Past

Recent advances in experimental imaging techniques have allowed us to observe the fine details of how droplets behave upon impact onto a substrate. However, these are highly non-linear, multiscale phenomena and are thus a formidable challenge to model. In addition, when the substrate is deformable, such as an elastic sheet, the fluid-structure interaction introduces an extra layer of complexity.

We present two modeling approaches for droplet impact onto deformable substrates: matched asymptotics and direct numerical simulations. In the former, we use Wagner's theory of impact to derive analytical expressions which approximate the behavior during the early time of impact. In the latter, we use the open source volume-of-fluid code Basilisk to conduct simulations designed to give insight into the later times of impact.

We conclude by showing how these methods are complementary, and a combination of both can give a thorough understanding of the droplet impact across timescales. 

Identifying systemically important countries is crucial for global financial stability. In this work we use (multilayer) network methods to identify systemically important countries. We study the financial system as a multilayer network, where each layer represent a different type of financial investment between countries. To rank countries by their systemic importance, we implement MultiRank, as well a simplistic model of financial contagion. In this first model, we consider that each country has a capital buffer, given by the capital to assets ratio. After the default of an initial country, we model financial contagion with a simple rule: a solvent country defaults when the amount of assets lost, due to the default of other countries, is larger than its capital. Our results show that when we consider that there are various types of assets the ranking of systemically important countries changes. We make all our methods available by introducing a python library. Finally, we propose a more realistic model of financial contagion that merges multilayer network theory and the contingent claims sectoral balance sheet literature. The aim of this framework is to model the banking, private, and sovereign sector of each country and thus study financial contagion within the country and between countries. 

A subset of the integers larger than 1 is called $\textit{primitive}$ if no member divides another. Erdős proved in 1935 that the sum of $1/(n \log n)$ over $n$ in a primitive set $A$ is universally bounded for any choice of $A$. In 1988, he famously asked if this universal bound is attained by the set of prime numbers. In this talk we shall discuss some recent progress towards this conjecture and related results, drawing on ideas from analysis, probability, & combinatorics.

The Gibbons-Hawking ansatz is a powerful method for constructing a large family of hyperkaehler 4-manifolds (which are thus Ricci-flat), which appears in a variety of contexts in mathematics and theoretical physics. I will describe work in progress to understand the theory of minimal surfaces and mean curvature flow in these 4-manifolds. In particular, I will explain a proof of a version of the Thomas-Yau Conjecture in Lagrangian mean curvature flow in this setting. This is joint work with G. Oliveira.

The smooth mapping class group of the 4-sphere is pi_0 of the space of orientation preserving self-diffeomorphisms of S^4. At the moment we have no idea whether this group is trivial or not. Watanabe has shown that higher homotopy groups can be nontrivial. Inspired by Watanabe's constructions, we'll look for interesting self-diffeomorphisms of S^4. Most of the talk will be an outline for a program to find a nice geometric generating set for this mapping class group; a few small steps in the program are actually theorems. The point of finding generators is that if they are explicit enough then you have a hope of either showing that they are all trivial or finding an invariant that is well adapted to obstructing triviality of these generators.

We reconsider the approximations of the Black-Scholes model by discrete time models such as the binominal or the trinominal model.

We show that for continuous and bounded claims one may approximate the replication in the Black-Scholes model by trading in the discrete time models. The approximations holds true in measure as well as "with bounded risk", the latter assertion being the delicate issue. The remarkable aspect is that this result does not apply to the well-known binominal model, but to a much wider class of discrete approximating models, including, eg.,the trinominal model. by an example we show that we cannot do the approximation with "vanishing risk".

We apply this result to portfolio optimization and show that, for utility functions with "reasonable asymptotic elasticity" the solution to the discrete time portfolio optimization converge to their continuous limit, again in a wide class of discretizations including the trinominal model. In the absence of "reasonable asymptotic elasticity", however, surprising pathologies may occur.

Joint work with David Kreps (Stanford University)

We revisit portfolio selection models by considering a distributionally robust version, where the region of distributional uncertainty is around the empirical measure and the discrepancy between probability measures is dictated by optimal transport costs. In many cases, this problem can be simplified into an empirical risk minimization problem with a regularization term. Moreover, we extend a recently developed inference methodology in order to select the size of the distributional uncertainty in a data-driven way. Our formulations allow us to inform the distributional uncertainty region using market information (e.g. via implied volatilities). We provide substantial empirical tests that validate our approach.
(This presentation is based on the following papers: https://arxiv.org/pdf/1802.04885.pdf and https://arxiv.org/abs/1810.024….)

