Past

Genome replication is a stochastic process whereby each cell exhibits different patterns of origin activation and replication fork movement.  Despite this heterogeneity, replication is a remarkably stable process that works quickly and correctly over hundreds of thousands of iterations. Existing methods for measuring replication dynamics largely focus on how a population of cells behave on average, which precludes the detection of low probability errors that may have occurred in individual cells.  These errors can have a severe impact on genome integrity, yet existing single-molecule methods, such as DNA combing, are too costly, low-throughput, and low-resolution to effectively detect them.  We have created a method that uses Oxford Nanopore sequencing to create high-throughput genome-wide maps of DNA replication dynamics in single molecules.  I will discuss the informatics approach that our software uses, our use of mathematical modelling to explain the patterns that we observe, and questions in DNA replication and genome stability that our method is uniquely positioned to answer.

Given a stability condition defined over a category, every object in this category
is filtered by some distinguished objects called semistables. This
filtration, that is unique up-to-isomorphism, is know as the
 Harder-Narasimhan filtration.
One less studied property of stability conditions, when defined over an
 abelian category, is the fact that each of them induce a chain of torsion
classes that is naturally indexed.
 In this talk we will study arbitrary indexed chain of torsion classes. Our
first result states that every indexed chain of torsion classes induce a
 Harder-Narasimhan filtration. Following ideas from Bridgeland we
 show that the set of all indexed chains of torsion classes satisfying a mild 
 technical condition forms a topological space. If time we
 will characterise the neighbourhood or some distinguished points. 

Landmark-based human action recognition in videos is a challenging task in computer vision. One crucial step is to design discriminative features for spatial structure and temporal dynamics. To this end, we use and refine the path signature as an expressive, robust, nonlinear, and interpretable representation for landmark-based streamed data. Instead of extracting signature features from raw sequences, we propose path disintegrations and transformations as preprocessing to improve the efficiency and effectiveness of signature features. The path disintegrations spatially localize a pose into a collection of m-node paths from which the signatures encode non-local and non-linear geometrical dependencies, while temporally transform the evolutions of spatial features into hierarchical spatio-temporal paths from which the signatures encode long short-term dynamical dependencies. The path transformations allow the signatures to further explore correlations among different informative clues. Finally, all features are concatenated to constitute the input vector of a linear fully-connected network for action recognition. Experimental results on four benchmark datasets demonstrated that the proposed feature sets with only linear network achieves comparable state-of-the-art result to the cutting-edge deep learning methods. 

Starting with a countable transitive model of V=L, we show that by 
adding a single Cohen real, c, most intermediate models do no satisfy choice. In 
fact, most intermediate models to L[c] are not even definable.

The key part of the proof is the Bristol model, which is intermediate to L[c], 
but is not constructible from a set. We will give a broad explanation of the 
construction of the Bristol model within the constraints of time.

Gallagher's theorem is a strengthening of the Littlewood conjecture that holds for almost all pairs of real numbers. I'll discuss some recent refinements of Gallagher's theorem, one of which is joint work with Niclas Technau. A key new ingredient is the correspondence between Bohr sets and generalised arithmetic progressions. It is hoped that these are the first steps towards a metric theory of multiplicative diophantine approximation on manifolds. 

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

Here Z is Zermelo’s set theory of 1908, as later formulated: full separation, but no replacement or collection among its axioms. PROVI was presented in lectures in Cambridge in 2010 and later published with improvements by Nathan Bowler, and is, I claim, the weakest subsystem of ZF to support a recognisable theory of set forcing: PROV is PROVI shorn of its axiom of infinity. The provident sets are the transitive non-empty models of PROV. The talk will begin with a presentation of PROV, and then discuss more recent applications and problems: in particular an answer in the system Z + PROV to a question posed by Eugene Wesley in 1972 will be sketched, and two proofs (fallacious, I hope) of 0 = 1 will be given, one using my slim models of Z and the other applying the Spector–Gandy theorem to certain models of PROVI. These “proofs”, when re-interpreted, supply some arguments of Reverse Mathematics.

