Past

Classically, basin modelling is undertaken with very little a priori knowledge. Alongside the challenge of improving the general fidelity and utility of the modelling systems, is the challenge of constraining these systems with unknowns and uncertainties in such a way that models (and derived simulation results) can be readily regenerated/reevaluated in the light of new empirical data obtained during the course of exploration, development and production activities.

A result of Jeffrey Brock states that, given a hyperbolic 3-manifold which is a mapping torus over a surface $S$, its volume can be expressed in terms of the distance induced by the monodromy map in the pants graph of $S$. This is an abstract graph whose vertices are pants decompositions of $S$, and edges correspond to some 'elementary alterations' of those.
I will show how this theorem gives an estimate for the volume of hyperbolic complements of closed braids in the solid torus, in terms of braid properties. The core piece of such estimate is a generalization of a result of Masur, Mosher and Schleimer that train track splitting sequences (which I will define in the talk) induce quasi-geodesics in the marking graph.

In this talk, we aim to provide a valuation framework for counterparty credit risk based on a structural default model which incorporates jumps and dependence between the assets of interest. In this framework default is caused by the firm value falling below a prespecified threshold following unforeseeable shocks, which deteriorate its liquidity and ability to meet its liabilities. The presence of dependence between names captures wrong-way risk and right-way risk effects. The structural model traces back to Merton (1974), who considered only the possibility of default occurring at the maturity of the contract; first passage time models starting from the seminal contribution of Black and Cox (1976) extend the original framework to incorporate default events at any time during the lifetime of the contract. However, as the driving risk process used is the Brownian motion, all these models suffers of vanishing credit spreads over the short period - a feature not observed in reality. As a consequence, the Credit Value Adjustment (CVA) would be underestimated for short term deals as well as the so-called gap risk, i.e. the unpredictable loss due to a jump event in the market. Improvements aimed at resolving this issue include for example random default barriers, time dependent volatilities, and jumps. In this contribution, we adopt Lévy processes and capture dependence via a linear combination of two independent Lévy processes representing respectively the systematic risk factor and the idiosyncratic shock. We then apply this framework to the valuation of CVA and DVA related to equity contracts such as forwards and swaps. The main focus is on the impact of correlation between entities on the value of CVA and DVA, with particular attention to wrong-way risk and right-way risk, the inclusion of mitigating clauses such as netting and collateral, and finally the impact of gap risk. Particular attention is also devoted to model calibration to market data, and development of adequate numerical methods for the complexity of the model considered.

 Let $X$ be a smooth cubic hypersurface of dimension $m$ defined over a global field $K$. A conjecture of Colliot-Thelene(02) states that $X$ satisfies the Hasse Principle and Weak approximation as long as $m\geq 3$. We use a global version of Hardy-Littlewood circle method along with the theory of global $L$-functions to establish this for $m\geq 6$, in the case $K=\mathbb{F}_q(t)$, where $\text{char}(\mathbb{F}_{q})> 3$.

In this seminar I will present a semi-langrangian discretisation of the Monge-Ampère operator, which is of interest in optimal transport 
and differential geometry as well as in related fields of application.

I will discuss the proof of convergence to viscosity solutions. To address the challenge of uniqueness and convexity we draw upon the classical relationship with Hamilton-Jacobi-Bellman equations, which we extend to the viscosity setting. I will explain that the monotonicity of semi-langrangian schemes implies that they possess large stencils, which in turn requires careful treatment of the boundary conditions.

The contents of the seminar is based on current work with X Feng from the University of Tennessee.

Joint work with Anders Hansen (Cambridge), Olavi Nevalinna (Aalto) and Markus Seidel (Zwickau).             

Expansion, rank gradient and virtual splitting are all concepts of great interest in asymptotic group theory. We discuss a result of Marc Lackenby which demonstrates a surprising relationship between then, and give examples exhibiting different combinations of asymptotic behaviour.

