Past

The FEniCS system [1] allows the description of finite element discretisations of partial differential equations using a high-level syntax, and the automated conversion of these representations to working code via automated code generation. In previous work described in [2] the high-level representation is processed automatically to derive discrete tangent-linear and adjoint models. The processing of the model code at a high level eases the technical difficulty associated with management of data in adjoint calculations, allowing the use of optimal data management strategies [3].

This previous methodology is extended to enable the calculation of higher order partial differential equation constrained derivative information. The key additional step is to treat tangent-linear
equations on an equal footing with originating forward equations, and in particular to treat these in a manner which can themselves be further processed to enable the derivation of associated adjoint information, and the derivation of higher order tangent-linear equations, to arbitrary order. This enables the calculation of higher order derivative information -- specifically the contraction of a Kth order derivative against (K - 1) directions -- while still making use of optimal data management strategies. Specific applications making use of Hessian information associated with models written using the FEniCS system are presented.

[1] "Automated solution of differential equations by the finite element method: The FEniCS book", A. Logg, K.-A. Mardal, and  G. N. Wells (editors), Springer, 2012
[2] P. E. Farrell, D. A. Ham, S. W. Funke, and M. E. Rognes, "Automated derivation of the adjoint of high-level transient finite element programs", SIAM Journal on Scientific Computing 35(4), C369--C393, 2013
[3] A. Griewank, and A. Walther, "Algorithm 799: Revolve: An implementation of checkpointing for the reverse or adjoint mode of computational differentiation", ACM Transactions on Mathematical Software 26(1), 19--45, 2000

For the North Analysis British Seminar, see: https://www.maths.ox.ac.uk/node/29687

A core problem in the study of manifolds and their topology is that of telling them apart. That is, when can we say whether or not two manifolds are homeomorphic? In two dimensions, the situation is simple, the Classification Theorem for Surfaces allows us to differentiate between any two closed surfaces. In three dimensions, the problem is a lot harder, as the century long search for a proof of the Poincaré Conjecture demonstrates, and is still an active area of study today.
As an early pioneer in the area of 3-manifolds Seifert carved out his own corner of the landscape instead of attempting to tackle the entire problem. By reducing his scope to the subclass of 3-manifolds which are today known as Seifert fibred spaces, Seifert was able to use our knowledge of 2-manifolds and produce a classification theorem of his own.
In this talk I will define Seifert fibred spaces, explain what makes them so much easier to understand than the rest of the pack, and give some insight on why we still care about them today.
 

The aim of this talk is to tell the story of Non-Abelian Hodge Theory for curves. The starting point is the space of representations of the fundamental group of a compact Riemann surface. This space can be endowed with the structure of a complex algebraic variety in three different ways, giving rise to three non-algebraically isomorphic moduli spaces called the Betti, de Rham and Dolbeault moduli spaces respectively. 

After defining and outlining the construction of these three moduli spaces, I will describe the (non-algebraic) correspondences between them, collectively known as Non-Abelian Hodge Theory. Finally, we will see how the rich structure of the Dolbeault moduli space can be used to shed light on the topology of the space of representations.

The core idea behind metric spaces is the triangular inequality. Metrics have been generalized in many ways, but the most tempting way to alter them would be to "flip" the triangular inequality, obtaining an "anti-metric". This, however, only allows for trivial spaces where the distance between any two points is 0. However, if we intertwine the concept of antimetrics with the structures of partial linear--and cyclic--orders, we can define a structure where the anti-triangular inequality holds conditionally. We define this structure, give examples, and show an interesting result involving metrics and antimetrics.

Roger Penrose is the ultimate scientific all-rounder.  He started out in algebraic geometry but within a few years had laid the foundations of the modern theory of black holes with his celebrated paper on gravitational collapse. His exploration of foundational questions in relativistic quantum field theory and quantum gravity, based on his twistor theory, had a huge impact on differential geometry. His work has influenced both scientists and artists, notably Dutch graphic artist M. C. Escher.

