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.

Recent advances in experimental imaging techniques have allowed us to observe the fine details of how droplets behave upon impact onto a substrate. However, these are highly non-linear, multiscale phenomena and are thus a formidable challenge to model. In addition, when the substrate is deformable, such as an elastic sheet, the fluid-structure interaction introduces an extra layer of complexity.

We present two modeling approaches for droplet impact onto deformable substrates: matched asymptotics and direct numerical simulations. In the former, we use Wagner's theory of impact to derive analytical expressions which approximate the behavior during the early time of impact. In the latter, we use the open source volume-of-fluid code Basilisk to conduct simulations designed to give insight into the later times of impact.

We conclude by showing how these methods are complementary, and a combination of both can give a thorough understanding of the droplet impact across timescales. 

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

Recently, Joyce constructed a Ringel-Hall style graded vertex algebra on the homology of moduli stacks of objects in certain categories of algebro-geometric and representation-theoretic origin. The construction is most natural for 2n-Calabi-Yau categories. We present this construction and explain the geometric reason why it exists. If time permits, we will explain how to compute the homology of the moduli stack of objects in the derived category of a smooth complex projective variety and to identify it with a lattice-type vertex algebra.

Random graphs were introduced as a convenient example for demonstrating the impossibility of ‘complete disorder’ by Erdos, who also thought that these objects will never become useful in the applied areas outside of pure mathematics. In this talk, I will view random graphs as objects in the field of applied mathematics and discus how the application-driven objectives have set new directions for studying random graphs. I will focus on characterising the sizes of connected components in graphs with a given degree distribution, on the percolation-like processes on such structures, and on generalisations to the coloured graphs. These theoretical questions have interesting implications for studying resilience of networks with nontrivial structures, and for materials science where they explain kinetics-driven phase transitions. Even more surprisingly, the results reveal intricate connections between random graphs and non-linear partial differential equations indicating new possibilities for their analysis.

Spatial networks are often the most natural way to represent spatial information of different kinds. One of the outstanding problems in current spatial network research is to effectively quantify the heterogeneity of the discrete-valued spatial distributions underlying a spatial graph. In this talk we will presentsome recent alternative approaches to estimate heterogeneity in spatial networks based on simple dynamical processes running on them.

 In Classification Theory, Shelah defined several cardinal invariants of a complete theory which detect the presence of certain trees among the definable sets, which in turn quantify the complexity of forking.  In later model-theoretic developments, local versions of these invariants were recognized as marking important dividing lines - e.g. simplicity and NTP2.  Around these dividing lines, a dichotomy theorem of Shelah states that a theory has the tree property if and only if it is witnessed in one of two extremal forms--the tree property of the first or second kind--and it was asked if there is a 'quantitative' analogue of this dichotomy in the form of a certain equation among these invariants.  We will describe these model-theoretic invariants and explain why the quantitative version of the dichotomy fails, via a construction that relies upon some unexpected tools from combinatorial set theory. 

 

The principal ideal theorem (1930) guaranteed that any number field K would embed into a finite extension, called the Hilbert class field of K, in which every ideal of the original field became principal -- however the Hilbert class field itself will not necessarily have class number 1. The class field tower problem asked whether iteratively taking Hilbert class fields must stabilize after finitely many steps. In 1964, it was finally answered in the negative by Golod and Shafarevich who produced infinitely many examples and pioneered the framework that is still the most common setting for deciding when a number field will have an infinite class field tower.

In this talk, I will finish the proof of their cohomological result and thus fully justify how it settled the class field tower problem.