In this talk I will discuss a new proposal for constructing quantizations of holomorphic Poisson structures, and generalized complex manifolds more generally, which is based on using the A model of an associated symplectic manifold known as a Morita equivalence. This construction will be illustrated through the example of toric Poisson structures.

https://arxiv.org/pdf/1906.11251.pdf

The puzzling phenonenon of wave localization refers to unexpected confinement of waves triggered by disorder in the propagating medium. Localization arises in many physical and mathematical systems and has many important implications and applications. A particularly important case is the Schrödinger equation of quantum mechanics, for which the localization behavior is crucial to the electrical properties of materials. Mathematically it is tied to exponential decay of eigenfunctions of operators instead of their expected extension throughout the domain. Although localization has been studied by physicists and mathematicians for the better part of a century, many aspects remain mysterious. In particular, the sort of deterministic quantitative results needed to predict, control, and exploit localization have remained elusive. This talk will focus on major strides made in recent years based on the introduction of the landscape function and its partner, the effective potential. We will describe these developments from the viewpoint of a computational mathematician who sees the landscape theory as a completely unorthodox sort of a numerical method for computing spectra.

In my talk I will discuss the use of topological methods in the analysis of neural data. I will show how to obtain good state spaces for Head Direction Cells and Grid Cells. Topological decoding shows how neural firing patterns determine behaviour. This is a local to global situation which gives rise to some reflections.

This week's Fridays@2 will be a panel discussion focusing on what it is like to pursue a research degree. The panel will share their thoughts and experiences in a question-and-answer session, discussing some of the practicalities of being a postgraduate student, and where a research degree might lead afterwards. Participants include:

Jono Chetwynd-Diggle (Smith Institute)

Victoria Patel (PDE CDT, Mathematical Institute)

Robin Thompson (Christ Church)

Rosemary Walmsley (DPhil student Health Economics Research Centre, Oxford) 

The spatial coordination of cellular differentiation enables functional organogenesis. How coordination results in specific patterns of differentiation in a robust manner is a fundamental question for all developmental systems in biology. Theoreticians such as Turing and Wolpert have proposed the importance of specific mechanisms that enable certain types of patterns to emerge, but these mechanisms are often difficult to identify in natural systems. Therefore, we have started using synthetic biology to ask whether specific mechanisms of pattern formation can be engineered into a simple cellular background. In this talk, I will show several examples of emergent spatial patterning that results from the insertion of synthetic signalling pathways and transcriptional logic into E. coli. In all cases, we use computational modelling to initially design circuits with a desired outcome, and improve the selection of biological components (DNA sub-sequences) that achieve this outcome according to a quantifiable measure. In the specific case of Turing patterns, we have yet to produce a functional system in vivo, but I will describe new analytical tools that are helping to guide the design of synthetic circuits that can produce a Turing instability.

Optical super-resolution microscopy enables the observations of individual bio-molecules. The arrangement and dynamic behaviour of such molecules is studied to get insights into cellular processes which in turn lead to various application such as treatments for cancer diseases. STFC's Central Laser Facility provides (among other) public access to super-resolution microscope techniques via research grants. The access includes sample preparation, imaging facilities and data analysis support. Data analysis includes single molecule tracking algorithms that produce molecule traces or tracks from time series of molecule observations. While current algorithms are gradually getting away from "connecting the dots" and using probabilistic methods, they often fail to quantify the uncertainties in the results. We have developed a method that samples a probability distribution of tracking solutions using the Metropolis-Hastings algorithm. Such a method can produce likely alternative solutions together with uncertainties in the results. While the method works well for smaller data sets, it is still inefficient for the amount of data that is commonly collected with microscopes. Given the observations of the molecules, tracking solutions are discrete, which gives the proposal distribution of the sampler a peculiar form. In order for the sampler to work efficiently, the proposal density needs to be well designed. We will discuss the properties of tracking solutions and the problems of the proposal function design from the point of view of discrete mathematics, specifically in terms of graphs. Can mathematical theory help to design a efficient proposal function?

Recently, Joyce constructed a Ringel-Hall style graded vertex algebra on the homology of moduli stacks of objects in certain categories of algebro-geometric and representation-theoretic origin. The construction is most natural for 2n-Calabi-Yau categories. We present this construction and explain the geometric reason why it exists. If time permits, we will explain how to compute the homology of the moduli stack of objects in the derived category of a smooth complex projective variety and to identify it with a lattice-type vertex algebra.

