Past

The Poincare Conjecture was first formulated over a century ago and states that there is only one closed simply connected 3-manifold, hinting at a link between 3-manifolds and their fundamental groups. This seemingly basic fact went unproven until the early 2000s when Perelman proved Thurston's much more powerful Geometrisation Conjecture, providing us with a powerful structure theorem for understanding all closed 3-manifolds.
In this talk I will introduce the results developed throughout the 20th century that lead to Thurston and Perelman's work. Then, using Geometrisation as a black box, I will present a proof of the Poincare Conjecture. Throughout we shall follow the crucial role that the fundamental group plays and hopefully demonstrate the geometric and group theoretical nature of much of the modern study of 3-manifolds.
As the title suggests, no prior understanding of 3-manifolds will be expected.
 

The Kronecker-Weber theorem states that every finite abelian extension of the rationals is contained in some cyclotomic field. I will present a proof that emphasizes the standard local-global philosophy by first proving it for the p-adics and then deducing the global case.

Many time-dependent PDEs -- KdV, Burgers, Gray-Scott, Allen-Cahn, Navier-Stokes and many others -- combine a higher-order linear term with a lower-order nonlinear term.  This talk will review the method of exponential integrators for solving such problems with better than 2nd-order accuracy in time.

A defect-based a posteriori error estimate for Krylov subspace approximations to the matrix exponential is introduced. This error estimate constitutes an upper norm bound on the error and can be computed during the construction of the Krylov subspace with nearly no computational effort. The matrix exponential function itself can be understood as a time propagation with restarts. In practice, we are interested in finding time steps for which the error of the Krylov subspace approximation is smaller than a given tolerance. Finding correct time steps is a simple task with our error estimate. Apart from step size control, the upper error bound can be used on the fly to test if the dimension of the Krylov subspace is already sufficiently large to solve the problem in a single time step with the required accuracy.

I will review the idea that entanglement must ultimately be understood in terms of modes, rather than in terms of particles. The most striking instance of mode entanglement is a single particle entangled state, which I will discuss both in the case of bosons as well as in the case of fermions. I then proceed to show that the Aharonov-Bohm effect can be understood by using a single electron entangled state. Finally, I will argue that this demonstrates beyond doubt that the Aharonov-Bohm effect is non non-local, contrary to what is frequently claimed in the literature.

The advent of social media and the resulting ability to instantaneously communicate ideas and messages to connections worldwide is one of the great consequences arising from the telecommunications revolution over the last century. Individuals do not, however, communicate only upon a single platform; instead there exists a plethora of options available to users, many of whom are active on a number of such media. While each platform offers some unique selling point to attract users, e.g., keeping up to date with friends through messaging and statuses (Facebook), photo sharing (Instagram), seeing information from friends, celebrities and numerous other outlets (Twitter) or keeping track of the career paths of friends and past colleagues (Linkedin), the platforms are all based upon the fundamental mechanisms of connecting with other users and transmitting information to them as a result of this link.

In this talk a model for the spreading of online information or “memes" on multiplex networks is introduced and analyzed using branching-process methods. The model generalizes that of [Gleeson et al., Phys. Rev. X., 2016] in two ways. First, even for a monoplex (single-layer) network, the model is defined for any specific network defined by its adjacency matrix, instead of being restricted to an ensemble of random networks. Second, a multiplex version of the model is introduced to capture the behavior of users who post information from one social media platform to another. In both cases the branching process analysis demonstrates that the dynamical system is, in the limit of low innovation, poised near a critical point, which is known to lead to heavy-tailed distributions of meme popularity similar to those observed in empirical data.

[1] J. P. Gleeson et al. “Effects of network structure, competition and memory time on social spreading phenomena”. Physical Review X 6.2 (2016), p. 021019.

[2] J. D. O’Brien et al. "Spreading of memes on multiplex networks." New Journal of Physics 21.2 (2019): 025001.

For measuring families of curves, or, more generally, of measures, $M_p$-modulus is traditionally used. More recent studies use so-called plans on measures. In their fundamental paper Ambrosio, Di Marino and Savare proved that these two approaches are in some sense equivalent within $1<p<\infty$. We consider the limiting case $p=1$ and show that the $AM$-modulus can be obtained alternatively by the plan approach. On the way, we demonstrate unexpected behavior of the $AM$-modulus in comparison with usual capacities.

