What is the tropical Robinson-Schensted-Knuth correspondence? And what are it's applications? A conceptual explanation would be great! Is there an expository note about this somewhere? 
Some references have already appeared in the answers and comments below. To make the question more specific, classical RSK has combinatorial interpretations in terms of symmetric functions, for example. If RSK gives the Cauchy identity:
$$\sum_{\lambda} s _{\lambda}(x)s _{\lambda}(y)=\prod _{i,j} \frac{1}{1-x_iy_j}$$
what is an analogous interpretation for tropical RSK? (From some buzzwords I've heard in a few talks recently, it probably has something to do with shifted or elliptic Schur functions.)
 A: I believe this paper is the starting point for this topic:
MR1872253 (2003j:05128)
Kirillov, Anatol N.
Introduction to tropical combinatorics.

A: I think the clearest write up is in Danilov and Koshevoy. Let me try to get you off on the right foot by explaining what it is we are proving.
RSK (for these purposes) is a bijection between $n \times n$ nonnegative integer matrices and pairs of SSYT of the same shape, filled with entries from $1$ to $n$. By the bijection between SSYT and Gelfand-Tsetlin patterns, this is the same as pairs of GT-patterns of size $n$ with the same bottom row. Remember that a GT-pattern is a triangular array of numbers so, if we glue these two GT-patterns together along their common entries, we get a square. 
Thus, we have encoded both sides of the RSK correspondence as $n \times n$ arrays of integers obeying certain inequalities. On one side, the inequality is that the entries are nonnegative, on the other side we have the GT inequalities. Then the key facts are the following: 


*

*This map is piece-wise linear.

*As a piece-wise linear function, it is (once you get all the coordinates right) the tropicalization of the LU factorization map: The birational map which takes an $n \times n$ matrix and writes it as a product of a lower triangular and an upper triangular matrix.
