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There are many open problems in the young field of tropical geometry; my work raises the following questions. (1) In my construction, 
the branched covers with connected and disconnected domains are considered simultaneously. Can I construct a tropical moduli space 
for only the connected covers and reproduce my original object by exponentiation of this new one (in the combinatorial sense)? (2) My 
tropical Hurwitz spaces are abstract (as opposed to embedded). Does my construction give a natural embedding? (3) The weights and 
the gluing both take their most natural form in the language of the topological quantum field theory that computes Hurwitz numbers. 
Although this construction will not work for an arbitrary TQFT, I conjecture that it will generalize to a large class of them. What are 
the right hypotheses to put on these TQFTs to make my construction generalize? (4) One goal of the synthetic approach to tropical 
geometry is to inform our investigation into tropicalization methods. Does my construction suggest any techniques for tropicalizing 
the classical Hurwitz spaces? And finally, (5) there are many other classical objects that have a structural morphism like Hurwitz spaces. 
Can I define tropical versions of these classical objects as well?

Research Ideas

Many of these research goals rely on combinatorics or computing that could be done by an advanced undergraduate. For example, for a 
fixed degree, computing Hurwitz numbers for arbitrary ramification profiles could be programmed. An undergraduate could certainly do 
this for a few small choices of the covering degree and use this to say more specific things about the other techniques. Conversely, many 
of the useful formulae for computing certain Hurwitz numbers are recursive with a strong combinatorial flavor. An undergraduate could 
also use these recursive structures to produce constructions similar to mine.

Project Ideas

There is also a lot of work to be done making tropical geometry approachable. I would love to co-author a document with a student 
about some aspects of tropical geometry for a broader audience. For example, many of the basic constructions from classical geometry 
make perfect sense in tropical geometry, but the details use a different sort of intuition than we usually employ in modern algebra. 
Similarly, tropicalization makes sense for any object defined over a non-Archimedean field, like Q_p. As such, tropical geometry has 
some applications in elementary number theory. Writing these ideas in a nice for and preparing general audience talks about them 
would be a fabulous way for a student to learn some math and practice crucial communication skills.