Past

The principal ideal theorem (1930) guaranteed that any number field K would embed into a finite extension, called the Hilbert class field of K, in which every ideal of the original field became principal -- however the Hilbert class field itself will not necessarily have class number 1. The class field tower problem asked whether iteratively taking Hilbert class fields must stabilize after finitely many steps. In 1964, it was finally answered in the negative by Golod and Shafarevich who produced infinitely many examples and pioneered the framework that is still the most common setting for deciding when a number field will have an infinite class field tower.

In this talk, I will finish the proof of their cohomological result and thus fully justify how it settled the class field tower problem.

After discussing the need for implicit constitutive relations to describe the response of both solids and fluids, I will discuss applications wherein such implicit constitutive relations can be gainfully exploited. It will be shown that such implicit relations can explain phenomena that have hitherto defied adequate explanation such as fracture and the movement of cracks in solids, the response of biological matter, and provide a new way to look at numerous non-linear phenomena exhibited by fluids. They provide a totally new and innovative way to look at the problem of Turbulence. It also turns out that classical Cauchy and Green elasticity are a small subset of the more general theory of elastic bodies defined by implicit constitutive equations. 

We discuss some results on integration of ``rough differential forms'', which are generalizations of classical (smooth) differential forms to similar objects involving Hölder continuous functions that may be nowhere differentiable. Motivations arise mainly from geometric problems related to irregular surfaces, and the techniques are naturally related to those of Rough Paths theory. We show in particular that such a geometric integration can be constructed substituting appropriately differentials with more general asymptotic expansions (of Stratonovich or Ito type) and by summing over a refining sequence of partitions, leading to a two-dimensional extension of the classical Young integral, that coincides with the integral introduced recently by R. Züst. We further show that Stratonovich sums gives an advantage allowing to weaken the requirements on Hölder exponents, and discuss some work in progress in the stochastic case. Based on joint works with E. Stepanov, G. Alberti and I. Ballieul.

For any word w in a free group of rank r>0, and any compact group G, w induces a `word map' from G^r to G by substitutions of elements of G for the letters of w. We may also choose the r elements of G independently with respect to Haar measure on G, and then apply the word map. This gives a random element of G whose distribution depends on w. An interesting observation is that this distribution doesn't change if we change w by an automorphism of the free group. It is a wide open question whether the measures induced by w on compact groups determine w up to automorphisms.
My talk will be mostly about the case G = U(n), the n by n complex unitary matrices. The technical tool we use is a precise formula for the moments of the distribution induced by w on U(n). In the formula, there is a surprising appearance of concepts from infinite group theory, more specifically, Euler characteristics of mapping class groups of surfaces. I'll explain how our formula allows us to make progress on the question described above.
This is joint work with Doron Puder (Tel Aviv).
 

Gibbs measures of nonlinear Schrödinger equations are a fundamental object used to study low-regularity solutions with random initial data. In the dispersive PDE community, this point of view was pioneered by Bourgain in the 1990s. We study the problem of the derivation of Gibbs measures as high-temperature limits of thermal states in many-body quantum mechanics.

In our work, we apply a perturbative expansion in the interaction. This expansion is then analysed by means of Borel resummation techniques. In two and three dimensions, we need to apply a Wick-ordering renormalisation procedure. Moreover, in one dimension, our methods allow us to obtain a microscopic derivation of the time-dependent correlation functions for the cubic nonlinear Schrödinger equation. This is based partly on joint work with Jürg Fröhlich, Antti Knowles, and Benjamin Schlein.

Moduli spaces of polarised varieties (varieties together with an ample line bundle) are not Hausdorff in general. A basic goal of algebraic geometry is to construct a Hausdorff moduli space of some nice class of polarised varieties. I will discuss how one can achieve this goal using the theory of canonical Kähler metrics. In addition I will discuss some fundamental properties of this moduli space, for example the existence of a Weil-Petersson type Kähler metric. This is joint work with Philipp Naumann.

