ESP Biography

In this course we will introduce the key ideas and concepts in quantum mechanics using no mathematics beyond high school algebra. We will focus on three simple quantum mechanical systems, the photon in a box, the electron in a box, and Schrodinger's cat. Focus will be on the interpretation and meaning of quantum mechanics, from Copenhagen to Everett (many-worlds). We will attempt to correct several common misconceptions regarding quantum mechanics, including several regarding the role of consciousness in the theory.

Suppose you have $$k$$ integers. Can you, for arbitrary remainder $$r=0,...,k-1$$, find some subset of those integers so that their sum has remainder $$r$$ upon division by $$k$$? Why is it the case that every rational number has a decimal expansion that eventually terminates or repeats? Can you write a computer program using finite memory which can recognize all primes? These sort of questions can be answered using the ubiquitous pigeonhole principle. The pigeonhole principle is one of the first problem solving techniques that young math students learn, but that doesn't mean it is always easy to apply! This class will contain a short lecture with several examples followed by a longer problem solving section in which students are given a list of problems to try to solve (with varying difficulties and prerequisite knowledge), and ending with a section where students share some of their solutions to the more difficult problems. (Modeled after the Polya problem solving seminar.)

This class will cover a few essential concepts from General Relativity. Our goal is to introduce students to the metric tensor, a key object in the study of GR. We will discuss how physicists relate mathematical formulation to physical understanding. Subtopics will include: coordinate transformations, tensor notation, the Einstein equations, and spacetime curvature.

How can we provide a powerful and secure foundation for all of mathematics? This was one of the key questions facing mathematicians at the turn of the 20th century. We will first define the notions of a formal system, as well as the notions of "semantics" and "syntanctics". We will then briefly develop a very simple formal system, that of first order logic with set theory, in which essentially all of mathematics may be expressed (using just three logical symbols nonetheless!) and proved. We will then discuss the limitations of such a system, especially Godel's super awesome incompleteness theorems.