Famous Robinson Schensted Knuth correspondence gives a correspondence between the matrices with non-negative integer entries and pair of semi standard tableaux. The proof that I have seen is highly combinatorial e.g. in Knuth's paper [Permutations, matrices, and generalized young tableaux]. Does there exist a geometrical proof of this correspondence? 



Geometric proof of Robinson-Schensted-Knuth correspondence?
This depends on the meaning of the word "geometric". If you are thinking of RSK and want the geometry in the way the algorithm is *presented* (in the case of permutations only), Viennot's paper [Une forme geometrique de la correspondance de Robinson–Schensted](https://doi.org/10.1007/BFb0090011) [mentioned](https://mathoverflow.net/a/30932) by PeterR is your answer. If you are thinking of geometry of flag varieties, you might like Steinberg's theorem (see van Leeuwen's thesis [A Robinson–Schensted algorithm in the geometry of flags for Classical Groups](http://www-math.univ-poitiers.fr/~maavl/pdf/thesis.pdf) and his *J. Algebra* paper [An Application of Hopf Algebra techniques to Representations of Finite Classical Groups](http://www-math.univ-poitiers.fr/%7Emaavl/pdf/Hopf.pdf). Finally, if you want RSK to be a map between integer points in polytopes, there are several versions of that, going back to Gansner in 1981 (you can find a description of that and references in the second half of the paper [Hook length formula and geometric combinatorics](https://www.mat.univie.ac.at/~slc/wpapers/s46pak.pdf) of mine).


