IAP 2021 Classes

The classical heat equation describes how temperature evolves over time as it seeks an equilibrium. The equation is linear and smoothing - solutions become more smooth over time as heat diffuses and becomes more evenly distributed.

Geometric flows, such as mean curvature flow and the Ricci flow, describe the evolution of geometric objects over time. These equations are highly nonlinear and, unlike the ordinary heat equation, even very smooth initial data can explode as a singularity forms. Understanding these singularities is crucial to understanding the flow.

I will introduce these geometric flows and explain the basic approach to understanding the singularities.

In this talk we will introduce and explore the notion of topological degree in the specific case of continuous mappings from the circle to itself, as well as the related notion of winding number. As an application, we will give a topological proof of the Fundamental Theorem of Algebra: every non-constant polynomial has a root in the field of complex numbers.

Starting with Schur's attempt from a hundred years ago at proving Fermat's Last Theorem, we explore connections between graph theory on one hand, and an area of number theory known as additive combinatorics on the other hand. I will show the proof of Schur's theorem, and take you on a tour through subsequent developments in the century since Schur's work.

Consider the following game. We first roll a fair six-sided die to determine a random number X. Then we flip X fair coins and count up the number of heads. In the long-run, what is the average number of heads that we get? You may be surprised that this question can be answered by a simple application of the familiar chain rule from 18.01. Interested? Come to this talk to learn how a discrete random variable can be encoded into a polynomial, and then how calculus can be used to give useful information about its distribution. No pre-requisite knowledge beyond 18.01 is needed.

As in Physics, the notion of an atom shows up in several fields of Mathematics, including measure theory, order theory, and commutative ring theory. In the fields where the notion of an atom makes sense, we can often define an atomic universe as one on which (almost) everything can be decomposed into atoms. In this talk, we will explore atomicity in the context of monoids, commutative rings, posets, and lattices.

We give a historical overview of knot theory, decorated with many pictures, and starting from the observation that Gauss knew a lot more topology than one might otherwise assume.

I will briefly describe the fascinating history of discovery of Kepler's laws of planetary motion which was an important precursor to Newton's laws and thus to the scientific revolution that shaped the world we live in. Here I will mostly follow the book "Sleepwalkers" by A. Koestler. The mathematical part of the lecture will be devoted to deducing Kepler's laws. Although there is a short derivation using standard tools from 18.03, 18.032 I will instead present a more geometric, elementary but elegant proof, eventually connecting the problem to non-Eucledian geometry.

I consider the problem of partitioning the vertices of a graph into two (non-trivial) parts so as to either minimize or maximize the number of cut edges. I will explain how these two seemingly similar problems behave widely differently; one is "easy" while the other is "hard" (guess which?). I will present various ways of finding either the best cut or an approximately good one. We'll discuss min-max relations, geometric embeddings, and approaches based on eigenvalues or convex functions.

This lecture will introduce the basic physics of COVID-19 transmission via respiratory aerosol droplets and derive a simple safety guideline for well-mixed indoor spaces. Rather than limiting social distance or occupancy, the guideline limits the probability of transmission by infected individual in a given time, which depends on ventilation, mask use, air filtration, and the type of human activity. The fluid mechanics of respiratory disease transmission will also be discussed, in order to understand corrections to the first approximation of a well mixed room.