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\section*{tropical geometry}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{geometry}{}\paragraph*{{Geometry}}\label{geometry}

[[!include higher geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{books_and_lecture_notes}{Books and Lecture Notes}\dotfill \pageref*{books_and_lecture_notes} \linebreak
\noindent\hyperlink{collections_of_articles}{Collections of articles}\dotfill \pageref*{collections_of_articles} \linebreak
\noindent\hyperlink{papers_and_preprints}{Papers and Preprints}\dotfill \pageref*{papers_and_preprints} \linebreak
\noindent\hyperlink{miscellaneous_mo_questions_discussions_etc}{Miscellaneous: MO questions, discussions, etc.}\dotfill \pageref*{miscellaneous_mo_questions_discussions_etc} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Tropical geometry is often thought of as the [[algebraic geometry]] over the [[tropical semiring]]. Many central results are combinatorial in nature, with relations to the (geometry and combinatorics of) polyhedra, [[matroid]]s, [[cluster algebra]]s and toric geometry. Recently it found applications in explaining [[mirror symmetry]] at a more fundamental level (see e.g. \hyperlink{Gross11}{Gross 11}).

Tropical algebraic geometry establishes and studies some very general principles to translate algebro-geometric problems into purely combinatorial ones.

\emph{Example} In algebraic geometry one often work with polynomials. In tropical geometry, these polynomials are ``tropicalized'' and this turns them into piecewise linear functions.

For instance: $f (x, y) = x^2 + y^2-1$.

This tropicalizes to $trop(f ) = x^2 \oplus y^2 \oplus 0 = 
min(2x, 2y, 0)$, and this is a piecewise linear curve.

(To see this remember that in the [[tropical semiring]], the sum of two numbers is their minimum, and their product is their sum. $x^2 = x\oplus x$, $y^2 = y\oplus y$ and so $x^2+y^2$ converts to $min(2x,2y)$. The 0 will remain mysterious for the moment. (If you cannot wait look at the AARMS notes listed below.))

\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{books_and_lecture_notes}{}\subsubsection*{{Books and Lecture Notes}}\label{books_and_lecture_notes}

Textbook accounts/lecture notes include

\begin{itemize}%
\item [[Diane Maclagan]], [[Bernd Sturmfels]] \emph{Introduction to tropical geometry}, \href{http://homepages.warwick.ac.uk/staff/D.Maclagan/papers/TropicalBook.pdf}{draft book}


\item [[Mark Gross]], \emph{Tropical geometry and mirror symmetry}, CBMS regional conf. ser. 114 (2011), based on the CBMS course in Kansas, \href{http://www.ams.org/bookstore-getitem/item=CBMS-114}{AMS book page}, \href{http://www.math.ucsd.edu/~mgross/kansas.pdf}{pdf}


\item G. Mikhalkin, \emph{Tropical geometry}, book, early draft, \href{http://www.scribd.com/doc/47771116/Tropical-geometry-Grigory-Mikhalkin}{scribd}; \emph{Tropical geometry and its applications}, Proc. Intern. Congr. Math., V. 2 (Madrid, 2006), Eur. Math. Soc., Z\"u{}rich, 2006, 827--852 \href{http://www.mathunion.org/ICM/ICM2006.2/Main/icm2006.2.0827.0852.ocr.djvu}{djvu} \href{http://www.mathunion.org/ICM/ICM2006.2/Main/icm2006.2.0827.0852.ocr.pdf}{pdf}


\item [[Diane Maclagan]], \href{http://homepages.warwick.ac.uk/staff/D.Maclagan/AARMS/AARMSTropical.pdf}{AARMS Tropical Geometry}:lecture notes from a four week graduate summer school on Tropical Geometry held at the University of New Brunswick in July/August 2008 under the auspices of the Atlantic Association for Research in the Mathematical Sciences (AARMS).