This is a joint work with Vendula Honzlov\'a Exnerov\'a, Ond\v{r}ej F.K. Kalenda and Olli Martio. Partially supported by the grant GA\,\v{C}R P201/18-07996S of the Czech Science Foundation.

X.S. Lin defined an invariant of knots in S^3 by counting represenations 
of the knot group into SU(2) with fixed meridinal holonomy. Lin's 
invariant was subsequently shown to coincide with the Levine-Tristam 
signature. I'll define an analogous total Lin invariant which counts 
repesentations into both SU(2) and SL_2(R). Unlike the SU(2) version, this 
invariant does not (as far as I know) coincide with other known 
invariants. I'll describe some applications to left-orderability of Dehn 
fillings and branched covers, as well as a curious connection with the 
Alexander polynomial. This is joint work with Nathan Dunfield.

We observe a noisy version of a large rank-one matrix. Depending on the strength of the noise, can we recover non-trivial information on the matrix? This problem, interesting on its own, will be motivated by its link with a "spin glass" model, which is a model of statistical mechanics where a large number of variables interact with one another, with random interactions that can be positive or negative. The resolution of the initial question will involve a Hamilton-Jacobi equation

I will discuss new results for the gradient field models with uniformly convex potential (also known as the Ginzburg-Landau field). A connection between the scaling limits of the field and elliptic homogenization was introduced by Naddaf and Spencer in 1997. We quantify the existing central limit theorems in light of recent advances in quantitative homogenization; and positively settle a conjecture of Funaki and Spohn about the surface tension. Joint work with Scott Armstrong. 

I will describe joint work with Bruno Premoselli which gives a new existence theorem for negatively curved Einstein 4-manifolds, which are obtained by smoothing the singularities of hyperbolic cone metrics. Let (M_k) be a sequence of compact 4-manifolds and let g_k be a hyperbolic cone metric on M_k with cone angle \alpha (independent of k) along a smooth surface S_k. We make the following assumptions:

1. The injectivity radius i(k) of M_k tends to infinity (where in defining injectivity radius we ignore those geodesics which hit the cone singularity)

2. The normal injectivity radius of S_k is at least i(k)/2.

3. The area of the singular locii satisfy A(S_k)\leq C \exp(5 i(k)/2) for some C independent of k.

When these assumptions hold, we prove that for all large k, M_k carries a smooth Einstein metric of negative curvature. The proof involves a gluing theorem and a parameter dependent implicit function theorem (where k is the parameter). As I will explain, negative curvature plays an essential role in the proof. (For those who may be aware of our arxiv preprint, https://arxiv.org/abs/1802.00608 [arxiv.org], the work
I will describe has a new feature, namely we now treat all cone angles, and not just those which are greater than 2\pi. This gives lots more examples of Einstein 4-manifolds.)

How do you create a self-sustaining, flourishing academic community in a developing country? What kind of challenges need to be overcome to ensure that quality education becomes available? What can we do to help make it happen? In this talk, we will describe our experience visiting the University of Yangon in Myanmar. During the visit, we delivered a course to the academic staff, and discussed future collaborations between Oxford and Yangon, as well as further directions for Mathematical education in Myanmar, all the while marvelling at the wonders of the Burmese culture.

I'll talk about smooth solutions of Euler equations with compactly supported velocities, and applications to other equations.

In this talk I want to outline the proofs our of main results, i.e. the localisation theorem and the identification of the homotopy type of Grothendieck-Witt theory in terms of K- and L-theory.
Finally, as a small application I want to present a refinement and extension of certain maps relating certain Madsen-Tillmann spectra and orthogonal/symplectic algebraic K-theory spectra of the integers.

All original material is joint work with B.Calmès, E.Dotto, Y.Harpaz, M.Land, K.Moi, D.Nardin, T.Nikolaus and W.Steimle.
 

I will start by briefly reviewing the Tate construction and in particular, the Tate diagonal. Using these I will then illustrate Lurie’s notion of Poincaré categories by considering Poincaré structures on module categories over a ring (spectrum) in detail. In particular, I will describe the somewhat subtle genuine Poincaré structure on the category of perfect complexes of an ordinary ring, which conjecturally links the classical notion of the Grothendieck-Witt spectrum to our derived version. Finally, I will compute its associated L-groups.

