Suppose we have a $2\times 2$ matrix
$$
M = \begin{pmatrix}
x_{11} & x_{12}\\
x_{21} & x_{22}
\end{pmatrix}
$$
with entries in a field $F$. The characteristic polynomial of this matrix is $p(t) := {\rm det}(tI_2 – M) = t^2 – (x_{11} + x_{22})t + x_{11}x_{22} – x_{21}x_{12}$. One might ask: how can we produce a matrix with a given characteristic polynomial? This can be accomplished using the rational canonical form:
$$
t^2 + at + b\mapsto
\begin{pmatrix}
0 & -b\\
1 & -a
\end{pmatrix}.
$$
We can calculate that the characteristic polynomial of this matrix to be $t^2 + at + b$. This map gives a bijection between quadratic monic polynomials in $F[t]$ and matrices of the above form. One way to understand this phenomenon is through algebraic groups. To explain, let’s stick with $F$ having characteristic zero, though much of what we do can be done in characteristic $p$ for $p$ sufficiently large as well using very different techniques.
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Posted by Jason Polak on 14. November 2014 · Write a comment · Categories: math

692783-alexander_grothendieck

Today Alexander Grothendieck, probably best known for his significant development of the theory of schemes, died this morning at the Saint-Girons hospital in Ariège. He was 86 years old. Let me just list my four favourite Grothendieck inventions as a personal tribute:

  1. The functor of points is the perspective that a scheme (also invented by Grothendieck) can be viewed as a contravariant functor from affine schemes to sets. This perspective is immensely useful and clarifying in the theory of algebraic groups especially, and the representability of such functors is a fascinating field of study.
  2. The definition of $K_0$ was invented for his Riemann-Roch theorem, Grothendieck defined this simple functor of an essentially small abelian category to be the free abelian group whose generators $[A]$ correspond to isomorphism classes of objects $A$ and with relations $[B] = [A] + [C]$ for every short exact sequence $0\to A\to B\to C\to 0$. It’s such a simple and elegant definition and today it forms part of the basis for algebraic $K$-theory.
  3. Grothendieck topologies are generalisations of open covers where the open inclusions are replaced with more general morphisms, so that things like etale cohomology can be defined. It’s a fairly simple idea with enormous applications.
  4. Universal homological $\delta$-functors are a clean and crisp way to talk about derived functors and constructing natural transformations between them and they are used in homological algebra everywhere.
Posted by Jason Polak on 13. November 2014 · Write a comment · Categories: math · Tags:

There are two StackExchange sites I regularly visit: MathOverflow and Mathematics. Roughly, the former is for research level questions and the latter is for almost any kind of mathematics question. Mathematics is very popular at 216 questions per day at this time of posting (I don’t know how this average is calculated). Unfortunately, that means most of the questions aren’t interesting to any given user. To help find the questions I might like, I used to use the ‘Favourite Tags’ and ‘Ignored Tags’ features, with ‘Ignored Tags’ set to not display. Unfortunately, this only works so well, since there are so many tags, and my ‘Ignored Tags’ list got very long:

calculus homework integration complex-numbers summation combinatorics statistics real-analysis notation education graph-theory matrices limits probability probability-theory induction differential-equations stochastic-processes differential-geometry logic inequality general-topology trigonometry svd pde computational-mathematics vector-spaces numerical-methods algebra-precalculus terminology arithmetic algorithms functions probability-distributions polynomials linear-algebra analytic-geometry logarithms sequences-and-series multivariable-calculus gradient-flows geometry game-theory elementary-number-theory functional-analysis systems-of-equations fourier-analysis optimization discrete-mathematics analysis finance matlab uniform continuity statistical-mechanics elementary-set-theory differential-topology limits-without-lhospital number-theory uniform-continuity group-theory trees metric-spaces absolute-value definition galois-theory rotations complex-geometry puzzle complex-analysis

To be clear, I like many of these topics, but most of the questions with these tags on math.stackexchange are not very interesting. However, I recently discovered the feature ‘Advanced Tag Subscriptions’ that can be accessed by clicking edit>advanced tag subscriptions on the right-hand side of any StackExchange site. There you can create various filters (think of them as custom StackExchange sites) that have options like this:
stacky
The effect is that you can make a whitelist, showing only the tags you like. Moreover these tags can be from different sites so your filter can display a questions listing from as many sites as you like. Unfortunately, there are still a few problems with this approach:

  • You can only add tags and not more advanced filtering (like answers:0, score:3, etc.)
  • You cannot search through the filtered questions, which would ameliorate the first point
  • The page displaying the filtered questions does not display the score of the question, only the number of answers
Posted by Jason Polak on 09. November 2014 · Write a comment · Categories: math · Tags: , ,

I keep a fair amount of notes on things I want to learn and results that I am working on. Usually, I keep these in a LaTeX file on my computer in a neatly arranged directory structure. Unfortunately, for some time I’ve been dissatisfied with this approach. For one, LaTeX compiles into a isolated document that can’t be linked easily to other documents. For another, LaTeX makes good looking papers and books but is mediocre at formatting small little notes that might be linked in strange ways.

A typical example is conference notes: I always take notes at conferences at varying degrees of completeness, usually more on the side of a jumbled mess. Sometimes I just jot down one or two things that I’d like to keep in my mind, such as a fact I didn’t know or something to look up later. In typing these notes, perhaps I’d like to highlight these things somehow, perhaps by putting them in a pretty coloured box. With LaTeX, although it’s possible to use coloured boxes for text (and I do), it requires some macro with weird syntax that I keep forgetting.
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Posted by Jason Polak on 05. November 2014 · Write a comment · Categories: math

Unfortunately for the subject of K-Theory, one of its major journals “The Journal of K-Theory”, has been shrouded in controversy for the past few years, and as of today been deserted by all but two of its editors. Originally, this Cambridge University Press journal was meant to take the torch from Springer’s “K-Theory”, which also ended under strange circumstances (more details here).

