Posted by Jason Polak on 28. August 2015 · Write a comment · Categories: math

The other day we were camping and I had to rinse my Nalgene bottle (post not sponsored by Nalgene).
JKCP1338_01
I only had a fixed amount of water, less than the capacity of the bottle, and I had to decide how to best use the water to minimise the concentration of contaminants. I could pour all the water in the bottle, give it a good shake, and then spill it out. Or, I could divide the rinsing water into several portions. I decided to choose amongst the methods, indexed by $n$, of dividing the rinsing water into $n$ portions, and then doing a sequence of $n$ separate rinses. The problem is, how should I choose $n$?
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Posted by Jason Polak on 25. August 2015 · Write a comment · Categories: algebraic-geometry, number-theory · Tags: ,

In the Arthur-Selberg trace formula and other formulas, one encounters so-called ‘orbital integrals’. These integrals might appear forbidding and abstract at first, but actually they are quite concrete objects. In this post we’ll look at an example that should make orbital integrals seem more friendly and approachable. Let $k = \mathbb{F}_q$ be a finite field and let $F = k( (t))$ be the Laurent series field over $k$. We will denote the ring of integers of $F$ by $\mathfrak{o} := k[ [t]]$ and the valuation $v:F^\times\to \mathbb{Z}$ is normalised so that $v(t) = 1$.

Let $G$ be a reductive algebraic group over $\mathfrak{o}$. Orbital integrals are defined with respect to some $\gamma\in G(F)$. Often, $\gamma$ is semisimple, and regular in the sense that the orbit $G\cdot\gamma$ has maximal dimension. One then defines for a compactly supported smooth function $f:G(F)\to \mathbb{C}$ the orbital integral
$$
\Ocl_\gamma(f) = \int_{I_\gamma(F)\backslash G(F)} f(g^{-1}\gamma g) \frac{dg}{dg_\gamma}.
$$
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The flat dimension of an $R$-module $M$ is the infimum over lengths of flat resolutions of $M$, and the weak dimension (or $\mathrm{Tor}$-dimension) of $R$ is the supremum over all possible flat dimensions of modules. Let’s use $\mathrm{w.dim}(R)$ to denote the weak dimension of $R$. As with the global dimension, the weak dimension of $R$ can be computed as the supremum over the set of flat dimensions of the modules $R/I$ for $I$ running over the set of all left-ideals or right-ideals, either is fine!

So, if every ideal is flat, then $\mathrm{w.dim}(R) \leq 1$. What about the converse? If $\mathrm{w.dim}(R) \leq 1$, is it true that every ideal is flat? Let’s make a side remark in that if we replace weak dimension with global dimension, and flat with projective, then the answer follows from Schanuel’s lemma. However, as far as I know there is no Schanuel’s lemma when ‘projective’ is replaced by ‘flat’.

However, we can get away with using part of the proof of Schanuel’s lemma. Before continuing, the reader may wish to check out the statement and proof of Schanuel’s lemma using a double complex spectral sequence.
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Posted by Jason Polak on 07. July 2015 · 1 comment · Categories: math

Reading a math book cover to cover is a rite of passage for some, but the process can be pleasant or a nightmare; it can be rewarding or a mindless verification of theorems. This post is a guide to reading mathematics books, and is aimed for someone who is new at the process such as undergraduate or graduate students.

1. Find a book that suits your style of exploration

Some authors will be closer in line with your way of thinking than others. Unfortunately, it’s often quite difficult to tell which is which. One way that might help is reading through the first few pages of a book to give you a better impression of the author’s style. That, together with looking at the table of contents and introduction ought to be good enough to choose from a list of possible books on a given topic.

For example, I originally starting learning homological algebra through Mac Lane’s book ‘Homology’. After a while, it seemed as though the author was spending too much time on explicit descriptions of different homological constructions and less time on the big picture, or organising principles, though I didn’t know exactly what those were at the time. After about forty pages, I stopped and switched over to Weibel’s ‘An Introduction to Homological Algebra’, and it was much more pleasant and I ended up reading the entire book. This is, by the way, one of the rare times that I’ve actually finished an entire book.
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Let $R$ be a ring and $M$ be an $R$-module. The flat dimension of $M$ is the infimum over all lengths of flat resolutions of $M$. Usually, the flat dimension of $M$ is denoted by $\mathrm{fd}_R(M)$. For example, $\mathrm{fd}_{\mathbb{Z}}(\mathbb{Q}) = 0$. Since $\mathbb{Q}$ has projective dimension $1$, the flat dimension and projective dimension of a module can be different. Sometimes they can be the same: $\mathbb{Z}/n$ for $n$ a positive integer has the same flat and projective dimension as $\mathbb{Z}$-modules.

