Past

Symplectic reduction is the natural quotient construction for symplectic manifolds. Given a free and proper action of a Lie group G on a symplectic manifold M, this process produces a new symplectic manifold of dimension dim(M) - 2 dim(G). For non-free actions, however, the result is usually fairly singular. But Sjamaar-Lerman (1991) showed that the singularities can be understood quite precisely: symplectic reductions by non-free actions are partitioned into smooth symplectic manifolds, and these manifolds fit nicely together in the sense that they form a stratification.

Symplectic reduction has an analogue in hyperkähler geometry, which has been a very important tool for constructing new examples of these special manifolds. In this talk, I will explain how Sjamaar-Lerman’s results can be extended to this setting, namely, hyperkähler quotients by non-free actions are stratified
spaces whose strata are hyperkähler.

In this talk we will discuss non-Abelian T-duality as a solution generating technique in type II Supergravity, briefly reviewing its potential to motivate, probe or challenge classifications of supersymmetric solutions, and focusing on the open problem of providing the newly generated AdS brackgrounds with consistent dual superconformal field theories. These can be seen as renormalization fixed points of linear quivers of increasing rank. As illustrative examples, we consider the non-Abelian T-duals of AdS5xS5, the Klebanov-Witten background, and the IIA reduction of AdS4xS7, whose proposed quivers are, respectively, the four dimensional N=2 Gaiotto-Maldacena theories describing the worldvolume dynamics of D4-NS5 brane intersections, its N=1 mass deformations realized as D4-NS5-NS5’, and the three dimensional N=4 Gaiotto-Witten theories, corresponding to D3-D5-NS5. Based on 1705.09661 and 1609.09061.

The so-called moonshine phenomenon relates modular forms and finite group representations. After the celebrated monstrous moonshine, various new examples of moonshine connection have been discovered in recent years. The study of these new moonshine examples has revealed interesting connections to K3 surfaces, arithmetic geometry, and string theory.  In this colloquium I will give an overview of these recent developments. 
 

The derivation of so-called `effective descriptions' that explicitly incorporate microscale physics into a macroscopic model has garnered much attention, with popular applications in poroelasticity, and models of the subsurface in particular. More recently, such approaches have been applied to describe the physics of biological tissue. In such applications, a key feature is that the material is active, undergoing both elastic deformation and growth in response to local biophysical/chemical cues.

Here, two new macroscale descriptions of drug/nutrient-limited tissue growth are introduced, obtained by means of two-scale asymptotics. First, a multiphase viscous fluid model is employed to describe the dynamics of a growing tissue within a porous scaffold (of the kind employed in tissue engineering applications) at the microscale. Secondly, the coupling between growth and elastic deformation is considered, employing a morpho-elastic description of a growing poroelastic medium. Importantly, in this work, the restrictive assumptions typically made on the underlying model to permit a more straightforward multiscale analysis are relaxed, by considering finite growth and deformation at the pore scale.

In each case, a multiple scales analysis provides an effective macroscale description, which incorporates dependence on the microscale structure and dynamics provided by prototypical `unit cell-problems'. Importantly, due to the complexity that we accommodate, and in contrast to many other similar studies, these microscale unit cell problems are themselves parameterised by the macroscale dynamics.

In the first case, the resulting model comprises a Darcy flow, and differential equations for the volume fraction of cells within the scaffold and the concentration of nutrient, required for growth. Stokes-type cell problems retain multiscale dependence, incorporating active cell motion [1]. Example numerical simulations indicate the influence of microstructure and cell dynamics on predicted macroscale tissue evolution. In the morpho-elastic model, the effective macroscale dynamics are described by a Biot-type system, augmented with additional terms pertaining to growth, coupled to an advection--reaction--diffusion equation [2].

[1] HOLDEN, COLLIS, BROOK and O'DEA. (2018). A multiphase multiscale model for nutrient limited tissue growth, ANZIAM (In press)

[2] COLLIS, BROWN, HUBBARD and O'DEA. (2017). Effective Equations Governing an Active Poroelastic Medium, Proceedings of the Royal Society A. 473, 20160755

In this talk we discuss the recent proof for the existence of $C^{1,1}$ isometric immersions of several classes of negatively curved surfaces into $\R^3$, including the Lobachevsky plane, metrics of helicoid type and a one-parameter family of perturbations of the Enneper surface. Our method, following Chen--Slemrod--Wang and Cao--Huang--Wang, is to transform the Gauss--Codazzi equations into a system of hyperbolic balance laws, and prove the existence of weak solutions by finding the invariant regions. In addition, we provide further characterisation of the $C^{1,1}$ isometrically immersed generalised helicoids/catenoids established in the literature.