\item MSRI introductory workshop on tropical geometry \href{http://www.msri.org/web/msri/scientific/workshops/show/-/event/Wm483}{page}, Aug 24-28, 2009 (with videos of the lectures)


\item Erwan Brugall\'e{}, Kristin Shaw, \emph{A bit of tropical geometry}, \href{http://arxiv.org/abs/1311.2360}{arxiv/1311.2360}



\end{itemize}
\hypertarget{collections_of_articles}{}\subsubsection*{{Collections of articles}}\label{collections_of_articles}

\begin{itemize}%
\item (CM580) Tropical geometry and integrable systems, \href{http://www.ams.org/bookstore-getitem/item=CONM-580}{Contemp. Math. \textbf{580}}, Amer. Math. Soc., Providence, RI, 2012
\item \emph{Tropical and idempotent mathematics}, \href{http://www.mccme.ru/tropical12/Tropics2012final.pdf}{pdf}, proceedings conf. Moscow 2012

\end{itemize}
\hypertarget{papers_and_preprints}{}\subsubsection*{{Papers and Preprints}}\label{papers_and_preprints}

\begin{itemize}%
\item [[Dan Abramovich]], \emph{Moduli of algebraic and tropical curves}, \href{http://arxiv.org/abs/1301.0474}{arxiv/1301.0474}


\item [[Dan Abramovich]], Lucia Caporaso, Sam Payne, \emph{The tropicalization of the moduli space of curves}, \href{http://arxiv.org/abs/1212.0373}{arxiv/1212.0373}


\item I. Itenberg, G. Mikhalkin, \emph{Geometry in the tropical limit}, \href{http://arxiv.org/abs/1108.3111}{arXiv: 1108.3111}


\item M. Einsiedler, [[M. Kapranov]], D. Lind, \emph{Non-archimedean amoebas and tropical varieties}, \href{http://arxiv.org/abs/math/0408311}{math.AG/0408311}


\item [[Mikhail Kapranov]], \emph{Thermodynamics and the moment map}, \href{http://arxiv.org/abs/1108.3472}{arxiv/1108.3472}


\item Walter Gubler, \emph{A guide to tropicalizations}, \href{http://arxiv.org/abs/1108.6126}{arxiv/1108.6126}


\item E. Katz, \emph{A tropical toolkit}, \href{http://arxiv.org/abs/math/0610878}{math.AG/0610878} (Expo. Math. 27 (2009), No. 1, 1-36)


\item Oleg Viro, \emph{Hyperfields for tropical geometry I. hyperfields and dequantization}, \href{http://arxiv.org/abs/1006.3034}{arxiv/1006.3034}; \emph{Tropical geometry and hyperfields}, talk at Mathematics - XXI century. PDMI 70th anniversary, \href{http://www.mathnet.ru/php/presentation.phtml?option_lang=eng&presentid=1286}{video}; \emph{On basic concepts of tropical geometry}, Trudy Mat. Inst. Steklova \textbf{273} (2011), 271--303


\item Patrick Popescu-Pampu, Dmitry Stepanov, \emph{Local tropicalization}, \href{http://arxiv.org/abs/1204.6154}{arxiv/1204.6154}


\item W. Gubler, \emph{Tropical varieties for non-Archimedean analytic spaces}, Invent. Math. \textbf{169} (2007), 321--376.


\item David Speyer, [[Bernd Sturmfels]], \emph{Tropical mathematics}, \href{http://arxiv.org/abs/math/0408099}{math.CO/0408099}; \emph{The tropical Grassmannian}, Adv. Geom. 4 (2004) 389–411 \href{https://arxiv.org/abs/math/0304218}{math.AG/0304218} \href{http://emis.ams.org/journals/AG/4-3/4_389.pdf}{emis}


\item Andreas Gathmann, \emph{Tropical algebraic geometry}, \href{http://arxiv.org/abs/math/0601322}{math.AG/0601322}


\item [[Diane Maclagan]], \emph{Polyhedral structures on tropical varieties}, \href{http://arxiv.org/abs/1302.5372}{arXiv:1302.5372}


\item Paul Johnson, \emph{Hurwitz numbers, ribbon graphs, and tropicalization}, \href{http://arxiv.org/abs/1303.1543}{arxiv/1303.1543} (pages 55-72 in CM580)