I will explain how Lurie‘s approach to L-theory via Poincaré categories can be extended to yield cobordism categories of Poincaré objects à la Ranicki. These categories can be delooped by an iterated Q-construction and the resulting spectrum is a derived version of Grothendieck-Witt-theory.  Its homotopy type can be described in terms of K- and L-theory as conjectured by Hesselholt-Madsen. Furthermore, it has a clean universal property analogous to that of K-theory, localisation sequences in much greater generality than classical Grothendieck-Witt theory, gives a cycle description of Weiss-Williams‘ LA-theory and allows for maps from the geometric cobordism category, refining and unifying various known invariants.

All original material is joint work with B.Calmès, E.Dotto, Y.Harpaz, M.Land, K.Moi, D.Nardin, T.Nikolaus and W.Steimle.

The large-scale features of groups and spaces are recorded by asymptotic invariants. Examples of asymptotic invariants are the asymptotic cone and, for hyperbolic groups, the Gromov boundary.
In his study of asymptotic cones of connected Lie groups, Yves Cornulier introduced a class of maps called sublinearly biLipschitz equivalences. Like the more traditionnal quasiisometries, sublinearly biLipschitz equivalences are biLipschitz on the large-scale, but unlike quasiisometries, they are generally not coarse. Sublinearly biLipschitz equivalences still induce biLipschitz homeomorphisms between asymptotic cones. In this talk, I will focus on Gromov-hyperbolic groups and show how the Gromov boundary can be used to produce invariants distinguishing them up to sublinearly biLipschitz equivalences when the asymptotic cones do not. I will especially give applications to the large-scale sublinear geometry of hyperbolic Lie groups.
 

Jacob Bernoulli is known for his studies of the curves, infinitesimal math- ematics and statistics. However, before being a professor in mathematics, he taught experimental physics at the University of Basel. This explains his high interest in solving physical problems with newly developed Leibnizian calculus. In his scientific notebook, Meditationes, there are more than thirty notes about various mechanical problems for solving of which Bernoulli has applied Leibnizian calculus and has advanced this method along the way. A discussion with a craftsman brought Bernoulli’s attention to the problem of the strength of a beam early in his career and occupied his mind until his death. The craftsman’s narration based on his experience highlighted the flaws in Galilean-Leibnizian theory of the strength of a beam. This was the starting point of Bernoulli’s quest to mathematically find the profile of a bent beam (the Elastica Problem) and the physical laws governing it. He started a challenge to encourage other mathematicians of the time to study the problem, providing a hint hidden in an anagram. Although he published his solution of the Elastica Problem in 1694, that was not the end of the quest for him. Studying his unpublished notes in Meditationes reveals that over the last decade of his life, Bernoulli has reconsidered the problem. In my project, I demonstrate that he has found remarkable concepts such as mean tensile stress, and the notion of local stress-strain relation, etc.

In my talk I will discuss modular properties of false theta functions. Due to a wrong sign factor these are not directly seen to be modular, however there are ways to repair this. I will report about this in my talk.

Energy transfers from small-scale turbulence and waves to large-scale flows are ubiquituous in oceans, atmospheres, planetary cores and stars.

Therefore, turbulence and waves have a direct effect on the large-scale organization of geophysical and astrophysical fluids and can affect their long-term dynamics.

In this talk I will discuss recent direct numerical simulation (DNS) results of two upscale energy transfer mechanisms that emerge from the dynamics of a fluid that is self-organized in a turbulent layer next to a stably-stratified one. This self-organization in an adjacent "two-layer" turbulent-stratified system is ubiquituous in nature and is representative of e.g. Earth's troposphere-stratosphere system, the oceans' surface mixed layer-thermocline system, and stars' convective-radiative interiors. The first set of DNS results will demonstrate how turbulent motions can generate internal waves, which then force a slowly-reversing large-scale flow, akin to Earth's Quasi-Biennial Oscillation (QBO). The second set of DNS results will show how the stratified layer regulates the emergence of large-scale vortices (LSV) in the turbulent layer under rapid rotation in the regime known as geostrophic turbulence. I will demonstrate why it is important to resolve both the turbulence and the waves, as otherwise the natural variability of the QBO is lost and LSV cannot form. I will discuss future works and highlight how the results may guide the implementation of upscale energy transfers in global earth system models.