Roger Penrose is one of the great ambassadors for science. In this lecture and in conversation with mathematician and broadcaster Hannah Fry he will talk about work and career.

This lecture is in partnership with the Science Museum in London where it will take place. Please email @email to register.

You can also watch online:

https://www.facebook.com/OxfordMathematics

https://livestream.com/oxuni/Penrose-Fry

The Oxford Mathematics Public Lectures are generously supported by XTX Markets.

In 1985, Babai and Sós asked whether there exists a constant c>0 such that every finite group of order n has a product-free set of size at least cn, where a product-free set of a group is a subset that does not contain three elements x,y and z  satisfying xy=z. Gowers showed that the answer is no in the early 2000s, by linking the existence of product-free sets of large density to the existence of low dimensional unitary representations.

In this talk, I will provide an answer to the aforementioned question by model theoretic means. Furthermore, I will relate some of Gowers' results to the existence of nontrivial definable compactifications of nonstandard finite groups.
 

I will talk about some basic facts about slope stable sheaves and the Bogomolov inequality.  New techniques from stability conditions will imply new stronger bounds on Chern characters of stable sheaves on some special varieties, including  Fano varieties, quintic threefolds and etc. I will discuss the progress in this direction and some related open problems.

Using non-minimal pure spinor superspace, Cederwall has constructed BRST-invariant actions for D=10 super-Born-Infeld and D=11 supergravity which are quartic in the superfields. But since the superfields have explicit dependence on the non-minimal pure spinor variables, it is non-trivial to show these actions correctly describe super-Born-Infeld and supergravity. In this talk, I will expand solutions to the equations of motion from the pure spinor action for D=10 abelian super Born-Infeld to leading order around the linearized solutions and show that they correctly describe the interactions expected. If I have time, I will explain how to generalize these ideas to D=11 supergravity.

How long a monotone path can one always find in any edge-ordering of the complete graph $K_n$? This appealing question was first asked by Chvatal and Komlos in 1971, and has since attracted the attention of many researchers, inspiring a variety of related problems. The prevailing conjecture is that one can always find a monotone path of linear length, but until now the best known lower bound was $n^{2/3−o(1)}$, which was proved by Milans. This talk will be
about nearly closing this gap, proving that any edge-ordering of the complete graph contains a monotone path of length $n^{1−o(1)}$. This is joint work with Bucic, Kwan, Sudakov, Tran, and Wagner.

In the first part of this talk, I will present an extension of Chebfun, called Ballfun, for computing with functions and vectors in the unit ball. I will then describe an algorithm for solving the incompressible Navier-Stokes equations in the ball. Contrary to projection methods, we use the poloidal-toroidal decomposition to decouple the PDEs and solve scalars equations. The solver has an optimal complexity (up to polylogarithmic terms) in terms of the degrees of freedom required to represent the solution.

I will explain how Dirac operators provide precious information about geometric and algebraic aspects of representations of real Lie groups. In particular, we obtain an explicit realisation of representations, leading terms in the asymptotics of characters and a precise connection with nilpotent orbits.

This talk features a self-contained introduction to persistent homology, which is the main ingredient of topological data analysis. 

As a rule of thumb, the dominant resistive force on a cyclist riding along a flat road at a speed above 10mph is aerodynamic drag; at higher speeds, this drag becomes even more influential because of its non-linear dependence on speed. Reducing drag, therefore, is of critical importance in bicycle racing, where winning margins are frequently less than a tyre's width (over a 200+km race!). I shall discuss a mathematical model of aerodynamic drag in cycling, present mathematical reasoning behind some of the decisions made by racing cyclists when attempting to minimise it, and touch upon some of the many methods of aerodynamic drag assessment.