When the Computing Laboratory discarded its hardcopy journals around 2008, I kept the first ten years or so of each of six classic numerical analysis journals, starting from volume 1, number 1.  This will not be a seminar in the usual sense but a mutual exploration.  Come prepared to look through a few of these old volumes yourself and perhaps to tell the group of something interesting you find.  Bring a pen and paper.  All are welcome.

Mathematics of Computation, from 1943
SIAM Journal, from 1953
Numerische Mathematik, from 1959
BIT, from 1961
SIAM Journal on Numerical Analysis, from 1964
Journal of the IMA, from 1965

We look at the construction of radial metrics with an isolated singularity for the constant fractional curvature equation. This is a semilinear, non-local equation involving the fractional Laplacian, and appears naturally in conformal geometry. 

I will review Bott's classical periodicity result about topological K-theory (with period 2 in the case of complex K-theory, and period 8 in the case of real K-theory), and provide an easy sketch of proof, based on the algebraic periodicity of Clifford algebras. I will then introduce the `higher real K-theory' of Hopkins and Miller, also known as TMF. I'll discuss its periodicity (with period 576), and present a conjecture about a corresponding algebraic periodicity of `higher Clifford algebras'. Finally, applications to physics will be discussed.

The role of surface-water flooding in controlling fluxes of water and carbon between the land and the atmosphere is increasingly recognized in studies of the Earth system. Simultaneous advances in remote earth observation and large-scale land-surface and hydrological modeling promise improvements in our ability to understand these linkages, and suggest that improvements in prediction of river flow and inundation extents may result. Here we present an analysis of newly-available observational estimates of surface water inundation obtained through satellite Earth observation with results from simulations produced by using the Joint UK Land Environment Simulator (JULES) land-surface model operating at 0.5 degree resolution over the African continent. The model was forced with meteorological input from the WATCH Forcing Data for the period 1981-2001 and sensitivity to various model configurations and parameter settings were tested. Both the PDM and TOPMODEL sub-grid scale runoff generation schemes were tested for parameter sensitivities, with the evaluation focussing on simulated river discharge in sub-catchments of the Congo, Nile, Niger, Orange, Okavango and Zambezi rivers. It was found that whilst the water balance in each of the catchments can be simulated with acceptable accuracy, the individual responses of each river vary between model configurations so that there is no single runoff parameterization scheme or parameter values that yields optimal results across all catchments. We trace these differences to the model’s representation of sub-surface flow and make some suggestions to improve the performance of large-scale land-surface models for use in similar applications. Our findings also demonstrate links between episodes of extensive surface water flooding and large-scale climatic indices, although the pattern of correlations contains a level of spatial and temporal detail that warrants careful attention to the climatology of individual situations. These findings suggest that the use of Earth observation data together with improved models of large-scale hydrology have the potential to improve our ability to predict surface-water flooding and to develop our understanding of the role of flooding in driving components of the water and carbon cycles.

Sudden cardiac arrhythmic death remains a major health challenge in Western Society. Recent advances in computational methods and technologies have made clinically-based cardiac modelling a reality. An important current focus is the use of modelling to understand the nature of arrhythmias in the setting of different forms of structural heart disease. However, many challenges remain regarding the best use of these models to inform clinical decision making and guide therapies. In this talk, I will introduce a diverse sample of applications of modelling in this context, ranging from basic science studies which aim to leverage a fundamental mechanistic understanding of different aspects of pathological cardiac function, to direct clinical-application projects which aim to use modelling to immediately inform a clinical therapy. I will also discuss the challenges involved in clinically-driven modelling, and how to both engage, and manage, the expectations of clinicians at the same time, particularly with respect to the potential uses of 'patient-specific' modelling.

In this talk we're going to discuss Hamilton's maximum principle for the Ricci flow. As an application, I would like to explain a technique due to Boehm and Wilking which provides a general tool to obtain new Ricci flow invariant curvature conditions from given ones. As we'll see, it plays a key role in Brendle and Schoen's proof of the differentiable sphere theorem.

