Past

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.

Polynomials have been at the heart of mathematics for a millennium, yet when it comes to applying them, there are many puzzles and surprises. Among others, our tour will visit Newton, Lagrange, Gauss, Galois, Runge, Bernstein, Clenshaw and Chebfun (with a computer demo).

Deep generative models provide powerful tools for fitting difficult distributions such as modelling natural images. But many of these methods, including  variational autoencoders (VAEs) and generative adversarial networks (GANs), can be notoriously difficult to fit.

One well-known problem is mode collapse, which means that models can learn to characterize only a few modes of the true distribution. To address this, we introduce VEEGAN, which features a reconstructor network, reversing the action of the generator by mapping from data to noise. Our training objective retains the original asymptotic consistency guarantee of GANs, and can be interpreted as a novel autoencoder loss over the noise.

Second, maximum mean discrepancy networks (MMD-nets) avoid some of the pathologies of GANs, but have not been able to match their performance. We present a new method of training MMD-nets, based on mapping the data into a lower dimensional space, in which MMD training can be more effective. We call these networks Ratio-based MMD Nets, and show that somewhat mysteriously, they have dramatically better performance than MMD nets.

A final problem is deciding how many latent components are necessary for a deep generative model to fit a certain data set. We present a nonparametric Bayesian approach to this problem, based on defining a (potentially) infinitely wide deep generative model. Fitting this model is possible by combining variational inference with a Monte Carlo method from statistical physics called Russian roulette sampling. Perhaps surprisingly, we find that this modification helps with the mode collapse problem as well.

I will talk about recent work with Jack Thorne in which we find the average size of the Selmer group for a family of genus-2 curves by analyzing a graded Lie algebra of type E_8. I will focus on the role representation theory plays in our proofs.

Even when confronted with the same data, agents often disagree on a model of the real-world. Here, we address the question of how interacting heterogenous agents, who disagree on what model the real-world follows, optimize their trading actions. The market has latent factors that drive prices, and agents account for the permanent impact they have on prices. This leads to a large stochastic game, where each agents' performance criteria is computed under a different probability measure. We analyse the mean-field game (MFG) limit of the stochastic game and show that the Nash equilibria is given by the solution to a non-standard vector-valued forward-backward stochastic differential equation. Under some mild assumptions, we construct the solution in terms of expectations of the filtered states. We prove the MFG strategy forms an \epsilon-Nash equilibrium for the finite player game. Lastly, we present a least-squares Monte Carlo based algorithm for computing the optimal control and illustrate the results through simulation in market where agents disagree on the model.
[ joint work with Philippe Casgrain, U. Toronto ]
 

Standard representation theory transforms groups=algebra into vector spaces = (linear) algebra. The modern approach, geometric representation theory constructs geometric objects from algebra and captures various algebraic representations through geometric gadgets/invariants on these objects. This field started with celebrated Borel-Weil-Bott and Beilinson-Bernstein theorems but equally is in rapid expansion nowadays. I will start from the very beginnings of this field and try to get to the recent developments (time permitting).

We consider a non-linear PDE on $\mathbb R^d$ with a distributional coefficient in the non-linear term. The distribution is an element of a Besov space with negative regularity and the non-linearity is of quadratic type in the gradient of the unknown. Under suitable conditions on the parameters we prove local existence and uniqueness of a mild solution to the PDE, and investigate properties like continuity with respect to the initial condition. To conclude we consider an application of the PDE to stochastic analysis, in particular to a class of non-linear backward stochastic differential equations with distributional drivers.

Let $G$ be a group which splits as $G = F_n * G_1 *...*G_k$, where every $G_i$ is freely indecomposable and not isomorphic to the group of integers.  Guirardel and Levitt generalised the Culler- Vogtmann Outer space of a free group by introducing an Outer space for $G$ as above, on which $\text{Out}(G)$ acts by isometries. Francaviglia and Martino introduced the Lipschitz metric for the Culler- Vogtmann space and later for the general Outer space. In a joint paper with Francaviglia and Martino, we prove that the group of isometries of the Outer space corresponding to $G$ , with respect to the Lipschitz metric, is exactly $\text{Out}(G)$. In this talk, we will describe the construction of the general Outer space and the corresponding Lipschitz metric in order to present the result about the isometries.