# Splash 2018

Splash is a program in which undergraduate students develop and teach short courses to local high school students. I gave seven lectures on various topics in mathematics and physics. Here are brief descriptions and the slides.

1. Alan Turing, World War II, and the Theory of Computation
The recent movie “The Imitation Game” brought attention to Alan Turing, a British mathematician who played a key role in the Allied codebreaking effort during World War II. In this class, we’ll explore the fascinating story of how the Enigma machine used by the Germans was compromised, and look at some of the mathematics involved. We’ll also talk about Turing’s broader impact on computer science, and how the “Turing machine” is still a crucial aspect of theoretical computer science. We’ll finish by discussing the most important outstanding problem in this field, the $P=NP$ question, and how it relates to everyday computational problems. Slides here.
2. Georg Cantor, Kurt Godel, and the Incompleteness of Mathematics
In a speech to the International Congress of Mathematicians in 1900, the eminent mathematician David Hilbert made a bold claim: “In mathematics there is no ignorabimus,” that is, nothing we cannot know. In 1930 he redoubled on this belief, stating that there is no question in mathematics that cannot eventually be answered, and famously claiming “We must know, we will know.” In 1931, the young logician Kurt Godel proved him to be incorrect, through two theorems which resoundingly demonstrate the formal incompleteness of mathematics. In this class, we’ll start by exploring Georg Cantor’s contributions to set theory and the understanding of infinity and the transfinite. We’ll then explore the exact meaning of Godel’s incompleteness theorems, and see how a seemingly benign problem which interested Cantor is actually unsolvable. We’ll also discuss the lives of both men; after enlightening the world on the foundations of mathematics, each of them eventually went insane. Slides here.
3. Albert Einstein, Black Holes, and Gravitational Waves
In response to an experiment (eventually shown to be faulty) which seemed at the time to contest the evidence for special relativity, Einstein famously quipped: “Subtle is the Lord, but malicious he is not.” A century after the discovery of general relativity, physicists have detected the most subtle signal of all, gravitational waves. In this class, we’ll explore the basic aspects of general relativity theory, and see how it leads naturally to the possibility of black holes. We’ll then review the current state of knowledge on black holes, including their formation and the existence of supermassive black holes. After this we’ll look at the gravitational wave solution to Einstein’s equations, and see how the LIGO collaboration manages to detect the extremely faint signals they produce on Earth. Slides here.
4. From Pierre Fermat to Andrew Wiles: the Last Theorem
In his copy of Diophantus’ Arithmetica, the amateur French mathematician Pierre de Fermat claimed he had proved a theorem, but did not give the proof. This was typical of Fermat, and other mathematicians found proofs for his other results. The theorem in question was the last one standing, and this became its name: Fermat’s Last Theorem. It took over 350 years before Andrew Wiles found a proof of this theorem. In this class, we’ll look at some of the fascinating history of the theorem and failed attempts to prove it in the intervening centuries. The general areas of mathematics that eventually proved successful, elliptic curves and modular forms, will be introduced.
5. Leonhard Euler: Master of us All
Pierre-Simon de Laplace, a great mathematician in his own right, is believed to have said “Read Euler, read Euler, he is the master of us all.” Such a claim would be no surprise. Euler was one of the most productive mathematicians in history; he left about 30,000 pages of work in mathematics, physics, engineering, astronomy, and even music theory. In this class, we’ll look at a few of Euler’s interesting and approachable breakthroughs. What is the sum 1+1/4+1/9+…? Can you walk each road in a town just once? How many regular solids are there (having all faces the same, like a cube)? How many pentagons are on a soccer ball? Euler introduced mathematics capable of answering each of these questions, and we will see how. Slides here.
6. Richard Feynman, the Path Integral, and Least Action Principles
Richard Feynman was a physicist of great importance in the 20th century. He had an amusing character, which famously led him to take up safecracking as a hobby while working at the highly secretive Manhattan Project to develop an atomic bomb. In this class, we’ll look at one of Feynman’s most important contributions to physics, the path integral formulation of quantum mechanics. This work was crucial because it connects quantum theory to the most important idea in classical physics, the least action principle. This class will enable you to understand these foundational and not-often-discussed ideas in theoretical physics, and will also be punctuated by more light-hearted anecdotes from Feynman’s life. Slides here.
7. Evariste Galois and the Solvability of Equations
Evariste Galois was a precocious young mathematician who was never recognized in his lifetime. As a teenager, he answered a question which had gone unanswered for hundreds of years: when can an algebraic equation be solved? Before dying at the age of 20 in a duel, he laid down foundations for areas now known as group theory and Galois theory, and his papers were posthumously discovered by mathematicians who could recognize the genius. In this class, we’ll look at Galois’s interesting but brief life, and then explore the basics of the theory he laid out. You’ll learn the basics of what mathematicians call algebra - completely different from what high schools call algebra - and how it can be used in some familiar problems. We’ll then see how these methods can be applied to the question of the solvability of equations, and give a rough idea of why some equations simply can’t be solved. We’ll also see how this relates to an ancient question: using compass and straightedge, can you trisect an angle? (The answer is no, and we’ll see a simple reason why.) Slides here.