Graeme Taylor, Lehmer’s conjecture for matrices over the ring of integers of some imaginary quadratic fields, Journal of Number Theory, Volume 132, Issue 4, April 2012, Pages 590-607, ISSN 0022-314X, 10.1016/j.jnt.2011.09.006.
(http://www.sciencedirect.com/science/article/pii/S0022314X11002289)

The paper covers the classification of the cyclotomic matrices/graphs for four of the six rings I considered in my thesis, but there have been some improvements to the methods. In particular, the proof that any maximal cyclotomic graph over those rings has all vertices of weighted degree four has been substantially streamlined; and there’s an explicit proof that any cyclotomic graph is contained in a maximal one. A follow-up paper proving Lehmer’s conjecture for polynomials arising from such graphs over the same rings is in preparation.

AMS Contributed Paper Sessions: Combinatorics and Graph Theory, I
Y. Kim, Cycle-saturated graphs with minimum number of edges.
D. Pragel, Algebraic and Graph-Theoretic Properties of the Box Product of Two Paths.
A. Barghi, Firefighting on Random Geometric Graphs.
J. Ellis-Monaghan, Ribbon Graphs and Twisted Duality.
J. Fierson, Some graph theoretical results for the task mapping problem for parallel computers.
S. Raval, Complex Contagions on Graph Dynamical Systems.

Although I’m officially a number theorist (honest, it says so right there in the sidebar!) much of my thesis topic and subsequent work has been more concerned with graphs, and there was plenty of interest on offer here.

From a research perspective the box product construction particularly caught my attention: in the presented work, products of paths were considered, which yield grids that can be sliced vertically into copies of one factor, and horizontally into the other. This carries over into some nice structural properties of the adjacency matrix, and they were able to come up with a particularly neat characterisation of its determinant based on the length of the paths. The obvious next step would be to try something more complicated than paths, and I wonder if some candidates from my own studies of cyclotomic graphs might be suitable.

On the other hand, the firefighting problem is something I’d have no idea how to solve, but it seems like I could make an undergrad project out of it – or a web game! Given a graph, some vertices are specified as being on fire. Each round, firefighters may be placed at any vertices that aren’t on fire, then the fire spreads to any neighbouring vertices that haven’t been protected in this way. On an infinite graph, the question is whether such a fire can be contained or could burn indefinitely.

AMS Colloquium Lectures
A. Lubotzky, Expander graphs in pure and applied mathematics, I.

More in the graph-theory line: unfortunately I was only able to attend this, the first of a series of three talks by Alex Lubotzky on the subject, but at least I now know what expander graphs are and why I might care! The original motivation was practical: in designing a communications network (be it mobile phones or multicore processors) you want short routes between nodes for speed and reliability, but as few connections between nodes as possible to minimise cost. Expander graphs are those which (remarkably) manage to balance these opposing properties, but they also find application in a surprising range of abstract mathematical topics.

MAA Contributed Paper Sessions: Cryptology for Undergraduates
D. Cabarcas, Algebraic Cryptanalysis as a tool for teaching Cryptology.
D. Spickler, Cryptography Tools: A Teaching Tool for the Investigation of Classical Cryptography and Cryptanalysis. (Cryptography Tools)
C. Beaver, Group Signature Schemes: How to share a secret without telling it.
A. Li, Cryptography, a Great Topic for Undergraduate Mathematics Courses.
T. Feil, A Cryptology Course for the Non-Mathematician.
R. Talbert, A Brief Fly-Through of Cryptology for First-Semester Students using Active Learning and Common Technology.
R. Beezer, A first-year seminar in cryptology. (slides).
S. Boersma, Student Codebooks: An in-depth writing assignment.
K. Smith, Codes in History, the Arts, and Literature.
K. Meyer, Making Cryptography Come Alive.
M. May, Using Cryptography to Show Students that Math is Everywhere.

