# Blog posts

## Easy cases of the volume conjecture?

##### 2013-11-18 10:00:00 +0000mathematics
The volume conjecture relates the hyperbolic volume of a knot complement to quantum invariants of the knot. Specifically, the conjecture is that where $J_N$ computes the colored Jones polynomial and $\xi_N = e^{2\pi i / N}$. For some knots $K$, there are nice formulas for $J_N(K;\xi_N)$. For instance, if $K$ is the figure eight knot, then $J_N(K;q)$ can be written as When $q = \xi_N$ and one takes the limit, this sum transforms quite nicely...

## Reflecting Triangles, live

##### 2011-02-23 08:05:21 +0000personalmathematics
A while back I made some movies which began with a triangle in the plane, reflected that triangle through its three sides, reflected those triangles through their sides, and so forth. The interesting result is that for only four shapes of triangles, the resulting set of triangle vertices is discrete. Using Raphael and a plane geometry package that I wrote, I quickly redid this visualization in Javascript; you can now move the vertices around to...

## Culturomics

##### 2010-12-18 09:02:45 +0000linguisticshistory
I have really fallen in love with Google Books Ngram Viewer, so I thought I'd do a little culturomics'' myself. Here's an image I made using Google's data: The brightness of the pixel at position $(x,y)$ is related to how frequently $x$'' appears in books published in the year $y$. Specifically, if $p$ is the number of times $x$'' appears in print during year $y$, divided by the number of times any number less than...

## Many more Lights Out

##### 2010-07-17 00:05:32 +0000mathematics
A very long while ago I posted some solutions to Lights Out; back then, I solved the $n$-by-$n$ board by row-reducing an $n^2$-by-$n^2$ matrix. Since then, both Boris Okun and Brent Werness pointed out to me that I should've solved Lights Out by using a scanning algorithm: propagating the button presses down one row at a time, and exponentiating the propagation matrix to make sure that I don't get stuck at the last row. This...

## Reflecting Triangles

##### 2010-03-16 05:23:23 +0000personalmathematics
My advisor, Shmuel Weinberger, was teaching Math 113, and asked for some pictures of the following procedure: Start with a triangle in the plane. Reflect that triangle across its three sides. And repeat, reflecting the resulting triangles through their sides, and so forth. I made a couple movies of this, illustrating this procedure as you move through the space of triangles. Observe how, for only four shapes of triangles, the resulting set of triangle vertices...

## Projector on Blackboard

##### 2010-01-19 07:30:13 +0000mathematics
I recently gave a beamer talk, which gave me the chance to point the beamer at my blackboard. Your browser does not support the video tag.

## My mathematical genealogy

##### 2009-06-11 20:06:38 +0000general
According to the Mathematics Genealogy Project, my mathematical genealogy is: Luca Pacioli Domenico Maria Novara da Ferrara Nicolaus Copernicus Georg Joachim von Leuchen Rheticus Caspar Peucer Salomon Alberti Ernestus Hettenbach Ambrosius Rhodius Christoph Notnagel Johann Andreas Quenstedt Michael Walther, Jr. Johann Pasch Johann Andreas Planer who doesn't have a Wikipedia page Christian August Hausen Abraham Gotthelf Kästner Johann Friedrich Pfaff Carl Friedrich Gauss Christian Ludwig Gerling Julius Plücker C. Felix (Christian) Klein William Edward Story...

## Möbius strip, and pairs of points on a circle.

##### 2009-01-28 18:33:51 +0000personalmathematics
Here's a little movie I made: Your browser does not support the video tag. I'm grading for the first year topology course at Chicago, and one of their homework problems asked them to show that pairs of (indistinguishable!) points on a circle correspond to points on the Möbius strip; in other words, the quotient of the torus $T^2 = S^1 \times S^1$ by the $\Z/2$-action which exchanges the two $S^1$ factors is a Möbius strip....

## I can drive!

##### 2008-09-26 21:46:39 +0000personal
I took my road test this morning—and I passed! After all these years, I am a licensed driver. Now, where should I drive to?

