Hyperbolic geometry is the geometry of surfaces and spaces that aren't flat. Think of a 2D surface which is curved like a saddle. A triangle drawn on this surfa

Hyperbolic geometry is the geometry of surfaces and spaces that aren't flat. Think of a 2D surface which is curved like a saddle. A triangle drawn on this surface will have angles that add up to less than 180 degrees. Ergodic theory is the study of dynamical systems (things that change over time) that are allowed to run for a long time. They are "ergodic" if the state after the system has run for a long time is like picking a state randomly from the set of all possible states. Pouring cream into coffee and then mixing it is ergodic - after a sufficiently long time, you might as well have just arranged all the molecules at random. Symplectic geometry is a bit more abstract. It's the study of dynamical systems that look similar to Newtonian mechanics, i.e. there are analogues of position and momentum. It encompasses all of Newtonian mechanics (i.e. mechanics without friction) but also a large set of other possible dynamics, like electrodynamics. Riemann surfaces are surfaces that look like the complex plane "up close" but might look more complicated "from a distance". For example, an infinitely tall spiral staircases (which extends to infinity in the x and y directions) is a Riemann surface. It's interesting to study functions on Riemann surfaces, because the limitation of behaving like the complex plane at small distances is quite restrictive.