Past

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.

In 2010 Coecke, Sadrzadeh, and Clark formulated a new model of natural language which operates by combining the syntactics of grammar and the semantics of individual words to produce a unified ''meaning'' of sentences. This they did by using category theory to understand the component parts of language and to amalgamate the components together to form what they called a ''distributional compositional categorical model of meaning''. In this talk I shall introduce the model of Coecke et. al., and use it to compare the meaning of sentences in Irish and in English (and thus ascertain when a sentence is the translation of another sentence) using a cosine similarity score.

The Irish language is a member of the Gaelic family of languages, originating in Ireland and is the official language of the Republic of Ireland.

Let $p$ be a fixed prime number and let $SCVF_p$ be the theory of separably closed non-trivially valued fields of
characteristic $p$. In the talk, we will see that, in many ways, the step from $ACVF_{p,p}$ to $SCVF_p$ is not more
complicated than the one from $ACF_p$ to $SCF_p$.

At a basic level, this is true for quantifier elimination (Delon), for which it suffices to add parametrized $p$-coordinate
functions to any of the usual languages for valued fields. It follows that all completions are NIP.

At a more sophisticated level, in finite degree of imperfection, when a $p$-basis is named or when one just works with
Hasse derivations, the imaginaries of $SCVF_p$ are not more complicated than the ones in $ACVF_{p,p}$, i.e., they are
classified by the geometric sorts of Haskell-Hrushovski-Macpherson. The latter is proved using prolongations. One may
also use these to characterize the stable part and the stably dominated types in $SCVF_p$, and to show metastability.

I will explain the notion of a singular ring and sketch how singular rings provide field and vertex algebras introduced by Borcherds and Kac. All of these notions make sense in general symmetric monoidal categories and behave nicely with respect to symmetric lax monoidal functors. I will provide a complete classification of singular rings if the tensor product is a cartesian product. This applies in particular to categories of topological spaces or (algebraic) stacks equipped with the usual cartesian product. Moduli spaces provide a rich source of examples of singular rings. By combining these ideas, we obtain vertex and field algebras for each reasonable moduli space and each choice of an orientable homology theory. This generalizes a recent construction of vertex algebras by Dominic Joyce.

Snapshot mosaic multispectral imagery acquires an undersampled data cube by acquiring a single spectral measurement per spatial pixel. Sensors which acquire p frequencies, therefore, suffer from severe 1/p undersampling of the full data cube.  We show that the missing entries can be accurately imputed using non-convex techniques from sparse approximation and matrix completion initialised with traditional demosaicing algorithms.

Formally, the Fourier coefficients of Eisenstein series on Kac-Moody groups contain as yet mysterious automorphic L-functions relevant to open conjectures such as that of Ramanujan and Langlands functoriality. In this talk, we will consider the constant Fourier coefficient, if it even makes sense rigorously, and its relationship to the geometry and combinatorics of a Kac-Moody group. Joint work with Kyu-Hwan Lee.

This talk is concerned with the mathematical and numerical analysis of a steady phase change problem for non-isothermal incompressible viscous flow. The system is formulated in terms of pseudostress, strain rate and velocity for the Navier-Stokes-Brinkman equation, whereas temperature, normal heat flux on the boundary, and an auxiliary unknown are introduced for the energy conservation equation. In addition, and as one of the novelties of our approach, the symmetry of the pseudostress is imposed in an ultra-weak sense, thanks to which the usual introduction of the vorticity as an additional unknown is no longer needed. Then, for the mathematical analysis two variational formulations are proposed, namely mixed-primal and fully-mixed approaches, and the solvability of the resulting coupled formulations is established by combining fixed-point arguments, Sobolev embedding theorems and certain regularity assumptions. We then construct corresponding Galerkin discretizations based on adequate finite element spaces, and derive optimal a priori error estimates. Finally, numerical experiments in 2D and 3D illustrate the interest of this scheme and validate the theory.

A recent calculation of the multi-Higgs boson production in scalar theories
with spontaneous symmetry breaking has demonstrated the fast growth of the
cross section with the Higgs multiplicity at sufficiently large energies,
called “Higgsplosion”. It was argued that “Higgsplosion” solves the Higgs
hierarchy and fine-tuning problems. The phenomena looks quite exciting,
however in my talk I will present arguments that: a) the formula for
“Higgsplosion” has a limited applicability and inconsistent with unitarity
of the Standard Model; b) that the contribution from “Higgsplosion” to the
imaginary part of the Higgs boson propagator cannot be re-summed in order to
furnish a solution of the Higgs hierarchy and fine-tuning problems.

Based on our recent paper https://arxiv.org/abs/1808.05641 (A. Belyaev, F. Bezrukov, D. Ross)

Let $\Omega\subseteq\mathbb{R}^2$ be a domain and let $f\in W^{1,1}(\Omega,\mathbb{R}^2)$ be a homeomorphism (between $\Omega$ and $f(\Omega)$). Then there exists a sequence of smooth diffeomorphisms $f_k$ converging to $f$ in $W^{1,1}(\Omega,\mathbb{R}^2)$ and uniformly. This is a joint result with A. Pratelli.
 