This session was one of my reasons for making the long trip, and was definitely worth it. Based on the enthuiasm of the speakers, the feedback they’ve received from their students, and the sheer number of people who turned up for this session, I think it’s safe to say that cryptography is definitely worth offering in the undergraduate syllabus. The American undergraduate experience is rather different to the English one I had, or the Scottish one I tutored for, and in particular there’s a need for mathematics courses for non-mathematics students. Several speakers were able to provide a cryptology course for such an audience, as the mathematical prerequisites can be made fairly modest and supplemented by the history of the subject, or its relevance today to topics like privacy and security online. One even managed to assess it through written projects, despite the protests of the more mathematically inclined students! The consensus seems to be that if you’re going to teach such a course, your starting point should be Cryptography by Trappe and Washington, and -despite my love of the discrete log problem – it’s probably best to stick to symmetric crypto and a bit of RSA. Various speakers had developed software to remove some of the computational grind (such as crypto tools, linked above), but the coolest contribution was probably instructions (PDF) on how to make an Enigma machine out of a pringles can!

AMS-SIAM Special Session on Mathematics of Computation: Algebra and Number Theory, I & II
M. O’Sullivan, The sum-product algorithm for binary codes having check nodes of degree two.
D. Harm, Complexity of the Graph Isomorphism Problem.
N. Boston, Combining Group Theory and Number Theory Computations.
M. Jacobson, Class Group and Regulator Computation in Quadratic Fields.
A. Sutherland, Genus 1 point counting in quadratic space and essentially quartic time.
A. Silverberg, Finding the rational points on a certain genus 12 curve.
R. Scheidler, Efficient Divisor Reduction on Hyperelliptic Curves.
D. Moulton, Finding small sets whose subset sums include a given set.
J. Silverman, Lehmer’s Conjecture and points on elliptic curves that are congruent to torsion points.
C. Smyth, Minimal polynomials of algebraic numbers with rational parameters.
K. Hare, Pisot and Salem polynomials dividing Newman polynomials.

This session was the other reason for my attendance – Mahler measure is quite a niche topic, so with two talks on the agenda here I felt I should turn up, but they weren’t the only draw. If you dig deep enough in this blog, you’ll find that I spent the start of my PhD thinking about point counting problems and hyperelliptic curve arithmetic, which both featured here. A particular highlight was Andrew Sutherland’s talk, which presented improvements to SEA which have led to a substantially larger record for point counting on elliptic curves.

MAA Session on New and Continuing Connections between Math and the Arts, I

Fractal Tree No. 3 by R. Fathauer

`Mathematical Art’ usually conjures up images of fractals, but there’s a lot more to it than that and several themes emerged from this session and the attached exhibition.

The Alhambra in Spain gets another bump up my list of potential mathematical tourism sites: although it seems that debate continues over whether all seventeen wallpaper tilings can be found there, it seems to have the best (and best known) collection. But other talks mentioned their appearance in everything from Tibetan sand mandalas to Norwegian rosemaling. I discovered that there’s such a thing as ethnomathematics, which aims to go beyond cataloguing such connections between mathematics and culture and attempt to explain them.

Also finding its way to the travel list is the Museum of Mathematics, although I’ll have to wait a bit as it doesn’t exist yet… hopefully it’ll open in 2012. Rather than focus on dry historical exhibits, their vision is for installation pieces like a race circuit for square-wheel tricycles, large geometric sculptures, and interactive digital art. The target audience might be schoolkids, but I suspect I’d walk around with a big smile on my face too!

Another exciting project I was oblivious to is the Bridges series of conferences on connections between maths and art: these combine invited talks and papers (with peer-reviewed proceedings) with hands-on activities, an art exhibition, film screenings, all in a location chosen to inspire! The next one is at the University of Coimbra, Portugal, in July.

AMS Special Session on Self-Organization in Human, Biological, and Artificial Systems, II
R. Niemeyer, Graphs, Dynamical Systems, Fractals: A Heuristic Framework for Modeling the Structure and Dynamics of Complex Interactions Across Multiple levels of Analysis.
L. Smith, An Agent-Based Approach to Modeling Gang Rivalries.