## Global Warming according to Google

##### 2008-08-22 00:04:32 +0000personal
Google Trends plots the search volume (or some other measure? search percentage?) for a given phrase over time. It’s ridiculously fun! As an example, let’s look at the number of times people search for the words hot and cold. I downloaded the CSV file offered by Google trends to make the following graph: The thick red and blue lines are the linear regressions on the number of searches for hot and cold, respectively. Behold!—people are...

## Ancient xerox technology.

##### 2008-07-28 23:02:57 +0000personal
The Romans (among others!) wrote in wax with a stylus; the wax was embedded in boards, which were bound together in pairs. If a Roman were to place clay between these boards, could they make a copy of their wax tablet in the clay? It strikes me as remarkable that coins were minted so long before books were printed—though I guess the motivation behind minting coins and printing books are rather different.

## Hyperbolization of Polyhedra

##### 2008-07-26 15:14:56 +0000talksmathematics
I gave a talk in the Farb and Friends Student Seminar (back in March!) on: MR1131435 This is an awesome paper—well-worth a few words on every blog! The construction is way easier than you might think. The ingredients: A model space $X$ with a map $f : X \to \Delta^n$ Any simplicial complex $K$ with a nondegenerate (edge-non-collapsing) map $K \to \Delta^n$ (if having a map to $\Delta^n$ seems like a bother, note that the...

## Solutions to Lights Out

##### 2008-07-21 21:32:40 +0000personalmathematics
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 the buttons are lit when the game begins, and the goal is to turn all the lights off. There are two key...

## Percolation.

##### 2008-07-20 06:02:52 +0000personalquestionsmathematics
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! I'll be a bit more explicit: a dot is drawn at a random location; if it does not overlap any previous dots, it gets a new color. Otherwise, the dot takes the color of the component it...

## Possible homology of closed manifolds.

##### 2008-03-09 01:10:21 +0000questionsmathematics
In this fun paper, MR1845679 it is pointed out that homology is a very basic invariant, and closed manifolds are very basic objects and so a very basic question is: what sequences of abelian groups are the homology groups of a closed simply connected manifold? It isn't very hard to realize any sequence of abelian groups up to the middle dimension, but that middle dimension is tricky (e.g., classify $(n-1)$-connected $2n$-manifolds). Anyway, I was wondering:...

## Political relationships hidden in markets.

##### 2008-03-08 23:44:38 +0000economicspersonal
I’m again applying Granger causality to time series data from Intrade. This time, however, I connect box A to box B with a green arrow if A becoming more likely causes B to become more likely, and with a red arrow if A becoming more likely causes B to become less likely. Shorter arrows suggest stronger relationships (technically, a lower p-value). Running the algorithm on the market data since January 1, 2008 with a lag...

## Granger causality and Intrade data.

##### 2008-03-06 19:31:09 +0000economicspersonalcomputersciencemathematics
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, and Intrade produces CSV market data. I fed the market data for various contracts since January 1, 2008 into...

## Movies of some neat cubical complexes.

##### 2008-02-25 05:27:32 +0000personalmathematics
I made some movies of some of my favorite complexes: let $I^n$ be the $n$-dimensional cube, and let $e_1, \ldots, e_n$ be the $n$ edges around the origin, and let $e_i e_j$ be the square face containing the edges $e_i$ and $e_j$. Define a subcomplex $\Sigma^2_n \subset I^n$ consisting of the squares and all the squares in $I^n$ parallel to these. It turns out that $\Sigma^2_n$ is a surface with a lot of symmetries. In...

## Books that are useless on a desert island.

##### 2008-02-01 05:36:57 +0000personal
Drew Hevle raises a very interesting question: suppose you are stranded on a desert island; what books would be entirely useless in this situation? Here are a few books that I wouldn’t want to be stranded on an island with: Federal Income Tax: Code and Regulations Selected Sections A Million Random Digits with 100,000 Normal Deviates Government Phone Book USA 2000 How to Build an Igloo: And Other Snow Shelters Do you have other ideas...

## Visualizing pineapple pancakes.

##### 2008-01-30 01:04:53 +0000personal
The pineapple sauce pancake graph has English words as vertices, and a directed edge from $a$ to $b$ if the concatenation $ab$ is also an English word. For instance, there is a vertex labeled pine, and a vertex labeled apple, and an edge from pine to apple. Anyway, the graph is huge; and the usual visualization tool (Graphviz) doesn’t work particularly well on the whole graph, so I took a few hundred vertices around pine,...

