I will give the first of two(?) expository talks on how toric and tropical geometry can be useful to study matroids.

In this talk, we will discuss the cohomology groups of the free loop space of a topological space $X$, which is the space of all continuous maps from $S^1$ to $X$. Using the Eilenberg–Moore spectral sequence, we will connect these cohomology groups and Hochschild homology and compute them in specific cases. We will mostly follow the article "On the Characteristic Zero Cohomology of the Free Loop Space" by L. Smith.

We review several known, but unfortunately not well-known results about the (singular) homology of fibre bundles. We use simplicial sets and in particular twisted Cartesian products, which are the simplicial analogues of fibre bundles. The central result is the twisted Eilenberg-Zilber theorem, which relates the chains on a bundle to a twisted tensor product of the chains on base and fibre. The Eilenberg-Moore theorems are easy consequences of it.

Given a variety X a natural question to ask is whether it is birational to projective space. In general, this is quite a very hard question to answer. We restrict ourselves to a subclass of objects known as algebraic tori. It turns out that a slightly weaker notion related to rationality can be "cleanly" understood in terms of the character lattice for a given torus. I will give some background to state this result and then discuss the work that has been done on the rationality question for algebraic tori.