I will talk about supersymmetric index of 4d N=1 supersymmetric theories on S^1xM_3 which counts supersymmetric states.  
In the first part, I will discuss a general formula to describe an asymptotic behaviour of the index in the limit of shrinking S^1
which we refer to as 4d (refined) supersymmetric Cardy formula. This part is based on arXiv:1611.00380 with Lorenzo Di Pietro.
In the second part, I will apply this formula to black hole physics. I will mainly focus on superconformal index of SU(N) N=4 super Yang-Mills theory
which is expected to be dual to type IIB superstring theory on AdS_5 x S^5. We will see that the index in the large-N limit reproduces the Bekenstein-Hawking entropy
of rotating charged BPS black hole on the gravity side. Our result for finite N makes a prediction to the black hole entropy with full quantum corrections.
The second part is based on arXiv:1901.08091.

Solidification processes of liquid metal alloys,  bubble dynamics (as in Taylor flows), pinch-offs of liquid-liquid jets, the formation of polymeric membranes, or the structure of high concentration photovoltaic cells are described by the dynamics of multiphase fluids. On the other hand, in applications such as mass spectrometry, lab-on-a-chip, and electro-fluidic displays, fluids on the micro-scale associated with a dielectric medium are of interest. Moreover, in many of these applications one is interested in influencing (or controlling) the underlying phenomenon in order to reach a desired goal. Examples for the latter could be the porosity structure of a polymeric membrane to achieve certain desired filtration properties of the membrane, or to optimize a microfluidic device for the transport of pharmaceutical agents.

A promising mathematical model for the behavior of multiphase flows associated with the applications mentioned above is given by a phase-field model of Cahn-Hilliard / Navier-Stokes (CHNS) type. Some strengths of phase field (or diffuse interface) approaches are due to their ability to overcome both, analytical difficulties of topological changes, such as, e.g., droplet break-ups or the coalescence of interfaces, and numerical challenges in capturing the interface dynamics between the fluid phases. Deep quenches in solidification processes of liquid alloys or rapid wall hardening in the formation of polymer membranes ask for non-smooth energies in connection with Cahn-Hilliard models. Analytically, this gives rise to a variational inequality coupled to the equations of hydrodynamics, thus yielding a non-smooth system (in the sense that the map associated with the underlying operator equation is not necessarily Frechet differentiable). In contrast to phase-field approaches,
one may consider sharp interface models. In view of this, our microfluidic applications alluded to above are formulated in terms of  sharp interface models and Hele-Shaw flows. In this context, we are particularly interested in applications of electrowetting on dielectric (EWOD) with contact line pinning. The latter phenomenon resembles friction, yields a variational inequality of the second kind, and – once again – it results in an overall nonsmooth mathematical model of the physical process.

   In both settings described above, optimal control problems are relevant in order to influence the underlying physical process to approach a desired system state.  The associated optimization problems are delicate as the respective constraints involve non-smooth structures which render the problems degenerate and prevent a direct application of sophisticated tools for the characterization of solutions. Such characterizations are, however, of paramount importance in the design of numerical solution schemes.

This talk addresses some of the analytical challenges associated with optimal control problems involving non-smooth structures, offers pathways to solutions, and it reports on numerical results for both problem classes introduced above.
 

In this talk I will introduce the concept of the path signature and motivate its recent use in analysis of time-ordered data. I will then describe a new feature map for barcodes in persistent homology by first realizing each barcode as a path in a vector space, and then computing its signature which takes values in the tensor algebra over that vector space. The composition of these two operations — barcode to path, path to tensor series — results in a feature map that has several desirable properties for statistical learning, such as universality and characteristicness.

Mechanobiology is a field of science that aims to understand how mechanics regulate biology. It focuses on how mechanical forces and alterations in mechanical properties of cell or tissues regulate biological processes in development, physiology and disease. In fact, all these processes occur in our body, which presents a clear structural and hierarchical organization that goes from the organism to the cellular level. To advance in the understanding of all these processes at different scales requires the use of simplified representations of our body, which is normally known as modelling or equivalently the creation of a model. Different types of models can be found in the literature: in-vitro, in-vivo and in-silico models.

