I'm worried this might not be fitting for this forum, but it's basically a literature and reference request.

I'm looking to do a project in algebra where we are supposed to research some topic (preferrably something fresh) for a few months and write about it.

Such a new subject that has catched my fancy is that of tropical geometry (in the sense of min/max-plus algebras), but upon researching it, it seems that the only applications to actual problems are in fairly advanced algebraic geometry, which I don't know much about.

So my question is: are there any open questions or recent theory in tropical settings or similar that is comprehensible for an undergraduate to understand, investigate and perhaps speculate a bit about?

I've heard that there is theory for transfering from normal geometry to tropical that maps multiplication to addition via the additivity of the logarithm and a limit process for the addition to maximum part, but I was unable to extract something along those lines from the articles google led me to.

Basically, I'm looking for something that is new, but not super technical or overly messy. Is it perhaps better to investigate some related aspect of algebraic geometry? Any literature reference or insight into these branches of mathematics would be greatly appreciated. If you have a wildly different, but interesting suggestion for a subject you could share that too.

For reference about the level I'm looking for, I'm familiar with subjects such as vector calculus, complex analysis, baby rudin level analysis, basic probability theory, linear algebra, basic fourier theory, basic abstract algebra and so on.

  • $\begingroup$ I wish tropical geometry was a thing when I was in school. Sounds so refreshing... $\endgroup$ – JohnD Nov 25 '14 at 0:10
  • $\begingroup$ I only read the wikipedia article on this Tropical Geometry...but it seems there are tropical analogs of classical theorems. Perhaps you could pick a theorem and search for its tropical analog? $\endgroup$ – Jonny Nov 25 '14 at 0:54
  • $\begingroup$ You could try emailing someone that does this for a living. The person that immediately comes to mind is Bernd Sturmfels. You could also email his student David Speyer (or ask on overflow, which he more oftenly frequents). $\endgroup$ – Alex Youcis Nov 25 '14 at 9:54

The August-September 2014 issue of the American Mathematical Monthly has an interesting article about tropical geometry. Before reading this, I had never known about the subject.

I found it by the Google search "tropical geometry site:maa.org". This is the link I got: http://jmobile.maa.org/i/342731/4

  • $\begingroup$ Thank you for the suggestion! Not a member myself, but the library at my university claims to have this journal, so I'll check it out as soon as possible. $\endgroup$ – neptun Nov 25 '14 at 12:32

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