SWiM 2018 Lectures

Colleen Robles

June 20 (Wednesday)

The Gauss-Bonnet Theorem

The Gauss-Bonnet Theorem is the "crown jewel" of surface geometry. I will explain this beautiful result, what makes it so remarkable, and how one can prove/verify it using calculus.

Rick Durrett

June 21 (Thursday)

Truth is stranger than fiction: A look at some improbabilities

Probability is full of surprises and paradoxes, most of which result from doing the calculation incorrectly. We will illustrate this using some familiar old stories and new ones: the Monty Hall problem, cognitive dissonance in Monkeys, the birthday problem, lottery coincidences, the sad story of Sally Clark, the 2016 election, and gerrymandering in North Carolina. Some of these topics can be found in my blog or in previous versions of this talk, which can be reached by links on my web page.

Irina Kogan

June 22 (Friday)

A story of two postulates

“I have traversed this bottomless night, which extinguished all light and joy of my life. I entreat you, leave the science of parallel alone”, wrote a Hungarian mathematician Farkas Bolyai to his son János, horrified at the thought that his son is attracted by the problem of parallels. János was not deterred, however, and discovered, simultaneously with Lobachevski, a consistent geometry in which the Euclidean parallel postulate does not hold. We will zoom through the path of mathematical thought that took over 2000 years, noticing which results from a high school geometry text-book do not rely on the parallel postulate, and which are altered completely in hyperbolic geometry.

Jayce R. Getz

June 25 (Monday)

An invitation to modern number theory via elliptic curves

We will define elliptic curves and use them as a touchstone to introduce a key idea in modern number theory relating analysis and algebra.

Katie Newhall

June 26 (Tuesday)

Applied Stochastic Dynamics

As an applied mathematician, I look to mathematics to help understand the physical world around us and how it behaves. A ping-pong ball dropped follows a mathematical function for the distance traveled. But imagine a strong wind randomly swirling around, knocking the ping-pong ball back and forth as it falls. Where will it end up? What’s the most likely place it will end up? A combination of probability and calculus can answer these questions! I will discuss not a ping-pong ball, but a magnetic model of spins and “active matter” as a simplified collection of micro-organisms. These are all example systems that exhibit Stochastic dynamics.

Alexander Kiselev

June 27 (Wednesday)

How fluids move

Attempts to understand the nature of fluid motion have occupied minds of researchers for many centuries. Fluids are all around us, and we can witness complexity and subtleness of their behavior in every day life, in ubiquitous technology, and in dramatic phenomena such as tornado or hurricane. I will discuss derivation of the fundamental equations describing fluid motion, and talk about some of the modern challenges in mathematical fluid mechanics.

Tom Witelski

June 28 (Thursday)

Complex systems from simple dynamics

Complicated and dramatic system-wide behaviors can be sparked by simple individual events. Examples include avalanches on mountains and messages "going viral" in social networks. Intricate behaviors like pattern formation, synchronization, and chaos will be shown to occur from problems as simple as the dynamics of a bouncing ball.