Title: The mathematics of quantum mechanics The development of quantum mechanics in the early 20th century was a revolution in physics, and had a transformational impact on mathematics, as well. Even as we still grapple with the interpretations of the theory, its mathematical foundations are rigorous, and rooted in simple concepts of linear algebra. I […]
Title: Ways to write a positive integer into sums of positive integers Abstract: Take a positive integer, say 4; we can write it as a sum of positive integers in the following five ways: 4, 3+1, 2+2, 2+1+1, 1+1+1+1. Among these five ways, how many have only odd numbers as summands? How many of them […]
Title: Frieze Patterns Abstract: As we are in the midst of a cold, dark winter... there is no better time to learn about frieze patterns. Frieze patterns were first defined by mathematicians to classify certain symmetries in ancient art and architecture. In this talk, we will explore arrays of mathematical objects that obey these classified frieze pattern […]
Title: The role of curvature in the production of sound: Music, Geometry and BeyondAbstract: When a string vibrates, it produces different pitches depending on its length. This is how guitarists can play several notes using only one string. But what happens when we have surfaces or 3D objects that vibrate, like when playing a flute, […]
Title: Integer Decomposition Property of lattice polytopes Abstract: Given a polytope (higher dimensional polygon) P and a non-negative integer k, we can multiply every point of P by k to obtain another polytope kP. One of the properties P can have is the integer decomposition property: if x is a point in kP, then there […]
Title: What is an automorphic representation? Abstract: Andrew Wiles' famous proof of Fermat's last theorem was crucially based on constructing a mysterious connection between two classes of mathematical objects: "modular forms" and "elliptic curves". However, if you try looking up what a modular form actually is, you'll find an elementary but very unmotivated definition involving functions on […]
A Brief Introduction to Gödel's Incompleteness Theorems In 1931 Kurt Gödel proved a pair of landmark results that limited the strength of formal theories of mathematics. These "incompleteness theorems" are central to modern understandings of logic, and also to numerous misunderstandings. So today, we'll explore these theorems - how they work, how they were developed […]
Title: A Geometric Introduction to Representation TheoryAbstract: What exactly does it mean for a group to represent the symmetries of an object? In this talk, I will attempt to answer this question by convincing you that the study of group actions, or representations, comes up even more naturally than the study of groups themselves. We […]