Are we there yet? A beginner’s guide to the infinite
Euler's Method and Numerical Differential Equations
In a first course in differential equations, one develops numerous techniques for finding "analytic" solutions to problems. This alchemical process involves fitting the equation into a known form, and applying a solution formula given the parameters. This technique has value, but betrays the actual nature of modern differential equations. Within the realm of ordinary differential equations, one need stray only slightly into the nonlinear side of things, and the vast majority of analytic solution techniques are useless. So, one commonly appeals to numerical solutions. In this talk, we will introduce explicit Euler's Method, a straightforward way of solving differential equations (ordinary or partial) numerically. We will use this technique to solve numerous interesting nonlinear ODE systems. Time permitting, we will see how this technique fails for certain linear systems, and introduce an implicit version to remedy this. No knowledge of differential equations is assumed in this talk.
The art of error: exploring coding theory
Coding theory, an area of mathematics with application in modern communication and data transmission, traces its roots back to ancient ciphers and codes. This talk aims to unravel the mysteries of coding theory in a fun and engaging manner. Beginning with a historical perspective, we explore the evolution of codes from their early inception to their role in today's digital age. We will also discuss the mathematical foundation for the fundamentals of coding theory and specific codes. Come explore what it means to embrace the error that occurs all around us!
All Tied Up in Knots
Knot theory is one of the foremost research areas in low-dimensional topology. Knot theorists study tangled circles in three dimensional spaces. In particular, it is helpful to represent knots using two-dimensional diagrams and identify when diagrams represent the same or different knots. We will take a quick tour of some basic tools for answering this question, using Reidemeister moves and tri-colorability to identify and distinguish knot diagrams.
Cracking the Surface
In the 1800s, Mary Queen of Scots plotted to assassinate Queen Elizabeth of England; the plot unfolded in a series of letters between Mary and her co-conspirators, encrypted with a cipher. When it came time for the trial, Mary's life depended on the assumption that no one could crack the cipher. Today, cryptography - the mathematics of hiding messages or recovering hidden messages - plays a crucial role in national security. RSA encryption is widely used for secure messaging over the internet, and the NSA has a large program devoted specifically to cryptography.
In this talk, I will provide some of the history of encryption and we will talk about two of the oldest forms of encryption - transcription and substitution ciphers. We will discuss how to crack these types of ciphers, and by the end of this talk, you should be able decode the following message: "YK YJ ODJUODBB JUDJFE DWDYE. WF GXYBBYUJ!"
Numbers Don’t Lie: p-adics and the Strange Behavior of Alternative Number Systems
The real numbers are viewed by natural and social scientists alike as a central tool in the enterprise of describing reality. In this talk we will explore an alternative: the p-adic number system. Though the p-adics enjoy some of the nice features afforded to us by the reals, they differ in many strange and interesting ways.
Though numbers don’t lie, the reality they describe may not be a familiar one.
Optimization Problems in Infinite Dimensions
We face optimization problems constantly in life. We want to minimize carbon emissions, maximize profit, find the fastest way to the grocery store, or choose the longest lasting battery. One reason that multivariable calculus is so powerful is because it gives us tools for handling these problems. But what about optimization in infinite dimensions? What does this even mean? Why should we care?
In this talk, I'll walk you through the exciting historical drama leading to the Calculus of Variations. We will see some applications to physics along the way, and I'll conclude by sharing some of my own research contributions. This will be an accessible overview of a deeply beautiful field, and we'll keep things lighthearted throughout.