One of the main open problems in loop quantum gravity is to reconcile the fundamental quantum discreteness of space with general relativity in the continuum. In this talk, I present recent progress regarding this issue: I will explain, in particular, how the discrete spectra of geometric observables that we find in loop gravity can be understood from a conventional Fock quantisation of gravitational edge modes on a null surface boundary. On a technical level, these boundary modes are found by considering a quasi-local Hamiltonian analysis, where general relativity is treated as a Hamiltonian system in domains with inner null boundaries. The presence of such null boundaries requires then additional boundary terms in the action. Using Ashtekar’s original SL(2,C) self-dual variables, I will explain that the natural such boundary term is nothing but a kinetic term for a spinor (defining the null flag of the boundary) and a spinor-valued two-form, which are both intrinsic to the boundary. The simplest observable on the boundary phase space is the cross sectional area two-form, which generates dilatations of the boundary spinors. In quantum theory, the corresponding area operator turns into the difference of two number operators. The area spectrum is discrete without ever introducing spin networks or triangulations of space. I will also comment on a similar construction in three euclidean spacetime dimensions, where the discreteness of length follows from the quantisation of gravitational edge modes on a one-dimensional cross section of the boundary.
The talk is based on my recent papers: arXiv:1804.08643 and arXiv:1706.00479.
 

Matrix completion is an area of great mathematical interest and has numerous applications, including recommender systems for e-commerce. The recommender problem can be viewed as follows: given a database where rows are users and and columns are products, with entries indicating user preferences, fill in the entries so as to be able to recommend new products based on the preferences of other users. Viewing the interactions between user and product as links in a bipartite graph, the problem is equivalent to approximating a partially observed graph using clusters. We propose a divide and conquer algorithm inspired by the work of [1], who use recursive rank-1 approximation. We make the case for using an LP rank-1 approximation, similar to that of [2] by a showing that it guarantees a 2-approximation to the optimal, even in the case of missing data. We explore our algorithm's performance for different test cases.

[1]  Shen, B.H., Ji, S. and Ye, J., 2009, June. Mining discrete patterns via binary matrix factorization. In Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining (pp. 757-766). ACM.

[2] Koyutürk, M. and Grama, A., 2003, August. PROXIMUS: a framework for analyzing very high dimensional discrete-attributed datasets. In Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining (pp. 147-156). ACM.
 

We consider vector-valued maps which minimize an energy with two terms: an elastic term penalizing high gradients, and a potential term penalizing values far away from a fixed submanifold N. In the scaling limit where the second term is dominant, minimizers converge to maps with values into the manifold N. If the elastic term is the classical Dirichlet energy (i.e. the squared L^2-norm of the gradient), classical tools show that this convergence is uniform away from a singular set where the energy concentrates. Some physical models (as e.g. liquid crystal models) include however more general elastic energies (still coercive and quadratic in the gradient, but less symmetric), for which these classical tools do not apply. We will present a new strategy to obtain nevertheless this uniform convergence. This is a joint work with Andres Contreras.

We present a construction which associates to a KdV equation the lamplighter group. 
In order to establish this relation we use automata and random walks on ultra discrete limits. 
It is also related to the L2 Betti numbers introduced by Atiyah which are homotopy 
invariants of closed manifolds.

SLE curves are an important family of random curves in the plane. They share many similarites with solutions of SDE (in particular, with Brownian motion). Any quesion asked for the latter can be asked for the former. Inspired by that, Yizheng Yuan and I investigate the support for SLE curves. In this talk, I will explain our theorem with more motivation and idea. 

In this seminar we will look at an extension of the well known sewing lemma from rough path theory to fields on [0; 1]k. We will first introduce a framework suitable to study such fields, and then find a criterion for convergence of multiple Riemann type sums of a class of abstract integrands. A simple application of this extension is construct the Young integral for fields.Furthermore, we will discuss the use of this theorem to study integration of fields of lower regularity by using ideas familiar from rough path theory. Moreover, we will discuss difficulties we face by looking at “multi-parameter ODE's” both from an existence and uniqueness point of view.

The Beilinson-Drinfeld Grassmannian, which classifies a G-bundle trivialised away from a finite set of points on a curve, is one of the basic objects in the geometric Langlands programme. Similar construction in higher dimensions in the algebraic and analytic settings are not very interesting because of Hartogs' theorem. In this talk I will discuss a differentiable version. I will also explain a theory of D-modules on differentiable spaces and use it
to define differentiable chiral and factorisation algebras. By linearising the Grassmannian we get examples of differentiable chiral algebras. This is joint work with Dennis Borisov.

