I am thinking of doing research in Tropical Geometry for my senior thesis, and I was wondering to what degree we can translate problems in Algebraic Geometry to Tropical Geometry.

We know that there are many analogous theorems from Algebraic Geometry in the Tropical realm, and that tropical curves arise naturally when we degenerate algebraic curves through the tropical log limit map $\log_q$, but I was wondering how well defined the translation layer between them is.

For instance, if we have a statement in the Algebraic Geometry world, is there (1) a way to directly find an analogous statement in the Tropical world (or say there is no analogous statement) and (2) a way to test if the statement should hold (like if the statement is compatible with the log map)?

I know these are very vague questions, but I was just wondering if anyone had any input. Also, what is this log limit map actually doing, or telling us? Are there any types of degeneration that can tell us about other properties? Is there a sense in which the log degeneration is the most natural one?

I am thinking of doing research in Tropical Geometry for my senior thesis, and I was wondering to what degree we can translate problems in Algebraic Geometry to Tropical Geometry.

We know that there are many analogous theorems from Algebraic Geometry in the Tropical realm, and that tropical curves arise naturally when we degenerate algebraic curves through the tropical log limit map $\log_q$, but I was wondering how well defined the translation layer between them is.

For instance, if we have a statement in the Algebraic Geometry world, is there

1. a way to directly find an analogous statement in the Tropical world (or say there is no analogous statement) and
2. a way to test if the statement should hold (like if the statement is compatible with the log map)?

I know these are very vague questions, but I was just wondering if anyone had any input. Also, what is this log limit map actually doing, or telling us? Are there any types of degeneration that can tell us about other properties? Is there a sense in which the log degeneration is the most natural one?

I am thinking of doing research in Tropical Geometry for my senior thesis, and I was wondering to what degree we can translate problems in Algebraic Geometry to Tropical Geometry.

We know that there are many analogous theorems from Algebraic Geometry in the Tropical realm, and that tropical curves arise naturally when we degenerate algebraic curves through the tropical log limit map $\log_q$, but I was wondering to what extent there is a way to directly translate problems.

For instance, if we have a statement in the Algebraic Geometry world, is there (1) a way to directly translate the statement to the Tropical world and (2) a way to test if the statement should hold (like if the statement is compatible with the log map)?

I know these are very vague questions, but I was just wondering if anyone had any input. Also, what is this log limit map actually doing, or telling us? Are there any types of degeneration that can tell us about other properties? Is there a sense in which the log degeneration is the most natural one?

I am thinking of doing research in Tropical Geometry for my senior thesis, and I was wondering to what degree we can translate problems in Algebraic Geometry to Tropical Geometry.

We know that there are many analogous theorems from Algebraic Geometry in the Tropical realm, and that tropical curves arise naturally when we degenerate algebraic curves through the tropical log limit map $\log_q$, but I was wondering how well defined the translation layer between them is.

For instance, if we have a statement in the Algebraic Geometry world, is there (1) a way to directly find an analogous statement in the Tropical world (or say there is no analogous statement) and (2) a way to test if the statement should hold (like if the statement is compatible with the log map)?

I know these are very vague questions, but I was just wondering if anyone had any input. Also, what is this log limit map actually doing, or telling us? Are there any types of degeneration that can tell us about other properties? Is there a sense in which the log degeneration is the most natural one?