Past

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). 
 

This work is devoted to the study of the following boundary value problem for compressible Navier-Stokes equations\begin{align*}&\begin{aligned}[b] \partial_t(\varrho \mathbf{u})+\text{div}(\varrho \mathbf u\otimes\mathbf u) &+\nabla p(\rho)\\&= \text{div} \mathbb S(\mathbf u)+\varrho\, \mathbf f\quad\text{ in }\Omega\times (0,T),\end{aligned} \\[6pt]&\partial_t\varrho+\text{div}(\varrho \mathbf u)=0\quad\text{ in }\Omega\times (0,T), \\[6pt]&\begin{aligned}[c] &\mathbf u=0\quad\text{ on }\partial \Omega\times( 0,T), \\ &\mathbf u(x,0)=\mathbf u_0(x)\quad\text{ in } \Omega,\\&\varrho(x,0)= \varrho_0(x) \quad\text{ in } \Omega, \end{aligned}\end{align*} where $\Omega$ is a bounded domain in $\mathbb R^d$, $d=2,3$, $\varrho_0>0$, $\mathbf u_0$, $\mathbf f$ are given functions, $p(\varrho)=\varrho^\gamma$, $\mathbb S(\mathbf u)=\mu(\nabla\mathbf{u}+\nabla\mathbf{u}^\top)+\lambda \text{div } \mathbf{u}$, $\mu, \lambda$ are positive constants. We consider the endpoint cases $\gamma=3/2$, $d=3$ and $\gamma=1$, $d=2$, when the energy estimate does not guarantee the integrability of the kinetic energy density with an exponent greater than 1, which leads to the so-called concentration problem. In order to cope with this difficulty we develop new approach to the problem. Our method is based on the estimates of the Newton potential of $p(\varrho)$. We prove that the kinetic energy density in 3-dimensional problem with $\gamma=3/2$ is bounded in $L\log L^\alpha$ Orlitz space and obtain new estimates for the pressure function. In the case $d=2$ and $\gamma=1$ we prove the existence of the weak solution to the problem. We also discuss the structure of concentrations for rotationally-symmetric and stationary solutions.

The concept of an acylindrically hyperbolic group, introduced by D. Osin, generalizes hyperbolic and relatively hyperbolic groups, and includes many other groups of interest: Out(F_n), n>1, most mapping class groups, directly indecomposable non-cyclic right angled Artin groups, most graph products, groups of deficiency at least 2, etc. Roughly speaking, a group G is acylindrically hyperbolic if there is a (possibly infinite) generating set X of G such that the Cayley graph \Gamma(G,X) is hyperbolic and the action of G on it is "sufficiently nice". Many global properties of hyperbolic/relatively hyperbolic groups have been also proved for acylindrically hyperbolic groups. 
In the talk I will discuss a method which allows to construct a common acylindrically hyperbolic quotient for any countable family of countable acylindrically hyperbolic groups. This allows us to produce acylindrically hyperbolic groups with many unexpected properties.(The talk will be based on joint work with Denis Osin.)
 

Backward Stochastic Differential Equations (BSDEs) provide a systematic way to obtain Feynman-Kac formulas for linear as well as nonlinear partial differential equations (PDEs) of parabolic and elliptic type, and the numerical approximation of their solutions thus provide Monte-Carlo methods for PDEs. BSDEs are also used to describe the solution of path-dependent stochastic control problems, and they further arise in many areas of mathematical finance. 

In this talk, I will discuss the numerical approximation of BSDEs when the nonlinear driver is not Lipschitz, but instead has polynomial growth and satisfies a monotonicity condition. The time-discretization is a crucial step, as it determines whether the full numerical scheme is stable or not. Unlike for Lipschitz driver, while the implicit Bouchard-Touzi-Zhang scheme is stable, the explicit one is not and explodes in general. I will then present a number of remedies that allow to recover a stable scheme, while benefiting from the reduced computational cost of an explicit scheme. I will also discuss the issue of numerical stability and the qualitative correctness which is enjoyed by both the implicit scheme and the modified explicit schemes. Finally, I will discuss the approximation of the expectations involved in the full numerical scheme, and their analysis when using a quasi-Monte Carlo method.

By viewing a stochastic process as a random variable taking values in a path space, the support of its law describes the set of all attainable paths. In this talk, we show that the support of the law of a solution to a path-dependent stochastic differential equation is given by the image of the Cameron-Martin space under the flow of mild solutions to path-dependent ordinary differential equations, constructed by means of the vertical derivative of the diffusion coefficient. This result is based on joint work with Rama Cont and extends the Stroock-Varadhan support theorem for diffusion processes to the path-dependent case.

I will discuss results relating different partially wrapped Fukaya categories.  These include a K\"unneth formula, a `stop removal' result relating partially wrapped Fukaya categories relative to different stops, and a gluing formula for wrapped Fukaya categories.  The techniques also lead to generation results for Weinstein manifolds and for Lefschetz fibrations.  The methods are mainly geometric, and the key underlying Floer theoretic fact is an exact triangle in the Fukaya category associated to Lagrangian surgery along a short Reeb chord at infinity.  This is joint work with Sheel Ganatra and Vivek Shende.

Costello Yamazaki and Witten have proposed a new understanding of quantum integrable systems coming from a variant of Chern-Simons theory living on a product of two Riemann surfaces. I’ll review their work, and show how it can be extended to the case of integrable systems with boundary. The boundary Yang-Baxter Equations, twisted Yangians and Sklyanin determinants all have natural interpretations in terms of line operators in the theory.