The mechanisms underlying the initiation and perpetuation of cardiac arrhythmias are inherently multi-scale: whereas arrhythmias are intrinsically tissue-level phenomena, they have a significant dependence cellular electrophysiological factors. Spontaneous sub-cellular calcium release events (SCRE), such as calcium waves, are a exemplars of the multi-scale nature of cardiac arrhythmias: stochastic dynamics at the nanometre-scale can influence tissue excitation  patterns at the centimetre scale, as triggered action potentials may elicit focal excitations. This latter mechanism has been long proposed to underlie, in particular, the initiation of rapid arrhythmias such as tachycardia and fibrillation, yet systematic analysis of this mechanism has yet to be fully explored. Moreover, potential bi-directional coupling has been seldom explored even in concept.

A major challenge of dissecting the role and importance of SCRE in cardiac arrhythmias is that of simultaneously exploring sub-cellular and tissue function experimentally. Computational modelling provides a potential approach to perform such analysis, but requires new techniques to be employed to practically simulate sub-cellular stochastic events in tissue-scale models comprising thousands or millions of coupled cells.

This presentation will outline the novel techniques developed to achieve this aim, and explore preliminary studies investigating the mechanisms and importance of SCRE in tissue-scale arrhythmia: How do independent, small-scale sub-cellular events overcome electrotonic load and manifest as a focal excitation? How can SCRE focal (and non-focal) dynamics lead to re-entrant excitation? How does long-term re-entrant excitation interact with SCRE to perpetuate and degenerate arrhythmia?

A defining feature of robotics today is the use of learning and autonomy in the inner loop of systems that are actually being deployed in the real world, e.g., in autonomous driving or medical robotics. While it is clear that useful autonomous systems must learn to cope with a dynamic environment, requiring architectures that address the richness of the worlds in which such robots must operate, it is also equally clear that ensuring the safety of such systems is the single biggest obstacle preventing scaling up of these solutions. I will discuss an approach to system design that aims at addressing this problem by incorporating programmatic structure in the network architectures being used for policy learning. I will discuss results from two projects in this direction.

Firstly, I will present the perceptor gradients algorithm – a novel approach to learning symbolic representations based on the idea of decomposing an agent’s policy into i) a perceptor network extracting symbols from raw observation data and ii) a task encoding program which maps the input symbols to output actions. We show that the proposed algorithm is able to learn representations that can be directly fed into a Linear-Quadratic Regulator (LQR) or a general purpose A* planner. Our experimental results confirm that the perceptor gradients algorithm is able to efficiently learn transferable symbolic representations as well as generate new observations according to a semantically meaningful specification.

Next, I will describe work on learning from demonstration where the task representation is that of hybrid control systems, with emphasis on extracting models that are explicitly verifi able and easily interpreted by robot operators. Through an architecture that goes from the sensorimotor level involving fitting a sequence of controllers using sequential importance sampling under a generative switching proportional controller task model, to higher level modules that are able to induce a program for a visuomotor reaching task involving loops and conditionals from a single demonstration, we show how a robot can learn tasks such as tower building in a manner that is interpretable and eventually verifiable.

References:

1. S.V. Penkov, S. Ramamoorthy, Learning programmatically structured representations with preceptor gradients, In Proc. International Conference on Learning Representations (ICLR), 2019. http://rad.inf.ed.ac.uk/data/publications/2019/penkov2019learning.pdf

2. M. Burke, S.V. Penkov, S. Ramamoorthy, From explanation to synthesis: Compositional program induction for learning from demonstration, https://arxiv.org/abs/1902.10657
 

The well known (and notoriously hard) P vs NP problem asks whether every Boolean function with polynomial-size proofs is also computable in
polynomial time.

The standard approach to the P vs NP problem is via circuit complexity. For progressively richer classes of Boolean circuits (networks of AND, OR and NOT
logic gates), one wishes to show super-polynomial lower bounds on the sizes of circuits (as a function of the size of the input) computing some Boolean
function known to be in NP, such as the Satisfiability problem.