Although it’s a long way from my research activities, emergent systems is one of the topics that first steered me towards mathematics and computer science. So with a spare hour to fill, I decided to indulge an old interest by sampling a couple of talks from this session. Laura Smith’s was particularly intriguing: based partly on geographic constraints, her team of mathematicians and criminologists was able to build a model of the (violent) interactions of LA’s numerous gangs. The hope is that such a model would be accurate enough to predict where best to focus police efforts to reduce conflict, although because I’ve been watching too much Castle lately I found myself dreaming up scenarios of mathematically-savvy gang bosses using optimization theory to maximise their territory instead…

MAA Invited Addresses
M. Matchett Wood, Binary quadratic forms: From Gauss to algebraic geometry
R. Bell, Lessons from the Netflix Prize

Melanie Matchett Wood’s talk was in the rare category of those from which I felt I’d gained some insight into abstract algebra. Whilst modern terminology is probably the best working language, I think there’s a lot to be said for tracing the historical roots of a topic, rather than simply overwriting it with what can be opaque notation. Gauss may have essentially being doing group theory, but he didn’t know that, and the motivation and inspiration is perhaps easier to understand without that abstraction.

The Netflix prize offered US$1million for a 10% improvement to their film recommendation algorithm. That might seem a lot easier than other million dollar prize problems, compared to the ferociously difficult millenium problems, for instance. But it also meant a lot more viable competition, especially as when Robert Bell’s team hit the required 10%, they didn’t simply win but triggered a 30 day endgame which saw alliances form and the leadership change hands repeatedly: in the end, “BellKor’s Pragmatic Chaos” triumphed by just a fraction of a percent and a twenty minute earlier submission time than their closest rivals. His talk captured this drama, entertained with some of the sub-problems encountered (Why is it so hard to tell who’ll like Napoleon Dynamite? What happens if a user gets a girlfriend? and just who has the time to rate 99% of the netflix database?), and also described plenty of the mathematics behind their algorithm. There’s a documentary film in there somewhere…

AMS-MAA-SIAM Gerald and Judith Porter Public Lecture and Special Film Presentation
R. Lang, From flapping birds to space telescopes: The mathematics of origami.
Film: Between the Folds.

…which leads me neatly to the final events. Robert Lang seems to have been central to the revolution in Origami caused by the mathematisation of the discipline. The ability to algorithmically create folding patterns of stick-figure skeletons has pushed forward the level of detail that can be achieved with a single sheet; but as with other media, the possibility of greater realism has led also to a reaction in the form of abstract works, from mathematically-inspired patterns to ‘single crease’ sculptures. But it’s not just about art: origami folding lends itself to the design of airbags and heart stents, or to the problem of getting large structures into space.

All of which appears in the film Between the Folds, that I’m going to recommend regardless of the contents of your netflix queue. Here’s the trailer:

So all in all I had an excellent time at the JMM; I’m certainly planning to attend the next one, which it seems will be held in Boston even earlier in January. Hopefully I’ll be able to give a talk too- the question is, in which session?

I try to avoid technical details of proofs in my talks, and to make the slides intelligible even if you weren’t there, so if you just want the motivation for, or results of, my PhD work then this is probably the best place to look. For all the proofs in tedious detail, there’s my thesis itself. I’ve since come up with a much more compact proof of the results in Chapter 5, which has lead to this draft paper classifying all cyclotomic L-graphs for d=-15,-11,-7 and -2: it’s far more compact than the corresponding sections of my thesis, but perhaps at the price of readability!

If you’d just like to know more about Mahler measure and Lehmer’s problem in general, then I’d recommend this survey by Chris Smyth. For various records related to small Mahler measure, see Mossinghof’s tables.

