For many partial differential equations and systems that arise in applications, solutions are critical points of corresponding functionals. One can solve such problems by finding the critical points. We discuss various techniques for finding them and apply the methods to specific problems. The talk can also be followed by nonspecialists and students.

In any casino, the craps table often has the loudest and most excited crowd. This is because when the shooter wins, everyone wins. Its also because craps offers some of the best odds in the entire casino, giving players the best chance to win money. In this talk, the rules of craps will be explained. Also, the concept of markov chains will be introduced, and a finite markov chain will be used to model the game of craps.

The unique solvability of second order elliptic and parabolic equations (in either divergence form or non-divergence form) in Sobolev spaces is well known if the leading coefficients are, for example, uniformly continuous.
However, in general, it is not possible to solve equations in Sobolev spaces unless the coefficients have some regularity assumptions.
In this talk we will discuss some possible classes of discontinuous coefficients with which elliptic and parabolic equations are uniquely solvable in Sobolev spaces.
Especially, as our main results, we will focus on the unique solvability of equations with coefficients only measurable in one spatial variable and having small mean oscillations in the other variables (called partially BMO coefficients).
We will also discuss some applications of our results as well as a new approach to a priori L_p estimates.
Most of the talk is based on joint work with Nicolai Krylov and with Hongjie Dong.