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...Notes by eric katz tropical geometry grigory mikhalkin introduction two applications let us begin with examples of questions where trop ical is useful example can we represent an untied trefoil knot as a genus curve degree in rp yes means it turns out that be represented rational but not greater than since such must sit on quadric surface kk figure enumerate real and complex curves simultaneously com binatorics for there way to count or cp through d g points using bipartite graphs algebraic over the tropi cal semi eld t s underlying set half open interval amoeba line x y c wemayinclude log family maps logt z zn will dilated hausdor limit lim v union rays at origin this contains homological information about behavior near ends little else instead consider n parameterized piecewise linear object called noticethatwhenweapplylog weloseinformationabouttheargumentofpoints canthereforestrengthenthe correspondencebetweenclassicalandtropical including some phases picture these give phase data u...