Having hit a bit of a wall trying to prove that a maximal cyclotomic matrix necessarily squares to 4I, I’ve been exploring related questions instead. For instance, it only took a couple of tweaks to my code to search for matrices that square to 3I instead of 4I; there turn out to be only finitely many, which isn’t particularly interesting. However, one of them is an 8 vertex cube which I recognised as ‘half’ the maximal cyclotomic graph S₁₆.

This got me thinking about the properties of graphs obtained by stitching together two graphs, and lead to an interesting construction. If M squares to nI, then by taking a second copy of its graph, negating and joining each vertex in the original to the corresponding one in the copy, you get a new matrix which squares to (n+1)I.

By iterating this process, many of the maximal cyclotomic graphs can be recovered; and since there are infinite families of maximal cyclotomic graphs, I can demonstrate infinitely many non-trivial integer matrices with all eigenvales of absolute value sqrt(n) for any integer n≥5 too.

A particularly nice example is the family of signed n-hypercubes generated by running this procedure on the ’1-cyclotomic’ graph consisting of just a line. The picture at the top of this post is of the 5th step, and I’ve put together an animation illustrating its construction:

No mathematical reason to stop at 5, it just gets harder and harder to draw these things!

But I can at least set the stage for more technically-minded readers (a friendlier explanation/illustration will hopefully follow once I truly understand all this!). Chris has characterised all symmetric integer matrices with the property that their eigenvalues are at most 2 in modulus; under a suitable transformation of their characteristic polynomials, these give cyclotomic polynomials and thus are referred to as cyclotomic matrices. Conveniently, any submatrix of a cyclotomic matrix is itself cyclotomic, so it suffices to find maximal examples. Although there are infinite families of these matrices, there are only a few ‘types’ possible.

These types are best understood by considering not the matrix, but an associated graph, where values in the matrix determine the weights on edges and nodes of the graph. This introduces a notion of equivalence, since many matrices will correspond to the same graph or certain well-defined variations on it. Further, we can adjoin nodes and edges to the corresponding graph to try and ‘grow’ towards maximal examples.

The motivation comes from finding polynomials of small Mahler measure- whilst a cyclotomic polynomial has measure 1, all others seem to be pushed away, with the smallest known value being 1.176… The question is how to generate small examples, and these matrices provide a way: by adjoining a single extra node to a maximal cyclotomic graph, a non-cyclotomic graph/matrix is obtained and thus a non-cyclotomic polynomial. The minimal graphs with this property (non-cyclotomic, but all subgraphs cyclotomic) often correspond to polynomials with some of the smallest known Mahler measures.

But some examples are not generated in this way, which is where I’ve stepped in. There is no reason to restrict attention to integer matrices, and I’ve established which imaginary quadratic extensions of the rationals give rise to rings of integers over which suitable matrices can be found. For a couple of fields, there are very few new (non rational-integer) cyclotomic matrices, and I have a complete description of them, but in others there are again infinite families as well as occasional examples that don’t generalise.

So I explore this behaviour by growing graphs/matrices, and try to spot patterns as they emerge from the fragments. I use the university’s parallel computing cluster Eddie for brute force work in SAGE, but such is the nature of the combinatorial explosion that even this doesn’t suffice without some mathematical insight along the way, as I try to refine my growing algorithms and capture equivalence as early as possible. I’m hopefully nearing the point where all examples fit into known families, at which point I’ll need to switch into serious mathematician mode and try and prove why this should be so. But for now I need to make sure that nothing unexpected tumbles out of each batch of calculations!

On a completely unrelated note, I’ve dragged modulo errors up to date with wordpress 2.5 and switched themes; please shout if you find I’ve broken something along the way.

Much of the talk discussed the validity of results on infinite graphs when applied to smaller (and hence computationally feasible) ones: in particular, obtaining good bounds on clique and chromatic numbers. Towards the end, the reduction of graph problems to the satisfiability problem in propositional logic was mentioned. Obviously, the possibility of such reductions is a defining characteristic of NP-hard problems (and one I’ve recently been studying), but it’s good to see that this is of practical merit rather than simply theoretical interest.

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).