DARPA (The Defense Advanced Research Projects Agency) recently released the DARPA Mathematical Challenges (Word document)—23 challenging mathematical problems “with the goal of dramatically revolutionizing mathematics and thereby strengthening DoD’s scientific and technological capabilities.” The titles of the challenges are: The Mathematics of the Brain The Dynamics of Networks Capture and Harness Stochasticity in Nature 21st Century Fluids…
Category: Math
Math quotes on Freakonomics blog
There’s a mathematical topic today on the New York Times’ Freakonomics blog. Their daily bleg (definition: using a blog to beg for information): What’s been said about math? Readers are invited to leave their favorite quotes about mathematics (or quotes by mathematicians) in the comments.
Kuratowski’s closure-complement theorem (solution)
Stop! This post contains spoilers. This page has the solution to the problem posed in yesterday’s post. We challenged you to find a set from which we can make as many new sets as possible using only the closure and complement operations. In 1922 Kuratowski proved the following theorem. Theorem. At most 14 sets can…
Kuratowski’s closure-complement theorem
One of my favorite theorems in elementary topology is Kuratowski’s closure-complement theorem. First some notation. For any set let denote the complement of and denote the closure of . (Recall that and is the union of and all the limit points of ). Here’s the problem. Find a set so that we can construct as many…
Not so elementary my dear Doyle
I loved Sherlock Holmes mysteries when I was a child. A few years ago I learned the disappointing news that Sir Arthur Conan Doyle was a spiritualist who believed in séances. Because of this, his once close friend Harry Houdini became a rival. Their falling out began when Houdini joined the Doyles for an intimate séance,…
Königsberg today
In 1736 Leonhard Euler solved the now-famous bridges of Königsberg problem. It is often hailed as the birth of topology and graph theory. Although no graph appears in Euler’s paper, his argument is topological in spirit. (He proved another famous topological theorem a decade and a half later.) The problem is stated as follows. If a resident of Königsberg…
What is the difference between a theorem, a lemma, and a corollary?
I prepared the following handout for my Discrete Mathematics class (here’s a pdf version). Definition — a precise and unambiguous description of the meaning of a mathematical term. It characterizes the meaning of a word by giving all the properties and only those properties that must be true. Theorem — a mathematical statement that is proved using rigorous mathematical reasoning. In…
Aggregating political polls
A colleague and I were talking about how fun it would be to be teaching a statistics class during this presidential election season—especially talking about the political polling process. Recently I came across Fivethirtyeight.com, an interesting website that takes published political polls, applies its own analysis to them, and puts out its own predictions. (It is named after the number of electors in…
They Might be Giants sing about polygons
Here is a fun video from They Might be Giants. It is from their children’s CD/DVD Here Come The 123s, the follow up to Here Come the ABCs. The song is called “Nonagon”. (I feel bad for Heptagon, who apparently wasn’t invited to the party.)
Learn the Greek alphabet
I’m a big fan of Sporcle, a website that has, as they describe it, “mentally stimulating diversions.” When I was in graduate school, one of my fellow graduate students had a terrible time learning the Greek letters. He forgot the easy ones—he repeatedly called lambda, gamma and phi, psi. His advisor joked (I think he…
David Richeson: Division by Zero 