The weak dimension of a ring $R$ is defined to be $\mathrm{w.dim}(R) = \sup_{M} \{ \mathrm{fd}_R(M) \}$ where $M$ runs over all left $R$-modules. Due to the symmetric nature of the tensor product, we can also take the supremum over all right $R$-modules, in contrast to the asymmetric nature of global dimension.
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In the post Examples: Projective Modules that are Not Free, we saw nine examples of projective modules that are not free. On in particular was ‘the’ submodule $M = \oplus_{i=1}^\infty \mathbb{Z}$ of $\prod_{i=1}^\infty\mathbb{Z}$. Now, that’s a cool example to be sure, but the way we showed that $M$ was not free was to cite that $\prod_{i=1}^\infty\mathbb{Z}$ is uncountable. Actually, I like the argument a lot, but it’s possible to use the idea of that example and choose $M$ instead to be finite and different from all the examples in the aforementioned post. In fact, we’ll see a large class of examples that can be constructed from the ideas here.

The idea is to take an abelian group $A$ an consider $A$ as a module over its endomorphism ring $E = \mathrm{Hom}(A,A)$, where the endomorphisms are just homomorphisms $A\to A$ of abelian groups. Sometimes, $A$ can be projective over $E$. Actually, for a while it was believed that the projective dimension of $A$ over $E$ could only be $0$ or $1$, but eventually I.V. Bobylev showed in [1] that $A$ could have any projective dimension over $E$, including infinity!
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A commutative ring $R$ can be non-Noetherian and have all of its localisations at prime ideals Noetherian, such as the infamous $\prod_{i=1}^\infty \mathbb{Z}/2$. So being Noetherian is not a local property. However, there is an interesting variant of ‘local’ that does work, which I learnt from Yves Lequain’s paper [1]. It goes like this:

Theorem. Let $R$ be a ring and fix a left maximal ideal $M$ of $R$. Then $R$ is left Noetherian if and only if every left ideal contained in $M$ is finitely generated.

The nice thing about this statement is that it avoids localisation so it’s easy to state for noncommutative rings.
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Let $R$ be a ring. The projective dimension $\mathrm{pd}_R(M)$ of an $R$-module $M$ is the infimum over the lengths of projective resolutions of $M$. The left global dimension of $R$ is the supremum over the projective dimensions of all left $R$-modules. There is a notion of right global dimension where left modules are replaced with right modules. Since we’ll be talking about commutative rings only, we’ll just use global dimension to refer to both kinds, and write $\mathrm{g\ell.dim}(R)$ for the global dimension of $R$.

As an example, if $R$ is a field then $\mathrm{g\ell.dim}(R) = 0$ because every module is free. If $R$ is a principal ideal domain (PID), then $\mathrm{g\ell.dim}(R) = 1$. This is because any module $M$ admits a surjection $F\to M$ where $F$ is a free module. But the kernel of this map is also free since over a PID, a submodule of a free module is free. One of the first results in the theory of global dimension is that $\mathrm{g\ell.dim}(R[x]) = 1 + \mathrm{g\ell.dim}(R)$. So far then we have examples of rings with any finite global dimension.
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Posted by Jason Polak on 01. April 2015 · Write a comment · Categories: math

The other day I learned a little about the pygame python module that provides access to sdl functions on Linux along with some higher level functions. Basically, this means that you can use pygame to access some pretty low level graphics to draw things. I thought it might be fun to use it to make pretty pictures. Here’s an example where points are plotted, with the colour of each point a function of the coordinates. These simple examples should indicate how this type of low-level drawing can be used, perhaps to visualise complicated data or make interesting animations.

In sdl as with many computer graphics systems, coordinates start from the top left with $(0,0)$, and proceed positively right and down. The equations presented below give the red-green-blue (RGB) values of the color of the point at $(x,y)$. Each value is taken modulo $255$ (actually, I should have taken the value modulo $256$, since each color goes from $0$ to $255$, but I didn’t think carefully about this when I wrote the program!).

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mod1 $\begin{align*}
r &= x\\
g &= x-y\\
b &= x + y
\end{align*}
$
Posted by Jason Polak on 25. March 2015 · Write a comment · Categories: algebraic-topology, books

Doug Ravenel has made his book Nilpotence and periodicity in stable homotopy theory available for free download along with a list of errata, also available at the same page as the book.

Here is the official description from Princeton University Press:

Nilpotence and Periodicity in Stable Homotopy Theory describes some major advances made in algebraic topology in recent years, centering on the nilpotence and periodicity theorems, which were conjectured by the author in 1977 and proved by Devinatz, Hopkins, and Smith in 1985. During the last ten years a number of significant advances have been made in homotopy theory, and this book fills a real need for an up-to-date text on that topic.

Ravenel’s first few chapters are written with a general mathematical audience in mind. They survey both the ideas that lead up to the theorems and their applications to homotopy theory. The book begins with some elementary concepts of homotopy theory that are needed to state the problem. This includes such notions as homotopy, homotopy equivalence, CW-complex, and suspension. Next the machinery of complex cobordism, Morava K-theory, and formal group laws in characteristic p are introduced. The latter portion of the book provides specialists with a coherent and rigorous account of the proofs. It includes hitherto unpublished material on the smash product and chromatic convergence theorems and on modular representations of the symmetric group.

At the official page of the book, you can also buy a paperback copy.