## Clustering texts with an obvious grouping.

##### 2008-01-28 00:22:30 +0000personallinguistics
It was pointed out to me by Kenny Easwaran that I ought to try clustering texts that already have a natural grouping. So I ran the clustering program on 15 texts written by three authors, and here is the result: The largest eigenvalue is 25 times bigger than the next largest eigenvalue, and picks out the author pretty well. The top pile consists of Jane Austen’s books (with Emma split into three volumes). The middle...

## Clustering Shakespeare.

##### 2008-01-23 04:23:57 +0000personallinguistics
I ran my clustering program (which I just ran on the New Testament) on Shakespeare’s plays—which were conveniently packaged into a text file by Open Source Shakespeare. The result was the following graph: I know little about Shakespeare, so I can’t say too much about the above image. I’d love to know what you think: does this arrangement of his plays make any sense? Given that modern processors are so good at vector and matrix...

## Clustering the New Testament.

##### 2008-01-22 06:23:43 +0000theology
During Bible study last week, it was mentioned that people have used statistics to “determine” authorship of books of the Bible. Having a couple free hours last night, I tried my own experiment on the New Testament. The procedure was easy: I downloaded the Nestle-Aland 26th edition of the New Testament; each book in the New Testament became a vector $v$, with $v_w$ counting the number of times word $w$ appears in the book. The...

## NPR and wedding dresses.

##### 2007-11-15 00:20:05 +0000personal
While we (meaning my wife and I) were filling out the forms for our marriage license, we were interviewed by NPR for Morning Edition! A copy of the broadcast is available online.

## National Bingo Night.

##### 2007-05-30 00:59:19 +0000personal
National Bingo Night (which seems to me to be very silly, but ignoring that…) has a “play along at home” game, where you print out a bingo card. How would I design this? I had hoped that the website generated a Bingo card, digitally signed it, and then sent the signed card to the user. If it had been designed that way, ABC wouldn’t even need to remember which cards had been generated, as long...

## Istanbul, not Constantinople, as a cover, in two senses.

##### 2007-04-20 18:24:09 +0000personal
I am frequently amazed to discover that songs which I had believed to have been original are actually covers. It turns out, for instance, that TMBG’s “Istanbul (not Constantinople)” is a cover of a song from the 1950s. Ironically, one might argue that Istanbul is itself a cover of Constantinople–and that argument (unifying form and content) reminds me of the language games played by Salt: Grain of Life, a book asserting that its very structure...

## Spectral rigidity.

##### 2007-04-19 21:55:50 +0000questionsmathematics
For $X$ a metric space, and $S \subset X$, define the length spectrum of S to be $D_S := { d(x,y) : x, y \in S }$. It might be better to call this the “distance spectrum” or “distance set.” Ian, during his Pizza seminar, gave the following definition: a set $S \subset \R^n$ is a $k$-distance set if $D_S$ has cardinality no greater than $k$. In words, the distances between points in a $k$-distance...

## Vitamin C and analyzing myself.

##### 2007-04-04 17:23:00 +0000personal
Most mammals produce their own vitamin C, but humans carry a mutated form of the gene responsible for one of four enzymes enzymes necessary for vitamin C production, and so we humans must find it in our diets. In effect, every human being has a metabolic deficiency! And in light of this wonderful news, why not ingest tremendously huge amounts of vitamin C? In fact, I’d like to make this into a double-blind study of...

## To feed oneself for a week.

##### 2007-03-27 19:36:17 +0000personal
The question is: how little can I spend to feed myself for one week? I ought to eat 2000 calories/day, so I’ll need to purchase 14,000 calories/week. Here’s a “healthy” option: just eat apples. One ounce of apple has 15 calories, so I’ll need to eat 58 pounds of apples per week; I might be able to get this many apples for 29 dollars. But I can do better! One “Take 5” candy bar is...

## Tasha’s new toy.

##### 2007-03-06 05:38:15 +0000personal
Tasha the Cat received a new toy–a plastic circle containing corrugated cardboard, with a ball stuck in a track. Watch her pounce!

