Wiles and Langlands

One of the best things that pops regularly into my pigeonhole is the European Mathematical Society Newsletter. The current issue, number 104, is particularly rich in interesting articles.

For example, there is the transcript of a talk by Christian Krattenthaler (without the musical interludes, sad to say) on “Mathematics AND Music?”. He avoids many of the usual clichés about the relationship between these two disciplines, and defends the thesis

Both Mathematics AND Music are food for the soul AND the brain.

It may be difficult to explain to the non-mathematician that there is soul in mathematics; Christian refers them to the moving moment in the BBC documentary on Fermat’s Last Theorem when Andrew Wiles describes how he finally overcame the difficulties in the proof.

This brings me to the article I really want to talk about: Henri Darmon on “Andrew Wiles’ Marvellous Proof”, in which he explains the relationship between what Wiles did and the Langlands programme, which he describes as “the imposing, ambitious edifice of results and conjectures that has come to dominate the number theorist’s view of the world”. He gives a “beginner’s tour” of the Langlands programme, first excusing his own shortcomings. I want to say a bit about this, but his shortcomings are insignificant compared to mine! I shall also be much briefer, and refer you to the article.

The context is the solution of Diophantine equations, solutions of a polynomial equation in several variables over the integers (or maybe the rational numbers). But rather than tackle this head-on, we first count solutions over the finite field of order pr. (These are the Galois fields, constructed and proved unique by Galois.) For fixed p and varying r, we get an infinite sequence of numbers, which can be represented by a zeta function. Dwork and Deligne proved two remarkable facts about such zeta-functions: first, they are rational functions whose numerators and denominators are independent of the prime p (if a few bad primes are avoided), and second, the reciprocals of their roots are powers of √p. The last is the Riemann hypothesis over function fields, part of the work for which Deligne won the Abel prize.

The new developments involve allowing p to vary. For degree 2 equations in a single variable, the relationship between different primes is precisely described by Gauss’ quadratic reciprocity law. For higher degrees, things get a bit more complicated. We must combine the local zeta functions into a single global zeta function, and show that it is a modular form, in the sense that it is transformed in a very simple manner by linear fractional transformations of its argument.

This brings us to the Shimura–Taniyama conjecture, stating (more or less) that the zeta function of an elliptic curve is a modular form of weight 2. This is what Wiles proved (in the semistable case) and which led, by earlier work of Frey and Ribet, to a proof of Fermat’s Last Theorem. (The semistability assumption was later removed.)

So what does this have to do with Langlands? We have to look at Galois representations, linear representations of the Galois group of the algebraic numbers over the rationals (more precisely, of quotients corresponding to extensions of the rationals unramified outside a finite set S of primes). One can define a zeta function of such a representation; work of Weil, Grothendieck and others shows that, if the diophantine equation has good reduction outside S, then its zeta function is the quotient of the zeta functions of two such Galois representations. Now representations can be decomposed into irreducible representations, and the corresponding zeta functions multiply; so we can look at irreducible representations. Now there are notions of “modular” and “geometric” for Galois representations (the latter corresponding to realisation in an étale cohomology group, as the representations involved in zeta functions of diophantine equations do); the “main conjecture of the Langlands programme” states:

All geometric Galois representations are modular.

One of the main ingredients of Wiles’ work is a lifting theorem allowing the proof of this under suitable (local-to-global) hypotheses.

One detail I have not mentioned is the connection of the Dedekind eta-function with the generating function for the partition numbers, which featured in the work of Ramanujan; Darmon says it “plays a starring role alongside Jeremy Irons and Dev Patel in a recent film about the life of Srinivasa Ramanujan”.

Which brings me back to Krattenthaler’s article. In explaining how mathematics, like music, can contain humour, he outlines the proof of “Ramanujan’s most beautiful theorem”, the statement that the number of partitions of 5n+4 is always divisible by 5. For this, a certain amount of detail about q-series and Jacobi’s triple product formula is required before we get to the punchline of the joke!

The message from all this is that there is are deep-level correspondences between some superficially very different parts of mathematics!

I do urge you to read these articles yourself. Better, why not join the European Mathematical Society and get the Newsletter?

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Mike O’Nan

I just heard the news that Mike O’Nan died on Monday.