We consider the problem of unveiling the implicit network structure of user interactions in a social network, based only on high-frequency timestamps. Our inference is based on the minimization of the least-squares loss associated with a multivariate Hawkes model, penalized by $\ell_1$ and trace norms. We provide a first theoretical analysis of the generalization error for this problem, that includes sparsity and low-rank inducing priors. This result involves a new data-driven concentration inequality for matrix martingales in continuous time with observable variance, which is a result of independent interest. The analysis is based on a new supermartingale property of the trace exponential, based on tools from stochastic calculus. A consequence of our analysis is the construction of sharply tuned $\ell_1$ and trace-norm penalizations, that leads to a data-driven scaling of the variability of information available for each users. Numerical experiments illustrate the strong improvements achieved by the use of such data-driven penalizations.

Classical anabelian geometry shows that for hyperbolic curves the etale fundamental group encodes the curve provided the base field is sufficiently arithmetic. In higher dimensions it is natural to replace the etale fundamental group by the etale homotopy type. We will report on progress obtained in this direction in a recent joint work with Alexander Schmidt.

**Joint seminar with Logic. 

Classical anabelian geometry shows that for hyperbolic curves the etale fundamental group encodes the curve provided the base field is sufficiently arithmetic. In higher dimensions it is natural to replace the etale fundamental group by the etale homotopy type. We will report on progress obtained in this direction in a recent joint work with Alexander Schmidt.

**Joint seminar with Number Theory. Note unusual time and place**

I will present a survey of the main results about first and second order models of swarming where repulsion and attraction are modeled through pairwise potentials. We will mainly focus on the stability of the fascinating patterns that you get by random data particle simulations, flocks and mills, and their qualitative behavior.
 

The Singular Value Decomposition (SVD) of matrices and the related Principal Components Analysis (PCA) express a matrix in terms of singular vectors, which are linear combinations of all the input data and lack an intuitive physical interpretation. Motivated by the application of PCA and SVD in the analysis of populations genetics data, we will discuss the notion of leverage scores: a simple statistic that reveals columns/rows of a matrix that lie in the subspace spanned by the top principal components (left/right singular vectors). We will then use the leverage scores to present matrix decompositions that express the structure in a matrix in terms of actual columns (and/or rows) of the matrix. Such decompositions are easier to interpret in applications, since the selected columns and rows are subsets of the data. We will also discuss extensions of the leverage scores to reveal influential entries of a matrix.

I will present two recent results concerning the stability of boundary layer asymptotic expansions of solutions of Navier-Stokes with small viscosity. First, we show that the linearization around an arbitrary stationary shear flow admits an unstable eigenfunction with small wave number, when viscosity is sufficiently small. In boundary-layer variables, this yields an exponentially growing sublayer near the boundary and hence instability of the asymptotic expansions, within an arbitrarily small time, in the inviscid limit. On the other hand, we show that the Prandtl asymptotic expansions hold for certain steady flows. Our proof involves delicate construction of approximate solutions (linearized Euler and Prandtl layers) and an introduction of a new positivity estimate for steady Navier-Stokes. This in particular establishes the inviscid limit of steady flows with prescribed boundary data up to order of square root of small viscosity. This is a joint work with Emmanuel Grenier and Yan Guo.

In this talk I will try to show how certain asymptotic properties of a random walk on a graph are related to geometric properties of the graph itself. A special focus will be put on spectral properties and isoperimetric inequalities, proving Kesten's criterion for amenability.

Given a vector space V over a field K, let End(V) denote the algebra of linear endomorphisms of V. If V is finite-dimensional, then it is well-known that the diagonalizable subalgebras of End(V) are characterized by their internal algebraic structure: they are the subalgebras isomorphic to K^n for some natural number n. 

In case V is infinite dimensional, the diagonalizable subalgebras of End(V) cannot be characterized purely by their internal algebraic structure: one can find diagonalizable and non-diagonalizable subalgebras that are isomorphic.  I will explain how to characterize the diagonalizable subalgebras of End(V) as topological algebras, using a natural topology inherited from End(V).  I will also illustrate how this characterization relates to an infinite-dimensional Wedderburn-Artin theorem that characterizes "topologically semisimple" algebras.