Posted by Jason Polak on 21. April 2016 · 1 comment · Categories: math

I was looking through the past episodes of Wild Spectral Sequences, where we did the snake lemma, five lemma, cohomological dimension formula, Schanuel’s lemma (my favourite), and an application of the LHS sequence to calculating H^1 of tori. However, I’m surprised we never did the 3×3 lemma:

Theorem.If we have a 3×3 diagram:

3x3diag

where all the rows are exact, and two out of the three columns are exact, then all three columns are exact.

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Posted by Jason Polak on 29. March 2016 · Write a comment · Categories: math

Let $V$ be a variety over a field $k$ with separable closure $\bar k$. A $\bar k/k$-form of $V$, or simply form, is a variety $W$ such that $W_{\bar k} = V_{\bar k}$, where $\bar k$ is the separable closure of $k$. If $V$ is quasiprojective then then forms of $V$ are in bijection with the Galois cohomology set $H^1(k,\mathrm{Aut}(V))$. Here, the Galois action on $\mathrm{Aut}(V)$ is given by $\sigma\circ f = (1\otimes \sigma^{-1})f(1\otimes\sigma)$.

Let’s illustrate this with an example: the multiplicative group $\G_m$. Here, we are looking for forms that are also algebraic groups. Are there any nontrivial ones? Yes indeed! Let $E/k$ be a quadratic extension with $\langle\sigma\rangle = \mathrm{Gal}(E/k)$. Then the algebraic $k$-group $F$ defined for each $k$-algebra $R$ by

$$F(R) = (\mathrm{Res}^1_{E/k}\G_m)(R) = \{ x \in E\otimes_k R : x\sigma(x) = 1\}$$

is a form of $\G_m$, because $F_E\cong \G_{m,E}$. It’s also not isomorphic to $\G_m$ over $k$ (why? though we’ll see one reason why later). In any case, we’ve found two kinds of one-dimensional tori: $\G_m$, and $F(R)$ for a quadratic extension $E/k$. Are there any other ones? In this post we’ll classify all one-dimensional tori.
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Posted by Jason Polak on 26. March 2016 · Write a comment · Categories: math

It’s time for another episode of Wild Spectral Sequences! I haven’t written about this in a long time! Truthfully, I didn’t want to prove any more diagram lemmas, even though I still think the spectral sequence approach is much more fun than the diagram chasing approach. Today however, I’m going to talk a little about the Lyndon-Hochschild-Serre (LHS) spectral sequence for group cohomology applied to the first cohomology of tori.

A torus $T$ over a field $F$ is an algebraic group such that $T_{\overline{F}}$ is isomorphic to $\G_{m,\overline{F}}^n$. In other words, it’s a form of $\G_m^n$ for some $n$. The cohomology I’m talking about is the Galois cohomology group $H^1(F,T)$. Where does this group show up? If $G$ acts on an $F$-variety $V$, and $T\subset G$ is an $F$-torus that is the stabiliser of some $v\in V(F)$, then the “stable class” $G(\overline{F})v\cap V(F)$ of $v$ in $V(F)$ may be different than the usual orbit $G(F)v$.
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Posted by Jason Polak on 29. February 2016 · 1 comment · Categories: math

In the last post we proved that $F_1\Ccl$ is a category. Consider now the category $F_1^+\Ccl$ whose objects are pairs $(A\to A’, A’/A)$ where $A\to A’$ is an object of $F_1\Ccl$ and $A’/A$ is a choice of pushout of $*\leftarrow A\to A’$. In other words, we can think of $F_1^+\Ccl$ to be the category of sequences $A\to A’\to A’/A$ where $A’/A$ is the pushout. Perhaps even better, we can think of $F_1^+\Ccl$ as the category whose objects are all commutative squares

pushoutAst

that are pushout squares. Here I used the symbol $B$ to denote an arbitrary pushout as there may be more than one choice of pushout. In general I’ll just write $A’/A$ as I’ll just be talking about one choice at a time.
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Posted by Jason Polak on 29. February 2016 · Write a comment · Categories: math

TikZ is a drawing language for LaTeX that can produce all sorts of diagrams, including commutative diagrams with the tikz-cd package, which is possibly the best package for commutative diagrams, at least with regards to typesetting quality and usability. Sometimes when you draw a diagram, though, you might want a little more programming language thrown in to make drawing easier. A classic example is a timeline. Though probably not useful for research math papers, a timeline could be helpful for a survey paper or history text.