I will explain a new proof of the non-linear stability of the Minkowski spacetime as a solution of the Einstein vacuum equation. The proof relies on an iteration scheme at each step of which one solves a linear wave-type equation globally. The analysis takes place on a suitable compactification of $\mathbb{R}^4$ to a manifold with corners whose boundary hypersurfaces correspond to spacelike, null, and timelike infinity; I will describe how the asymptotic behavior of the metric can be deduced from the structure of simple model operators at these boundaries. This talk is based on joint work with András Vasy.

The aim of applied topology is to use and develop topological methods for applied mathematics, science and engineering. One of the main tools is persistent homology, an adaptation of classical homology, which assigns a barcode, i.e., a collection of intervals, to a finite metric space. Because of the nature of the invariant, barcodes are not well adapted for use by practitioners in machine learning tasks. We can circumvent this problem by assigning numerical quantities to barcodes, and these outputs can then be used as input to standard algorithms. I will explain how we can use tropical-like functions to coordinatize the space of persistence barcodes. These coordinates are stable with respect to the bottleneck and Wasserstein distances. I will also show how they can be used in practice.

Phase transitions and critical phenomena are ubiquitous in Nature. They permeate physics, chemistry, biology and complex systems in general, and are characterized by the role of correlations and fluctuations of many degrees of freedom. From a mathematical viewpoint, in the vicinity of a critical point, thermodynamic quantities exhibit singularities and scaling properties. Theoretical attempts to describe classical phase transitions using tools from differential topology and Morse theory provided strong arguments pointing that a phase transition may emerge as a consequence of topological changes in the configuration space around the critical point.

On the other hand, much work was done concerning the topology of networks which spontaneously emerge in complex systems, as is the case of the genome, brain, and social networks, most of these built intrinsically based on measurements of the correlations among the constituents of the system.

We aim to transpose the topological methodology previously applied in n-dimensional manifolds, to describe phenomena that emerge from correlations in a complex system, in which case Hamiltonian models are hard to invoke. The main idea is to embed the network onto an n-dimensional manifold and to study the equivalent to level sets of the network according to a filtration parameter, which can be the probability for a random graph or even correlations from fMRI measurements as height function in the context of Morse theory.  By doing so, we were able to find topological phase transitions either in random networks and fMRI brain networks.  Moreover, we could identify high-dimensional structures, in corroboration with the recent finding from the blue brain project, where neurons could form structures up to eleven dimensions.The efficiency and generality of our methodology are illustrated for a random graph, where its Euler characteristic can be computed analytically, and for brain networks available in the human connectome project.  Our results give strong arguments that the Euler characteristic, together with the distributions of the high dimensional cliques have potential use as topological biomarkers to classify brain Networks. The above ideas may pave the way to describe topological phase transitions in complex systems emerging from correlation data.

Quiver varieties, as first studied by Grojnowski and Nakajima, form an interesting class of geometric objects, which can be constructed by an array of different techniques (GIT, symplectic and Hyperkaehler reduction). In this talk, we will explain how to construct these varieties, and how their homology gives rise to a categorification of the representations of Kac-Moody Lie algebras

After reviewing old work with Teitelbaum, in which we constructed the character variety X of the additive group o_L in a finite extension L/Q_p and established the Fourier isomorphism for the distribution algebra of o_L, I will briefly report on more recent work with Berger and Xie, in which we establish the theory of (\varphi_L,\Gamma_L)-modules over X and relate it to Galois representations. Then I will discuss an ongoing project with Venjakob. Our goal is to use this theory over X for Iwasawa theory.