## Divinity versus Humanity.

##### 2007-03-04 19:44:51 +0000theologypersonal
On a recent plane trip, I was reading a very abridged version of (the ten thousand page long!) Church Dogmatics by Karl Barth, and I found something totally beautiful. Believing God to be entirely “transcendent in contrast to all immanence” and “divine in contrast to everything human,” and reading (e.g., in Philippians 2:7) that Jesus is God having emptied himself, having made himself nothing, I concluded that God somehow hid his divinity in order that...

## Translating individual words.

##### 2007-03-04 01:42:25 +0000theologypersonal
Given a text in two languages, is it possible to uncover the meaning of individual words? The Bible is a particularly easy text to work with, since corresponding sentences are marked (i.e., with the same chapter and verse numbers). I downloaded a copy of the Hebrew Bible and the King James’ Version, and looked at Deuteronomy 6:4. For each word in Hebrew, I found all the other verses with that word, and gathered together all...

##### 2007-02-26 21:58:36 +0000personal
Today, I was about to sit down and read a paper (in French–I may not speak in tongues, but apparently I can read in tongues, so to speak!), and I thought to myself about how nice it would be to have a cookie. I went to Uncle Joe’s, I went to the Classics Cafe, I went to Cobb’s coffee shop, and then I gave up, for there were no cookies in any of those places,...

## Genesis clusters around the Akedah.

##### 2007-02-26 03:06:35 +0000theologypersonal
Someone contacted me with some questions about Bayesian document clustering; with that inspiration and a free afternoon a few weeks ago, I took a Hebrew bible and built a matrix $(A_{ij})$ where $A_{ij}$ equals the frequency of the $i$-th (Hebrew!) word in the $j$-th chapter of Genesis. I calculated its singular value decomposition (supposedly this is “latent semantic analysis”), and then took some dot products (calculating the “correlation” of chapters)… Anyhow, the result was astounding!...

## Sharpness of the Hurwitz 84(g-1) theorem.

##### 2007-02-23 21:23:53 +0000teachingmathematics
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 $-1/84$, immediately gives that a closed hyperbolic surface of genus $g$ has no more than $84(g-1)$ isometries (preserving orientation); this is “Hurwitz’ $84(g-1)$ theorem.” Just to show off this theorem, here is a cubical complex which is a surface with lots of symmetries (and the clever...

## Classifying manifolds is impossible.

##### 2007-02-12 16:35:45 +0000mathematics
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 algorithm for deciding when a group is trivial. Here’s the reduction: we are given a 4-manifold $M$, and we...

## Subgroups of products versus products of subgroups.

##### 2007-02-04 04:17:10 +0000mathematics
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 $G$ be a group, and $H$ a subgroup of $G \times G$. Is $H$ always isomorphic to $G_1 \times G_2$, for some subgroups $G_1, G_2 < G$? But beware!–I am not requiring (or expecting) any canonicity or naturality for the isomorphism: for instance...

## Building aspherical manifolds.

##### 2007-01-25 19:17:59 +0000talksmathematics
I gave a Farb student seminar talk on a lovely paper, > MR0690848 I also used some of the material in > MR1937019 which summarizes other the many applications of the "reflection group trick," and works through some examples with cubical complexes. The main result is Theorem. Suppose $B\pi = K(\pi,1)$ is a finite complex. Then there is a closed aspherical manifold $M^n$ and a retraction $\pi_1(M) \to \pi$. This manifold $M$ can be explictly...

## Modeling bus ridership.

##### 2007-01-25 18:23:29 +0000personal
While on public transportation, my mind wanders… And one might assume the following about me and my buses, The bus travels for one unit of time, I will get on the bus at a random time (uniformly distributed), I will leave the bus at a random time (independent, unformly distributed). Then the probability that I am on the bus at time $t$ is $p(t) = 2 \cdot t \cdot (1-t)$. So one might expect that...

## Classifying clothing: the quest for the non-orientable tank top.

##### 2007-01-22 22:36:48 +0000personal
When I walk down the street, I create patterns in how I walk, often by controlling my stride length so I will step on cracks every third sidewalk square, or whatnot. If I were a true master, my stride length would be incommensurable with respect to the sidewalk length–surely this was the problem that forced irrationalities upon the Greeks… Anyway, I was also happy to realize (at a recent retreat) that clothing is nicely categorized...

