## YRM16 report

The Young Researchers in Mathematics conference took place this week in St Andrews. Since I was a plenary speaker, I was signed up as a proper participant, so I felt entitled to go to as many talks as I wanted without feeling I was gatecrashing. As expected, it was a lively conference, with plenty of enthusiasm and unexpected connections. Below are comments on some of the young researchers’ talks. If you trouble to read it through, note the threads connecting talks in quite different areas.

A couple of talks by St Andrews students involved transducers, finite machines which read input, change state, and write output. Casey Donoven explained that the second world war Enigma machine is an example; it is synchronous, that is, it writes one output symbol for every input symbol it reads. A transducer can read an infinite input string, and so defines a map on the set of such strings, which is contninuous (in the product topology induced from the discrete topology on the alphabet).

The set of binary strings corresponds naturally to the “middle third” Cantor set in the unit interval, with 0 and 1 corresponding to taking the left or right subinterval at each step. There is also a map from the set of binary strings to the unit interval, given by the “binary decimal” representation, which is not quite bijective since countably many reals have two binary representations. A transducer which preserves these pairs of strings induces a continuous map on the unit interval, as Casey explained.

Shayo Olukoya considered the group defined by synchronous transducers which are also bi-synchronizing. This means that there is a number k such that, after reading k symbols, the machine is in a state which depends only on the symbols read, and not on the initial state, and the same holds for a transducer realising the inverse function. He explained how to solve algorithmically the problem of whether the induced map has finite order.

Experts will see Richard Thompson’s groups lurking in the background here. Nayab Khalid described a more general situation, a “substitution and glueing” procedure on directed graphs, which produces Thompson-like groups acting on interesting fractals like the basilica Julia set, and proving things about these groups.

Returning to Casey’s almost-bijection between Cantor space 2N and the unit interval, there is also a famous almost-bijection between Baire space NN (the set of infinite sequences of positive integers) and the unit interval, given by the continued fraction. The Gauss map Tx → 1/x (mod 1) on the unit interval corresponds to the left shift on NN. Natalia Jurga generalised this situation to maps with some of the properties of the Gauss map. Any probability vector on N induces a product measure on Baire space, which can be pulled back to the unit interval using the map T. Natalia was interested in the maximum Hausdorff dimension of such a measure.

Jason Dungca also used the Gauss map, with the probability of an integer n proportional to 1/n2; he was interested in phase changes in the multifractal spectrum of Gibbs measures. I won’t attempt further explanation.

Wojciech Ożański started his talk about singularities of the Navier–Stokes equation in fluid mechanics by reminding us of the definitions of Hausdorff and box dimension. He was interested in bounding these dimensions of the sets of singularities and blow-up times for a solution.

Douglas Howroyd defined these two dimensions and also Assouad dimension, the one he was chiefly interested in. He showed us how to compute the dimension of fractals defined by certain sets of affine contractions. These map the unit square onto various rectangular regions obtained from a dissection of the square, and go by the name of carpets.

Demi Allen used Hausdorff dimension to refine a result of Khintchine on diophantine approximation. Given a function ψ, let A(ψ) be the set of real numbers x in the unit interval which satisfy |xp/q| < ψ(q)/q for infinitely many rationals p/q. Khintchine showed that the Lebesgue measure of A(ψ) is equal to 1 if ∑ψ(q) diverges, 0 if it converges (and ψ is monotonic). So, for example, if ψ(q) = q−t, the Lebesgue measure is 0 if t > 1. In this case, its Hausdorff dimension turns out to be 2/(t+1). Demi was interested in multivariable versions of this. The key turned out to be a version of the “mass transference principle” relating Lebesgue measure on a ball to Hausdorff meaasure on a larger ball, extended from balls to neighbourhoods of hyperplanes.

Khavlah Mustafa considered the Möbius (linear fractional) groups over certain rings such as the complex, dual, and double numbers. (These are obtained from the real numbers by adjoining square roots of −1, 0 and 1 respectively.) She described fixed points and orbits of 1-parameter subgroups, and gave a dynamical classification of the fixed points.

For classical groups over finite fields, Daniel Rogers described how to find maximal subgroups in the catch-all Aschbacher class C9, which are themselves classical groups over fields of the same characteristic. (For ease of exposition he restricted his attention to the special linear groups.) He was particularly interested in dimension 16, the smallest for which the classification of maximal subgroups is not yet done, and showed us some interesting examples. He used results of Steinberg which made it all seem much easier than in fact it is.