Joint work with Josè E. Figueroa-Lòpez, Washington University in St. Louis

Abstract: We consider a univariate semimartingale model for (the logarithm 
of) an asset price, containing jumps having possibly infinite activity. The 
nonparametric threshold estimator\hat{IV}_n of the integrated variance 
IV:=\int_0^T\sigma^2_sds proposed in Mancini (2009) is constructed using 
observations on a discrete time grid, and precisely it sums up the squared 
increments of the process when they are below a  threshold, a deterministic 
function of the observation step and possibly of the coefficients of X. All the
threshold functions satisfying given conditions allow asymptotically consistent 
estimates of IV, however the finite sample properties of \hat{IV}_n can depend 
on the specific choice of the threshold.
We aim here at optimally selecting the threshold by minimizing either the 
estimation mean squared error (MSE) or the conditional mean squared error 
(cMSE). The last criterion allows to reach a threshold which is optimal not in 
mean but for the specific  volatility and jumps paths at hand.

A parsimonious characterization of the optimum is established, which turns 
out to be asymptotically proportional to the Lévy's modulus of continuity of 
the underlying Brownian motion. Moreover, minimizing the cMSE enables us 
to  propose a novel implementation scheme for approximating the optimal 
threshold. Monte Carlo simulations illustrate the superior performance of the 
proposed method.

Most motile bacteria are equipped with multiple helical flagella, slender appendages whose rotation in viscous fluids allow the cells to self-propel. We highlight in this talk two consequences of hydrodynamics for bacteria. We first show how the swimming of cells with multiple flagella is enabled by an elastohydrodynamic instability. We next demonstrate how interactions between flagellar filaments mediated by the fluid govern the ability of the cells to reorient. 

The development of reduced order models for complex applications, offering the promise for rapid and accurate evaluation of the output of complex models under parameterized variation, remains a very active research area. Applications are found in problems which require many evaluations, sampled over a potentially large parameter space, such as in optimization, control, uncertainty quantification and applications where near real-time response is needed.

However, many challenges remain to secure the flexibility, robustness, and efficiency needed for general large-scale applications, in particular for nonlinear and/or time-dependent problems.

After giving a brief general introduction to reduced order models, we discuss developments in two different directions. In the first part, we discuss recent developments of reduced methods that conserve chosen invariants for nonlinear time-dependent problems. We pay particular attention to the development of reduced models for Hamiltonian problems and propose a greedy approach to build the basis. As we shall demonstrate, attention to the construction of the basis must be paid not only to ensure accuracy but also to ensure stability of the reduced model. Time permitting, we shall also briefly discuss how to extend the approach to include more general dissipative problems through the notion of port-Hamiltonians, resulting in reduced models that remain stable even in the limit of vanishing viscosity and also touch on extensions to Euler and Navier-Stokes equations.

The second part of the talk discusses the combination of reduced order modeling for nonlinear problems with the use of neural networks to overcome known problems of on-line efficiency for general nonlinear problems. We discuss the general idea in which training of the neural network becomes part of the offline part and demonstrate its potential through a number of examples, including for the incompressible Navier-Stokes equations with geometric variations.

This work has been done with in collaboration with B.F. Afkram (EPFL, CH), N. Ripamonti EPFL, CH) and S. Ubbiali (USI, CH).

Many problems arising in Physics can be posed as minimisation of energy functionals under linear partial differential constraints. For example, a prototypical example in the Calculus of Variations is given by functionals defined on curl-free fields, i.e., gradients. Most work done subject to more general constraints met significant difficulty due to the lack of associated potentials. We show that under the constant rank assumption, which holds true of almost all examples of constraints investigated in connection with lower-semicontinuity, linear constraints admit a potential in frequency space. As a consequence, the notion of A-quasiconvexity, which involves testing with periodic fields leading to difficulties in establishing sufficiency for weak sequential lower semi-continuity, can be tested against compactly supported fields. We will indicate how this can simplify the general framework.

Euler’s equation, the ‘most beautiful equation in mathematics’, startlingly connects the five most important constants in the subject: 1, 0, π, e and i. Central to both mathematics and physics, it has also featured in a criminal court case and on a postage stamp, and has appeared twice in The Simpsons. So what is this equation – and why is it pioneering?

Robin Wilson is an Emeritus Professor of Pure Mathematics at the Open University, Emeritus Professor of Geometry at Gresham College, London, and a former Fellow of Keble College, Oxford.