We study a class of non linear integro-differential equations on the Wasserstein space related to the optimal control of McKean-Vlasov jump-diffusions. We develop an intrinsic notion of viscosity solutions that does not rely on the lifting to an Hilbert space and prove a comparison theorem for these solutions. We also show that the value function is the unique viscosity solution. Based on a joint work with V. Ignazio, M. Reppen and H. M. Soner

Let n be a random integer (sampled from {1,..,X} for some large X). It is a classical fact that, typically, n will have around (log n)^{log 2} divisors. Must some of these be close together? Hooley's Delta function Delta(n) is the maximum, over all dyadic intervals I = [t,2t], of the number of divisors of n in I. I will report on joint work with Kevin Ford and Dimitris Koukoulopoulos where we conjecture that typically Delta(n) is about (log log n)^c for some c = 0.353.... given by an equation involving an exotic recurrence relation, and then prove (in some sense) half of this conjecture, establishing that Delta(n) is at least this big almost surely.

The dynamics of many physical systems often evolve to asymptotic states that exhibit periodic spatial and temporal variations in their properties such as density, temperature, etc. Such regular patterns look the same when moved by a basic unit and/or rotated by certain special angles. They possess both translational and rotational symmetries giving rise to discrete spatial Fourier transforms. In contrast, an aperiodic crystal displays long range spatial order but no translational symmetry. 

Recently, quasicrystals which are related to aperiodic crystals have been observed to form in diverse physical systems such as metallic alloys (atomic scale) and dendritic-, star-, and block co-polymers (molecular scale). Such quasicrystals lack the lattice symmetries of regular crystals, yet have discrete Fourier spectra. We look to understand the minimal mechanism which promotes the formation of such quasicrystalline structures using a phase field crystal model. Direct numerical simulations combined with weakly nonlinear analysis highlight the parameter values where the quasicrystals are the global minimum energy state and help determine the phase diagram. 

By locating parameter values where multiple patterned states possess the same free energy (Maxwell points), we obtain states where a patch of one type of pattern (for example, a quasicrystal) is present in the background of another (for example, the homogeneous liquid state) in the form of spatially localized dodecagonal (in 2D) and icosahedral (in 3D) quasicrystals. In two dimensions, we compute several families of spatially localized quasicrystals with dodecagonal structure and investigate their properties as a function of the system parameters. The presence of such meta-stable localized quasicrystals is significant as they may affect the dynamics of the crystallisation in soft matter.

The fully coupled numerical simulation of different physical processes, which can typically occur
at variable time and space scales, is often a very challenging task. A common feature of such models is that
their discretization gives rise to systems of linearized equations with an inherent block structure, which
reflects the properties of the set of governing PDEs. The efficient solution of a sequence of systems with
matrices in a block form is usually one of the most time- and memory-demanding issue in a coupled simulation.
This effort can be carried out by using either iteratively coupled schemes or monolithic approaches, which
tackle the problem of the system solution as a whole.

This talk aims to discuss recent advances in the monolithic solution of coupled multi-physics problems, with
application to poromechanical simulations in fractured porous media. The problem is addressed either by proper
sparse approximations of the Schur complements or special splittings that can partially uncouple the variables
related to different physical processes. The selected approaches can be included in a more general preconditioning
framework that can help accelerate the convergence of Krylov subspace solvers. The generalized preconditioner
relies on approximately decoupling the different processes, so as to address each single-physics problem
independently of the others. The objective is to provide an algebraic framework that can be employed as a
general ``black-box'' tool and can be regarded as a common starting point to be later specialized for the
particular multi-physics problem at hand.

Numerical experiments, taken from real-world examples of poromechanical problems and fractured media, are used to
investigate the behaviour and the performance of the proposed strategies.

In the twentieth century we leant that the theory of communication is a mathematical theory. Mathematics is able to add to the value of data, for example by removing redundancy (compression) or increasing robustness (error correction). More famously mathematics can add value to data in the presence of an adversary which is my personal definition of cryptography. Cryptographers talk about properties of confidentiality, integrity, and authentication, though modern cryptography also facilitates transparency (distributed ledgers), plausible deniability (TrueCrypt), and anonymity (Tor).
Modern cryptography faces new design challenges as people demand more functionality from data. Some recent requirements include homomorphic encryption, efficient zero knowledge proofs for smart contracting, quantum resistant cryptography, and lightweight cryptography. I'll try and cover some of the motivations and methods for these.
 

In the seminar, I will talk about a parabolic toy-model for the incompressible Navier-Stokes equations, that satisfies the same energy inequality, same scaling symmetry and which is also super-critical in dimension 3. I will present some partial regularity results that this model shares with the incompressible model and other results that occur only for our model.