L2

Understanding the outcome of a collision between liquid drops (merge or bounce?) as well their impact and spreading over solid surfaces (splash or spread?) is key for a host of processes ranging from 3d printing to cloud formation. Accurate experimental observation of these phenomena is complex due to the small spatio-temporal scales or interest and, consequently, mathematical modelling and computational simulation become key tools with which to probe such flows.

Experiments show that the gas surrounding the drops can have a key role in the dynamics of impact and wetting, despite the small gas-to-liquid density and viscosity ratios. This is due to the formation of gas microfilms which exert their influence on drops through strong lubrication forces.  In this talk, I will describe how these microfilms cannot be described by the Navier-Stokes equations and instead require the development of a model based on the kinetic theory of gases.  Simulation results obtained using this model will then be discussed and compared to experimental data.

The ability to study the transcriptome, proteome – and other aspects – of many individual cells represents one of the most important technical breakthroughs and tools in biology and medical science of the past few years. They are revolutionising study of biological systems and human disease, enabling for example: hypothesis-free identification of rare pathogenic (or protective) cell subsets in chronic diseases, routine monitoring of patient immune phenotypes and direct discovery of mole cular targets in rare cell populations. In parallel, new computational and analytical approaches are being intensively developed to analyse the vast data sets generated by these technologies. However, there is still a huge gap between our ability to generate the data, analyse their technical soundness and actually interpret them. The QBIOX network may provide for a unique opportunity to complement recent investments in Oxford technical capabilities in single-cell technologies with the development of revolutionary, visionary ways of interpreting the data that would help Oxford researchers to compete as leaders in this field.

Please register via https://www.eventbrite.co.uk/e/qbiox-colloquium-trinity-term-2017-ticke…

The management of natural resources, from fisheries and climate change to gut bacteria colonies, all require the development of ecological models that represent the full spectrum of population interactions, from competition, through mixotrophy and mutualism, to predation.

Mixotrophic plankton, that both photosynthesise and eat other plankton, underpin all marine food webs and help regulate climate by facilitating gas exchange between the ocean and atmosphere. We show the recent discovery that their feeding preferences change with increasing temperature implies climate change could dramatically alter the structure of marine food webs.

We describe a theoretical framework that reveals the key role of mixotrophy in facilitating transitions between trophic interactions. Mixotrophy smoothly and stably links competition to predation, and extends this linkage to include mutualism in both facultative and obligate forms. Such smooth stable transitions further allow the development of eco-evolutionary theory at the population level through quantitative trait modelling.

Gauss invented Gaussian quadrature following an approach entirely different from the one we now find in textbooks. I will describe leisurely the contents of Gauss's original memoir on quadrature, an impressive piece of mathematics, based on continued fractions, Padé approximation, generating functions, the hypergeometric series and more.

Given a Calabi-Yau threefold X, one can count curves on X using various approaches, for example using stable maps or ideal sheaves; for any curve class on X, this produces an infinite sequence of invariants, indexed by extra discrete data (e.g. by the domain genus of a stable map).  Conjecturally, however, this sequence is determined by only a finite number of integer invariants, known as Gopakumar-Vafa invariants.  In this talk, I will propose a direct definition of these invariants via sheaves of vanishing cycles, building on earlier approaches of Kiem-Li and Hosono-Saito-Takahashi.  Conjecturally, these should agree with the invariants as defined by stable maps.  I will also explain how to prove the conjectural correspondence for irreducible curves on local surfaces.  This is joint work with Yukinobu Toda.

This document reviews the so called least square methodology (LSM) and its application for the valuation and risk of callable exotics and regulatory value adjustments (xVA). We derive valuation algorithms for xVA, both with or without collateral, that are particularly accurate, efficient and practical. These algorithms are based on a reformulation of xVA, designed by Jesper Andreasen and implemented in Danske Bank's award winning systems, that hasn't been previously published in full. We then investigate the matter of risk sensitivities, in the context of Algorithmic Automated Differentiation (AAD). A rather recent addition to the financial mathematics toolbox, AAD is presently generally acknowledged as a vastly superior alternative to the classical estimation of risk sensitivities through finite differences, and the only practical means for the calculation of the large number of sensitivities in the context of xVA. The theory and implementation of AAD, the related check-pointing techniques, and their application to Monte-Carlo simulations are explained in numerous textbooks and articles, including Giles and Glasserman's pioneering Smoking Adjoints. We expose an extension to LSM, and, in particular, we derive an original algorithm that resolves the matters of memory consumption and efficiency in differentiating simulations together with the LSM step.

We report on recent developments in the study of nonlocal operators. The central object of the talk are quadratic forms similar to those that define Sobolev spaces of fractional order. These objects are naturally linked to Markov processes via the theory of Dirichlet forms. We provide regularity results for solutions to corresponding integrodifferential equations. Our emphasis is on forms with singularand anisotropic measures. Some of the objects under consideration are related to the Boltzmann equation, which leads to an interesting question of comparability of quadrativ forms. The talk is based on recent results joint with B. Dyda and with K.-U. Bux and T. Schulze.