Lecture notes- Galois Theory

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Galois Theory

To save carrying the original paperwork about, and to give myself a recap on the material, I’ve written up the lecture notes from MA40037:Galois Theory as taught at the University of Bath by Geoff Smith.

The content is broadly as follows: Rings, Integral Domains, Fields of Fractions, Units, Ideals, Homomorphisms, The First Isomorphism Theorem, The Chinese Remainder Theorem, Irreducibles, Field Extensions, Characteristic, Minimal Polynomials and Algebraic Numbers, Galois Theory.

The notes very closely match those I made and hence the lectures given, except the section on the Chinese Remainder Theorem, which was adapted from problem sheets. There have been various minor linguistic tweaks, but few mathematical ones.

It should be noted (to avoid confusion under composition) that the convention of writing function arguments to the left (i.e., (x)f rather than f(x)) is adopted here; and that square brackets are sometimes used for factors in polynomials where these appear in expressions also featuring function or polynomial evaluations (which are denoted by round brackets).

Proof reading would be appreciated!

A second survey of game theory

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I’ve now converted the majority of my game theory project to E2nodes, through a dedicated account. The content has been significantly rearranged for some topics, to make better use of E2’s nodal structure; hopefully individual entries stand alone a bit better so that it isn’t necessary to read the entire report to understand what’s going on. I’ll also be adding more content to that account to clarify or expand upon the original document.

NP vs Co-NP

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view as PDF

The NP vs co-NP problem is related to the more famous P vs NP question. As part of CM30071:Logic and its applications (University of Bath Computer Science), my group gave a presentation on this topic, and its connections to propostional logic (through the satisfiability and validity problems). The attached file contains a two page summary of the main ideas from the talk.

A survey of game theory

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For my final year project (MA40128 at the University of Bath), I studied game theory, and now that I’ve completed it, I’m making the report available here. There’s a choice of [ps] or [pdf] format; I intend to node the whole thing at a later date. The report discusses Von Neumann’s minimax theorem; Nash’s theorem and equilibria; the Prisoner’s dilemma (including recent results on IPD tournaments); and Shapley value.

Iterated prisoner’s dilemma in MATALB

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To assist with my upcoming project on game theory, I’ve put together some MATLAB functions to run iterated prisoner’s dilemma tournaments. In particular, this demonstrates the effects of the Southampton strategy I described a while ago on E2. MATLAB’s vector operations turned out to be particularly well suited to this, especially for handling unknown competition length or numbers of participants. The files are reasonably well documented, but I thought I’d offer some additional guidance.

Drop the files below into the same directory and run the tournament script to generate some sample output. Each program plays every other for the specified number of rounds, including itself, then a total score is obtained for each. The matrix returned gives the scores of individual pairings in the natural way; the vector is the scores. You can adjust the identities of the players by altering the vector Q defined in tournament; any number of players and any choice of strategies should work (including, of course, repeated instances of any given strategy).

play handles any given play of the game, that is, one round between two strategies, with their respective histories available. I’ve implemented always cooperate, always defect, tit-for-tat, grim, and the southampton master/slaves. Any unrecognised strategy code is treated as a random 50/50 choice, and the addition of further strategies should be fairly smooth- add another if condition that tests for your chosen number, then add that number into the player vector Q to see it in action.

I don’t know the precise details of the Southampton programs, but mine work along the same lines. A sequence of 8 moves codes for each; during the first 8 rounds, each program continues to play its signal moves unless the opponent has signalled incorrectly. From round 9 onwards, the programs will have identified their opponent as southampton slave, master or non-southampton. In the case of the latter, both strategies switch to permanent defection to reduce the opponent score. In a master-slave pairing, the master defects and the slave cooperates to maximise return to the master at the cost of the slave. Master-master or slave-slave pairings play out as mutual cooperation to bolster their scores.

The payoff matrix I use is consistent with my report, but not exactly standard. This of course can also be changed.

Source:
[iteratedpd.m][play.m][rounds.m][tournament.m]

Prospects in Mathematics (Durham 2005)

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So, I attended my first mathematics conference last week; two days of pure mathematics talks to lure us into postgraduate study. There are very few ‘pure’ topics I wouldn’t enjoy a lecture on, and I’ve been attending my own university’s staff/postgrad colloquia series this semester simply out of mathematical curiosity and enthusiasm. But beyond this entertainment value, the Durham lectures helped confirm/deny some opinions on potential research areas, so the event was certainly worthwhile.