## Corrugated coffee cup holders.

##### 2007-01-17 22:10:21 +0000personal
I’ve been (not surprisingly) drinking quite a bit of coffee lately, and I’ve noticed that many corregated coffee cup holders include a bit of loose glue. At first, I thought this was a mistake, an oversight in the perfection of the coffee cup holder design. On the contrary, that bit of excess glue melts when the hot coffee is poured into the cup, adhering the corregated holder to the cup–brilliant!

## Estimating the speed of the plane.

##### 2007-01-16 22:45:14 +0000physics
I’m sometimes bored while flying, and I like looking out the window (though if I can, I usually pick aisle seats so I can exit more quickly). I realized something rather amusing. I closed one eye, and held two fingers about an inch apart and a foot away from my open eye. Then, I timed how long it took an object on the ground to move from the one finger to the other finger an...

## Most numbers are boring, asymptotically speaking.

##### 2006-12-10 08:03:55 +0000personal
Let $f(n)$ be the number of Google hits for the integer $n$. Then $f(578)$ is about 100 million, and $f(1156)$, that is, the number of hits for a number twice as big, is about 40 million, a bit less than half as big. Doubling the input continues to halve the output: $f(2312)$ is about 20 million (half again!), and $f(4624)$ is about 8 million, and $f(9248)$ is about 4 million. There are about half as...

## On the Popularity of Certain Numbers.

##### 2006-12-03 06:36:47 +0000personal
I searched for each number between 1 and 500 on Google, and recorded the (estimated) number of hits. I’m not aware of anyone having done this before; in any case, I made a chart: Click on the above chart to see a bigger version. You can also look more closely at the first hundred numbers, or look at the above data with a log scale on the y-axis. I have some observations and questions: There’s...

## Growth series.

##### 2006-11-30 05:32:12 +0000general
In seminar today, Okun pointed out the following interesting observation; for any finitely generated group $G$, you can define its growth series $G(t) = \sum_{g \in G} t^{\ell(g)}$, where $\ell(g)$ is the length of the shortest word for $g$. The first observation is that $G(t)$ is often a rational function, in which case $G(1)$ makes sense. The second observation is that $G(1)$ is “often” equal to $\chi(G)$. This is an example of weighted $L^2$ cohomology....

## Constructing a Lie group from a Lie algebra.

##### 2006-11-30 04:56:41 +0000mathematics
Cartan proved that every finite-dimensional real Lie algebra $\germ g$ comes from a connected, simply-connected Lie group $G$. I hadn't known the proof of this result (and apparently it is rather uglier than one might hope), but > MR0854249 gives a short proof of it, which I presented to the undergraduates in my Lie group seminar. I'll sketch the proof now. Theorem. For every Lie algebra $\mathfrak{g}$, there is a simply-connected, connected Lie group $G$...

## History of Static Electricity?

##### 2006-11-29 20:31:36 +0000personal
What can be said about the history of static electricity? Did Greek science know about it? Any medieval experiments with static electricity? It’s sort of interesting that people knew about magnetism and electricity for hundreds of years before finding many good uses for that knowledge (granted, compasses and potentially batteries for electroplating, but these things are trinkets in our modern world so dependent on electricity); in contrast, the span between radiation and harnessing nuclear power...

## Experiments in cooking.

##### 2006-11-28 06:16:13 +0000personal
I tried making bread, but with significantly less flour than neccessary (and therefore, far more water than needed). The result was very much like cooked paste. It was pointed out to me that since the essence of bread is flour, trying to get by with less flour was undermining the very essence of bread (and I find such arguments very satisfying). I also made baklava again, and that turned out much better than the first...

## Coxeter group visualization.

##### 2006-11-28 06:00:09 +0000mathematics
Jenn is a fabulous program for visualizing the Cayley graphs of finite Coxeter groups. The pictures are absolutely beautiful (oh, symmetry!).

## On forgetting to close parentheses.

##### 2006-11-22 23:27:23 +0000personal
Sometimes I’m scared that, at some point in my past, I opened a pair of parentheses without closing them. Even worse, I’m sure I’ve feared this very thing in the past. Then again, maybe this is the common fear of all schemers: that our whole lives might now be a parenthetical comment.