity, over all its immersions in the plane. We show that the tropical crossing number is at most quadratic in the number of edges and this bound is sharp. For curves of genus up to two, we systematically compute the crossing number. Finally, we use our immersed tropical curves to construct totally faithful nodal algebraic curves via lifting results of Mikhalkin and Shustin.

Abstract:
Tropical geometry gives a bound on the ranks of divisors on curves in terms of the combinatorics of the dual graph of a degeneration. We show that for a family of examples, curves realizing this bound might only exist over certain characteristics or over certain fields of definition. Our examples also apply to the theory of metrized complexes and weighted graphs. These examples arise by relating the lifting problem to matroid realizability. We also give a proof of Mn\"ev universality with explicit bounds on the size of the matroid, which may be of independent interest.

Abstract:
Duality of curves is one of the important aspects of the ``classical'' algebraic geometry. In this paper, using this foundation, the duality of tropical polynomials is constructed to introduce the duality of Non-Archimedean curves. Using the development of ``mechanism'' which is based on ``distortion'' values and their matrices, we discuss some aspects refereing to quadrics with respect to their dual objects. This topic includes also the induced dual subdivision of Newton Polytope and its compatible properties. Finally, a regularity of tropical curves in the duality sense is generally defined and, studied for families of tropical quadrics.

Abstract:
In this paper, we study the correspondence between tropical curves and holomorphic curves. The main subjects in this paper are superabundant tropical curves. First we give an effective combinatorial description of these curves. Based on this description, we calculate the obstructions for appropriate deformation theory, describe the Kuranishi map, and study the solution space of it. The genus one case is solved completely, and the theory works for many of the higher genus cases, too.

Abstract:
Harmonic amoebas are generalizations of amoebas of algebraic curves embedded in complex tori. Introduced in \cite{Kri}, the consideration of such objects suggests to enlarge the scope of classical tropical geometry of curves. In the present paper, we introduce the notion of harmonic morphisms from tropical curves to affine spaces, and show how they can be systematically described as limits of families of harmonic amoeba maps on Riemann surfaces. It extends the fact that tropical curves in affine spaces always arise as degenerations of amoebas of algebraic curves. The flexibility of this machinery gives an alternative proof of Mikhalkin's approximation theorem for regular phase-tropical morphisms to any affine space, as stated e.g. in \cite{Mikh06}. All the approximation results presented here are obtained as corollaries of a theorem on convergence of imaginary normalized differentials on families of Riemann surfaces.

Abstract:
This is mostly* a non-technical exposition of the joint work arXiv:1212.0373 with Caporaso and Payne. Topics include: Moduli of Riemann surfaces / algebraic curves; Deligne-Mumford compactification; Dual graphs and the combinatorics of the compactification; Tropical curves and their moduli; Non-archimedean geometry and comparison. * Maybe the last section is technical.

Abstract:
In this paper, we study completely faithful torsion $\mathbb{Z}_p[[G]]$-modules with applications to the study of Selmer groups. Namely, if $G$ is a nonabelian group belonging to certain classes of polycyclic pro-$p$ group, we establish the abundance of faithful torsion $\mathbb{Z}_p[[G]]$-modules, i.e., non-trivial torsion modules whose global annihilator ideal is zero. We then show that such $\mathbb{Z}_p[[G]]$-modules occur naturally in arithmetic, namely in the form of Selmer groups of elliptic curves and Selmer groups of Hida deformations. It is interesting to note that faithful Selmer groups of Hida deformations do not seem to appear in literature before. We will also show that faithful Selmer groups have various arithmetic properties. Namely, we show that faithfulness is an isogeny invariant, and we prove "control theorem" results on the faithfulness of Selmer groups over a general admissible $p$-adic Lie extension.

Abstract:
We introduce the notion of families of n-marked smooth rational tropical curves over smooth tropical varieties and establish a one-to-one correspondence between (equivalence classes of) these families and morphisms from smooth tropical varieties into the moduli space of n-marked abstract rational tropical curves.