28 February 2018, 5pm-6pm, Mathematical Institute, Oxford

Please email @email to register

It is a classical theorem of Magnus that the word problem for one-relator groups is solvable; its precise complexity remains unknown. A geometric characterization of the complexity is given by the Dehn function. I will present joint work with Daniel Woodhouse showing that one-relator groups have a rich collection of Dehn functions, including the Brady--Bridson snowflake groups on which our work relies.
 

It is a classical theorem of Magnus that the word problem for one-relator groups is solvable; its precise complexity remains unknown. A geometric characterization of the complexity is given by the Dehn function. I will present joint work with Daniel Woodhouse showing that one-relator groups have a rich collection of Dehn functions, including the Brady--Bridson snowflake groups on which our work relies.

We present some recent results on the regularity criteria for weak solutions to the incompressible Navier--Stokes equations (NSE) in 3 dimensions. By the work of Constantin--Fefferman, it is known that the alignment of vorticity directions is crucial to the regularity of NSE in $\R^3$.  In this talk we show a boundary regularity theorem for NSE on curvilinear domains with oblique derivative boundary conditions. As an application, the boundary regularity of incompressible flows on balls, cylinders and half-spaces with Navier boundary condition is established, provided that the vorticity is coherently aligned up to the boundary. The effects of  vorticity alignment on the $L^q$, $1<q<\infty$ norm of the vorticity will also be discussed.

We develop a structure  theory for del Pezzo surfaces that are regular but geometrically non-normal, based on work of Reid, but now independence on the p-degree of the ground field. This leads to existence results, as well as non-existence results for ground fields  of p-degree one. In turn, we  settle questions arising from Koll'ar's analysis on the structure of Mori fiber spaces in dimension three. This is joint work with Andrea Fanelli.

Low-rank plus sparse matrices arise in many data-oriented applications, most notably in a foreground-background separation from a moving camera. It is known that low-rank matrix recovery from a few entries (low-rank matrix completion) requires low coherence (Candes et al 2009) as in the extreme cases when the low-rank matrix is also sparse, where matrix completion can miss information and be unrecoverable. However, the requirement of low coherence does not suffice in the low-rank plus sparse model, as the set of low-rank plus sparse matrices is not closed. We will discuss the relation of non-closedness of the low-rank plus sparse model to the notion of matrix rigidity function in complexity theory.

Let $F$ be a non-Archimedean local field with ring of integers $\mathcal O$ and maximal ideal $\mathfrak p$. T. Shintani and G. Hill independently introduced a large class of smooth representations of $GL_N(\mathcal O)$, called regular representations. Roughly speaking they correspond to elements in the Lie algebra $M_N(\mathcal O)$ which are regular mod $\mathfrak p$ (i.e, having centraliser of dimension $N$). The study of regular representations of $GL_N(\mathcal O)$ goes back to Shintani in the 1960s, and independently and later, Hill, who both constructed the regular representations with even conductor, but left the much harder case of odd conductor open. In recent simultaneous and independent work, Krakovski, Onn and Singla gave a construction of the regular representations of $GL_N(\mathcal O)$ when the residue characteristic of $\mathcal O$ is not $2$.

In this talk I will present a complete construction of all the regular representations of $GL_N(\mathcal O)$. The approach is analogous to, and motivated by, the construction of supercuspidal representations of $GL_N(F)$ due to Bushnell and Kutzko. This is joint work with Shaun Stevens.
 

The object of this talk is a class of generalised Newtonian fluids with implicit constitutive law.
Both in the steady and the unsteady case, existence of weak solutions was proven by Bul\'\i{}\v{c}ek et al. (2009, 2012) and the main challenge is the small growth exponent qq and the implicit law.
I will discuss the application of a splitting and regularising strategy to show convergence of FEM approximations to weak solutions of the flow. 
In the steady case this allows to cover the full range of growth exponents and thus generalises existing work of Diening et al. (2013). If time permits, I will also address the unsteady case.
This is joint work with Endre Suli.

In this talk, I will present the construction of a family of solutions to
vacuum Einstein equations which consist of an arbitrary number of high
frequency waves travelling in different directions. In the high frequency
limit, our family of solutions converges to a solution of Einstein equations
coupled to null dusts. This construction is an illustration of the so called
backreaction, studied by physicists (Isaacson, Burnet, Green, Wald...) : the
small scale inhomogeneities have an effect on the large scale dynamics in
the form of an energy impulsion tensor in the right-hand side of Einstein
equations. This is a joint work with Jonathan Luk (Stanford).