Michael Drischel (Nottingham) gave the first talk, on Sums of Squares, which you can find online, so I won’t discuss the content too much.

Bill Jackson (Queen Mary London) presented a talk on Rigidity of Graphs concerning combinatorics and graph theory. The first section was presented using the geometry package Cinderella with which I was working for my summer research, demonstrating its many applications. This isn’t a field I’ve studied at all, but the ideas are both accessible and interesting so the talk was one of my favourites. There were even some connections to organic chemistry, which I haven’t thought about for a long time!

Patrick Dorey (Durham) gave a talk entitled Surprises in Quantum Mechanics; sadly I doubt I can ever get to grips with this topic (I can only remember abandoning two books partly read, and both were on Quantum Physics). However (ignoring a talk on funding) the next talk managed to overcome even my general dislike of Physics- Nina Snaith (Bristol)‘s talk Every moment brings a treasure: how physicists came to the rescue of number theory. This was one of the more entertaining presentations anyway, but the central result was genuinely intriguing- how random matrix theory, a topic developed in the context of mathematical physics, was able to back up conjectures related to the Riemann zeta function arrived at by traditional number theoretic approaches. The method has turned out to have applications in other areas, and even features as a plot device in the film Proof!

The first day closed with a traditional talk-and-chalk on Geometry and integrability by David Calderbank. Due to a quirk of the MMath structure, I wasn’t allowed to take our differential geometry course. So this was a topic I knew very little about; the talk itself was interesting but I don’t think the field holds much appeal for me. Playing with surfaces is fun, but I prefer my analysis to be more topological rather than heavily connected to calculus.

Some of the ideas of the previous talk were picked up in the first of day two; Michael Singer (Edinburgh) giving an outline of a popular example of an integrable systems in a talk entitled The geometry of nonlinear waves. I’m hoping to track down the Maple worksheet for this one; you really have to see the graphs (or perform experiments with canals!) to appreciate what’s going on.

The most influential talk for me was John Cremona (Nottingham)’s Explicit methods in Number Theory. This was more accurately subtitled Rational points on curves and has cemented my interest in Algebraic Number Theory. For some time I’ve been deliberating between algebraic geometry and algebraic number theory; hindered by our lack of a number theory course at Bath! Based on this talk (and fortunate discussions with John at breakfast) it seems that the aspects of the algebraic geometry course I particularly liked more naturally fall within the remit of number theory; as do the bits of computer algebra that I enjoyed.

Norbert Peyerimhoff (Durham) spoke on averaging and equidistribution problems in geometry; I think this was another talk-and-chalk but I didn’t make notes because the content didn’t really appeal (didn’t help that it got very difficult very quickly!). Similarly Cobordism and groups of formal power series by Neil Strickland (Sheffield) confirmed that algebraic topology, whilst utterly fascinating, is really really difficult. Maple users can find the talk itself at this address, a non-interactive pdf version is also available if you can’t read Maple10 (it seems that, with Maple9.5, I can’t).

The final talk, given by Farid Tari (Durham) Singularities and the imagination offered more diffgeo, and an opportunity for me to demonstrate my complete lack of spatial awareness during the ‘practical’ component where we attempted to build some surface out of a piece of paper! Again, interesting as a talk but not as a career (although Farid was a very good speaker and well suited to holding our attention at the end of a demanding two days).

On the Geometry of Triangle Centres- Report

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During the summer of 2005 I carried out research into the geometry of the triangle, exploring consequences of a result on statics and barycentric coordinate systems first demonstrated by my supervisor Geoff Smith. This project was financed by a Nuffield Undergraduate Research Bursary, specifically URB/02196/G. More on that scheme can be found at The Nuffield Foundation site.

The main results of this work are contained in my report, Applications of the GPAT to Triangle Geometry, which you can access in [ps] or [pdf] formats according to preference.

The latter part of that work refers to conjectures based on experimentation with the Cinderella geometry package; from this a java applet has been generated which can be accessed here.

Additional applets demonstrating other geometric results are also available on: The Euler Line, The Brocard Points, The Fermat and Nagel points.

For reference, the original papers from which I was working are as follows (all via Forum Geometricorum):