# How should one approach tropical mathematics?

Let me preface this by saying that my background is pretty meagre (i.e. solid undergrad). However, a few months ago I came across Litvinov - The Maslov dequantization, idempotent and tropical mathematics: A brief introduction which presented an idea that struck me as really remarkable. One can develop a theory of analysis of functions taking values in an idempotent semiring (e.g. the max+ algebra), which happens to be naturally suited towards some traditionally nonlinear problems. Under this formulation (specialized to the case of max+), the integral of a good function corresponds to the supremum, the Fourier transform roughly corresponds to the Legendre transform (!), and it seems that one can develop a theory of "linear" (e.g. in terms of the max+ operations) PDE analogous to the traditional linear theory (!!). For example, the HJB equation is a nonlinear first order PDE, but linear in the max+ sense of the word. This all blew my mind, but after trying to read a few more papers on the subject I decided to put it on the back burner for later thought.

Then, a few days ago I was reading something written by Gian-Carlo Rota in which he makes a remark about developing an "algebra" for multisets. I guess distributive lattices model the "algebra" of sets well enough (in fact there is Birkhoff's theorem), but the quantitative aspect of multisets make this seem inappropriate. So just playing around a bit, I realize that if one models multisets on elements in X by functions from X to the nonnegative integers (the multiplicity), then multiset union corresponds to pointwise addition and multiset intersection corresponds to pointwise min. The min+ algebra on the nonnegative integers! Perhaps this points in the direction of why I think of tropical mathematics as something of interest to people in algebraic combinatorics (maybe this generalization is wrong).

Ok, sorry for that ramble. Essentially, I have 2 questions. First, aside from references to "dequantization", how should I envision the role of tropical mathematics? My lack of background makes it hard for me to get an idea of what is going on here (especially on the geometry side of things), but it seems like there are some big ideas lurking around.

Second, if I wanted to learn more about this stuff, what would be the best route to take? What expository papers should I look at / save for later? It's a bit intimidating that it seems like one needs background in algebraic geometry before one can seriously approach such ideas, but maybe that's just the way it is.

• Given the roots of tropical mathematics in non-Archimedean fields, the only thing one can say for sure is that you should not approach tropical mathematics by many small steps, since otherwise you will never get there. 😁 Sep 13 '21 at 17:50
• What strikes me about this picture is that this science was actively developed back in the 80s, when I was a student, and later when Maslov became my scientific advisor. More than 30 years passed, but still this looks like a novelty for many people. Sep 13 '21 at 18:31
• @SergeiAkbarov It seems that mathematical theories developed "back in the 80s" are usually novelties for many people, in contrary to fashionable technologies.
– Z. M
Sep 13 '21 at 20:08
• @LSpice There are. I heard that maybe during some talks. Recently, I find that some of this is described in Chambert-Loir's Lecture notes, in particular, Chapter 3, but I am looking for a shorter summary (than around 50 pages).
– Z. M
Sep 13 '21 at 20:43
• @LSpice, I presume you would be gratified to get some applause for your non-archimedean remark... :) ... Koan-like, really. :) Sep 13 '21 at 23:21

Sturmfels/Speyer may be a good start. Otherwise I would look around on Arxiv particularly things that Dr. Bernd Sturmfels has written. Also, if you peruse his website (just look up Bernd Sturmfels + UC Berkeley), you'll probably get something interesting.

For an overview of tropical geometry perhaps: A. Gathmann, Tropical algebraic geometry, Jahresbericht der DMV 108

It's available here.

If you're worried that you won't understand the geometric aspects, it might be worthwhile to check out the original papers of Simon -- it looks like most of the tropical ones are in English. Simon was, of course, primarily a computer scientist and combinatorialist, so while I haven't read those papers I suspect that the hardcore algebraic geometry stuff won't be there.

There is a short introduction by Mikhalkin: Introduction to Tropical Geometry (notes from the IMPA lectures in Summer 2007) expanded in more detail in this book: http://www.math.toronto.edu/mikha/book.pdf (the link is now outdated; this may be the same book: Mikhalkin and Rau - Tropical Geometry.)

The Oberwolfach lecture notes by Itenberg, Mikhalkin and Shustin are also nice.

• The book link is outdated; I edited what initially seems likely to be the book, but can't currently find the original to make sure. Sep 13 '21 at 18:22
• @user44191, re, the Wayback Machine reports that link as archived, but it's not currently loading for me. Sep 13 '21 at 20:41

From a completely different perspective, there is Jean-Eric Pin's article Tropical semirings, and then Stéphane Gaubert's short introduction Methods and Applications of $$(\operatorname{max},+)$$ Linear Algebra, Report 3088, January 1997, INRIA and a longer set of lecture notes by the same author, Introduction aux Systèmes Dynamiques à Événements Discrets (these are in French but are excellent lecture notes). The point of view of Gaubert is to describe discrete event systems which are great fun and very approachable. Later when discussing the applications in that area some of the points that you made in the original question can be seen but with much greater emphasis on the applicability. See also Jean-Pierre Quadrat, Max–Plus Algebra and Applications to System Theory and Optimal Control, Proccedings of the International Congress of Mathematicians, Zurich 1994, Birkhauser, 1995.

I learnt a lot from a talk by Jeremy Gunawardena many years ago, and for further information you could look at his Idempotent Semi-rings and his website (which is easy to find).

Another useful website is http://www.maxplus.org/ linking into several groups working on the (max,+)-algebra.

Note none of this really needs the algebraic-geometric side as such, at least to start with, and is much more linked to ‘systems theory’, and applications such as combinatorics, scheduling, and dynamic programming problems in Operational Research. It is also great fun both to learn and to teach.