I never knew him well, but I know for sure that his name will not be forgotten: a sporadic simple group, a theorem about maximal subgroups of symmetric groups (depending on a result on the socle of a primitive permutation group) that he proved simultaneously with Leonard Scott, a technique for showing non-existence of sharply 2-transitive subsets of certain 2-transitive groups …

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All kinds of mathematics: workshop and conclusion

Conference photograph

The final day of the conference was devoted to a satellite workshop on “Symmetry in finite and infinite structures”. The morning was a solo performance by Laci Babai, in which he explained in detail his quasi-polynomial algorithm for graph isomorphism. In the afternoon there were short contributed talks by participants, at which I had to make invidious choices and miss many talks I wanted to hear. I did listen to Milan Stehlík using root systems in statistics (in quite a different way from my one venture into this), and also celebrating the fact that, in New Zealand, the Maori have succeeded in having the Whanganui River declared a legal entity (with the rights, duties and liabilities of such) after 140 years. Then Shayo Olukoya introduced us to the rational group (homeomorphisms of Cantor space induced by transducers acting on strings) and some of its subgroups; Fatma Al-Kharousi gave us some combinatorics related to semigroups of order-preserving maps of a finite set; Justine Falque described nearly complete work to show that if the number of orbits on n-sets of an oligomorphic permutation group is polynomially bounded, then the orbit algebra is finitely generated, and so the orbit-counting sequence is polynomial on residue classes; Shichang Song showed that the automorphism group of Philip Hall’s universal locally finite group has ample generics; and Wilf Wilson talked about algorithms for finding maximal subsemigroups, which will be included in the Semigroups package for GAP. What a feast!

Altogether an amazing conference, certainly one of the most memorable experiences of my life. You can see the short film that was made to open the conference; if you watch, you will see João Araújo’s sense of humour shining through. Thank you, João, for this, and for so much else you contributed to the conference, and to your many kindnesses to me over the last ten years; the theorems, the crazy trips all over Portugal, and much more! (João is my equal top coauthor; we have ten papers published or in press, even though the first one only appeared in 2013.) And thank you to all the other organisers as well: Teresa Oliveira (Aberta), Mário Edmundo (Lisboa), Maria Elisa Fernandes (Aveiro), Gracinda Gomes (Lisboa), Ana Paula Santana (Coimbra), and Ivan Yudin (Coimbra). I hope that everyone who was there took away good memories and good new mathematics to think about and work on.

As I said in my talk, in seventy years time I want these problems solved!

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All kinds of mathematics, 4

The fourth day began with a talk by László Babai on his quasi-polynomial algorithm for graph isomorphism. Two longer and more detailed accounts were scheduled for the workshop; this talk gave the context, and an important lemma which underlies the proof.

Then Cheryl Praeger talked about work with Stephen Glasby, Kyle Ross, and Gabriel Verret on the composition length of a primitive permutation group of degree n. Laci Pyber had outlined a c log n bound, but Cheryl and her coauthors have made this precise, finding the best possible constant c (which turns out to be 8/3) and determining the groups which meet this bound. I suggest an exercise for anyone interested: find a good bound for the composition length of a 2-transitive group – this will be much easier! In the discussion, the result that João and I found last year for the number of generators of a 2-transitive group (namely 2) was recalled. I still find it hard to believe that nobody noticed this before!

Francesco Matucci told us about hyperbolic groups (those for which a Cayley graph is δ-hyperbolic in the sense of Gromov), and presented a theorem which has as a corollary that torsion-free hyperbolic groups embed in the rational group (the group of maps of Cantor space generated by transducers acting on infinite strings).

Dugald Macpherson came next. He and his colleagues have been investigating very wide model-theoretic generalisations of the Hasse–Weil estimates for definable sets over finite fields. Depending on whether the approximations are exact or approximate, very different results are obtained.

Michael Giudici and his colleagues have been investigating semiprimitive permutation groups, those in which every non-trivial normal subgroup is either transitive or semiregular. This condition is very relevant to things related to the Weiss conjecture, which Pablo Spiga discussed. The role of minimal normal subgroups and the socle is taken by plinths for such a group (minimal transitive normal subgroups); remarkably, they have good control over these.

Ben Steinberg’s talk was about random walks on a module M over a finite ring R, where the transitions are affine maps of the form x → ax+b. Here, a is not required to be a unit, so the maps may be non-invertible. For example, if R is the 2-element field, the steps are random translations or jumps to random points. He uses the representation theory of semigroups (the subject of his recent book) to work out the eigenvalues of the transition matrix.

Wolfram Bentz told us about permutation groups, transformation semigroups, and graphs; at least this was material which I knew about, although there were probably others who didn’t.

Misha Volkov gave a very nice talk about completely reachable automata, those for which every non-empty subset of the states is the image of some composition of transitions. Clearly the semigroup generated by such an automaton has size at least 2n−1, where n is the number of states. This is a much stronger condition than synchronization, and Misha was able to use binary trees to construct and characterise such automata.

The day, and the conference proper, ended with my talk; you can find the slides in the usual place.

