In recent years, tropical geometry has developed as a theory on its own. Its two main aims are to answer open questions in algebraic geometry and to give new proofs of celebrated classical results. The main subject of this thesis is concerned with the former: the solution of implicitization problems via tropical geometry. We develop new and explicit techniques that completely solve these challenges in four concrete examples.

We start by studying a family of challenging examples inspired by algebraic statistics and machine learning: the restricted Boltzmann machines F(n,k). These machines are highly structured projective varieties in tensor spaces. They correspond to a statistical model encoded by the complete bipartite graph K(k,n), by marginalizing k of the n+k binary random variables. In Chapter 2, we investigate this problem in the most general setting. We conjecture a formula for the expected dimension of the model, verifying it in all relevant cases. We also study inference functions and their interplay with tropicalization of polynomial maps.

In Chapter 3, we focus on the particular case F(4,2), answering a question by Drton, Sturmfels and Sullivant regarding the degree (and Newton polytope) of the homogeneous equation in 16 variables defining this model. We show that its degree

is 110 and compute its Newton polytope. Along the way, we derive theoretical results in tropical geometry that are crucial for other examples in this thesis, as well as novel computational methods.

In Chapter 4, we study the first secant varieties of monomial projective curves from a tropical perspective. Our main tool is the theory of geometrical tropicalization developed by Hacking, Keel and Tevelev. Their theory hinges on computing the tropicalization of subvarieties of tori by analyzing the combinatorics of their boundary in a suitable (tropical) compactification. We enhance this theory by providing a formula for computing

multiplicities on tropical varieties. We believe that this construction will give insight to understand higher secants of monomial projective curves, which are key objects in toric and birational geometry.

In Chapter 5, we answer the general question of implicitization of parametric surfaces in 3-space via geometric tropicalization. The generic case, together with its higher-dimensional analog, was studied by Sturmfels, Tevelev and Yu. We address this problem for non-generic surfaces. This involves understanding the combinatorics of the intersection of irreducible algebraic curves in the two-dimensional torus and explicitly resolving singularities of points in curves by blow-ups. We conclude with a brief discussion on open problems, including connections to Berkovich spaces, extension of the theory to non-archimedean valued fields and applications of tropical implicitization to classifying tropical surfaces in three-space.

Methane hydrates are crystalline solids of water that contain methane molecules trapped inside their molecular cavities. Gas hydrates with methane as a guest molecule form Structure I hydrates with a unit cell containing 46 water molecules arranged on 2 small dodecahedral cages and 6 tetra decahedral large cages. An ideal Structure I methane hydrate unit cell contains 8 methane molecules, with one in each cage. Methane molecules are classified according to whether they occupy the large tetra decahedral or the small dodecahedra cells. The influence of occupation and the difference between the behavior of methane release during the dissociation process for the different cage types is the major interest of this work.To assess and analyze the structure evolution during the dissociation of methane hydrates, a series of molecular dynamics simulations using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) is conducted. The dissociation conditions examined include different heating rates at 0.8 TK/s, 4TK/s, 40 TK/s and 400 TK/s, and different temperature increments, ∆T, as steps of 80 K, 85 K, 90 K, 95 K and 100 K above hydrate equilibrium stability conditions for 5 ns. Both simulated systems were first equilibrated at 270 K and 5 MPA. The potential energy of the system, mean-squared displacement (MSD), and the radial distribution function were analyzed to determine the full process of dissociation, temperature changes, molecular diffusive behavior, and structure evolution. Temperature step results showed the earliest dissociation starting 50 ps into the simulation at a ∆T of 100 K, while at a ∆T of 80 K, dissociation was not observed. There was not a clear dissociation preference observed between large and small cages, so it appears that the dissociation affects the entire structure uniformly when temperature increases are applied throughout the system rather than transported from a boundary. Temperature ramping simulations showed that the dissociation temperature increased with an increased heating rate. The mean-squared displacement results for the oxygen atoms in the water molecules at a high heating rate of 400 TK/s showed a similar behavior to that for methane gas. This behavior may be an indicator of fast evaporation for water molecules, while at slower heating rates methane molecules showed much higher MSD values, indicating diffusive behavior. As in the temperature step simulation there were not clear differences in dissociation between large and small cages, which suggests homogeneous dissociation in all cases. While this study showed that a total occupied hydrate will experience homogeneous dissociation independent of the heating method, (i.e., either gradual temperature increases or constant high temperature exposure), future studies can examine the stability of the hydrate at different occupancies. Additionally, since hydrate is exposed to liquid water in natural settings, it is important for future studies to analyze the impacts of a liquid water interface in the stability and dissociation dynamics of methane hydrate.