Oxford University

Set theory exhibits a truly robust mutual interpretability phenomenon: in any model of one set theory we can define models of diverse other set theories and vice versa. In any model of ZFC, we can define models of ZFC + GCH and also of ZFC + ¬CH and so on in hundreds of cases. And yet, it turns out, in no instance do these mutual interpretations rise to the level of bi-interpretation. Ali Enayat proved that distinct theories extending ZF are never bi-interpretable, and models of ZF are bi-interpretable only when they are isomorphic. So there is no nontrivial bi-interpretation phenomenon in set theory at the level of ZF or above.  Nevertheless, for natural weaker set theories, we prove, including ZFC- without power set and Zermelo set theory Z, there are nontrivial instances of bi-interpretation. Specifically, there are well-founded models of ZFC- that are bi-interpretable, but not isomorphic—even $\langle H_{\omega_1}, \in \rangle$ and $\langle H_{\omega_2}, \in \rangle$ can be bi-interpretable—and there are distinct bi-interpretable theories extending ZFC-. Similarly, using a construction of Mathias, we prove that every model of ZF is bi-interpretable with a model of Zermelo set theory in which the replacement axiom fails. This is joint work with Alfredo Roque Freire.

We discuss two recent extensions of work on Reynolds-robust preconditioners for the Navier-Stokes equations, to non-Newtonian fluids and to the equations of magnetohydrodynamics.  We model non-Newtonian fluids by means of an implicit constitutive relation between stress and strain. This framework is broadly applicable and allows for proofs of convergence under quite general assumptions. Since the stress cannot in general be solved for in terms of the strain, a three-field stress-velocity-pressure formulation is adopted. By combining the augmented Lagrangian approach with a kernel-capturing space decomposition, we derive a preconditioner that is observed to be robust to variations in rheological parameters in both two and three dimensions.  In the case of magnetohydrodynamics, we consider the stationary incompressible resistive Newtonian equations, and solve a four-field formulation for the velocity, pressure, magnetic field and electric field. A structure-preserving discretisation is employed that enforces both div(u) = 0 and div(B) = 0 pointwise. The basic idea of the solver is to split the fluid and electromagnetic parts and to employ our existing Navier-Stokes solver in the Schur complement. We present results in two dimensions that exhibit robustness with respect to both the fluids and magnetic Reynolds numbers, and describe ongoing work to extend the solver to three dimensions.

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Given two metric spaces $X$ and $Y$, it is natural to ask how faithfully, from the point of view of the metric, one can embed $X$ into $Y$. One way of making this precise is asking whether there exists a coarse embedding of $X$ into $Y$. Positive results are plentiful and diverse, from Assouad's embedding theorem for doubling metric spaces to the elementary fact that any finitely generated subgroup of a finitely generated group is coarsely embedded with respect to word metrics. Moreover, the consequences of admitting a coarse embedding into a sufficiently nice space can be very strong. By contrast, there are few invariants which provide obstructions to coarse embeddings, leaving many seemingly elementary geometric questions open.
I will present new families of invariants which resolve some of these questions. Highlights of the talk include a new algebraic dichotomy for connected unimodular Lie groups, and a method of calculating a lower bound on the conformal dimension of a compact Ahlfors-regular metric space.
 

The genus of a knot in a 3-manifold is defined to be the minimum genus of a compact, orientable surface bounding that knot, if such a surface exists. We consider the computational complexity of determining knot genus. Such problems have been studied by several mathematicians; among them are the works of Hass--Lagarias--Pippenger, Agol--Hass--Thurston, Agol and Lackenby. For a fixed 3-manifold the knot genus problem asks, given a knot K and an integer g, whether the genus of K is equal to g. In joint work with Lackenby, we prove that for any fixed, compact, orientable 3-manifold, the knot genus problem lies inNP, answering a question of Agol--Hass--Thurston from 2002. Previously this was known for rational homology 3-spheres by the work of Lackenby.

The automorphism group of a d-regular tree is a topological group with many interesting features. A nice thing about this group is that while some of its features are highly non-trivial (e.g., the existence of infinitely many pairwise non-conjugate simple subgroups), often the ideas involved in the proofs are fairly intuitive and geometric. 
I will present a proof for the fact that the outer automorphism group of (Aut(T)) is trivial. This is original joint work with Gil Goffer, but as is often the case in this area, was already proven by Bass-Lubotzky 20 years ago. I will mainly use this talk to hint at how algebra, topology and geometry all play a role when working with Aut(T).
 

Whitehead published two papers in 1936 on free groups. Both concerned decision problems for equivalence of (sets of) elements under automorphisms. The first focused on primitive elements (those that appear in some basis), the second looked at arbitrary sets of elements. While both of the resulting algorithms are combinatorial, Whitehead's proofs that these algorithms actually work involve some nice manipulation of surfaces in 3-manifolds. We will have a look at how this works for primitive elements. I'll outline some generalizations due to Culler-Vogtmann, Gertsen, and Stallings, and if we have time talk about how it fits in with some of my current work.

A broad challenge in the theory of finitely generated groups is to understand their subgroups. A group is commensurably coHopfian if its finite index subgroups are distinct from its infinite index subgroups (that is to say not abstractly isomorphic). We will focus primarily on hyperbolic groups, and give the first examples of one-ended hyperbolic groups that are not commensurably coHopfian.
This is joint work with Emily Stark.
 

We build on the literature on financial contagion using models of cross-holdings of equity participations and debt in different seniority classes, and extend them to include bail-ins and contingent convertible debt instruments, two mechanisms of debt-to-equity conversion. We combine these with recently proposed methods of network valuation under stochastic external assets, allowing for the pricing of debt instruments in each seniority layer and the calculation of default probabilities. We show that there exist well-defined valuations for all financial assets cross-held within the system. The full model constitutes an extension of classic asset pricing models that accounts for cross-holdings of debt securities. Our contribution is to add convertible debt to this framework.

In this talk, I will introduce the notion of a sheaf on a topological space. I will then explain why "topological spaces" are an artificial limitation on enjoying life (esp. cohomology) to the fullest and what to do about that (answer: sites). Sheaves also fail our needs, but they have a suitable natural upgrade (i.e. stacks).
This talk will be heavily peppered with examples that come from the world around you (music, torsors, etc.).