Past

I will give an overview of the Nielsen-Thurston theory of the mapping class group and its connection to hyperbolic geometry and dynamics. Time permitting, I will discuss the surface entropy conjecture and a theorem of Hamenstadt on entropies of `generic' elements of the mapping class group. No prior knowledge of the concepts involved is required.

Injection of a gas into a gas/liquid foam is known to give rise to instability phenomena on a variety of time and length scales. Macroscopically, one observes a propagating gas-filled structure that can display properties of liquid finger propagation as well as of fracture in solids. Using a discrete model, which incorporates the underlying film instability as well as viscous resistance from the moving liquid structures, we describe brittle cleavage phenomena in line with experimental observations. We find that  the dimensions of the foam sample significantly affect the speed of the  cracks as well as the pressure necessary to sustain them: cracks in wider samples travel faster at a given driving stress, but are able to avoid arrest and maintain propagation at a lower pressure (the  velocity gap becomes smaller). The system thus becomes a study case for stress concentration and the transition between discrete and continuum systems in dynamical fracture; taking into account the finite dimensions of the system improves agreement with experiment.

Driven by its numerous applications in computational science, the approximation of smooth, high-dimensional functions via sparse polynomial expansions has received significant attention in the last five to ten years.  In the first part of this talk, I will give a brief survey of recent progress in this area.  In particular, I will demonstrate how the proper use of compressed sensing tools leads to new techniques for high-dimensional approximation which can mitigate the curse of dimensionality to a substantial extent.  The rest of the talk is devoted to approximating functions defined on irregular domains.  The vast majority of works on high-dimensional approximation assume the function in question is defined over a tensor-product domain.  Yet this assumption is often unrealistic.  I will introduce a method, known as polynomial frame approximation, suitable for broad classes of irregular domains and present theoretical guarantees for its approximation error, stability, and sample complexity.  These results show the suitability of this approach for high-dimensional approximation through the independence (or weak dependence) of the various guarantees on the ambient dimension d.  Time permitting, I will also discuss several extensions.

Stein ($1981$) proved the borderline Sobolev embedding result which states that for $n \geq 2,$ $u \in L^{1}(\mathbb{R}^{n})$ and $\nabla u \in L^{(n,1)}(\mathbb{R}^{n}; \mathbb{R}^{n})$ implies $u$ is continuous. Coupled with standard Calderon-Zygmund estimates for Lorentz spaces, this implies $u \in C^{1}(\mathbb{R}^{n})$ if $\Delta u \in L^{(n,1)}(\mathbb{R}^{n}).$ The search for a nonlinear generalization of this result culminated in the work of Kuusi-Mingione ($2014$), which proves the same result for $p$-Laplacian type systems. \paragraph{} In this talk, we shall discuss how these results can be extended to differential forms. In particular, we can prove that if $u$ is an $\mathbb{R}^{N}$-valued $W^{1,p}_{loc}$ $k$-differential form with $\delta \left( a(x) \lvert du \rvert^{p-2} du \right) \in L^{(n,1)}_{loc}$ in a domain of $\mathbb{R}^{n}$ for $N \geq 1,$ $n \geq 2,$ $0 \leq k \leq n-1, $ $1 < p < \infty, $ with uniformly positive, bounded, Dini continuous scalar function $a$, then $du$ is continuous.

I will give a self-contained introduction to the theory of cross ratios on boundaries of Gromov hyperbolic and CAT(-1) spaces, focussing on the connections to the following two questions. When are two spaces with the 'same' Gromov boundary isometric/quasi-isometric? Are closed Riemannian manifolds completely determined (up to isometry) by the lengths of their closed geodesics?

Young diagrams frequently appear in the study of partitions and representations of the symmetric group. By filling these diagrams with numbers, we obtain Young tableau, combinatorial objects onto which we can define the structure of a monoid via insertion algorithms. We will explore this structure and it's connection to a basis of the ring of symmetric polynomials. If we have time, we will mention alternative monoid structures and their corresponding bases.

Polar classes are very classical objects in Algebraic Geometry. A brief introduction to the subject will be presented and ideas and preliminarily results towards generalisations will be explained. These ideas can be applied towards variety sampling and relevant applications. 
 

This talk will feature a brief introduction to persistent homology, the vanguard technique in topological data analysis. Nothing will be required of the audience beyond a willingness to row-reduce enormous matrices (by hand if we can, by machine if we must).

In combinatorics, the 'nicest' way to prove that two sets have the same size is to find a bijection between them, giving more structure to the seeming numerical coincidences. In representation theory, many of the outstanding conjectures seem to imply that the characteristic p of the ground field can be allowed to vary, and we can relate different groups and different primes, to say that they have 'the same' representation theory. In this talk I will try to make precise what we could mean by this

For decades, researchers have been studying efficient numerical methods to solve differential equations, most of them optimised for one-core processors. However, we are about to reach the limit in the amount of processing power we can squeeze into a single processor. This explains the trend in today's computing industry to design high-performance processors looking at parallel architectures. As a result, there is a need to develop low-complexity parallel algorithms capable of running efficiently in terms of computational time and electric power.

Parallelisation across time appears to be a promising way to provide more parallelism. In this talk, we will introduce the main algorithms, following (Gander, 2015), with a particular focus on the parareal algorithm.

Unintended low energy thermal or mechanical stimuli can lead to the accidental ignition of explosive materials. During such events, described as ‘insults’ in the literature, ignition of the explosive is caused by localised regions of high temperature known as ‘hot spots’. We develop a model which helps us to understand how highly localised shear deformation, so-called shear banding, acts as a mechanism for hot spot generation. Through a boundary layer analysis, we give a deeper insight into how the additional self heating caused by chemical reactions affects the initiation and development of shear bands,  and highlight the key physical properties which control this process.

