Past

Congruence monoids in the ring of integers are given by certain unions of arithmetic progressions. To each congruence monoid, there is a canonical way to associate a semigroup C*-algebra. I will explain this construction and then discuss joint work with Xin Li on K-theoretic invariants. I will also indicate how all of this generalizes to congruence monoids in the ring of integers of an arbitrary algebraic number field.

Enumerative algebraic geometry counts the solutions to certain geometric constraints. Numerical algebraic geometry determines these solutions for any given 
instance. This lecture illustrates how these two fields complement each other, especially in the light of emerging new applications. We start with a gem from
the 19th century, namely the 3264 conics that are tangent to five given conics in the plane. Thereafter we turn to current problems in statistics, with focus on 
maximum likelihood estimation for linear Gaussian covariance models.
 

The Approximate Message Passing (AMP) algorithm is a powerful iterative method for reconstructing undersampled sparse signals. Unfortunately, AMP is sensitive to the type of sensing matrix employed and frequently encounters convergence problems. One case where AMP tends to fail is compressed sensing MRI, where Fourier coefficients of a natural image are sampled with variable density. An AMP-inspired algorithm constructed specifically for MRI is presented that exhibits a 'state evolution', where at every iteration the image estimate before thresholding behaves as the ground truth corrupted by Gaussian noise with known covariance. Numerical experiments explore the practical benefits of such effective noise behaviour.
 

Boundary integral equations (BIEs) are well established for solving scattering at bounded infinitely thin objects, so-called screens, which are modelled as “open surfaces” in 3D and as “open curves” in 2D. Moreover, the unknowns of these BIEs are the jumps of traces across $\Gamma$. Things change considerably when considering scattering at multi-screens, which are arbitrary arrangements of thin panels that may not be even locally orientable because of junction points (2D) or junction lines (3D). Indeed, the notion of jumps of traces is no longer meaningful at these junctions. This issue can be solved by switching to a quotient space perspective of traces, as done in recent work by Claeys and Hiptmair. In this talk, we present the extension of the quotient-space approach to the Galerkin boundary element (BE) discretization of first-kind BIEs. Unlike previous approaches, the new quotient-space BEM relies on minimal geometry information and does not require any special treatment at junctions. Moreover, it allows for a rigorous numerical analysis.
 

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