However, there is a more logic-oriented approach initiated by Cook and Reckhow, going through proof complexity rather than circuit complexity. For
progressively richer proof systems, one wishes to show super-polynomial lower bounds on the sizes of proofs (as a function of the size of the tautology) of
some sequence of propositional tautologies.

I will give a brief overview on known results along these two directions, and on their limitations. Somewhat surprisingly, similar techniques have been found
to be useful for these seemingly different approaches. I will say something about known connections between the approaches, and pose the question of
whether there are deeper connections.

Finally, I will discuss how the perspective of proof complexity can be used to formalize the difficulty of proving lower bounds on the sizes of computations
(or of proofs).

 

After a brief introduction to the theory of transcendental numbers, I will discuss Nesterenko's 1996 celebrated theorem on the algebraic independence of values of Eisenstein series, and some related open problems. This motivates the second part of the talk, in which I will report on a recent geometric generalization of Nesterenko's method.

In this talk, I will sketch a geometrically flavoured proof of the 
Madsen-Weiss theorem based on work by Eliashberg-Galatius-Mishachev.
In order to prove the triviality of appropriate relative bordism groups, 
in a first step a variant of the wrinkling theorem shows
that one can reduce to consider fold maps (with additional structure). 
In a subsequent step, a geometric version of the Harer stability
theorem is used to get rid of the folds via surgery. I will focus on 
this second step.

Transformation theory has long been known to be a mechanism for 
the design of metamaterials. It gives rise to the required properties of the 
material in order to direct waves in the manner desired.  This talk will 
focus on the mathematical theory underpinning the design of acoustic and 
elastodynamic metamaterials based on transformation theory and aspects of 
the experimental confirmation of these designs. In the acoustics context it 
is well-known that the governing equations are transformation invariant and 
therefore a whole range of microstructural options are available for design, 
although designing materials that can harness incoming acoustic energy in 
air is difficult due to the usual sharp impedance contrast between air and 
the metamaterial in question. In the elastodynamic context matters become 
even worse in the sense that the governing equations are not transformation 
invariant and therefore we generally require a whole new class of materials.

In the acoustics context we will describe a new microstructure that consists 
of rigid rods that is (i) closely impedance matched to air and (ii) slows 
down sound in air. This is shown to be useful in a number of configurations 
and in particular it can be employed to half the resonant frequency of the 
standard quarter-wavelength resonator (or alternatively it can half the size 
of the resonator for a specified resonant frequency) [1].

In the elastodynamics context we will show that although the equations are 
not transformation invariant one can employ the theory of waves in 
pre-stressed hyperelastic materials in order to create natural elastodynamic 
metamaterials whose inhomogeneous anisotropic material properties are 
generated naturally by an appropriate pre-stress. In particular it is shown 
that a certain class of hyperelastic materials exhibit this so-called 
“invariance property” permitting the creation of e.g. hyperelastic cloaks 
[2,3] and invariant metamaterials. This has significant consequences for the 
design of e.g. phononic media: it is a well-known and frequently exploited 
fact that pre-stress and large deformation of hyperelastic materials 
modifies the linear elastic wave speed in the deformed medium. In the 
context of periodic materials this renders materials whose dynamic 
properties are “tunable” under pre-stress and in particular this permits 
tunable band gaps in periodic media [4]. However the invariant hyperelastic 
materials described above can be employed in order to design a class of 
phononic media whose band-gaps are invariant to deformation [5]. We also 
describe the concept of an elastodynamic ground cloak created via pre-stress 
[6].

[1] Rowley, W.D., Parnell, W.J., Abrahams, I.D., Voisey, S.R. and Etaix, N. 
(2018) “Deepening subwavelength acoustic resonance via metamaterials with 
universal broadband elliptical microstructure”. Applied Physics Letters 112, 
251902.
[2] Parnell, W.J. (2012) “Nonlinear pre-stress for cloaking from antiplane 
elastic waves”. Proc Roy Soc A 468 (2138) 563-580.
[3] Norris, A.N. and Parnell, W.J. (2012) “Hyperelastic cloaking theory: 
transformation elasticity with pre-stressed solids”. Proc Roy Soc A 468 
(2146) 2881-2903
[4] Bertoldi, K. and Boyce, M.C. (2008)  “Mechanically triggered 
transformations of phononic band gaps in periodic elastomeric structures”. 
Phys Rev B 77, 052105.
[5] Zhang, P. and Parnell, W.J. (2017) “Soft phononic crystals with 
deformation-independent band gaps” Proc Roy Soc A 473, 20160865.
[6] Zhang, P. and Parnell, W.J. (2018) “Hyperelastic antiplane ground 
cloaking” J Acoust Soc America 143 (5)