Prof. Helen Byrne shares her academic path and experience as Director of Equality and Diversity.

More information will appear later.

 In this seminar I will discuss a recently-found class of RG flows in four dimensions exhibiting enhancement of supersymmetry in the infrared, which provides a lagrangian description of several strongly-coupled N=2 SCFTs. The procedure involves starting from a N=2 SCFT, coupling a chiral multiplet in the adjoint representation of the global symmetry to the moment map of the SCFT and turning on a nilpotent expectation value for this chiral. We show that, combining considerations based on 't Hooft anomaly matching and basic results about the N=2 superconformal algebra, it is possible to understand in detail the mechanism underlying this phenomenon and formulate a simple criterion for supersymmetry enhancement. 

In this session we discuss various different routes for promoting your research through a panel discussion with Dawn Gordon (Project Manager, Oxford University Innovation), Dyrol Lumbard (External Relations Manager, Mathematical Institute), James Maynard (Academic Faculty, Mathematical Institute) and Ian Griffiths, and chaired by Frances Kirwan. The panel discussion will include the topics of outreach, impact, and strategies for promoting aspects of mathematics that are less amenable to public engagement. 

New undergraduates often find that they have a lot more time to spend on independent work than they did at school or college.  But how can you use that time well?  When your lecturers say that they expect you to study your notes between lectures, what do they really mean?  There is research on how mathematicians go about reading maths effectively.  We'll look at a technique that has been shown to improve students' comprehension of proofs, and in this interactive workshop we'll practise together on some examples.  Please bring a pen/pencil and paper! 

This session is likely to be most relevant for first-year undergraduates, but all are welcome, especially those who would like to improve how they read and understand proofs.

Atherosclerosis is a manifestation of cardiovascular disease consisting of the buildup of inflamed arterial plaques. Because most heart attacks are caused by the rupture of unstable "vulnerable" plaque, the characterization of plaques and their vulnerability remains an outstanding problem in medicine.

Morphoelasticity is a mathematical framework commonly employed to describe tissue growth.

Its central premise is the decomposition of the deformation gradient into the product of an elastic tensor and a growth tensor.

In this talk, I will present some recent efforts to simulate intimal thickening -- the precursor to atherosclerosis -- using morphoelasticity theory.

The arterial wall is composed of three layers: the intima, media and adventitia. 

The intima is allowed to grow isotropically while the area of the media and adventitia is approximately conserved. 

All three layers are modeled as anisotropic hyperelastic materials, reinforced by collagen fibers.

We explore idealized axisymmetric arteries as well as more general geometries that are solved using the finite element method.

Results are discussed in the context of balloon-injury experiments on animals and Glagovian remodeling in humans.

One occasionally encounters computational problems that work just fine on ill-posed inputs, even though they should not. One example is polynomial eigenvalue problems, where standard algorithms such as QZ can find a desired solution to instances with infinite condition number to machine precision, while being completely oblivious to the ill-conditioning of the problem. One explanation is that, intuitively, adversarial perturbations are extremely unlikely, and "for all practical purposes'' the problem might not be ill-conditioned at all. We analyse perturbations of singular polynomial eigenvalue problems and derive methods to bound the likelihood of adversarial perturbations for any given input in different stochastic models.

Joint work with Vanni Noferini
 

Understanding how shifts of multiplicative functions correlate with each other is a central question in multiplicative number theory. A well-known conjecture of Elliott predicts that there should be no correlation between shifted multiplicative functions unless the functions involved are ‘pretentious functions’ in a certain precise sense. The Elliott conjecture implies as a special case the famous Chowla conjecture on shifted products of the Möbius function.

