On mathematical aspects of the most recent Nobel Prize in economics winners' work
The mathematical content of Fama and Shiller's research is pretty minimal. In the short run, returns are unpredictable, while in the long run it seems to be predictable. The short run unpredictability is an important input to mathematical finance, since it helps motivate the geometric Brownian motion model for returns, but Fama and Shiller's research is straightforward statistics.
Hansen's work is more mathematical. There is a long tradition in statistics of fitting models by matching moments. For example, you can fit a Gaussian to data by matching the mean and standard deviation. To do this, you pick as many moments as you have parameters to fit. Hansen's proposed the generalized method of moments, where you choose more moments than parameters, and you test statistically for the ability of the model to fit all of the moments simultaneously. Because of estimation error, it won't match all the moments exactly, but with some error, but we can test whether that error is "too large" for the model. For example, a Gaussian has skewness and excess kurtosis of exactly zero, while in a sample these will not be exactly zero. With some work you can adapt this into a test for Gaussianity, such as described in this paper by Richardson and Smith.
This is particularly interesting for economics, because one class of models that economists consider are infinite-horizon rational-expectations models. These models assume that agents solve an infinite-horizon planning problem. Prices in the model enter as a kind of Lagrange multiplier. (The details can be found in Stokey and Lucas, Recursive Methods in Economic Dynamics.) These models lead to an analogue of Euler-Lagrange equations in the calculus of variations (economists call these Euler equations). For example, suppose there is a stock that pays a random dividend $D_t$ at every time $t$. Then under a large class of rational-expectations models, the return $R_t$ on the stock satisfies an equation of the form $$ E_t( m R_t ) = 1, $$ where $m$ depends on the specific model. The Generalized Method of Moments allows you to test whether this equation holds in the data.
To step back from the economics a bit, the mathematical interest is this: a large-class of infinite-horizon stochastic optimization problems will produce an expectational equation of the form $$ E_t F(x_t, x_{t+1}). $$ for some $F$. If $x_t$ is some observable data, then we can use GMM test whether the data comes from the optimization problem. So Hansen's work is in the intersection of the calculus of variations and statistics.
Just to add a little something to arsmath's very good answer: The mathematics in Fama's main idea that returns are impredictable are indeed simple, and moreover, not due to him. What Fama did is a huge empirical study to support that claim. For the mathematical argument itself, which is simple but not absolutely trivial, in other words which has some mathematical content, I recommend any interested reader this very nice article of Samuelson (1965), Proof that properly anticipated prices fluctuate randomly.