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All kinds of mathematics, 3

Wednesday morning’s first speaker was Bob Gray, who told us about his quest, with Igor Dolinka, on a semigroup analogue for Philip Hall’s countable universal homogeneous locally finite group. He started with a clear introduction to Hall’s group. For semigroups, it turns out that no such example can exist; said otherwise, a countable universal locally finite semigroup cannot be homogeneous. So how homogeneous can it be? If the automorphism group of such a semigroup acts transitively on the copies of the finite semigroup S it contains, then S must be an amalgamation base; they call a semigroup maximally homogeneous if the converse holds. There is a unique such semigroup which is a limit of full transformation semigroups (as Hall’s group is a limit of symmetric groups, though the analogue of Hall’s construction fails). It has lots of nice properties; for example, its Graham–Houghton graph is the universal homogeneous graph with bipartition.

Bob started by talking about some of my books and papers which have influenced him. A reminder that any piece of writing, once it leaves the writer’s desk, has a life of its own.

Then António Malheiro told us about the plactic monoid (connected to Young tableaux) and some generalisations, the hypoplactic monoid (related to ribbon tableaux) and the Sylvester monoid (related to binary search trees). He was interested in finding analogues of Kashiwara chrystal bases for these objects.

James Mitchell spoke about semigroups generated by digraphs (where we regard a directed arc (ab) as a rank n−1 idempotent which maps a to b and fixes everything else. He surveyed many old and new results about these semigroups, and remarked that unlike many types of transformation semigroup the correlation of properties between semigroup and digraph is very close. There are connections with sliding-block puzzles (indeed, he quoted the work of Richard Wilson on these). Sliding-block puzzles are more usually represented by a groupoid than a semigroup; there must be connections!

Alexander Hulpke has new code for computing interval lattices in a group G (essentially, all subgroups between H and G for some subgroup H and their inclusions), which is much faster than the current GAP code. I am sure I will find a use for this code, which will be included in GAP 4.9 (hopefully before the end of the year).

And finally Peter Mayr talked about counting finite (universal) algebras up to term equivalence. He introduced the class of algebras with few subpowers (the number of subalgebras of An grows only exponentially with n rather than doubly exponentially, and showed us that such algebras are finitely related, have a polynomial time solution for the constraint satisfaction problem, and various other nice properties.

After lunch, we left for the excursion. First we went to the National Gallery, where the highlights included a polyptych of the generation of Henry the Navigator and the following generation of Portuguese royalty whose interpretation is very controversial, and a wonderful Hieronymus Bosch triptych. I also passed an old acquaintance whom my colleagues will certainly recognise:

St Andrew

Then to the Ajuda palace gardens, where we had a very enjoyable dinner in a tent in the garden, interspersed with some embarrassing recollections (which, fortunately, were not recorded), and so home.

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All kinds of mathematics, 2

Day 2 began with Gordon Royle telling us how he had almost completely finished the proof that roots of chromatic polynomials of planar graphs are dense in the interval (3,4). Now, on this topic, only the Birkhoff–Lewis conjecture that there are no chromatic roots of planar graphs in (4,5) remains. Gordon showed us a map with a line joining Perth to Lisbon; I think it was not a rhumb line.

Boris Zilber described his work trying to make sense, from a logician’s point of view, of the limits used by physicists in going from the discrete to the continuous, a problem which (as he pointed out) goes back to Hilbert’s 6th problem in 1900. He has some deep results, not all positive. For example, compact Lie groups are not approximable (in his sense) by finite groups.

Leonard Soicher has written a GAP program, using his GRAPE package, to find the chromatic number of a graph, with the GRAPE philosophy of using symmetry of the graph to cut down the search. Among other things, he can determine, for all primitive groups of degree at most 255, whether or not they are synchronizing; the total time taken is quite small. (There are a few difficult groups where the program needs a hint; we hope that understanding these better will lead to theoretical advances.)

John Meakin, the person I have known longest at this conference (we were undergraduates together in Brisbane from 1964 to 1967), started with the observation that connected covers of a connected topological space are classified by subgroups of the fundamental group. He has defined a gadget which classifies immersions in a similar way. Perhaps not surprisingly, it is an inverse semigroup. He showed us the outline of 15 steps in the argument.

Pedro Silva, with John Rhodes, introduced a class of objects more general than matroids, indeed general enough to include bases of permutation groups (a dream of mine for a long time): these are Boolean representable simplicial complexes, or BRSCs. He gave us a survey of this and mentioned software for computing with these objects. I do hope to say more someday.

