Oxford University

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.).
 

The 2007-2008 financial crisis forced governments to choose between the unattractive alternatives of either bailing out a systemically important bank (SIBs) or having it fail in a disruptive manner. Bail-in has been put forward as the primary tool to resolve a failing bank, which would end the too-big-to-fail problem by letting stakeholders shoulder the losses, while minimising the calamitous systemic impact of a bank failure. Though the aptness of bail-in has been evinced in relatively minor idiosyncratic bank failures, its efficacy in maintaining stability in cases of large bank failures and episodes of system-wide crises remains to be practically tested. This paper investigates the financial stability implications of the bail-in design, in all these cases. We develop a multi-layered network model of the European financial system that captures the prevailing endogenous-amplification mechanisms: exposure loss contagion, overlapping portfolio contagion, funding contagion, bail-inable debt revaluations, and bail-inable debt runs. Our results reveal that financial stability hinges on a set of `primary' and `secondary' bail-in parameters, including the failure threshold, recapitalisation target, debt-to-equity conversion rate, loss absorption requirements, debt exclusions and bail-in-design certainty – and we uncover how. We also demonstrate that the systemic footprint of the bail-in design is not properly understood without the inclusion of multiple contagion mechanisms and non-banks. Our evidence fortunately suggests that the pivot for stability is in the hands of policymakers. It also suggests, however, that the current bail-in design might be in the regime of instability.

The goal of the talk is to present a proof of the following statement:
Let (K,v) be an algebraic extension of (Q_p,v_p) whose completion is perfectoid. We show that K is relatively decidable to its tilt K^♭, i.e. if K^♭ is decidable in the language of valued fields, then so is K. 
In the first part [of the talk], I will try to cover the necessary background needed from model theory and the theory of perfectoid fields.

In this talk, I will report about a joint work with D. Ciubotaru, in which we investigate the Dunkl version of the classical Howe-duality (O(k),spo(2|2)). Similar Fischer-type decompositions were studied before in the works of Ben-Said, Brackx, De Bie, De Schepper, Eelbode, Orsted, Soucek and Somberg for other Howe-dual pairs. Our work builds on the notion of a Dirac operator for Drinfeld algebras introduced by Ciubotaru, which was inspired by the analogous theory for Lie algebras, as well as the work of Cheng and Wang on classical Howe dualities.

I will talk about a way of building graded Lie algebras from certain Heisenberg groups. The input for this construction arises naturally when studying families of algebraic curves, and we'll look at some examples in which Lie theory interacts with number theory in an illuminating way. 

 I will review some of the major research themes in Quantum Chaos over the past 50 years, and some of the questions currently attracting attention in the mathematics and physics literatures.

A mathematical structure is `axiomatizable' if it is completely determined by some family of sentences in a suitable first-order language. This idea has been explored for various kinds of structure, but I will concentrate on groups. There are some general results (not many) about which groups are or are not axiomatizable; recently there has been some interest in the sharper concept of 'finitely axiomatizable' or FA - that is, when only a finite set of sentences (equivalently, a single sentence) is allowed.

While an infinite group cannot be FA, every finite group is so, obviously. A profinite group is kind of in between: it is infinite (indeed, uncountable), but compact as a topological group; and these groups share many properties of finite groups, though sometimes for rather subtle reasons. I will discuss some recent work with Andre Nies and Katrin Tent where we prove that certain kinds of profinite group are FA among profinite groups. The methods involve a little model theory, and quite a lot of group theory.

The notion of colour Lie algebra, introduced by Ree (1960), generalises notions of Lie algebra and Lie superalgebra. From an orthogonal representation V of a quadratic colour Lie algebra g, we give various ways of constructing a colour Lie algebra g’ whose bracket extends the bracket of g and the action of g on V. A first possibility is to consider g’=g⊕V and requires the cancellation of an invariant studied by Kostant (1999). Another construction is possible when the representation is ``special’’ and in this case the extension is of the form g’=g⊕sl(2,k)⊕V⊗k^2. Covariants are associated to special representations and satisfy to particular identities generalising properties studied by Mathews (1911) on binary cubics. The 7-dimensional fundamental representation of a Lie algebra of type G_2 and the 8-dimensional spinor representation of a Lie algebra of type so(7) are examples of special representations.