In the last few years, there has been a lot of exciting progress on the Chowla and Elliott conjectures, and we give an overview of this. Nearly all of the previously obtained results have concerned correlations that are weighted logarithmically, and it is an interesting question whether one can remove these logarithmic weights. We show that one can indeed remove logarithmic averaging from the known results on the Chowla and Elliott conjectures, provided that one restricts to almost all scales in a suitable sense.

This is joint work with Terry Tao.

Morse theory explores the topology of a smooth manifold $M$ by looking at the local behaviour of a fixed smooth function $f : M \to \mathbb{R}$. In this talk, I will explain how we can construct ordinary homology by looking at the flow of $\nabla f$ on the manifold. The talk should serve as an introduction to Morse theory for those new to the subject. At the end, I will state a new(ish) proof of the functoriality of Morse homology.

Welfare economics argues that competitive markets lead to efficient allocation of resources. The classical theorems are based on the Walrasian market model which assumes the existence of market clearing prices. The emergence of such prices remains debatable. We replace the Walrasian market model by double auctions and show that the conclusions of welfare economics remain largely the same. Double auctions are not only a more realistic description of real markets but they explain how equilibrium prices and efficient allocations emerge in practice. 

Caustics are places where the light intensity diverges, and where the wave front has a singularity. We use a self-similar description to derive the detailed spatial structure of a cusp singularity, from where caustic lines originate. We also study singularities of higher order, which have their own, uniquely three-dimensional structure. We use this insight to study shock formation in classical compressible Euler dynamics. The spatial structure of these shocks is that of a caustic, and is described by the same similarity equation.

$$
\def\curl#1{\left\{#1\right\}}
\def\vek#1{\mathbf{#1}}
$$
lll-posed problems arise often in the context of scientific applications in which one cannot directly observe the object or quantity of interest. However, indirect observations or measurements can be made, and the observable data $y$ can be represented as the wanted observation $x$ being acted upon by an operator $\mathcal{A}$. Thus we want to solve the operator equation \begin{equation}\label{eqn.Txy} \mathcal{A} x = y, \end{equation} (1) often formulated in some Hilbert space $H$ with $\mathcal{A}:H\rightarrow H$ and $x,y\in H$. The difficulty then is that these problems are generally ill-posed, and thus $x$ does not depend continuously on the on the right-hand side. As $y$ is often derived from measurements, one has instead a perturbed $y^{\delta}$ such that ${y - y^{\delta}}_{H}<\delta$. Thus due to the ill-posedness, solving (1) with $y^{\delta}$ is not guaranteed to produce a meaningful solution. One such class of techniques to treat such problems are the Tikhonov-regularization methods. One seeks in reconstructing the solution to balance fidelity to the data against size of some functional evaluation of the reconstructed image (e.g., the norm of the reconstruction) to mitigate the effects of the ill-posedness. For some $\lambda>0$, we solve \begin{equation}\label{eqn.tikh} x_{\lambda} = \textrm{argmin}_{\widetilde{x}\in H}\left\lbrace{\left\|{b - A\widetilde{x}} \right\|_{H}^{2} + \lambda \left\|{\widetilde{x}}\right\|_{H}^{2}} \right\rbrace. \end{equation} In this talk, we discuss some new strategies for treating discretized versions of this problem. Here, we consider a discreditized, finite dimensional version of (1), \begin{equation}\label{eqn.Axb} Ax =  b \mbox{ with }  A\in \mathbb{R}^{n\times n}\mbox{ and } b\in\mathbb{R}^{n}, \end{equation} which inherits a discrete version of ill conditioning from [1]. We propose methods built on top of the Arnoldi-Tikhonov method of Lewis and Reichel, whereby one builds the Krylov subspace \begin{equation}
\mathcal{K}_{j}(\vek A,\vek w) = {\rm span\,}\curl{\vek w,\vek A\vek w,\vek A^{2}\vek w,\ldots,\vek A^{j-1}\vek w}\mbox{ where } \vek w\in\curl{\vek b,\vek A\vek b}
\end{equation}
and solves the discretized Tikhonov minimization problem projected onto that subspace. We propose to extend this strategy to setting of augmented Krylov subspace methods. Thus, we project onto a sum of subspaces of the form $\mathcal{U} + \mathcal{K}_{j}$ where $\mathcal{U}$ is a fixed subspace and $\mathcal{K}_{j}$ is a Krylov subspace. It turns out there are multiple ways to do this leading to different algorithms. We will explain how these different methods arise mathematically and demonstrate their effectiveness on a few example problems. Along the way, some new mathematical properties of the Arnoldi-Tikhonov method are also proven.

