Posted by Jason Polak on 08. June 2013 · Write a comment · Categories: number-theory · Tags: , ,


Have you ever tried to visualise the graph of a complex function $ f:\mathbb{C}\to\mathbb{C}$? The problem with complex functions is that usually we graph a complex number as an ordered pair $ (x,y)$ on a Euclidean plane, which corresponds to $ z = x + iy$. Unfortunately, this means that if we want to graph complex functions as we do real functions, we need to draw the graph in four-dimensional space! Some people have actually claimed the ability to visualise this, but I do not!

However, if we use time as a dimension, we could represent four dimensions as a moving three-dimensional image in time, like a movie. Sometimes, it’s hard to draw three dimensions in two dimensions, though we don’t actually lose too much because we can only see a two-dimensional picture of a three-dimensional scene at any given time.

There are a few animations in this post; they may take a few seconds to load, or a few minutes on a slower connection.
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Mostly to take a break from marking exams, I thought I’d start a new recurring series here about mathematics papers and books that I find, both new and old. The “new” will consist mainly of preprints that look interesting (to encourage me to browse the arXiv) and the “old” will consists of papers I will likely read (to encourage me to read, or at least skim, more papers).

New Preprints

  1. Clark Barwick, “On the algebraic K-theory of higher categories”: For a while I’ve wanted to learn a bit about higher category theory, but I’ve not yet found any application or motivation for it in something I’m already really interested in (including the stuff here). Perhaps this paper by Barwick will change my mind: algebraic K-theory may be best viewed as a functor with a universal property that generalises the well-known universal properties known for the classical K groups.
  2. Booker, Hiary, and Keating, Detecting squarefree numbers”: Under the Generalised Riemann Hypothesis the authors propose an algorithm to test whether an integer is squarefree, without needing the number’s factorisation.
  3. Shalit, “A sneaky proof of the maximum modulus principle”: This is a proof of the maximum modulus principle in complex analysis, now with even less complex analysis! This paper also appeared in the American Mathematical Monthly and the author wrote a blog post about it as well.

View the whole post to reveal the hidden classic:
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This is a fairly recent picture of the McGill campus:


However, soon the spell of unbearable heat will dawn on the city and there will be plenty of fun things to do outside. Despite the hot sun we shouldn’t neglect the indoor activities, such as the many awesome conferences and workshops that will be going on. This post is a list of a few of them that I think would be interesting to people who like algebra and number theory. They are listed in chronological order.
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Posted by Jason Polak on 29. January 2013 · Write a comment · Categories: number-theory · Tags: , ,

The 3n+1 conjecture (a.k.a. the Collatz conjecture) is easy to state, unsolved, probably difficult to prove if true, and can provide us with some pretty pictures. I’m only going to state it and show some pretty pictures.

First, what’s this conjecture? It’s about a simple algorithm: choose a positive integer $ n$. If $ n=1$, don’t do anything. Otherwise follow these instructions:

  1. If $ n$ is even, divide by $ 2$.
  2. If $ n$ is odd, multiply by $ 3$ and add $ 1$.

Do this to the result, and keep doing this until the result is 1. For instance, choose $ n=6$. We keep applying the steps above to get the numbers 6,3,10,5,16,8,4,2,1. The 3n+1 conjecture is that for any positive integer $ n$, this sequence always gets to 1.
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Posted by Jason Polak on 30. December 2012 · Write a comment · Categories: books, number-theory · Tags: , ,

Posting has slowed a little bit this month because of holidays, but in the last couple weeks during my visit home I decided to refresh some basic knowledge of valuation theory by going through thoroughly the book “Introduction to p-adic Numbers and Valuation Theory” by George Bachman. Naturally, I wrote this quick review.

Bachman’s book is designed to be a leisurely introduction to valuation theory and p-adic numbers. It has only 152 pages and naturally cannot be comprehensive. It is, rather, an enjoyable read that does not require much advanced knowledge, though some experience with metric spaces is certainly required to fully appreciate the later chapters on the extension of valuations.
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Posted by Jason Polak on 09. October 2012 · Write a comment · Categories: number-theory · Tags: , ,

(Or would “The Number 1, Part e” be more interesting?!)

Let’s talk about the number $ e$, my favourite number. Of course, to talk about it we need a definition, so we define $ e$ as

$ e = 1/0! + 1/1! + 1/2! + 1/3! + \cdots.$

Exercise: check that this converges! In Part 2, we shall see Hermite’s argument that $ e$ is transcendental. In order to warm up, let us prove that $ e$ is not an integer first, and then let us prove that $ e$ irrational.
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The Fields Medal Symposium (see link for program) is a newly created annual gathering of mathematicians from around the world at the Fields Institute in Toronto, Canada to celebrate the work of a fields medalist.

The first Symposium will occur this year on October 15-October 18 on the topic of Ngô’s work on the fundamental lemma. This event will include lectures at a variety of levels, including a panel for high school students and undergraduates and public lectures on the Langlands program.

This is tremendously exciting because so many famous people in the field will be at this Symposium, and thus students working on various aspects of the Langlands program and the fundamental lemma will benefit greatly from attending.

Also, I hope to write a few blog posts while I am in Toronto covering some aspects of the Symposium. Stay tuned!

Some Links

Those who have registered should also get the free tickets for the public opening.

Aside from the schedule, the Symposium is on Twitter and there is the Symposium Blog.

In Strasbourg Part 2, I gave a bit of motivation for rigid cohomology, but I skirted defining anything substantial, except for the zeta function. Recall that we have an smooth algebraic variety $ X$ of pure dimension $ d$ defined over the finite field $ \mathbb{F}_q$, and initially we were interested in the rational points $ X(\mathbb{F}_q)$ of $ X$. This led us to define the zeta function

$ Z(X,t) = \mathrm{exp}( \sum_{r=1}^\infty N_r(X)t^r/r)$.

of $ X$. Furthermore, I asked the reader to have faith that there are rigid cohomology spaces $ H_\mathrm{rig}^i(X)$ of $ X$, that the Frobenius extends to an operator $ F$ on each of them, and that we have a product expansion

$ Z(X,t) = \prod_{i=0}^{2d}\mathrm{det}(1 – tq^dF^{-1}|_{H_\mathrm{rig}^i(X)})^{{-1}^{i+1}}$.

Obviously many details have been left out, but this will suffice for continuing. I should state the disclaimer that from now on things will be a bit more sketchy since I’m not familiar with the more specialized material, and eventually I will just give references. As I have mentioned previously, I am not an expert in these areas, and I ask the patience of the reader since these summaries may lack some of the polish of my usual posts.
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