Past

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.