A very nice talk from Michael Kinyon followed, about the multiplication group of a loop. A loop is a set with a binary operation having an identity and unique left and right division, in other words, one whose Cayley table is a normalised Latin square; the multiplication group is generated by rows and columns of the Latin square. Many interesting results and conjectures connect properties of the loop with those of its multiplication group; in particular, for non-commutative simple loops the group is primitive. Colva Roney-Dougal (who unfortunately is not here) and Michael Giudici are working on the problem of classifying simple automorphic loops (those for which the stabiliser of the identity in the multiplication group consists of loop automorphisms). The conjecture is that these must be (non-abelian simple) groups. According to Michael, there is light at the end of the tunnel.

Brendan McKay talked about enumeration of three classes of matrices of non-negative integers: rectangular matrices with constant row and column sums; symmetric square matrices with zero diagonal and constant row sums; and adjacency matrices of regular graphs (the previous case with all entries zero and one). He gave exact formulae, asymptotics, and accurate simulations.

Robert Bailey told us about bases and resolving sets for permutation groups, coherent configurations, and metric spaces. We wrote a survey on this in 2011, to try to throw some light on the confusion caused by re-invention of the same concepts in many different fields; this is now his most cited paper, and is on the first page of my papers on Google scholar.

To end the day, Péter Pál Pálfy talked about twisted wreath products, and the role they play in the lattice representation problem. A famous question asks whether every finite lattice is the congruence lattice of a finite universal algebra; an equivalent question is whether every finite lattice is isomorphic to an interval in a subgroup lattice. Twisted wreath products have settled new cases of this conjecture.

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All kinds of mathematics, 1

There is far too much, and far too diverse, mathematics going on here for me to describe all or even most of it. Nine plenary lectures on the first day! The talks are being live streamed at https://videocast.fccn.pt/live/idprovider_uab/kinds_mathematics_remind, though the organisers admit that it doesn’t always work (there is only a fixed camera, and speakers do not always stand in front of it).

Henrique Leitão opened the conference with a talk about Pedro Nunes, covering the retrograde motion of the shadow of a sundial, his work on rhumb lines which was very likely used by Mercator as the basis of his map projection, and his algebra book. On the first topic, Nunes wrote,

This is most surprising but cannot be denied because it is demonstrated with mathematical certainty

(in Henrique’s opinion one of the first uses of science to prove the existence of a totally new phenomenon, and one which was not observed for well over 300 years). On algebra, he described a problem easily solved by algebra but where no geometric construction is known. Given the base a, the height h, and the ratio b/c of the other two sides of a triangle, find b and c. Nunes said of this problem and others, “Who knows by Algebra, knows scientifically”.

Peter Neumann told us about Galois’ work on transitive permutation groups of prime degree (in his language, irreducible equations of prime degree), and wondered if we will ever have the classification of these groups without using CFSG.

Then Cameron Freer talked about exchangeable measures concentrated on a single isomorphism class of countable relational structures. The random graph is the prototype, but is too easy; it wasn’t until much later that Petrov and Vershik handled Henson’s universal triangle-free graph, and Cameron with his colleagues Nate Ackerman and Rehana Patel extended this to all structures with trivial algebraic closure. At one point he showed us a picture by M. C. Escher, “Square Limit”, which can be regarded as a graphon for the random graph. The picture was from 1964, which was the same year that Richard Rado gave his explicit description of the graph; should it be re-named the Escher–Rado graph?

Rosemary Bailey told us about weak neighbour balance for designs in circular blocks, which ties in Hadamard matrices, doubly regular tournaments, something that Laci Babai and I called “S-digraphs”, and many other interesting things.

After lunch, Pablo Spiga told us about conjectures of Sims and Weiss and his work on the latter. Dimitri Leemans spoke about the epic battle that he, Maria Elisa Fernandes, Mark Mixer and I had with the maximal rank of a regular polytope with automorphism group the alternating group An, published in the Proceedings of the London Mathematical Society last month.Gareth Jones reported on his constructions for “doubly Beauville Hurwitz groups”, together with a very nice description of the algebraic geometry and group theory behind it.

Greg Cherlin had found an old unpublished paper that Sam Tarzi and I wrote on the structure of the automorphism group of G, the automorphism group of (yes, that repetition is intended) the random m-edge-colouring of the countable complete graph: all the automorphisms are induced by permutations, so the group is an extension of G by the symmetric group Sm, and the extension splits if and only if m is odd. Greg has in mind a far-reaching generalisation of this, particularly for homogeneous metric spaces, with unexpected and mysterious connections with the work of Bannai and Ito on P-polynomial structures on association schemes.

To finish the day, Jarik Nešetřil described recent work on Ramsey classes, and its connection with EPPA.

I didn’t mention the film that the organisers have made about me (based mostly on old photographs). I am really not used to being in the spotlight to this extent!

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