Optimal Collateralization with Bilateral Default Risk
Abstract
We consider over-the-counter (OTC) transactions with bilateral default risk, and study the optimal design of the Credit Support Annex (CSA). In a setting where agents have access to a trading technology, default penalties and collateral costs arise endogenously as a result of foregone investment opportunities. We show how the optimal CSA trades off the costs of the collateralization procedure against the reduction in exposure to counterparty risk and expected default losses. The results are used to provide insights on the drivers of different collateral rules, including hedging motives, re-hypothecation of collateral, and close-out conventions. We show that standardized collateral rules can have a detrimental impact on risk sharing, which should be taken into account when assessing the merits of standardized vs. bespoke CSAs in non-centrally cleared OTC instruments. This is joint work with D. Bauer and L.R. Sotomayor (GSU).
Sheafy matrix factorizations and bundles of quadrics
Abstract
A Landau-Ginzburg B-model is a smooth scheme $X$, equipped with a global function $W$. From $(X,W)$ we can construct a category $D(X,W)$, which is called by various names, including ‘the category of B-branes’. In the case $W=0$ it is exactly the derived category $D(X)$, and in the case that $X$ is affine it is the category of matrix factorizations of $W$. There has been a lot of foundational work on this category in recent years, I’ll describe the most modern and flexible approach to its construction. I’ll then interpret Nick Addington’s thesis in this language. We’ll consider the case that $W$ is a quadratic form on a vector bundle, and the corresponding global version of Knorrer periodicity. We’ll see that interesting gerbe structures arise, related to the bundle of isotropic Grassmannians.
Sheafy matrix factorizations and bundles of quadrics
Abstract
A Landau-Ginzburg B-model is a smooth scheme $X$, equipped with a global function $W$. From $(X,W)$ we can construct a category $D(X,W)$, which is called by various names, including ‘the category of B-branes’. In the case $W=0$ it is exactly the derived category $D(X)$, and in the case that $X$ is affine it is the category of matrix factorizations of $W$. There has been a lot of foundational work on this category in recent years, I’ll describe the most modern and flexible approach to its construction.
I’ll then interpret Nick Addington’s thesis in this language. We’ll consider the case that $W$ is a quadratic form on a vector bundle, and the corresponding global version of Knorrer periodicity. We’ll see that interesting gerbe structures arise, related to the bundle of isotropic Grassmannians.
C-groups
Abstract
Toby Gee and I have proposed the definition of a "C-group", an extension of Langlands' notion of an L-group, and argue that for an arithmetic version of Langlands' philosophy such a notion is useful for controlling twists properly. I will give an introduction to this business, and some motivation. I'll start at the beginning by explaining what an L-group is a la Langlands, but if anyone is interested in doing some background preparation for the talk, they might want to find out for themselves what an L-group (a Langlands dual group) is e.g. by looking it up on Wikipedia!
Classicality for overconvergent eigenforms on some Shimura varieties.
Abstract
A well known theorem of Coleman states that an overconvergent modular eigenform of weight k>1 and slope less than k-1 is classical. This theorem was later reproved and generalized using a geometric method very different from Coleman's cohomological approach. In this talk I will explain how one might go about generalizing the cohomological method to some higher-dimensional Shimura varieties.
p-adic functoriality for inner forms of unitary groups.
Abstract
In this talk I will explain a notion of p-adic functoriality for inner forms of definite unitary groups. Roughly speaking, this is a morphism between so-called eigenvarieties, which are certain rigid analytic spaces parameterizing p-adic families of automorphic forms. We will then study certain properties of classical Langlands functoriality that allow us to prove p-adic functoriality in some "stable" cases.
Automated parallel adjoints for model differentiation, optimisation and stability analysis
Abstract
The derivatives of PDE models are key ingredients in many
important algorithms of computational science. They find applications in
diverse areas such as sensitivity analysis, PDE-constrained
optimisation, continuation and bifurcation analysis, error estimation,
and generalised stability theory.
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These derivatives, computed using the so-called tangent linear and
adjoint models, have made an enormous impact in certain scientific fields
(such as aeronautics, meteorology, and oceanography). However, their use
in other areas has been hampered by the great practical
difficulty of the derivation and implementation of tangent linear and
adjoint models. In his recent book, Naumann (2011) describes the problem
of the robust automated derivation of parallel tangent linear and
adjoint models as "one of the great open problems in the field of
high-performance scientific computing''.
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In this talk, we present an elegant solution to this problem for the
common case where the original discrete forward model may be written in
variational form, and discuss some of its applications.
Shaping swimmers and their environments for better locomotion
Double Field Theory and the Geometry of Duality
Abstract
String theory on a torus requires the introduction of dual coordinates
conjugate to string winding number. This leads to physics and novel geometry in a doubled space. This will be
compared to generalized geometry, which doubles the tangent space but not the manifold.
For a d-torus, string theory can be formulated in terms of an infinite
tower of fields depending on both the d torus coordinates and the d dual
coordinates. This talk focuses on a finite subsector consisting of a metric
and B-field (both d x d matrices) and a dilaton all depending on the 2d
doubled torus coordinates.
The double field theory is constructed and found to have a novel symmetry
that reduces to diffeomorphisms and anti-symmetric tensor gauge
transformations in certain circumstances. It also has manifest T-duality
symmetry which provides a generalisation of the usual Buscher rules to
backgrounds without isometries. The theory has a real dependence on the full
doubled geometry: the dual dimensions are not auxiliary. It is concluded
that the doubled geometry is physical and dynamical.