Past

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.

By using Malliavin calculus, Bismut type formulas are established for the Lions derivative of , where  0,  is a bounded measurable function,  and  solves a distribution dependent SDE with initial distribution . As applications, explicit estimates are derived for the Lions derivative and the total variational distance between distributions of   solutions with different initial data. Both degenerate and non-degenerate situations are considered. Due to the lack of the semi-group property  and the invalidity of the formula =  , essential difficulties are overcome in the study.

Joint work with Professor Feng-Yu Wang

In recent joint work with Lorenzo Foscolo and Johannes Nordstr\”om we gave an analytic construction of large families of complete circle-invariant $G_2$
holonomy metrics on the total space of circle bundles over a complete noncompact Calabi—Yau 3-fold with asymptotically conical geometry. The
asymptotic models for the geometry of these $G_2$ metrics are circle bundles with fibres of constant length $l$, so-called asymptotically local conical
(ALC) geometry. These ALC $G_2$ metrics can Gromov—Hausdorff collapse with bounded curvature to the given asymptotically conical Calabi—Yau 3-fold as the fibre length $l$ goes to $0$. A natural question is: what happens to these families of $G_2$ metrics as we try to make $l$ large? In general the answer to this question is not known, but in cases with sufficient symmetry we have recently been able to give a complete picture.  

We give an overview of all these results and discuss some analogies with the class of asymptotically locally flat (ALF) hyperkaehler 4-manifolds. In
particular we suggest that a particular $G_2$ metric we construct should be regarded as a $G_2$ analogue of the Euclidean Taub—NUT metric on the complex plane.