Whether it is a bottle of soda, the tires of a car or the human body: Objects in everyday life can be described by only a few parameters, like temperature, pressure or volume. But how is this possible, if each of these systems are complex assemblies of atoms and molecules giving rise to a vast number of  coordinates, on the order of 10^23?

Ergodic theory is the mathematical attempt to provide an answer to this fundamental question. In this talk we will tackle the problem based on a prominent example - the Einstein model for a solid. It will be shown that this problem is reduced to studying rotations on a circle, for which we will prove a version of the ergodic theorem.

Linear Algebra and Analysis approach mathematical abstraction from two seemingly different perspectives. However, once we start talking about normed linear spaces or, more concretely, Hilbert spaces, these two subjects readily connected to each other and the new theory ultimately became the playground for physicists in the 1900s. In this talk, we present some background material on both linear algebra and elementary analysis, discuss their roles within the concept of a Hilbert Space, and why a Hilbert space is important to Quantum Mechanics.

What does it mean for a collection to be finite? On the one hand, we have our preschool notion that a collection is finite when can be counted with natural numbers in a way that terminates. On the other hand, there is a definition due to Dedekind that a set is finite if and only if it cannot be put in one-to-one correspondence with a proper subset. Intuitively these two notions should be equivalent, but can we prove it? I will argue that to avoid a circular argument, one direction requires more care than one would initially think. Further, the other direction is true only by virtue of the Axiom of Choice. To outline the proof of this fact, we will examine formal notions of definiabilty and the set-theoretic technique of forcing.