Protein interaction networks (PINs) allow the representation and analysis of biological processes in cells. Because cells are dynamic and adaptive, these processes change over time. Thus far, research has focused either on the static PIN analysis or the temporal nature of gene expression. By analysing temporal PINs using multilayer networks, we want to link these efforts. The analysis of temporal PINs gives insights into how proteins, individually and in their entirety, change their biological functions. We present a general procedure that integrates temporal gene expression information with a monolayer PIN to a temporal PIN and allows the detection of modular structure using multilayer modularity maximisation.

I will present some recent results obtained in collaboration with V. Banica and F. de la Hoz on the evolution of vortex filaments according to the so called Localized Induction Approximation  (LIA). This approximation is given by a non-linear geometric partial differential equation, that is known under the name of the Vortex Filament Equation (VFE). The aim of the talk is threefold. First, I will recall the Talbot effect of linear optics.  Secondly, I will give some explicit solutions of VFE where this Talbot effect is also present. Finally, I will consider some questions concerning the transfer of energy and momentum for these explicit solutions.

We present a support theorem for subcritical parabolic stochastic partial differential equations (SPDEs) driven by Gaussian noises. In the spirit of the classical theorem by Stroock and Varadhan for ordinary stochastic differential equations, we identify the support of the solution to singular SPDEs with the closure of the union of the support of solutions to approximate and renormalized equations. We implement our approach in the setting of regularity structures and obtain a general result covering a range of singular SPDEs (including $\Phi^4_3$, $\Phi^d_2$, KPZ, PAM (2D+3D), SHE, ...). As a Corollary to our result we obtain the uniqueness of invariant measures for various interesting SPDEs. This is a joint work with Martin Hairer.

We present a numerical investigation of stochastic transport for the damped and driven incompressible 2D Euler fluid flows. According to Holm (Proc Roy Soc, 2015) and Cotter et al. (2017), the principles of transformation theory and multi-time homogenisation, respectively, imply a physically meaningful, data-driven approach for decomposing the fluid transport velocity into its drift and stochastic parts, for a certain class of fluid flows. We develop a new methodology to implement this velocity decomposition and then numerically integrate the resulting stochastic partial differential equation using a finite element discretisation. We show our numerical method is consistent.
Numerically, we perform the following analyses on this velocity decomposition. We first perform uncertainty quantification tests on the Lagrangian trajectories by comparing an ensemble of realisations of Lagrangian trajectories driven by the stochastic differential equation, and the Lagrangian trajectory driven by the ordinary differential equation. We then perform uncertainty quantification tests on the resulting stochastic partial differential equation by comparing the coarse-grid realisations of solutions of the stochastic partial differential equation with the ``true solutions'' of the deterministic fluid partial differential equation, computed on a refined grid. In these experiments, we also investigate the effect of varying the ensemble size and the number of prescribed stochastic terms. Further experiments are done to show the uncertainty quantification results "converge" to the truth, as the spatial resolution of the coarse grid is refined, implying our methodology is consistent. The uncertainty quantification tests are supplemented by analysing the L2 distance between the SPDE solution ensemble and the PDE solution. Statistical tests are also done on the distribution of the solutions of the stochastic partial differential equation. The numerical results confirm the suitability of the new methodology for decomposing the fluid transport velocity into its drift and stochastic parts, in the case of damped and driven incompressible 2D Euler fluid flows. This is the first step of a larger data assimilation project which we are embarking on. This is joint work with Colin Cotter, Dan Crisan, Darryl Holm and Igor Shevchenko.

 

3d N=4 supersymmetric gauge theories provide a method for constructing HyperK\”ahler singularities, known as the Coulomb branch.
This method is complementary to the more traditional way of construction using HyperK\”ahler quotients, known in physics as the “Higgs branch”.
Out of all possible gauge theories there is an interesting subclass of quiver varieties, where the Coulomb branch has been studied in some detail.
Some examples are moduli spaces of classical and exceptional instantons and closures of nilpotent orbits. An interesting feature of Coulomb and Higgs branches is the phenomenon of "3d mirror symmetry” where for a pair of gauge theories, the Higgs branch and Coulomb branch exchange.
There is a large class of “mirror pairs” which I will discuss in some detail.

