Instead of the typical hour long talk we will have a series of shorter talks given by UMS members. Topics presented will include Weyl's equidistribution theorem, a result on the products of a finite number of primes, and more!

A group is an algebraic object, but it also possible (and useful!) to study the geometry of a group. To do this, we need to understand what a group looks like. In this talk, I will introduce the Cayley graph as a way to draw a picture of a group, and we will discuss what can be said about the group depending on the geometry of this graph.

Given a continuous map f from a topological space X to itself, one can ask whether it has fixed points (points such that f(x) = x), and whether f can be modified to remove the fixed points. I will discuss classical invariants that answer these questions, the Lefschetz number and the Reidemeister trace.

The classical Sylvester-Gallai theorem says that, given any finite set of points in the Euclidean plane, not all on the same line, there exists a line passing through exactly two of the points. In this talk, I'll talk about similar statements over the complex numbers and discuss the ideas from algebraic geometry that are used to prove such statements.

In this talk we'll discuss the continued fraction expansion of a real number--a useful expression for obtaining approximations. We'll begin by showing all real numbers have a continued fraction expansion and derive some basic formulas for computing said expansions. Following this, we'll introduce Birkhoff's Ergodic Theorem and explain how it can be used to derive some results about the distribution of the coefficients of the continued fraction expansion.

In algebraic number theory one tries to solve equations over the rationals and these attempts naturally lead to the study of number fields, which are finite-dimensional rational vector spaces with field structure. What about the opposite direction? Can one study number fields by solving equations over the rationals? For quadratic number fields, this is essentially the content of the quadratic reciprocity law of Gauss. The study of higher reciprocity laws (now synonymous with class field theory (CFT)) shows that such ‘congruence conditions’ can only occur if the number field has an abelian Galois group. In this lecture, we explain this connection and provide some explicit examples of abelian CFT. Time permitting, we may include an explicit example of non-abelian CFT.