Scott Harper talked about 2-generated groups (a class including the finite simple groups, as we know from the classification of these groups). His basic question goes back to Netto, who conjectured in 1882 that two random permutations generate the symmetric or alternating group with high probability (tending to 1 as n→∞). The spread of such a group is the maximum k for which, given any k non-identity elements x1,…xk, there is an element y with the property that xi and y generate the group for all i. Thus “spread at least 1” means that any non-identity element is in a 2-element generating set, a property also called “3/2-generation”. Remarkably, no finite group having spread precisely 1 is known, and it is conjectured that none exists.

I was delighted by Alex Rogers’ talk, which took me back to things from my distant past, forty years ago. She was calculating the PI-degree (the smallest degree of a polynomial identity) for quantum matrix algebras. (For ordinary matrix algebras of n×n matrices, the answer is 2n, given by the Amitsur–Levitski Theorem.) My first real job was at Bedford College (a part of the University of London which no longer exists); the head of department was Paul Cohn, and his big idea was to develop “non-commutative algebraic geometry”, not so far from modern quantum groups, and PI-degree played a role in this. Then, at the end, Alex revealed that the formula for PI-degree in some cases depends on the invariant factors of a specific matrix, and these are powers of 2. This suggested to me the chains of binary codes that my second DPhil student, Eric Lander, invented and studied.

Christian Bean demonstrated the program Struct which he and his colleagues are developing for studying (and finding patterns in) sets of permutations, and if possible enumerating these sets. As he said, the program is not yet completely user-friendly.

Waring’s problem concerns expressing positive integers as sums of kth powers: how many powers are required, and how many solutions are there? Kirsti Biggs extended this to Waring’s problem with shifts, where we are given small irrational numbers θi and a number η, and we want to make the difference between the given n and a sum of s terms (xi−θi)k smaller than η. The tool was replacing the count by an integral and using the Davenport–Heilbronn method.

Adelina Mânzățeanu found rational curves on a smooth cubic hypersurface over a finite field, passing through two given points. She made much use of one of my favourite theorems, the Chevalley–Warning Theorem. Her methods involved analysis in Fq(t) in some sense parallel to analogous arguments over Q, and she gave us a dictionary for comparing the two situations. (What is the analogue of a torus, of the Fourier transform, in the completion of Fq(t)?)

Thanatkrit Kaewtem talked about γ-Banach spaces (these are like Banach spaces but the unit ball need not be convex; a typical example is lγ for γ < 1). He was considering inner and outer entropy measures on a bounded operator between two such spaces, involving covering the image of a unit ball by translates of a small ball, an idea going back to Kolmogorov. Various inequalities connect these numbers to the operator norm and (in the case of an operator from a space to itself) its eigenvalues. He showed us why some of these inequalities are tight.

The first time I went to the YRM (in Warwick in 2011), there were no Queen Mary students there. Things were a little better this time. The sole QM student, Wan Nur Fairuz Alwani Wan Rozali, was considering a discrete dynamical system analogoes to the pendulum, given by iteration of a function on Z2 (I didn’t catch the exact formula for the function), proving a conjecture about its first return time to the positive X-axis.

With great self-restraint (there were some great talks, authoritative and exciting), I won’t say much about the plenary and keynote talks, except for one remarkable coincidence. Both Clément Mouhout, talking about partial differential equations, and Philip Welch, talking about axiomatic set theory, saw fit to tell us the story of the famous “Scottish Book” in which mathematicians of the Lwów school such as Banach, Kuratowski, and Ulam recorded problems. It has no direct connection with the location of the conference, but was named after the Scottish Café in in Lwów which the mathematicians used to meet. There could hardly be a better demonstration of the unity of mathematics than this!

And here are a couple of quotes from keynote speakers, brazenly taken out of context:

• “You are allowed to do that”
• “There are other dimensions also”

For me it was an excellent conference, with a higher than usual proportion of talks which completed a circuit in my brain by connecting to something quite different. I have plenty of new things to think about as a result. So thanks to Oliver, Daniel, Tom, Cristina, Zoë, and Sascha for putting on such a good show.

Mathematics is in good hands! 