Abstract: The edge reinforced random walk is a self-interacting process, in which the random walker prefer visited edges with a bias proportional to the number of times the edges were visited. We will gently introduce this model and talk about some of its histories and recent progresses.

 In the late seventies, Casson and Gordon developed several knot invariants that obstruct a knot from being slice, i.e. from bounding a disc in the 4-ball. In this talk, we use twisted Blanchfield pairings to define twisted generalisations of the Levine-Tristram signature function, and describe their relation to the Casson-Gordon invariants. If time permits, we will present some obstructions to algebraic knots being slice. This is joint work with Maciej Borodzik and Wojciech Politarczyk.

We will consider some problems on calculating  the average  angles of random polytopes. Some of them are open.

In this talk, we will explain how to construct embedded closed Lagrangian submanifolds in mirror quintic threefolds using tropical curves and the toric degeneration technique. As an example, we will illustrate the construction for tropical curves that contribute to the Gromov–Witten invariant of the line class of the quintic threefold. The construction will in turn provide many homologous and non-Hamiltonian isotopic Lagrangian
rational homology spheres, and a geometric interpretation of the multiplicity of a tropical curve as the weight of a Lagrangian. This is a joint work with Helge Ruddat.

Finding Riemannian metrics with holonomy G_2 is a challenging problem with links in mathematics to Einstein metrics and area-minimizing submanifolds, and to M-theory in theoretical physics.  I will provide a brief survey on recent progress towards studying this problem using a geometric flow approach, including connections to calibrated fibrations.

Partially molten materials resist shearing and compaction. This resistance

is described by a fourth-rank effective viscosity tensor. When the tensor

is isotropic, two scalars determine the resistance: an effective shear and

an effective bulk viscosity. In this seminar, calculations are presented of

the effective viscosity tensor during diffusion creep for a 3D tessellation of

tetrakaidecahedrons (truncated octahedrons). The geometry of the melt is

determined by assuming textural equilibrium.  Two parameters

control the effect of melt on the viscosity tensor: the porosity and the

dihedral angle. Calculations for both Nabarro-Herring (volume diffusion)

and Coble (surface diffusion) creep are presented. For Nabarro-Herring

creep the bulk viscosity becomes singular as the porosity vanishes. This

singularity is logarithmic, a weaker singularity than typically assumed in

geodynamic models. The presence of a small amount of melt (0.1% porosity)

causes the effective shear viscosity to approximately halve. For Coble creep,

previous modelling work has argued that a very small amount of melt may

lead to a substantial, factor of 5, drop in the shear viscosity. Here, a

much smaller, factor of 1.4, drop is obtained.

How do humans process information? What are their strengths and limitations? This crash course in cognitive psychology will provide the background necessary to think realistically about how learning works.

Computational nucleic acid devices show great potential for enabling a broad range of biotechnology applications, including smart probes for molecular biology research, in vitro assembly of complex compounds, high-precision in vitro disease diagnosis and, ultimately, computational therapeutics inside living cells. This diversity of applications is supported by a range of implementation strategies, including nucleic acid strand displacement, localisation to substrates, and the use of enzymes with polymerase, nickase and exonuclease functionality. However, existing computational design tools are unable to account for these different strategies in a unified manner. This talk presents a programming language that allows a broad range of computational nucleic acid systems to be designed and analysed. We also demonstrate how similar approaches can be incorporated into a programming language for designing genetic devices that are inserted into cells to reprogram their behaviour. The language is used to characterise the genetic components for programming populations of cells that communicate and self-organise into spatial patterns. More generally, we anticipate that languages and software for programming molecular and genetic devices will accelerate the development of future biotechnology applications.

Generative adversarial networks (GANs) use neural networks as generative models, creating realistic samples that mimic real-life reference samples (for instance, images of faces, bedrooms, and more). These networks require an adaptive critic function while training, to teach the networks how to move improve their samples to better match the reference data. I will describe a kernel divergence measure, the maximum mean discrepancy, which represents one such critic function. With gradient regularisation, the MMD is used to obtain current state-of-the art performance on challenging image generation tasks, including 160 × 160 CelebA and 64 × 64 ImageNet. In addition to adversarial network training, I'll discuss issues of gradient bias for GANs based on integral probability metrics, and mechanisms for benchmarking GAN performance.

If G is a topological group, a G-flow X is a non-empty, compact, Hausdorff space on which G acts continuously; it is minimal if all G-orbits are dense. By a theorem of Ellis, there is a (unique) minimal G-flow M(G) which is universal: there is a continuous G-map to every other G-flow. 

Here, we will be interested in the case where G = Aut(K) for some structure K, usually omega-categorical. Work of Kechris, Pestov and Todorcevic and others gives conditions on K under which structural Ramsey Theory (due to Nesetril - Rodl and others) can be used to compute M(G). 

