Graduate Student Advisory Committee (GSAC) Colloquium Schedule:

Abstract: Finding an action of a group on a particularly nice object can yield a wealth of information about its structure. Graphs are particularly easy to grasp, and trees are exceptionally nice among graphs. We will discuss a handful of surprisingly powerful structure theorems can be obtained once we know that a group acts on a tree, and give an example proof of a known result in number theory using one of these theorems.

Abstract: Representation theory studies the structure of groups by transcribing questions in group theory to the context of linear algebra. We will begin with a specific application of this technique to the study of periodic functions, which will yield a foundation for Fourier analysis. Then we will generalize our example, which will give us a powerful tool in the study of harmonic analysis. This approach reveals an interesting connection between two important and seemingly unrelated areas in pure and applied mathematics.

Abstract: A major accomplishment of representation theory has been its predictive power in quantum physics. In this talk, we will discuss a fundamental example of this powerful connection: using representation theory to describe the structure of the hydrogen atom. Using only some basic tools from representation theory applied to the underlying symmetry group of our hydrogen system, we can extract large amounts of information about the intricate structure of the four quantum numbers describing a given state of the hydrogen atom.