I will talk about the implications of UV completion of quantum gravity on the low energy spectrums. I will introduce the constraints on low-energy effective theory due to unitarity and analyticity of scattering amplitudes, in particular an infinite set of new unitarity constraints on the forward-limit limit of four-point scattering amplitudes due to the work of Arkani-Hamed et al. In three dimensions, we find the constraints imply that light states with charge-to-mass ratio z greater than 1 can only be consistent if there exists other light states, preferably neutral. Applied to the 3D Standard Model like spectrum, where the low energy couplings are dominated by the electron with z \sim 10^22, this provides a novel understanding of the need for light neutrinos.

Many real-world networks contain small recurring connectivity patterns also known as network motifs. Although network motifs are widely considered to be important structural features of networks that are closely connected to their function methods for characterizing and modelling the local connectivity structure of complex networks remain underdeveloped. In this talk, we will present a non-parametric approach that is based on generative models in which networks are generated by adding not only single edges but also but also copies of larger subgraphs such as triangles to the graph. We show that such models can be formulated in terms of latent states that correspond to subgraph decompositions of the network and derive analytic expressions for the likelihood of such models. Following a Bayesian approach, we present a nonparametric prior for model parameters. Solving the resulting inference problem results in a principled approach for identifying atomic connectivity patterns of networks that do not only identify statistically significant connectivity patterns but also produces a decomposition of the network into such atomic substructures. We tested the presented approach on simulated data for which the algorithm recovers the latent state to a high degree of accuracy. In the case of empirical networks, the method identifies concise sets atomic subgraphs from within thousands of candidates that are plausible and include known atomic substructures.

Ginzburg–Landau type functionals provide a relaxation scheme to construct harmonic maps in the presence of topological obstructions. They arise in superconductivity models, in liquid crystal models (Landau–de Gennes functional) and in the generation of cross-fields in meshing. For a general compact manifold target space we describe the asymptotic number, type and location of singularities that arise in minimizers. We cover in particular the case where the fundamental group of the vacuum manifold in nonabelian and hence the singularities cannot be characterized univocally as elements of the fundamental group. The results unify the existing theory and cover new situations and problems.

This is a joint work with Antonin Monteil and Rémy Rodiac (UCLouvain, Louvain- la-Neuve, Belgium)

We recall the impact of stabilizing a 4-manifold with $S^2 \times S^2$. The corresponding local situation concerns knots in the 3-sphere which bound (nullhomotopic) discs in a stabilized 4-ball. We explain how these discs arise, and discuss bounds on the minimal number of stabilizations needed. Then we compare this minimal number to the 4-genus.
This is joint work with A. Conway.

In this talk I present work in progress with Wolfgang König and Nicolas Perkowski on the parabolic Anderson model (PAM) with white noise potential in 2d. We show the behavior of the total mass as the time tends to infinity. By using partial Girsanov transform and singular heat kernel estimates we can obtain the mass-asymptotics by using the eigenvalue asymptotics that have been showed in another work in progress with Khalil Chouk. 

We prove an a priori bound for solutions of the dynamic $\Phi^4_3$ equation.

This bound provides a control on solutions on a compact space-time set only in terms of the realisation of the noise on an enlargement of this set, and it does not depend on any choice of space-time boundary conditions.

We treat the  large and small scale behaviour of solutions with completely different arguments.For small scales we use bounds akin to those presented in Hairer's theory of regularity structures. We stress immediately that our proof is fully self-contained, but we give a detailed explanation of how our arguments relate to Hairer's. For large scales we use a PDE argument based on the maximum principle. Both regimes are connected by a solution-dependent regularisation procedure.

The fact that our bounds do not depend on space-time boundary conditions makes them useful for the analysis of large scale properties of solutions. They can for example be used in a compactness argument to construct solutions on the full space and their invariant measures

We consider the two-dimensional Euler flow for an incompressible fluid confined to a smooth domain. We construct smooth solutions with concentrated vorticities around $k$ points which evolve according to the Hamiltonian system for the Kirkhoff-Routh energy,  using an outer-inner solution gluing approach. The asymptotically singular profile  around each point resembles a scaled finite mass solution of Liouville's equation.
We also discuss the {\em vortex filament conjecture} for the three-dimensional case. This is joint work with Juan D\'avila, Monica Musso and Juncheng Wei.

In this talk I will present a large class of non-supersymmetric AdS7 solutions of IIA supergravity, and their (in)stabilities. I will start by reviewing supersymmetric AdS7 solutions of 10D supergravity dual to 6D (1,0) SCFTs. I will then focus on their non-supersymmetric counterpart, discussing how they are related. The connection between supersymmetric and non-supersymmetric solutions leads to a hint for the SUSY breaking mechanism, which potentially allows to evade some of the assumptions of the Ooguri-Vafa Conjecture about the AdS landscape. A big subset of these solutions shows a curious pattern of perturbative instabilities whenever many open-string modes are considered. On the other hand an infinite class remains apparently stable.

Would you employ you? What are employers looking for in Mathematical graduates? What kind of work can use your skills? This workshop will get your minds thinking about the possibilities after you have finished studying and will cover:

·         General careers’ information starting from a mathematical sciences degree

·         Things to think about at CV and interview stage

·         How membership of a professional body (the IMA) supports your applications and career development.

·         Information about the Mathematics Teacher Training Scholarships