We consider a price-mediated contagion framework in which each bank, after an exogenous shock, may have to sell assets in order to comply with regulatory constraints. Interaction between banks takes place only through price impact. We characterize the equilibrium of the strategic deleveraging problem and we calibrate our model to publicly-available data, the US banks that were part of the 2015 regulatory stress-tests. We then consider a more sophisticated model in which each bank is exposed to two risky assets (marketable and not marketable) and is only able to sell the marketable asset. We calibrate our model using the six banks with significant trading operations and we show that, depending on the price impact, the contagion of failures may be significant. Our results may be used to refine current stress testing frameworks by incorporating potential contagion mechanisms between banks. This is joint work with Yann Braouezec.

Small block overlapping, and non-overlapping, Schwarz methods are theoretically highly attractive as multilevel smoothers for a wide variety of problems that are not amenable to point relaxation methods.  Examples include monolithic Vanka smoothers for Stokes, overlapping vertex-patch decompositions for $H(\text{div})$ and  $H(\text{curl})$ problems, along with nearly incompressible elasticity, and augmented Lagrangian schemes.

 While it is possible to manually program these different schemes,  their use in general purpose libraries has been held back by a lack   of generic, composable interfaces. We present a new approach to the   specification and development such additive Schwarz methods in PETSc  that cleanly separates the topological space decomposition from the  discretisation and assembly of the equations. Our preconditioner is  flexible enough to support overlapping and non-overlapping additive  Schwarz methods, and can be used to formulate line, and plane smoothers, Vanka iterations, amongst others. I will illustrate these new features with some examples utilising the Firedrake finite element library, in particular how the design of an approriate computational interface enables these schemes to be used as building blocks inside block preconditioners.

This is joint work with Patrick Farrell and Florian Wechsung (Oxford), and Matt Knepley (Buffalo).

We consider a well-studied optimal execution problem under little assumptions on the underlying midprice process. We do so by using signatures from rough path theory, that allows converting the original problem into a more computationally tractable problem. We include a few numerical experiments where we show that our methodology is able to retrieve the theoretical optimal execution speed for several problems studied in the literature, as well as some cases not included in the literatture. We also study some estensions of our framework to other settings.
 

Motivated by recent problems on mixing flows, it is useful to characterize Besov spaces via oscillation of functions (averages) and minimization problems for bounded variation functions (Bianchini-type norms). In this talk, we discuss various descriptions of Besov spaces in terms of different kinds of averages, as well as Bianchini-type norms. Our method relies on the K-functional of the theory of real interpolation. This is a joint work with S. Tikhonov (Barcelona).

Buildings are geometric objects, originally introduced by Tits to study Lie groups that act on their corresponding building. Apart from their significance for Lie groups, buidings and their automorphism groups are a rich source of examples for groups with interesting properties (for example, it is a result of Caprace that some buildings admit an automorphism group which is compactly generated, abstractly simple and locally compact). Right Angled Buildings (RABs) are a specific kind of building whose geometry can be well understood as it resembles the geometry of a tree. This allows one to generalise ideas like the Burger-Mozes universal groups to the setting of RABs.
I plan to give an introduction to RABs. As a complete formal introduction to buildings would take more than an hour, I will instead present various illustrative examples to give you an idea of what you should have in mind when you think of a (right-angled) building. I will be as formal as I can in presenting the basic features of buildings - Coxeter complexes, chambers, apartments, retractions and residues.  In the remaining time I will say as much as I can about the geometry of RABs, and explain how to use this geometry to derive a structure theorem for the automorphism group of a RAB, towards a definition of Burger-Mozes universal groups for RABs.
 