A topic of recent interest is the notion of implosions. I will argue that there is a simple operation on the quiver which leads to implosion. In other words, given a quiver such that its Coulomb branch is moduli space A, a simple operation of the quiver (making a bouquet) provides the implosion of A.
This has been tested on closures of nilpotent orbits of A type and on nilpotent cones of orthogonal groups and found to agree with the expected results.
If time permits, I will discuss isometries of Coulomb branches

The horizon conjecture, proved in a case by case basis, states that every supersymmetric smooth horizon admits an sl(2, R) symmetry algebra. However it is unclear how string corrections modify the statement. In this talk I will present the analysis of supersymmetric near-horizon geometries in heterotic supergravity up to two loop order in sigma model perturbation theory, and show the conditions for the horizon to admit an sl(2, R) symmetry algebra. In the second part of the talk, I shall consider the inverse problem of determining all extreme black hole solutions associated to a prescribed near-horizon geometry. I will expand the horizon fields in the radial co-ordinate, the so-called moduli, and show that the moduli must satisfy a system of elliptic PDEs, which implies that the moduli space is finite dimensional.

The talk is based on arXiv:1605.05635 [hep-th] and arXiv:1610.09949 [hep-th].

Who are you? What motivates you? What's important to you? How do you react to challenges and adversities? In this session we will explore the power of self-awareness (understanding our own characters, values and motivations) and introduce assertiveness skills in the context of building positive and productive relationships with colleagues, collaborators, students and others.
 

In this talk, I show how concepts from non-equilibrium statistical physics can be employed in the study of climate. The specific problem addressed is the geophysical-scale evolution of Arctic sea ice. Using an analogy with Brownian motion, the original evolution equation for the sea ice thickness distribution function by Thorndike et al. (J. Geophys. Res. 80(33), pp. 4501 — 4513, 1975) is transformed to a Fokker-Planck-like conservation law. The steady solution is $g(h) = {\cal N}(q) h^q \mathrm{e}^{-~ h/H}$, where $q$ and $H$ are expressible in terms of moments over the transition probabilities between thickness categories. The solution exhibits the functional form used in observational fits and shows that for $h \ll 1$, $g(h)$ is controlled by both thermodynamics and mechanics, whereas for $h \gg 1$ only mechanics controls $g(h)$. We also derive the underlying Langevin equation governing the dynamics of the ice thickness $h$, from which we predict the observed $g(h)$. Further, seasonality is introduced by using the Eisenman-Wettlaufer model (Proc. Natl. Acad. Sci. USA 106, pp. 28-32, 2009) for the thermal growth of sea ice. The time-dependent problem is studied by numerically integrating the Fokker-Planck equation. The results obtained from these numerical integrations and their comparison with satellite observations are discussed.

Dr Nicola Beer heads up the Department of Stem Cell Engineering at the new Novo Nordisk Research Centre Oxford. Her team will use human stem cells to derive metabolically-relevant cells and tissues such as islets, hepatocytes, and adipocytes todiscover novel secreted factors and corresponding signalling pathways which modify cell function, health, and viability. Bycombining in vitro-differentiated human stem cell-derived models with CRISPR and other genomic targeting techniques, the teamassay cell function from changes in a single gene up to a genome-wide scale. Understanding the genes and pathways underlying cell function (and dysfunction) highlights potential targets for new Type 2 Diabetes therapeutics. Dr Beer will talk about the work ongoing in her team, as well as more broadly about the role of human stem cells in drug discovery and patient treatment.

This talk will discuss efficient numerical methods for estimating the
probability of a large portfolio loss, and associated risk measures such
as VaR and CVaR.  These involve nested expectations, and following
Bujok, Hambly & Reisinger (2015) we use the number of samples for the
inner conditional expectation as the key approximation parameter in the
Multilevel Monte Carlo formulation.  The main difference in this case is
the indicator function in the definition of the probability. Here we
build on previous work by Gordy & Juneja (2010) who analyse the use of a
fixed number of inner samples , and Broadie, Du & Moallemi (2011) who
develop and analyse an adaptive algorithm.  I will present the
algorithm, outline the main theoretical results and give the numerical
results for a representative model problem.  I will also discuss the
extension to real portfolios with a large number of options based on
multiple underlying assets.