Given a "random" polynomial over the integers, it is expected that, with high probability, it is irreducible and has a big Galois group over the rationals. Such results have been long known when the degree is bounded and the coefficients are chosen uniformly at random from some interval, but the case of bounded coefficients and unbounded degree remained open. Very recently, Emmanuel Breuillard and Peter Varju settled the case of bounded coefficients conditionally on the Riemann Hypothesis for certain Dedekind zeta functions. In this talk, I will present unconditional progress towards this problem, joint with Lior Bary-Soroker and Gady Kozma.

Many fluid flows in natural systems are highly complex, with an often beguilingly intricate and confusing detailed structure. Yet, as with many systems, a good deal of insight can be gained by testing the consequences of simple mathematical models that capture the essential physics.  We’ll tour two such problems.  In the summer melt seasons in Greenland, lakes form on the surface of the ice which have been observed to rapidly drain.  The propagation of the meltwater in the subsurface couples the elastic deformation of the ice and, crucially, the flow of water within the deformable subglacial till.  In this case the poroelastic deformation of the till plays a subtle, but crucial, role in routing the surface meltwater which spreads indefinitely, and has implications for how we think about large-scale motion in groundwater aquifers or geological carbon storage.  In contrast, when magma erupts onto the Earth’s surface it flows before rapidly cooling and crystallising.  Using analogies from the kitchen we construct, and experimentally test, a simple model of what sets the ultimate extent of magmatic intrusions on Earth and, as it turns out, on Venus.  The results are delicious!  In both these cases, we see how a simplified mathematical analysis provides insight into large scale phenomena.

Since differentiation generally lowers exponents, it is straightforward that the space of Laurent polynomials $\mathbb{C}[x, x^{-1}]$ is a finitely generated module over the ring of differential operators $\mathbb{C}[x, \mathrm{d}/\mathrm{d}x]$. This innocent looking fact has been vastly generalized to a statement about holonomic D-modules, using the beautiful theory of b-functions (or Bernstein—Sato polynomials). I will give an overview of the classical theory before discussing some recent developments concerning a $p$-adic analytic analogue, which is joint work with Thomas Bitoun.

When the linear system in Newton’s method is approximately solved using an iterative method we have an inexact or truncated Newton method. The outer method is Newton’s method and the inner iterations will be the iterative method. The Inexact Newton framework is now close to 30 years old and is widely used and given names like Newton-Arnoldi, Newton-CG depending on the inner iterative method. In this talk we will explore convergence properties when the outer iterative method is Gauss-Newton, the Halley method or an interior point method for linear programming problems.

In this talk, which is based on joint work with Benjamin Gess, I will describe a pathwise well-posedness theory for stochastic porous media and fast diffusion equations driven by nonlinear, conservative noise. Such equations arise in the theory of mean field games, as an approximation to the Dean–Kawasaki equation in fluctuating hydrodynamics, to describe the fluctuating hydrodynamics of a zero range process, and as a model for the evolution of a thin film in the regime of negligible surface tension.  Our methods are motivated by the theory of stochastic viscosity solutions, which are applied after passing to the equation’s kinetic formulation, for which the noise enters linearly and can be inverted using the theory of rough paths.  I will also mention the application of these methods to nonlinear diffusion equations with linear, multiplicative noise.

When groups may be built up as graphs of 'simpler' groups, it is often 
of interest to study how good residual finiteness properties of simpler 
groups can imply residual properties of the whole. The essential case of 
this theory is the study of residual properties of finite groups. In 
this talk I will discuss the question of when a graph of finite 
$p$-groups is residually $p$-finite, for $p$ a prime. I describe the 
previous theorems in this area for one-edge and finite graphs of groups, 
and their method of proof. I will then state my recent generalisation of 
these theorems to potentially infinite graphs of groups, together with 
an alternative and more natural method of proof. Finally I will briefly 
describe a usage of these results in the study of accessibility -- 
namely the existence of a finitely generated inaccessible group which is 
residually $p$-finite.