Suppose you have a tab-delimited text file of dates of birthdays:

1809-02-12	Charles Darwin
1822-12-27	Louis Pasteur
1826-09-17	Bernhard Riemann
1877-02-07	Godfrey Harold Hardy
1845-03-03	Georg Cantor

You could draw them on a timeline in TikZ directly, but the problem is you’d have to worry about manually calculating how far down the line these dates are. This is a good case where writing a program in Python 3 that outputs the TikZ picture drawing commands is much easier and painless, especially if you need several timelines:
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Posted by Jason Polak on 19. February 2016 · 1 comment · Categories: math

MathJobs.org by the AMS has been the standard for most North American jobs for years. Two PhD students Daniel Luetgehetmann and Sebastian Meinert have launched a new platform called MathHire.

So, how does it compare to MathJobs? Obviously one comparison is the price point. Both sites have two types of listings: advertising only, and a application management system. For MathJobs, the current pricing is, quoted from their page:

Regular Account (for up to seven ads, full functioning, 12 months from date of sign-up), US610
Regular Account (for one ad only, full functioning, 12 months from date of sign-up), US415
Advertising-only (for up to seven ads, no online applications, 12 months from date of sign-up), US495
Advertising-only (for one ad only, no online applications, 12 months from date of sign-up), US305
Upgrade from existing single-ad account to seven ad account, US295

For MathHire (for academic jobs), the ‘advertising only option is free, and the advertising and application management system is supposed to be 200 euros (currently about 220USD), but all jobs posted before April 1, 2016 will be free as well. This makes sense — the system is new and it needs users to be sustainable.

MathJobs, despite its antiquated 90’s look, is definitely the more polished looking of the two in terms of usability. For example, the UI of MathJobs is much better at presenting information in a succint way. The MathHire website does not take up the full browser screen width and requires lots of scrolling to see a small amount of information. Also, it was difficult to tell from the job listings page which ones had the feature for applying on MathHire, whereas on MathJobs this is evident. The top bar on MathHire takes up too much screen space, especially on small-screened laptops.

However, I admire MathHire and I think it’s an interesting creation that deserves attention. For one, as we’ve seen, posting only is free. Given the extremely low technical requirements for managing a posting-only system, it seems a bit exorbitant to charge 305 dollars for one ad, as MathJobs does. MathHire also promises more features for reviewing applications from their website. I can’t comment too much on this, since I’ve not reviewed jobs on either sites.

If MathHire would work on making an intuitve, polished UI (which is one of the most important factors for the success of an online community, cf. stackexchange), I believe it would have a chance to overtake MathJobs eventually.

Update: since this post, the MathHire website user interface has been updated in response to some of the comments in this post and looks good. I encourage you to check it out!

Posted by Jason Polak on 13. February 2016 · Write a comment · Categories: math

Let $\Ccl$ be a category with cofibrations. Recall that we have introduced a subcategory of the arrow category of $\Ccl$ as follows:

Definition. Define a new category $F_1\Ccl$ to be the full subcategory of ${\rm Ar}\Ccl$ whose objects are the cofibrations of $\Ccl$, and whose cofibrations $(A\to A’)\to (B\to B’)$ are those pairs $(\varphi:A\to B,\psi:A’\to B’)$ that satisfy the following two properties:

  1. $A\to B$ is a cofibration.
  2. $A’\cup_A B\to B’$ is a cofibration.