In the first part of the talk I will give a description of the above theory and when it applies (the 'tame case'). In the second part, I will describe joint work with J. Hubicka and J. Nesetril which shows that the omega-categorical structures constructed in the late 1980's by Hrushovski as counterexamples to Lachlan's conjecture are not tame and moreover, minimal flows of their automorphism groups have rather different properties to those in the tame case. 

When S is a finite set of natural numbers, a GCD-sum is a particular kind of double-sum over the elements of S, and they arise naturally in several settings. In particular, these sums play a role when one studies the local statistics of point sequences on the unit circle. There are known upper bounds for the size of a GCD-sum in terms of the size of the set S, most recently due to de la Bretèche and Tenenbaum, and these bounds are sharp. Yet the known examples of sets S for which the GCD-sum over S provides a matching lower bound all possess strong multiplicative structure, whereas in applications the set S often comes with additive structure. In this talk I will describe recent joint work with Thomas Bloom in which we apply an estimate from sum-product theory to prove a much stronger upper bound on a GCD-sum over an additively structured set. I will also describe an application of this improvement to the study of the distribution of points on the unit circle, with a further application to arbitrary infinite subsets of squares. 

We study the value of a zero-sum stopping game in which the terminal payoff function depends on the underlying process and on an additional randomness (with finitely many states) which is known to one player but unknown to the other. Such asymmetry of information arises naturally in insider trading when one of the counterparties knows an announcement before it is publicly released, e.g., central bank's interest rates decision or company earnings/business plans. In the context of game options this splits the pricing problem into the phase before announcement (asymmetric information) and after announcement (full information); the value of the latter exists and forms the terminal payoff of the asymmetric phase.

The above game does not have a value if both players use pure stopping times as the informed player's actions would reveal too much of his excess knowledge. The informed player manages the trade-off between releasing information and stopping optimally employing randomised stopping times. We reformulate the stopping game as a zero-sum game between a stopper (the uninformed player) and a singular controller (the informed player). We prove existence of the value of the latter game for a large class of underlying strong Markov processes including multi-variate diffusions and Feller processes. The main tools are approximations by smooth singular controls and by discrete-time games.

After considering motivations in symplectic geometry, I’ll give a summary of $C^\infty$-Algebraic Geometry and how to extend these concepts to manifolds with corners. 

This talk will concern the problem of inference when the posterior measure involves continuous models which require approximation before inference can be performed. Typically one cannot sample from the posterior distribution directly, but can at best only evaluate it, up to a normalizing constant. Therefore one must resort to computationally-intensive inference algorithms in order to construct estimators. These algorithms are typically of Monte Carlo type, and include for example Markov chain Monte Carlo, importance samplers, and sequential Monte Carlo samplers. The multilevel Monte Carlo method provides a way of optimally balancing discretization and sampling error on a hierarchy of approximation levels, such that cost is optimized. Recently this method has been applied to computationally intensive inference. This non-trivial task can be achieved in a variety of ways. This talk will review 3 primary strategies which have been successfully employed to achieve optimal (or canonical) convergence rates – in other words faster convergence than i.i.d. sampling at the finest discretization level. Some of the specific resulting algorithms, and applications, will also be presented.

Classical methods for X-ray computed tomography (CT) are based on the assumption that the X-ray source intensity is known. In practice, however, the intensity is measured and hence uncertain. Under normal circumstances, when the exposure time is sufficiently high, this kind of uncertainty typically has a negligible effect on the reconstruction quality. However, in time- or dose-limited applications such as dynamic CT, this uncertainty may cause severe and systematic artifacts known as ring artifacts.
By modeling the measurement process and by taking uncertainties into account, it is possible to derive a convex reconstruction model that leads to improved reconstructions when the signal-to-noise ratio is low. We discuss some computational challenges associated with the model and illustrate its merits with some numerical examples based on simulated and real data.

In this talk, we demonstrate that formation of Type I singularities of suitable weak solutions of the Navier-Stokes equations occur if there exists non-zero mild bounded ancient solutions satisfying a 'Type I' decay condition. We will also discuss some new Liouville type Theorems. Joint work with Dallas Albritton (University of Minnesota).

I will discuss Agol's proof of the Virtually Fibred Conjecture of
Thurston, focusing on the role played by the `RFRS' property. I will
then show how one can modify parts of Agol's proof by replacing some
topological considerations with a group theoretic statement about
virtual fibring of RFRS groups.
 

In geometric group theory we study groups by their actions on metric spaces. Although a given group might admit many actions on different metric spaces, on a large scale these spaces will all look similar, and so the large scale properties of a space on which a group acts are intrinsic to the group. One particularly natural example of a large scale property used in this way is the Gromov boundary of a hyperbolic metric space. This is a topological space that can be thought of as compactifying the metric space at infinity. 

In this talk I will describe some constructions of spaces occurring in this way with nasty, fractal-like properties. On the other hand, there are limits to how pathological these spaces can be: theorems of Bestvina and Mess, Bowditch and Swarup imply that boundaries of hyperbolic groups are locally path connected whenever they are connected. I will discuss these results and some generalisations.