An informal session for DPhil students, ECRs and undergraduates with an interest in probability. The aim is to gain exposure to areas outside of your own research interests in an informal and accessible way.

We will talk about the Cauchy problem of the three-dimensional isentropic compressible Navier-Stokes equations. When viscosity coefficients are given as a constant multiple of density's power, based on some analysis  of  the nonlinear structure of this system, by introducing some new variables and the initial layer compatibility conditions, we identify the class of initial data admitting a local regular solution with far field vacuum and  finite energy  in some inhomogeneous Sobolev spaces, which solves an open problem of degenerate viscous flow partially mentioned by Bresh-Desjardins-Metivier (2006, Anal. Simi. Fluid Dynam.),  Jiu-Wang-Xin (2014, JMFM) and so on. Moreover, in contrast to the classical well-posedness theory in the case of  the constant viscosity,   we show   that one can not obtain any global classical solution whose $L^\infty$  norm of $u$ decays to zero as time $t$ goes to infinity under the assumptions on the conservation laws of total mass and momentum.

"Fibre theorems" in the style of Quillen's fibre lemma are versatile tools used to study the topology of partially ordered sets. In this talk, I will formulate two of them and explain how these can be used to determine the homotopy type of the complex of (conjugacy classes of) free factors of a free group.
The latter is joint work with Radhika Gupta (see https://arxiv.org/abs/1810.09380).

TBA

In this talk we will introduce quantifier elimination and give various examples of theories with this property. We will see some very useful applications of quantifier elimination to algebra and geometry that will hopefully convince you how practical this property is to other areas of mathematics.

 
We study continuous theories of classes of finite dimensional Hilbert spaces expanded by 
a finite family (of a fixed size) of unitary operators. 
Infinite dimensional models of these theories are called pseudo finite dimensional dynamical Hilbert spaces. 
Our main results connect decidability questions of these theories with the topic of approximations of groups by metric groups. 

In the classical theory of fluid mechanics, a linear relationship between the stress and rate of strain is often assumed. Even when this relationship is non-linear, it is typically formulated in terms of an explicit relation. Implicit constitutive theories provide a theoretical framework that generalises this, allowing a, possibly multi-valued, implicit constitutive relation. Since it is not possible to solve explicitly for the stress in the constitutive relation, a more natural approach would be to include the stress as a fundamental unknown in the formulation of the problem. In this talk I will present a formulation with this feature and a proof of convergence of the finite element approximations to a solution of the original problem.

A fundamental task in numerical computation is the solution of large linear systems. The conjugate gradient method is an iterative method which offers rapid convergence to the solution, particularly when an effective preconditioner is employed. However, for more challenging systems a substantial error can be present even after many iterations have been performed. The estimates obtained in this case are of little value unless further information can be provided about the numerical error. In this paper we propose a novel statistical model for this numerical error set in a Bayesian framework. Our approach is a strict generalisation of the conjugate gradient method, which is recovered as the posterior mean for a particular choice of prior. The estimates obtained are analysed with Krylov subspace methods and a contraction result for the posterior is presented. The method is then analysed in a simulation study as well as being applied to a challenging problem in medical imaging.