Although published by CUP, “The Journal of K-Theory” is owned by A. Bak’s company which has yet to hand over the journal to the four-year old K-Theory Foundation as promised. The editors, except for Bak and one other, have resigned in protest, and issued a final announcement today. The now-resigned editors have promised to help authors with pending submissions.

Luckily for the K-Theory world, the K-Theory Foundation will launch the new “Annals of K-Theory” next year, and it is already accepting submissions. The resignation announcement in particular encourages libraries to cancel their subscriptions to the old journal in the fear that, in the words of the editors, “a phantom of JKT might actually still exist” by the time the new journal is published.

Here’s a wish of good luck to the new “Annals of K-Theory”.

Let $R$ be a ring and $M$ an $R$-module with a finite free resolution (an “FFR module”). That is, there exists an exact sequence $0\to F_n\to F_{n-1}\to \cdots\to F_0\to M\to 0$ with each $F_i$ a finitely generated free $R$-module. If we denote by $r_i$ the rank of $F_i$, then the Euler characteristic of $M$ is defined to be $\chi(M) = r_0 – r_1 + \cdots + (-1)^nr_n$. One can easily prove that this is independent of the finite free resolution chosen and hence is a well-defined integer. In the post The Alternating Binomial Sum Vanishes we saw that it’s possible to prove that the alternating sum $\sum (-1)^i\binom{n}{i}$ vanishes by using some facts about Euler characteristics together with the Koszul complex.

Today’s problem shows that the Euler characteristic ins’t that interesting for ideals or homomorphic images of rings. First, here are some facts about the Euler characteristic, some of which might be useful for solving the problem:

  1. Any FFR module over a commutative ring has nonnegative Euler characteristic
  2. If an FFR module over a commutative ring has nonzero Euler characteristic, then the annihilator of $A$ is nil
  3. (Stallings) If $A$ is an FFR module with annihilator $I$ then $\chi(A)I = 0$.

We refer to these as Fact 1, Fact 2, and Fact 3.

Problem. If $I$ is a nonzero (!) ideal of a ring $R$ with a finite free resolution, show that $R/I$ also has a finite free resolution and that $\chi(I) = 1$ and $\chi(R/I) = 0$.

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Posted by Jason Polak on 15. October 2014 · Write a comment · Categories: math

If $R$ is a commutative ring and $A$ an $R$ module, let $Z(A)$ denote the zero divisors of $R$ on $A$. In other words, $Z(A) = \{ r\in R : \exists_a(ra = 0) \}$. If $I$ is the annihilator of $A$, then it is known that any prime minimal over $I$ is contained in $A$. In particular, any minimal prime of $R$ is contained in $Z(R)$. (Recall that a prime $P\subset R$ minimal if it does not properly contain any other prime, and $P$ is said to be minimal over $I$ if there is no inclusion $I\subseteq Q\subset P$ with $Q$ another prime.)

Problem. If $R$ is a ring with no nonzero nilpotent elements, then $Z(R) = \cup_P P$ where $P$ ranges over all the minimal primes of $R$.

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Posted by Jason Polak on 11. October 2014 · Write a comment · Categories: math

Let $R$ be a commutative ring. We say that $R$ is a valuation ring if for any $a,b\in R$, either $a | b$ or $b | a$. For an integral domain $R$, this is equivalent to say that for any nonzero $r$ in the fraction field of $R$, either $r\in R$ or $r^{-1}\in R$.

Examples of valuation rings include discrete valuation rings such as the ring of formal power series $k[[t]]$ for a field $k$ and the ring $\mathbb{Z}_p$ of $p$-adic integers for a prime $p$. Valuation rings also have unique maximal ideals (exercise). Can you think of a valuation ring that is not a discrete valuation ring and not a field? Anyways, here’s the main problem for today:

Problem. Show that every radical ideal in a valuation ring is a prime ideal.

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Posted by Jason Polak on 11. October 2014 · Write a comment · Categories: math, opensource · Tags: ,

Sage is a neat bundle of mathematical software that can be used to do anything from finding class numbers of number fields (like I did to make this graph) to testing whether a finitely presented group is trivial (sometimes…). Typically one works with Sage in a “worksheet” in a web browser, so that it’s easy to organise large numbers of calculations for later reference. Navigating large worksheets, however, might be cumbersome without a table of contents, just like long math papers without a table of contents are hard to read.

In this post, we’ll go over how to (manually) make a table of contents in your Sage worksheet with HTML, with the following example: More »

Posted by Jason Polak on 08. October 2014 · Write a comment · Categories: math

In an effort to keep this blog active, I’ve decided to post some old solutions to various problems I’ve done from books, starting with Kaplansky’s “Commutative Rings”. Hopefully they might be useful to someone learning the subject. I’ve done about half the problems, so it might take a while to post them all, and perhaps posting them will motivate me to do the other half. I won’t necessarily post them all in order, but I will start with the first problem in the book. I encourage readers to try the problem first, and post alternative solutions in the comments! The wording is changed slightly to avoid copyright issues, and I will attempt to provide some background comments to make the post self-contained. All the solutions will be linked to in the Solutions Page for easy reference.

Problem. If every proper ideal in a commutative ring $R$ is prime, show that $R$ is a field.

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