Joint work with Abdul-Lateef Haji-Ali

I will present Vidit Nanda's paper "Local homology and stratification" (https://arxiv.org/abs/1707.00354), and briefly explain how in my master thesis I am applying ideas from the paper to study word embedding problems.

Abstract of the paper:  We outline an algorithm to recover the canonical (or, coarsest) stratification of a given regular CW complex into cohomology manifolds, each of which is a union of cells. The construction proceeds by iteratively localizing the poset of cells about a family of subposets; these subposets are in turn determined by a collection of cosheaves which capture variations in cohomology of cellular neighborhoods across the underlying complex. The result is a finite sequence of categories whose colimit recovers the canonical strata via (isomorphism classes of) its objects. The entire process is amenable to efficient distributed computation.
 

A theorem of Gromov states that the number of generators of the fundamental group of a manifold with nonnegative 
curvature is bounded by a constant which only depends on the dimension of the manifold. The main ingredient 
in the proof is Toponogov’s theorem, which roughly speaking says that the triangles on spaces with positive 
curvature, such as spheres, are thick compared to triangles in the Euclidean plane. In the talk I shall explain 
this more carefully and deduce Gromov’s result.

I will give a gentle introduction to joint work in progress with George Boxer, Frank Calegari, and Vincent Pilloni, in which we prove that all abelian surfaces over totally real fields are potentially modular. We also prove that infinitely many abelian surfaces over Q are modular.

What if one desires to have a World perfectly slippery to water? What are the strategies that can be adopted? And how can smart slippery surfaces be created? In this seminar, I will outline approaches to creating slippery surfaces, which all involve reducing or removing droplet contact with the solid, whilst still supporting the droplet. The first concept is to decorate the droplet surface with particles, thus creating liquid marbles and converting the droplet-solid contact into a solid-solid contact. The second concept is to use the Leidenfrost effect to instantly vaporize a layer of water, thus creating a film of vapor and converting the droplet-solid contact into vapor-solid contact. The third concept is to infuse oil into the surface, thus creating a layer of oil and converting the droplet-solid contact into a lubricant-solid contact. I will also explain how we design such to have smart functionality whilst retaining and using the mobility of contact lines and droplets. I will show how Leidenfrost levitation can lead to new types of heat engines [1], how a microsystems approach to the Leidenfrost effect can reduce energy input and lead to a new type of droplet microfluidics [2] (Fig. 1a) and how liquid diodes can be created [3]. I will explain how lubricant impregnated surfaces lead to apparent contact angles [4] and how the large retained footprint of the droplet allows droplet transport and microfluidics using energy coupled via a surface acoustic wave (SAW) [5]. I will argue that droplets confined between reconfigurable slippery boundaries can be continuously translated in an energy invariant manner [6] (Fig. 1b). I will show that a droplet Cheerios effect induced by the menisci arising from structuring the underlying lubricated surface or by droplets in close proximity to each other can be used to guide and position droplets [7] (Fig. 1c). Finally, I will show that active control of droplet spreading by electric field induced control of droplet spreading, via electrowetting or dielectrowetting, can be achieved with little hysteresis [8] and can be a new method to investigate the dewetting of surfaces [9].

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Figure 1 Transportation and positioning of droplets using slippery surfaces: (a) Localized Leidenfrost effect, (b) Reconfigurable boundaries, and (c) Droplet Cheerio’s effect.

Acknowledgements The financial support of the UK Engineering & Physical Sciences Research Council (EPSRC) and Reece Innovation ltd is gratefully acknowledged. Many collaborators at Durham, Edinburgh, Nottingham Trent and Northumbria Universities were instrumental in the work described.

[1] G.G. Wells, R. Ledesma-Aguilar, G. McHale and K.A. Sefiane, Nature Communications, 2015, 6, 6390.

[2] L.E. Dodd, D. Wood, N.R. Geraldi, G.G. Wells, et al., ACS Applied & Materials Interfaces, 2016, 8, 22658.

[3] J. Li, X. Zhou , J. Li, L. Che, J. Yao, G. McHale, et al., Science Advances, 2017, 3, eaao3530.

[4] C. Semprebon, G. McHale, and H. Kusumaatmaja, Soft Matter, 2017, 13, 101.

[5] J.T. Luo, N.R. Geraldi, J.H. Guan, G. McHale, et al., Physical Review Applied, 2017, 7, 014017.