In this work, study the mean first saturation time (MFST), a generalization to the mean first passage time, on networks and show an application to the 2015 Burundi refugee crisis. The MFST between a sink node j, with capacity s, and source node i, with n random walkers, is the average number of time steps that it takes for at least s of the random walkers to reach a sink node j. The same concept, under the name of extreme events, has been studied in previous work for degree biased-random walks [2]. We expand the literature by exploring the behaviour of the MFST for node-biased random walks [1] in Erdős–Rényi random graph and geographical networks. Furthermore, we apply MFST framework to study the distribution of refugees in camps for the 2015 Burundi refugee crisis. For this last application, we use the geographical network of the Burundi conflict zone in 2015 [3]. In this network, nodes are cities or refugee camps, and edges denote the distance between them. We model refugees as random walkers who are biased towards the refugee camps which can hold s_j people. To determine the source nodes (i) and the initial number of random walkers (n), we use data on where the conflicts happened and the number of refugees that arrive at any camp under a two-month period after the start of the conflict [3]. With such information, we divide the early stage of the Burundi 2015 conflict into two waves of refugees. Using the first wave of refugees we calibrate the biased parameter β of the random walk to best match the distribution of refugees on the camps. Then, we test the prediction of the distribution of refugees in camps for the second wave using the same biased parameters. Our results show that the biased random walk can capture, to some extent, the distribution of refugees in different camps. Finally, we test the probability of saturation for various camps. Our model suggests the saturation of one or two camps (Nakivale and Nyarugusu) when in reality only Nyarugusu camp saturated.

[1] Sood, Vishal, and Peter Grassberger. ”Localization transition of biased random walks on random
networks.” Physical review letters 99.9 (2007): 098701.
[2] Kishore, Vimal, M. S. Santhanam, and R. E. Amritkar. ”Extreme event-size fluctuations in biased
random walks on networks.” arXiv preprint arXiv:1112.2112 (2011).
[3] Suleimenova, Diana, David Bell, and Derek Groen. ”A generalized simulation development approach
for predicting refugee destinations.” Scientific reports 7.1 (2017): 13377.

A dedicated search of the CMB sky, driven by implications of conformal
cyclic cosmology (CCC), has revealed a remarkably strong signal, previously
unobserved, of numerous small regions in the CMB sky that would appear to be
individual points on CCC's crossover 3-surface from the previous aeon, most
readily interpreted as the conformally compressed Hawking radiation from
supermassive black holes in the previous aeon, but difficult to explain in
terms of the conventional inflationary picture.

We consider solutions to the two-dimensional incompressible Euler system with only integrable vorticity, thus with possibly locally infinite energy. With such regularity, we use the recently developed theory of Lagrangian flows associated to vector fields with gradient given by a singular integral in order to define Lagrangian solutions, for which the vorticity is transported by the flow. We prove strong stability of these solutions via strong convergence of the flow, under the only assumption of $L^1$ weak convergence of the initial vorticity. The existence of Lagrangian solutions to the Euler system follows for arbitrary $L^1$ vorticity. Relations with previously known notions of solutions are shown.

In directed algebraic topology, a topological space is endowed 
with an extra structure, a selected subset of the paths called the 
directed paths or the d-structure. The subset has to contain the 
constant paths, be closed under concatenation and non-decreasing 
reparametrization. A space with a d-structure is a d-space.
If the space has a partial order, the paths increasing wrt. that order 
form a d-structure, but the circle with counter clockwise paths as the 
d-structure is a prominent example without an underlying partial order.
Dipaths are dihomotopic if there is a one-parameter family of directed 
paths connecting them. Since in general dipaths do not have inverses, 
instead of fundamental groups (or groupoids), there is a fundamental 
category. So already at this stage, the algebra is less desirable than 
for topological spaces.
We will give examples of what is currently known in the area, the kind 
of methods used and the problems and questions which need answering - in 
particular with applications in computer science in mind.
 

The physics literature has for a long time posited a connection between the geometry of continuous random fields and discrete percolation models. Specifically the excursion sets of continuous fields are considered to be analogous to the open connected clusters of discrete models. Recent work has begun to formalise this relationship; many of the classic results of percolation (phase transition, RSW estimates etc) have been proven in the setting of smooth Gaussian fields. In the first part of this talk I will summarise these results. In the second I will focus on the number of excursion set components of Gaussian fields in large domains and discuss new results on the mean and variance of this quantity.