There are many examples of linear recurrences, from the sequence 1,1,0,1,1,0 to the Fibbonacci numbers. We will characterize their zero sets using nothing more complicated than some elementary linear algebra.

This talk will assume no prerequisites beyond a standard calculus course. The work of William Thurston exposed the rich and complicated structure behind hyperbolic 3-manifolds. Such work often uses various techniques involving knots. This talk serves as an introduction to topics in knot theory, hyperbolic geometry, and low-dimensional topology that show up often when studying hyperbolic 3-manifolds. Topics will include knots, Dehn surgery, and time permitting, virtual knots. As the title of the talk suggests, emphasis will be put on all of the aforementioned topics but this time with the word "hyperbolic" before it.

Diophantine Approximation is the systematic treatment of approximating irrational numbers by successively better rational numbers. We will discover a surprising relationship between how good these approximations can be made and the minimal algebraic definition of an irrational number. This will lead to the definition of an "irrationality measure" which tells us, in some sense, how irrational a number is. We will then study the special case of Liouville numbers which are infinitely irrational in this sense leading to the conclusion that such numbers cannot be roots of any polynomials. This observation will allow us to conclude by explicitly constructing an uncountable everywhere, dense, measure zero subset of the transcendental numbers.

In this talk, we work through basic notions of probability theory with the aim of proving Wigner's semicircle law, a celebrated result in both math and physics that exemplifies a discrete, random distribution converging to a continuous, deterministic distribution. This result is among the first in random matrix theory, a field which today has connections and collaborations with a vast array of fields, including number theory, representation theory, machine learning, data science, stochastic PDEs, and combinatorics.