I gave a talk in the Farb and Friends Student Seminar (back in March!) on: Davis, Michael W.; Januszkiewicz, Tadeusz Hyperbolization of polyhedra. J. Differential Geom. 34 (1991), no. 2, 347—388. This is an awesome paper—well-worth a few words on every blog! The construction is way easier than you might think. […]

I’ll briefly introduce the Lights Out puzzle: the game is played on an n-by-n grid of buttons which, when pressed, toggle between a lit and unlit state. The twist is that toggling a light also toggles the state of its neighbors (above, below, right, left—although, on the boundary, lights have fewer neighbors). All […]

I made a movie recently for my advisor. The movie is so pretty, that I thought I’d share it here: may I present to you randomly drawn dots, where two dots are the same color when they touch! Get the latest Flash Player to see this player. [Javascript required to view Flash movie, please turn it […]

In this fun paper, Kreck, Matthias An inverse to the Poincaré conjecture.Festschrift: Erich Lamprecht. Arch. Math. (Basel) 77 (2001), no. 1, 98—106. it is pointed out that homology is a very basic invariant, and closed manifolds are very basic objects and so a very basic question […]

Granger causality is a technique for determining whether one time series can be used to forecast another; since the Intrade market provides time series data for political questions, we can look at whether political outcomes can be used to forecast other political outcomes. There’s a library for the statistical package R to do the Granger test, […]

I made some movies of some of my favorite complexes: let be the -dimensional cube, and let be the edges around the origin, and let be the square face containing the edges and . Define a subcomplex consisting of the squares and all the squares in […]

For a metric space, and , define the length spectrum of S to be . It might be better to call this the “distance spectrum” or “distance set.” Ian, during his Pizza seminar, gave the following definition: a set is a -distance set if has cardinality no greater than . In […]

There are usually courses at Mathcamp about surfaces; there should be courses about orbifolds! For instance, knowing that the smallest hyperbolic orbifold is the (2,3,7)-orbifold, having orbifold Euler characteristic , immediately gives that a closed hyperbolic surface of genus has no more than isometries (preserving orientation); this is “Hurwitz’ theorem.” Just to […]

At a recent Pizza Seminar, Matt Day gave a lovely talk explaining why it isn’t possible to classify 4-manifolds. An algorithm for deciding whether two closed 4-manifolds are homeomorphic gives an algorithm for deciding whether a closed 4-manifold is simply connected, and therefore (since every finitely presented group is the fundamental group of a 4-manifold), and […]

This is a question I wandered into accidentally years ago now, which I think other people might be amused to think about (or more likely, put on an abstract algebra exam). Let be a group, and a subgroup of . Is always isomorphic to , for some subgroups ? But beware!—I […]