One of the most studied probabilistic objects is the 1D Brownian
motion. It can be seen as a natural probability measure on the space
of continuous functions, indexed by time. There are several natural
generalizations of Brownian motion to higher dimensions. For example,
one could allow the range to be d-dimensional and obtain a certain
random trajectory in higher dimensions - the d-dimensional Brownian
motion. Alternatively, one could ask what happens if you take the
indexing set to be d-dimensional. The arising random height function
is called the continuum Gaussian free field (GFF). I would like to
discuss its geometry in 2 dimensions. It comes out that the 2D GFF is
not defined pointwise - it is just a random generalized function. Yet,
we can give sense to some geometric structures like level sets, or
excursions off the level sets. This reveals interesting connections to
the 2D Brownian motion, but also to other 2D structures like
Schramm-Loewner Evolution.