L5

We characterise the solutions to a continuous-time optimal liquidity provision problem in a market populated by informed and uninformed traders. In our model, the asset price exhibits fads -- these are short-term deviations from the fundamental value of the asset. Conditional on the value of the fad, we model how informed traders and uninformed traders arrive in the market. The market maker knows of the two groups of traders but only observes the anonymous order arrivals. We study both, the complete information and the partial information versions of the control problem faced by the market maker. In such frameworks, we characterise the value of information, and we find the price of liquidity as a function of the proportion of informed traders in the market. Lastly, for the partial information setup, we explore how to go beyond the Kalman-Bucy filter to extract information about the fad from the market arrivals.

The class of henselian valued fields with non-discrete value group is not well-understood. In 2018, Koenigsmann conjectured that a list of seven natural axioms describes a complete axiomatisation of Q_p^ab, the maximal extension of the p-adic numbers Q_p with abelian Galois group, which is an example of such a valued field. Informed by the recent work of Jahnke-Kartas on the model theory of perfectoid fields, we formulate an eighth axiom (the discriminant property) that is not a consequence of the other seven. Revisiting work by Koenigsmann (the Galois characterisation of Q_p) and Jahnke-Kartas, we give a uniform treatment of their underlying method. In particular, we highlight how this method yields short, non-standard model-theoretic proofs of known results (e.g. finite extensions of perfectoid fields are perfectoid).

We study the spectrum of the Laplacian on finite-area hyperbolic surfaces of large volume, focusing on small eigenvalues i.e. those below 1/4. I will discuss some recent results and open problems in this area. Based on joint works with Michael Magee and with Joe Thomas.
 

Gauge theory excels at solving minimal genus problems for 3- and 4-manifolds.  A notable triumph is its resolution of the Thom conjecture, asserting that the genus of a smooth complex curve in the complex projective plane is no larger than any smooth submanifold homologous to it.  Gauge theoretic techniques have also been used to verify analagous conjectures for Kähler surfaces or, more generally, symplectic 4-manifolds.  One can formulate versions of these conjectures for surfaces with boundary lying in a 3-manifold, and I'll discuss work in progress with Katherine Raoux which attempts to extend these "relative" Thom conjectures outside the complex (or even symplectic) realm using tools from Floer homology.

The Farrell--Jones conjecture predicts that the algebraic K-theory of a group ring is isomorphic to a certain equivariant homology theory, and there are also versions for L-theory and Waldhausen's A-theory. In principle, this provides a way to calculate these K-groups, and has many applications. These include classifying manifolds admitting a given fundamental group and a positive resolution of the Borel conjecture.

I will discuss work with Yassine Guerch and Sam Hughes on the Farrell--Jones conjecture for extensions of relatively hyperbolic groups, as well as an application to their automorphism groups in the one-ended case. The methods are from geometric group theory: we go via the theory of JSJ decompositions to produce acylindrical actions on trees.

When tensor products of N minimal models accumulate at central charge N, they also admit relevant operators arbitrarily close to marginality. This raises the tantalizing possibility that they can be use to reach purely Virasoro symmetric CFTs where the breaking of extended chiral symmetry can be seen in a controlled way. This talk will give an overview of the theories where this appears to be the case, according to a brute force check at low lying spins. We will also encounter an interesting non-example where the same type of analysis can be used to give a simpler proof of integrability.