L4

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.

In this paper we extend the existing literature on xVA along three directions. First, we enhance current BSDE-based xVA frameworks to include initial margin by following the approach of Crépey (2015a) and Crépey (2015b). Next, we solve the consistency problem that arises when the front- office desk of the bank uses trade-specific discount curves that differ from the discount curve adopted by the xVA desk. Finally, we address the existence of multiple aggregation levels for contingent claims in the portfolio between the bank and the counterparty, providing suitable extensions of our proposed single-claim xVA framework. 

This is a joint work with: Francesca Biagini and Immacolata Oliva

Preprint available at: https://arxiv.org/abs/1905.11328

Conical Symplectic Resolutions form a broad family of holomorphic symplectic manifolds that are of interest to mathematical physicists, algebraic geometers, and representation theorists; Nakajima Quiver Varieties and Hypertoric Varieties are known as their special cases. In this talk, I will be focused on the Symplectic Geometry of Conical Symplectic Resolutions, and its non-symplectic applications. More precisely, I will talk about my work on finding Exact Lagrangian Submanifolds inside CSRs, and work in progress (joint with Alexander Ritter) about the construction of Symplectic Cohomology on CSRs.

In the seminar, I will talk about a parabolic toy-model for the incompressible Navier-Stokes equations, that satisfies the same energy inequality, same scaling symmetry and which is also super-critical in dimension 3. I will present some partial regularity results that this model shares with the incompressible model and other results that occur only for our model.

Formation of singularities for the relativistic Euler equations (N. Athanasiou): An archetypal phenomenon in the study of hyperbolic systems of conservation laws is the development of singularities (in particular shocks) in finite time, no matter how smooth or small the initial data are. A series of works by Lax, John et al confirmed that for some important systems, when the initial data is a smooth small perturbation of a constant state, singularity formation in finite time is equivalent to the existence of compression in the initial data. Our talk will address the question of whether this dichotomy persists for large data problems, at least for the system of the Relativistic Euler equations in (1+1) dimensions. We shall also give some interesting studies in (3+1) dimensions. This is joint work with Dr. Shengguo Zhu.

Global Well-Posedness for a Class of Stochastic McKean-Vlasov Equations in One Dimension (A. Mayorcas): We show global well-posedness for a family of parabolic McKean--Vlasov SPDEs with additive space-time white noise. The family of interactions we consider are those given by convolution with kernels that are at least integrable. We show that global well-posedness holds in both the repulsive/defocussing and attractive/focussing cases. Our strategy relies on both pathwise and probabilistic techniques which leverage the Gaussian structure of the noise and well known properties of the deterministic PDEs.

The AO-model describes crystalline solids in the presence of defects like dislocation lines. We demonstrate that the model supports low-energy structures like grains and determine for simple geometries the grain boundary energy density. At small misorientation angles we recover the well-known Read-Shockley law. Due to the atomistic nature of the model it is possible to consider the the Boltzmann-Gibbs distribution at non-zero temperature. Using ideas by Froehlich and Spencer we prove rigorously the presence of long-range order if the temperature is sufficiently small.
 

In this talk we consider a mean field optimal control problem with an aggregation-diffusion constraint, where agents interact through a potential, in the presence of a Gaussian noise term. Our analysis focuses on a PDE system coupling a Hamilton-Jacobi and a Fokker-Planck equation, describing the optimal control aspect of the problem and the evolution of the population of agents, respectively. We will discuss the existence and regularity of solutions for the aforementioned system. We notice this model is in close connection with the theory of mean-field games systems. However, a distinctive feature concerns the nonlocal character of the interaction; it affects the drift term in the Fokker-Planck equation as well as the Hamiltonian of the system, leading to new difficulties to be addressed.

McDuff and Schlenk determined when a four-dimensional symplectic ellipsoid can be symplectically embedded into a four-dimensional ball. They found that if the ellipsoid is close to round, the answer is given by an ``infinite staircase" determined by the odd index Fibonacci numbers, while if the ellipsoid is sufficiently stretched, all obstructions vanish except for the volume obstruction. Infinite staircases have also been found when embedding ellipsoids into polydisks (Frenkel - Muller, Usher) and into the ellipsoid E(2, 3) (Cristofaro-Gardiner - Kleinman). In this talk, we will see how the sharpness of ECH capacities for embedding of ellipsoids implies the existence of infinite staircases for these and three other target spaces.  We will then discuss the relationship with toric varieties, lattice point counting, and the Philadelphia subway system. This is joint work with Dan Cristofaro-Gardiner, Alessia Mandini,
and Ana Rita Pires.

Tautological bundles on Hilbert schemes of points often enter into enumerative and physical computations. I will explain how to use the Donaldson-Thomas theory of toric threefolds to produce combinatorial identities that are expressed geometrically using tautological bundles on the Hilbert scheme of points on a surface. I'll also explain how these identities can be used to study Euler characteristics of tautological bundles over Hilbert schemes of points on general surfaces.

I will discuss some recent results around Hilbert schemes of points on singular surfaces, obtained in joint work with Craw, Gammelgaard and Gyenge, and their connection to combinatorics (of coloured partitions) and representation theory (of affine Lie algebras and related algebras such as their W-algebra).