ACORN generators represents an approach to generating uniformly distributed pseudo-random numbers which is straightforward to implement for arbitrarily large order $k$ and modulus $M=2^{30t}$ (integer $t$). They give long period sequences which can be proven theoretically to approximate to uniformity in up to $k$ dimensions, while empirical statistical testing demonstrates that (with a few very simple constraints on the choice of parameters and the initialisation) the resulting sequences can be expected to pass all the current standard tests .

The standard TestU01 Crush and BigCrush Statistical Test Suites are used to demonstrate for ACORN generators with order $8≤k≤25$ that the statistical performance improves as the modulus increases from $2^{60}$ to $2^{120}$. With $M=2^{120}$ and $k≥9$, it appears that ACORN generators pass all the current TestU01 tests over a wide range of initialisations; results are presented that demonstrate the remarkable consistency of these results, and explore the limits of this behaviour.

This contrasts with corresponding results obtained for the widely-used Mersenne Twister MT19937 generator, which consistently failed on two of the tests in both the Crush and BigCrush test suites.

There are other pseudo-random number generators available which will also pass all the TestU01 tests. However, for the ACORN generators it is possible to go further: we assert that an ACORN generator might also be expected to pass any more demanding tests for $p$-dimensional uniformity that may be required in the future, simply by choosing the order $k>p$, the modulus $M=2^{30t}$ for sufficiently large $t$, together with any odd value for the seed and an arbitrary set of initial values. We note that there will be $M/2$ possible odd values for the seed, with each such choice of seed giving rise to a different $k$-th order ACORN sequence satisfying all the required tests.

This talk builds on and extends results presented at the recent discussion meeting on “Numerical algorithms for high-performance computational science” at the Royal Society London, 8-9 April 2019, see download link at bottom of web page http://acorn.wikramaratna.org/references.html.

I will give an introduction to the theory of definable parameterization of definable sets in the o-minimal context and its application to diophantine problems. I will then go on to discuss uniformity issues with particular reference to the subanalytic case. This is joint work with Jonathan Pila and Raf Cluckers

 For a semisimple modular tensor category the Reshetikhin-Turaev construction yields an extended three-dimensional topological field theory and hence by restriction a modular functor. By work of Lyubachenko-Majid the construction of a modular functor from a modular tensor category remains possible in the non-semisimple case. We explain that the latter construction is the shadow of a derived modular functor featuring homotopy coherent mapping class group actions on chain complex valued conformal blocks and a version of factorization and self-sewing via homotopy coends. On the torus we find a derived version of the Verlinde algebra, an algebra over the little disk operad (or more generally a little bundles algebra in the case of equivariant field theories). The concepts will be illustrated for modules over the Drinfeld double of a finite group in finite characteristic. This is joint work with Christoph Schweigert (Hamburg).

This talk is a brief introduction to the analysis of Donaldson theory, a branch of gauge theory. Roughly, this is an area of differential topology that aims to extract smooth structure invariants from the geometry of the space of solutions (moduli space) to a system of partial differential equations: the Yang-Mills equations.

I will start by discussing the differential geometric background required to talk about Yang-Mills connections. This will involve introducing the concepts of principal fibre bundles, connections and curvature. In the second half of the talk I will attempt to convey the flavour of the mathematics used to address technical issues in gauge theory. I plan to do this by presenting a sketch of the proof of Uhlenbeck's compactness theorem, the main technical tool involved in the compactification of the moduli space.

Property (FA) is one of the `rigidity properties’ defined for groups, concerning the way a group can act on trees. We’ll take a look at why you might be interested in an action on a tree, what the property is, and then investigate which automorphism groups of free products have it.

The last decade or so has seen substantial progress in the theory of (outer) automorphism groups of right-angled Artin groups (RAAGs), spearheaded by work of Charney and Vogtmann. Many of the techniques used for RAAGs also apply to a wider class of groups, graph products of finitely generated abelian groups, which includes right-angled Coxeter groups (RACGs). In this talk, I will give an introduction to automorphism groups of such graph products, and describe recent developments surrounding the outer automorphism groups of RACGs, explaining the links to what we know in the RAAG case.