An Analysis Conundrum

Long time, no see! I’ve been busy with other projects, including my rewrite (with some new ideas from collaborators) of my PhD program. But enough of that, some maths!

So my colleagues brought up a conundrum that we didn’t resolve satisfactorily. I’m no good at analysis, so maybe this is a known issue.

We’re going to get the length of a line going from the origin to (1,1):

A line from (0,0) to (1,1)

By Pythagoras (aka the Euclidean metric), we know the length of this line is  \sqrt{2}. We want to work this out via an approximation. We start with a dumb approximation, a line from (0,0) to (0,1) and a line from (0,1) to (1,1):

This approximation has length 2. Not a bad approximation for a first go. So we refine it:

This is closer to the goal. If we add up the length, it’s still 2. Hmmm. Well, it’s touching the true line at more places, so let’s keep going.

Closer again, but if you add up the verticals, that’s always 1. And the same for horizontals, so the distance is still 2. Again, more points intersect the line, but the approximation gets no better. Even with a much finer approximation:

The length is still 2, not \sqrt{2}! (Apologies for my dodgy drawing, it’s slightly off, but you get the idea). Following this logic, no matter how many times we iterate this, we never get any closer to the true value.

So what’s the deal? As a sequence of points, this appears to converge to the line, but the distance never does. Is this basically saying that the taxicab metric never converges to the Euclidean metric? Is the infinite iterate of this process some kind of Cantor set?

Anyone got a good answer?

How to construct finite fields

As a primer to some of the finite fields stuff I’ve been talking about (and will talk about a lot more), I thought it’d be nice if we had a few concrete examples to play with. Let’s go!

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Restarting the inductive proof

A good proof will teach you one clear lesson. After combining my mathematical and non-mathematical interests on my general blog for about ten months, it was clear that it’d be best if I didn’t bug the non-mathies with maths stuff, and the mathies with non-maths stuff. So I’ve split the original blog into this maths-oriented one and one on my general creative pursuits. I’ve ported all the old pages across, comments and all. I hope you can enjoy this blog with its new clarity of purpose.

Enumerating p-groups VII – The main target

(This is part of a series. Read Parts I, II, III, IV, V and VI if you want to get the whole story)

In our previous installment we had established the main gist of my thesis: we want to enumerate groups of a certain kind by finding PORC functions that give the number of groups of that size and structure. Specifically we were looking at p-groups G of order p^n for a fixed n but variable p, such that we have a lower exponent-p central series:

 G = P_0(G) \geq P_1(G) \geq P_2(G) = 1

where P_i(G) = [P_{i-1}(G), G] P_{i-1}(G)^p. This means we have a group G that has a subgroup [G,G]G^p and for that subgroup, its only subgroup of the similar structure is the trivial subgroup. We now want to be able to do something with this structure, to justify why we are looking at these kinds of groups.

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Enumerating p-groups VI – A central structure

(This is part of a series. Read Parts I, II, III, IV and V if you want to get the whole story)

When we last talked about my thesis, I had introduced the problem that was central to my PhD thesis: can we give as a function of p the number of groups of order p^n of a certain structure? Because I was focussing on providing an expository introduction for non-mathematicians, I danced around what this certain structure is. In this blog post, I’ll talk about that structure. There’ll be more technical mathematics as a necessity. Read more »

Enumerating p-groups V – My thesis

(This is part of a series. Read Parts I, II, III and IV if you want to get the whole story)

At the end of the last installment we saw some examples of our desiderata – functions that counted the number of p-groups of order p^n for fixed n and variable (but implicitly prime) p. These functions all seemed to follow the same sort of form: a few small exceptions (where we explicitly calculate the results), and a function made up of terms of the form “a gcd involving p” multiplied by a polynomial in p. We don’t know that this is always the case. The conjecture that claims it to be so is called “Higman’s PORC Conjecture” named after Graham Higman. So what’s this got to do with yours truly? Read more »

Enumerating p-groups IV – Previous results

A mathematical result is not just the result of a bunch of theorems. Taken in isolation, perhaps from a textbook, you get tricked that this is how maths goes. But there’s usually a long history of seemingly silly questions, false starts, fads and fallow years. I found it interesting to poke at the history of where my PhD question came from. And not just the PhD’s provenence, but my own mathematical ancestry. Using the Mathematics Genealogy Project, I can trace my mathematical family tree, up to the illustrious Gauss! To be honest, this is no big deal – my supervisor more-or-less came from the German school of group theory.

This post I thought I might look at the bedrock of results that I could lay the foundations of my thesis on. Some of these I mentioned in the previous post, but I’ll be approaching it from a different direction. Read more »

Enumerating p-groups III – The Question

Every research project and thesis is driven by an interesting question or problem. I want to talk a bit about the line of thinking that drove my thesis. It’s a simple question but leads to some interesting and difficult mathematics. I’ll ramp up the maths jargon a little, but they you won’t need to understand it to get the general gist.

In my previous post I made mention of the usefulness of coming up with examples of groups. When group theorists were getting serious about studying these things, they’d come up with all sorts of examples, finite or infinite, weird or straightforward. Suppose we wanted to make a gold-standard, definitive list of finite groups, perhaps of a given order. Something to hang on the wall of the Group Theory Hall of Fame for all to admire and reference. How would you go about it? Read more »

Enumerating p-groups II – Group theory primer

In this installment, I thought I might give a gentle introduction to group theory and some of the jargon. Don’t fret too much about the details, just try to get the gist. I encourage you to grab a piece of paper and explore by working on examples.

One of the original points of investigating these things called groups was to investigate sets of symmetries that were related to one another.  Take, for example, a square. We recognize instinctively that a square has a reasonable amount of symmetry. Butterflies have a symmetry to their wings. But what do we mean by symmetry?

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Enumerating p-groups I – Introduction

In the old days, if you wanted to get someone to read your PhD thesis you’d try to prove something really neat and build a career on it. Your body of work could rest on the foundations laid by your thesis. If you weren’t so lucky, your thesis would hang out thousands of others in a infrequently-visited section of the university library. In the last few decades, a great way to get more people to witness your thesis was to put it online. It’d be at the bottom of your list of papers, but it’d be there. Both of these approaches hoped that you had a career to at least guide a reader to the website. Nowadays, people can do good work but go on to do other things. But people can know of your work, so long as you bring the work to the people. This series of blog posts is my attempt at this :) If you’re more traditional or want to see the details, you can still view my thesis online. I’m also hoping to make these blog posts a little more expository and fun. I’ve also learned a few things since my thesis, so it’d be nice to bring them along.

Here’s how I’ve envisioned the series:

  1. Introduction to the whole shebang.
  2. Some preliminary group theory to get people up to speed.
  3. The question that drives the entire thesis.
  4. Previous approaches people have made.
  5. The problem I chose to attack.
  6. The plan of attack on the problem
  7. A few posts on the technical components of the thesis (types, degeneracy sets, and conjugacy classes)
  8. Navigating the prickly path of implementing the solution in software.
  9. Results I’ve gotten and new developments.

My thesis focussed on enumerating certain types of groups of prime-power order and involved a large amount of computational algebra. I’ll be talking a lot about what these terms mean, but I’ll also take time to explain how the rubber hits the road in the code and how I dealt with computational issues that are quite separate from the mathematics. I’ve also been working on the problem a bit more, so I can explain those developments. I’m flexible, however, so if you have any questions or suggestions, let me know in the comments section.