[6] É. Ruiz-Gutiérrez, J.H. Guan, B.B. Xu, G. McHale, et al., Physical Review Letters, 2017, 118, 218003.

[7] J.H. Guan, É. Ruiz-Gutiérrez, B.B. Xu, D. Wood, G. McHale, et al., Soft Matter, 2017, 13, 3404.

[8] Z. Brabcová, G. McHale, G.G. Wells, et al., Applied Physics Letters, 2017, 110, 121603.

[9] A.M.J. Edwards, R. Ledesma-Aguilar, et al., Science Advances, 2016, 2, e1600183

A classical construction principle for dependent failure times is to consider shocks that destroy components within a system. The arrival times of shocks can destroy arbitrary subsets of the system, thus introducing dependence. The seminal model – based on independent and exponentially distributed shocks - was presented by Marshall and Olkin in 1967, various generalizations have been proposed in the literature since then. Such models have applications in non-life insurance, e.g. insurance claims caused by floods, hurricanes, or other natural catastrophes. The simple interpretation of multivariate fatal shock models is clearly appealing, but the number of possible shocks makes them challenging to work with, recall that there are 2^d subsets of a set with d components. In a series of papers we have identified mixture models based on suitable stochastic processes that give rise to a different - and numerically more convenient - stochastic interpretation. This representation is particularly useful for the development of efficient simulation algorithms. Moreover, it helps to define parametric families with a reasonable number of parameters. We review the recent literature on multivariate fatal shock models, extreme-value copulas, and related dependence structures. We also discuss applications and hierarchical structures. Finally, we provide a new characterization of the Marshall-Olkin distribution.

Authors: Mai, J-F.; Scherer, M.;

The rapidly increasing number of cores in high-performance computing systems causes a multitude of challenges for developers of numerical methods. New parallel algorithms are required to unlock future growth in computing power for applications and energy efficiency and algorithm-based fault tolerance are becoming increasingly important. So far, most approaches to parallelise the numerical solution of partial differential equations focussed on spatial solvers, leaving time as a bottleneck. Recently, however, time stepping methods that offer some degree of concurrency, so-called parallel-in-time integration methods, have started to receive more attention.

I will introduce two different numerical algorithms, Parareal (by Lions et al., 2001) and PFASST (by Emmett and Minion, 2012), that allow to exploit concurrency along the time dimension in parallel computer simulations solving partial differential equations. Performance results for both methods on different architectures and for different equations will be presented. The PFASST algorithm is based on merging ideas from Parareal, spectral deferred corrections (SDC, an iterative approach to derive high-order time stepping methods by Dutt et al. 2000) and nonlinear multi-grid. Performance results for PFASST on close to half a million cores will illustrate the potential of the approach. Algorithmic modifications like IPFASST will be introduced that can further reduce solution times. Also, recent results showing how parallel-in-time integration can provide algorithm-based tolerance against hardware faults will be shown.

When nematic liquid crystal droplets are produced in the form or tori (or such is the shapes of confining cavities), they may be called toroidal nematics, for short. When subject to degenerate planar anchoring on the boundary of a torus, the nematic director acquires a natural equilibrium configuration within the torus, irrespective of the values of Frank's elastic constants. That is the pure bend arrangement whose integral lines run along the parallels of all inner deflated tori. This lecture is concerned with the stability of such a universal equilibrium configuration. Whenever its stability is lost, new equilibrium configurations arise in pairs, the members of which are symmetric and exhibit opposite chirality. Previous work has shown that a rescaled saddle-splay constant may be held responsible for such a chiral symmetry breaking. We shall show that that is not the only possible instability mechanism and, perhaps more importantly, we shall attempt to describe the qualitative properties of the equilibrium nematic textures that prevail when the chiral symmetry is broken.

Cube Complexes with Coupled Links (CLCC) are a special family of non-positively curved cube complexes that are constructed by specifying what the links of their vertices should be. In this talk I will introduce the construction of CLCCs and try to motivate it by explaining how one can use information about the local geometry of a cube complex to deduce global properties of its fundamental group (e.g. hyperbolicity and cohomological dimension). On the way, I will also explain what fly maps are and how to use them to deduce that a CAT(0) cube complex is hyperbolic.