We have already discussed this definition and hopefully motivated it. Now comes the hard part: we need to prove the next theorem! The tricky part of course will be the third axiom (isn’t it always the third axiom?) because it requires more three-dimensional-ish diagram reasoning.

Theorem. The category $F_1\Ccl$ is a category with cofibrations.

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Posted by Jason Polak on 13. February 2016 · Write a comment · Categories: computer-science

Author: Feng-Hsiung Hsu
Title: Behind Deep Blue: Building the Computer that Defeated the World Chess Champion

Photo by Jason Polak (A chess set I received when I was sixteen).

Photo by Jason Polak (A chess set I received when I was sixteen).

I love battles of skill and stories of seemingly impossible goals. That’s the stuff of Bruce Lee, the Riemann hypothesis, and getting a tenure-track position. And then there’s the computer chess problem: create a machine that can beat the world chess champion at tournament-time chess. This happened nearly twenty years ago, when Deep Blue defeated Garry Kasparov in a six-game match. This is the story told in Behind Deep Blue by Feng-Hsiung Hsu.

Today in 2016, far more advanced chess programs like Stockfish running on a laptop can easily vanquish world-class human players. Hsu’s book however, has lost none of its intrigue or charm. Of course, there are several books written about Deep Blue, many of them chess analyses. Behind Deep Blue however is not a chess book. After all, if it were I wouldn’t be reviewing it on a mathematics blog. Instead, Behind Deep Blue is a story about a bunch of guys solving a computer science and hardware engineering problem.
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Posted by Jason Polak on 04. February 2016 · Write a comment · Categories: math

Fermat’s little theorem says that $p | a^{p-1} – 1$ whenever $p$ is a prime and $a$ is any integer not divisible by $p$. For example, $13 | 2^{12} – 1$. Fermat’s little theorem doesn’t tell you how many times $p$ divides into $a^{p-1}-1$, however. In fact, $p^k | a^{p-1}-1$ for $k > 1$ is pretty rare. For $a = 2$ these are called Wieferich primes and are quite rare. In fact, the only $p < 10^6$ for which $p^2 | 2^{p-1} - 1$ are 1093 and 3511 -- and I believe even more effective searches have failed to turn up other primes. For $a = 3$, there is only one prime under two hundred thousand that works: 11. Indeed, 3^11 - 1 = 2^3 * 11^2* 61. For 31 there are only two primes under a hundred thousand: 79 and 6451. More »

Posted by Jason Polak on 29. January 2016 · Write a comment · Categories: math

In Waldhausen Cats 4, we gained some intuition on the arrow category. Recall that given any category $\Ccl$, we can define the arrow category ${\rm Ar}\Ccl$ to be the category whose objects are the morphisms of $\Ccl$. A morphism $(A\to A’)\to (B\to B’)$ in ${\rm Ar}\Ccl$ is a pair of morphisms $(\varphi:A\to B,\psi:A’\to B’)$ making the resulting square commute. If $\Ccl$ is a category with cofibrations, then ${\rm Ar}\Ccl$ can be made into a category with cofibrations by declaring $(\varphi,\psi)$ a cofibration exactly when $\varphi$ and $\psi$ are cofibrations back in $\Ccl$. The content of Waldhausen Cats 4 was a proof that this is so.

Now we level up, and do something a little cooler and more complicated. We start with a category of cofibrations $\Ccl$, and make ${\rm Ar}\Ccl$ into a category with cofibrations like we just did.

Definition. Define a new category $F_1\Ccl$ to be the full subcategory of ${\rm Ar}\Ccl$ whose objects are the cofibrations of $\Ccl$, and whose cofibrations $(A\to A’)\to (B\to B’)$ are those pairs $(\varphi:A\to B,\psi:A’\to B’)$ that satisfy the following two properties:

  1. $A\to B$ is a cofibration.
  2. The natural map $A’\cup_A B\to B’$ is a cofibration.

In the next post, we shall prove that $F_1\Ccl$ is indeed a category with cofibrations. In this post, we’ll just play around with this definition a little to get some intuition for it.
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