It is known that every continuum X is a weakly confluent image of a continuum Y which is hereditarily indecomposable and of covering dimension one.  We use the ultracoproduct construction to gain information about the number of composants of Y.  For example, in ZFC, we can ensure that this number is arbitrarily large.  And if we assume the GCH, we can arrange for Y to have as many composants at it has points.

I will present some basic concepts in Large Cardinal theory. A Large Cardinal axiom is the assertion of the existence of a cardinal so large that it entails the existence of set-sized models of ZFC, something which we know ZFC alone does not do. Large Cardinal axioms are therefore strengthenings of ZFC. We believe them to be consistent with ZFC, but this is a touchy subject. Nevertheless, Large Cardinal axioms have become an essential tool in set theory, providing insight into the fine structure of the set theoretic universe. In my talk, I will focus on inaccessible and measurable cardinals, and, if time permits, I will discuss supercompact cardinals.

When solving partial differential equations driven by additive spatial white noise, the efficient sampling of white noise realizations can be challenging. In this talk we focus on the efficient sampling of white noise using quasi-random points in a finite element method and multilevel Quasi Monte Carlo (MLQMC) setting. This work is an extension of previous research on white noise sampling for MLMC.

We express white noise as a wavelet series expansion that we divide in two parts. The first part is sampled using quasi-random points and contains a finite number of terms in order of decaying importance to ensure good QMC convergence. The second part is a correction term which is sampled using standard pseudo-random numbers.

We show how the sampling of both terms can be performed in linear time and memory complexity in the number of mesh cells via a supermesh construction. Furthermore, our technique can be used to enforce the MLQMC coupling even in the case of non-nested mesh hierarchies. We demonstrate the efficacy of our method with numerical experiments.

In this talk, we analyze a virtual element method (VEM) for solving a non-selfadjoint fourth-order eigenvalue problem derived from the transmission eigenvalue problem. We write a variational formulation and propose a $C^1$-conforming discretization by means of the VEM. We use the classical approximation theory for compact non-selfadjoint operators to obtain optimal order error estimates for the eigenfunctions and a double order for the eigenvalues. Finally, we present some numerical experiments illustrating the behavior of the virtual scheme on different families of meshes.

The development of an effective method of targeting delivery of stem cells to the site of an injury is a key challenge in regenerative medicine. However, production of stem cells is costly and current delivery methods rely on large doses in order to be effective. Improved targeting through use of an external magnetic field to direct delivery of magnetically-tagged stem cells to the injury site would allow for smaller doses to be used.
We present a model for delivery of stem cells implanted with a fixed number of magnetic nanoparticles under the action of an external magnetic field. We examine the effect of magnet geometry and strength on therapy efficacy. The accuracy of the mathematical model is then verified against experimental data provided by our collaborators at the University of Birmingham.

Six-dimensional theories provide a unification of four-dimensional theories via dimensional reduction  together with access to some of the novel features arising from M-theory.  Ambitwistor strings directly generate S-matrices for massless theories in terms of formulae that localize on the solutions to the scattering equations; algebraic equations that determine n points on the Riemann sphere from n massless momenta.  These are sufficient to provide compact formulae for tree-level S-matrices for bosonic theories. This talk introduces their extension to the polarized scattering equations which arise from twistorial versions on ambitwistor-strings.  These lead to simple explicit formulae for superamplitudes in 6D for super Yang-Mills, supergravity, D5 and M5 branes and massive superamplitudes in 4D.  The framework extends also to 10 and 11 dimensions.  This is based on joint work with Yvonne Geyer, arxiv:1812.05548 and 1901.00134. 

With the introduction of supermarket loyalty cards in recent decades, there has been an ever-growing body of customer-level shopping data. A natural way to represent this data is with a bipartite network, in which customers are connected to products that they purchased. By predicting likely edges in these networks, one can provide personalised product recommendations to customers.
In this talk, I will first discuss a basic approach for recommendations, based on network community detection, that we have validated on a promotional campaign run by our industrial collaborators. I will then describe a multilayer network model that accounts for the fact that customers tend to buy the same grocery items repeatedly over time. By modelling such correlations explicitly, link-prediction accuracy improves considerably. This approach is also useful in other networks that exhibit significant edge correlations, such as social networks (in which people often have repeated interactions with other people), airline networks (in which popular routes are often served by more than one airline), and biological networks (in which, for example, proteins can interact in multiple ways).