\item Brugalle Erwan, Markwig Hannah, \emph{Deformation of tropical Hirzebruch surfaces and enumerative geometry}, \href{http://arxiv.org/abs/1303.1340}{arxiv/1303.1340}


\item Qingchun Ren, Steven V Sam, Bernd Sturmfels, \emph{Tropicalization of classical moduli spaces}, \href{http://arxiv.org/abs/1303.1132}{arxiv/1303.1132}


\item Martin Ulirsch, \emph{Functorial tropicalization of logarithmic schemes: The case of constant coefficients}, \href{http://arxiv.org/abs/1310.6269}{arxiv/1310.6269}


\item Luis Felipe Tabera, \emph{On real tropical bases and real tropical discriminants}, \href{http://arxiv.org/abs/1311.2211}{arxiv/1311.2211}


\item Simon Hampe, \emph{Tropical linear spaces and tropical convexity}, \href{http://arxiv.org/abs/1505.02045}{arxiv/1505.02045}


\item Tyler Foster, \emph{Introduction to adic tropicalization}, \href{http://arxiv.org/abs/1506.00726}{arxiv/1506.00726}


\item G. Mikhalkin, \emph{Quantum indices of real plane curves and refined enumerative geometry}, \href{http://arxiv.org/abs/1505.04338}{arxiv/1505.04338}



\end{itemize}
\begin{quote}%
We associate a half-integer number, called the quantum index, to algebraic curves in the real plane satisfying to certain conditions. The area encompassed by the logarithmic image of such curves is equal to $\pi^2$ times the quantum index of the curve. We use the quantum index to refine real enumerative geometry in a way consistent with the Block-G"ottsche invariants from tropical enumerative geometry.


\end{quote}
\begin{itemize}%
\item Diane Maclagan, Felipe Rinc\'o{}n, \emph{Tropical ideals}, \href{http://arxiv.org/abs/1609.03838}{arxiv/1609.03838}


\item Kiumars Kaveh, Christopher Manon, \emph{Khovanskii bases, Newton-Okounkov polytopes and tropical geometry of projective varieties}, \href{https://arxiv.org/abs/1610.00298}{arxiv/1610.00298}


\item [[Alain Connes]], [[Caterina Consani]], \emph{Geometry of the scaling site}, \href{https://arxiv.org/abs/1603.03191}{arxiv/1603.03191}


\item Keyvan Yaghmayi, \emph{Geometry over the tropical dual numbers}, \href{https://arxiv.org/abs/1611.05508}{arxiv/1611.05508}



\end{itemize}
An alternative algebraic framework for tropical mathematics (not based on semirings), ``more compatible with valuation theory'' has been proposed in

\begin{itemize}%
\item Zur Izhakian, Manfred Knebusch, Louis Rowen, \emph{Algebraic structures of tropical mathematics}, \href{http://arxiv.org/abs/1305.3906}{arxiv/1305.3906}

\end{itemize}
Connections to diophantine integration (involving p-adic integration):

\begin{itemize}%
\item Eric Katz, Joseph Rabinoff, David Zureick-Brown, \emph{Diophantine and tropical geometry, and uniformity of rational points on curves}, \href{http://arxiv.org/abs/1606.09618}{arxiv/1606.09618}

\end{itemize}
\hypertarget{miscellaneous_mo_questions_discussions_etc}{}\subsubsection*{{Miscellaneous: MO questions, discussions, etc.}}\label{miscellaneous_mo_questions_discussions_etc}

\begin{itemize}%
\item MathOverflow : \href{http://mathoverflow.net/questions/72613/mikhalkins-tropical-schemes-versus-durovs-tropical-schemes}{Mikhalkin's tropical schemes versus Durov's tropical schemes}, \href{http://mathoverflow.net/questions/84629/learning-tropical-geometry}{learning-tropical-geometry}


\item \href{http://en.wikipedia.org/wiki/Tropical_geometry}{wikipedia page}


\item $n$Cafe: \href{http://golem.ph.utexas.edu/category/2013/01/tight_spans_isbell_completions.html}{Tight spans, Isbell completions and semi-tropical modules}



\end{itemize}
category: geometry



\end{document}