There is a rich and highly-developed combinatorial theory for Schur functions (Young tableaux, the Littlewood - Richardson Rule, etc), but one can argue that it suffers from a few too many seemingly arbitrary choices and miracles.
On the other hand, Kashiwara\'s theory of crystal bases for quantum groups comes close to subsuming this theory, and at the same time is (a) canonical and (b) has a much greater range of applicability (namely, to the representations of semisimple Lie groups and algebras and their quantum analogues).
The main goal of our talk will be to explain that Kashiwara\'s theory can be developed at a purely combinatorial level, and need not rely on any of the representation theory of quantum groups. Even in type A, this leads to a more natural understanding of the combinatorics of Schur functions.

Motivated by the problems of optimal compensation of executives and of investment fund managers, we consider principal-agent problems in continuous time, when the principal\'s and the agent\'s risk-aversion are modeled by standard utility functions. The agent can control both the drift (the ``mean\") and the volatility (the ``variance\"") of the underlying stochastic process. The principal decides what type of contract/payoff to give to the agent. We use martingale/duality methods familiar from the theory of continuous-time optimal portfolio selection. Our results depend on whether the agent can control the drift independently of the volatility