University of Oxford

An anomalous symmetry of an operator algebra A is a mapping from a group G into the automorphism group of A which is multiplicative up to inner automorphisms. To any anomalous symmetry, there is an associated cohomology invariant in H^3(G,T). In the case that A is the Hyperfinite II_1 factor R and G is amenable, the associated cohomology invariant is shown to be a complete invariant for anomalous actions on R by the work of Connes, Jones, and Ocneanu.

In this talk, I will introduce anomalous actions from the basics discussing examples and the history of their study in the literature. I will then discuss two obstructions to possible cohomology invariants of anomalous actions on simple C*-algebras which arise from considering K-theoretic invariants of the algebras. One of the obstructions will be of algebraic flavour and the other will be of topological flavour. Finally, I will discuss the classification question for certain classes of anomalous actions.

The pro-C completion of a free profinite group on an infinite set of generators is a profinite group of a greater rank. However, it is still not known whether it is a free profinite group too.  We will discuss this question, present a positive answer for some special varieties, and show partial results regarding the general case. In addition, we present the infinite tower of profinite completions, which leads to a generalisation for completions of higher orders. 

 Wide Neural Networks are well known for their Gaussian Process behaviour. Based upon this fact, an initialisation scheme for the weights and biases of a network preserving some geometrical properties of the input data is presented — The edge-of-chaos. This talk will introduce such a scheme before briefly mentioning a recent contribution related to the edge-of-chaos dynamics of wide randomly initialized low-rank feedforward networks. Formulae for the optimal weight and bias variances are extended from the full-rank to low-rank setting and are shown to follow from multiplicative scaling. The principle second order effect, the variance of the input-output Jacobian, is derived and shown to increase as the rank to width ratio decreases. These results inform practitioners how to randomly initialize feedforward networks with a reduced number of learnable parameters while in the same ambient dimension, allowing reductions in the computational cost and memory constraints of the associated network.

Placeholder entry; date+time TBC. 

Abstract for talk: In this talk, we present a systematic discretization of the Bernstein-Gelfand-Gelfand (BGG) diagrams and complexes over cubical meshes of arbitrary dimension via the use of tensor-product structures of one-dimensional piecewise-polynomial spaces, such as spline and finite element spaces. We demonstrate the construction of the Hessian, the elasticity, and div-div complexes as examples for our construction.

The Stokes–Onsager–Stefan–Maxwell (SOSM) equations model the flow of concentrated mixtures of distinct chemical species in a common thermodynamic phase. We derive a novel variational formulation of these nonlinear equations in which the species mass fluxes are treated as unknowns. This new formulation leads to a large class of high-order finite element schemes with desirable linear-algebraic properties. The schemes are provably convergent when applied to a linearization of the SOSM problem.

COURSE_PROPOSAL (12).pdf

The space of unordered tuples. The notion of differentiability and the theory of metric Sobolev in the context of multi-valued functions. Multivalued maximum principle and Holder regularity. Estimates on the Hausdorff dimension of the singular set of Dir-minimizing functions. If time permits: mass minimizing currents and their link with Dir-minimizers. 

In this talk we will study the graph structure of supersingular isogeny graphs. These graphs are known to have very few loops and multi-edges. We formalize this idea by studying and finding bounds for their number of loops and multi-edges. We also find conditions under which these graphs are simple. To do so, we introduce a method of counting the total number of collisions (which are special endomorphisms) based on a trace formula of Gross and a known formula of Kronecker, Gierster and Hurwitz. 

The method presented in this talk can be used to study many kinds of collisions in supersingular isogeny graphs. As an application, we will see how this method was used to estimate a certain number of collisions and then show that isogeny graphs do not satisfy a certain cryptographic property that was falsely believed (and proven!) to hold.

The joint spectral radius of a finite family S of matrices measures the rate of exponential growth of the maximal norm of an element from the product set S^n as n grows. This notion was introduced by Rota and Strang in the 60s. It arises naturally in a number of contexts in pure and applied mathematics. I will discuss its basic properties and focus on a formula of Berger and Wang and results of J. Bochi that extend to several matrices the classical for formula of Gelfand that relates the growth rate of the powers of a single matrix to its spectral radius. I give new proofs and derive explicit estimates with polynomial dependence on the dimension, refining these results. If time permits I will also discuss connections with the Tits alternative, the notion of joint spectrum, and a geometric version of these results regarding groups acting on non-positively curved spaces.

In his pioneering and celebrated 1968 paper on the elementary theory of finite fields Ax asked if the theory of the class of all the finite rings $\mathbb{Z}/m\mathbb{Z}$, for all $m>1$, is decidable. In that paper, Ax proved that the existential theory of this class is decidable via his result that the theory of the class of all the rings $\mathbb{Z}/p^n\mathbb{Z}$ (with $p$ and $n$ varying) is decidable. This used Chebotarev’s Density Theorem and model theory of pseudo-finite fields.

I will talk about a recent solution jointly with Angus Macintyre of Ax’s Problem using model theory of the ring of adeles of the rational numbers.

I'll briefly explain quantum groups and $R$-matrices and why they're the same thing. Then we'll see how to construct various $R$-matrices from Nakajima quiver varieties and some possible applications.