The Helmholtz equation models waves propagating with a fixed frequency. Discretising the Helmholtz equation for high frequencies via standard finite-elements results in linear systems that are large, non-Hermitian, and indefinite. Therefore, when solving these linear systems, one uses preconditioned iterative methods. When one considers uncertainty quantification for the Helmholtz equation, one will typically need to solve many (thousands) of linear systems corresponding to different realisations of the coefficients. At face value, this will require the computation of many preconditioners, a potentially expensive task.

Therefore, we investigate how well a preconditioner for one realisation of the Helmholtz equation works as a preconditioner for another realisation. We prove that if the two realisations are 'nearby' (with a precise meaning of 'nearby'), then the preconditioner is robust (that is, preconditioned GMRES converges in a number of iterations that is independent of frequency). We also give some preliminary computational results indicating the speedup one obtains in uncertainty quantification calculations.

A well-known conjecture in computer science and statistical physics is that Glauber dynamics on the set of k-colorings of a graph G on n vertices with maximum degree \Delta is rapidly mixing for k \ge \Delta+2. In 1999, Vigoda showed rapid mixing of flip dynamics with certain flip parameters on the set of proper k-colorings for k > (11/6)\Delta, implying rapid mixing for Glauber dynamics. In this paper, we obtain the first improvement beyond the (11/6)\Delta barrier for general graphs by showing rapid mixing for k > (11/6 - \eta)\Delta for some positive constant \eta. The key to our proof is combining path coupling with a new kind of metric that incorporates a count of the extremal configurations of the chain. Additionally, our results extend to list coloring, a widely studied generalization of coloring. Combined, these results answer two open questions from Frieze and Vigoda’s 2007 survey paper on Glauber dynamics for colorings. 

This is joint work with Michelle Delcourt and Luke Postle.

The focus of this talk is how to tackle huge linear least square problems via sketching, a dimensionality reduction technique from randomised numerical linear algebra. The technique allows us to project the huge problem to a smaller dimension that captures essential information of the original problem. We can then solve the projected problem directly to obtain a low accuracy solution or using the projected problem to construct a preconditioner for the original problem to obtain a high accuracy solution. I will survey the existing projection techniques and evaluate the performance of sketching for linear least square problems by comparing it to the state-of-the-art traditional solution methods. More than ten-fold speed-up has been observed in some cases.

There have been two main attempts so far to forecast the level of development of artificial intelligence (or ‘computerisation’) over time, Frey and Osborne (2013, 2017) and Manyika et al (2017). Unfortunately, their methodology seems to be flawed. Their results depend upon expert predictions of which occupations will be automatable in 2050, but these predictions are notoriously unreliable. Therefore, we develop an alternative which does not depend upon these expert predictions. We build a dataset of all the start-ups, firms, and university research laboratories working on automating different types of tasks, and use this to build a dynamic network model of them and how they interact. How automatable each type of task is ‘emerges’ from the model. We validate it, predicting the level of development of supervised learning in 2017 using data from the year 2000, and use it to forecast of the automatability of each of these task types from 2018 to 2050. Finally, we discuss extensions for our model; how it could be used to test the impact of public policy decisions or forecast developments in other high-technology industries.

Progress in developing a consistent theory that describes physical phenomena
at scales where quantum and general relativistic effects are large is
hindered by the lack of experiments. In this talk, we present a proposal
that would overcome this experimental obstacle by using a Bose-Einstein
condensate (BEC) to test for possible conflicts between quantum theory and
general relativity. Recent developments in large BEC systems allows us to
verify if gravitationally-induced wave function collapse occurs at the
timescales predicted by Roger Penrose. BECs with high particle numbers
(N>10^9) can also be used to demonstrate quantum field theory in curved
spacetime by observing how changes in the spacetime affect the phononic
quantum field of a BEC. These effects will enable the development of a new
generation of instruments that will be able to probe scales where new
physics might emerge, with applications including gravitational wave
detectors, gravimeters, gradiometers and dark energy probes.