Here, I will present our modelling strategy in which we integrate different mathematical models and experiments in order to tackle relevant mechanical-based mechanisms in wound healing and cancer metastasis progression [1,2]. In fact, we have focused our research on individual [3] and collective cell migration [4], because it is a crucial event in all these mechanisms. Therefore, unravelling the intrinsic mechanisms that cells use to define their migration is an essential element for advancing the development of new technologies in regenerative medicine and cancer.

Due to the complexity of all these mechanisms, mathematical modelling is a relevant tool for providing deeper insight and quantitative predictions of the mechanical interplay between cells and extracellular matrix during cell migration. To assess the predictive capacity of these models, we will compare our numerical results with microfluidic-based experiments [2], which provide experimental information to test and refine the main assumptions of our models.

Actually, we design and fabricate multi-channel 3D microfluidics cell culture chips, which allow recreating the physiology and disease of one organ or any biological process with a precise control of the micro environmental factors [5]. Therefore, this kind of organ-on-a-chip experiments constitutes a novel modelling strategy of in vitro multicellular human systems that in combination with mathematical simulations provide a relevant tool for research in mechanobiology.

References

Escribano J, Chen M, Moeendarbary E, Cao X, Shenoy V, Garcia-Aznar JM, Kamm RD, Spill F.  Balance of Mechanical Forces Drives Endothelial Gap Formation and May Facilitate Cancer and Immune-Cell Extravasation. PLOS Computational Biology, in press.

The Kelvin wave is perhaps the most important of the equatorially trapped waves in the terrestrial atmosphere and ocean, and plays a role in various phenomena such as tropical convection and El Nino. Theoretically, it can be understood from the linear dynamics of a stratified fluid on an equatorial beta plane, which, with simple assumptions about the disturbance structure, leads to wavelike solutions propagating along the equator, with exponential decay in latitude. However, when the simplest possible background flow is added (with uniform latitudinal shear), the Kelvin wave (but not the other equatorial waves) becomes unstable. This happens in an extremely unusual way: there is instability for arbitrarily small nondimensional shear p, and the growth rate is proportional to exp(-1/p^2) as p->0. This in contrast to most hydrodynamic instabilities, in which the growth rate typically scales as a positive power of p-p_c as the control parameter p passes through a critical value p_c.

This Kelvin wave instability has been established numerically by Natarov and Boyd, who also speculated as to the underlying mathematical cause. Here we show how the growth rate and full spatial structure of the instability may be derived using matched asymptotic expansions applied to the (linear) equations of motion. This involves an adventure with Whittaker functions in the exponentially-decaying tails of the Kelvin waves, and a trick to reveal the exponentially small growth rate from a formulation that only uses regular perturbation expansions. Numerical verification of the analysis is also interesting and challenging, since special high-precision solutions of the governing ODE are required even when the nondimensional shear is not that small (circa 0.5).

Data analysis techniques are often highly domain specific - there are often certain patterns which should be in certain types of data but may not be apparent in data. The first part of the talk will cover a technique for finding such patterns through a tool which combines visual analytics and machine learning to provide insight into temporal multivariate data. The second half of the talk will discuss recent work on imposing high level geometric  structure into continuous optimizations including deep neural networks.
 

For a family of proper hyperbolic complex curves $f: X \longrightarrow \Delta^*$ over a puntured disc $\Delta^*$ with semistable reduction at the center, Oda proved, with transcendental methods, that the outer monodromy action of $\pi_1(\Delta^*) \cong \mathbb{Z}$ on the classical unipotent fundamental group of the generic fiber of $f$ is trivial if and only if $f$ has good reduction at the center. In this talk I explain a joint work with B. Chiarellotto and A. Shiho in which we give a purely algebraic proof of Oda's result.

Topological quantum field theories (TQFTs) are an extensively studied scheme for constructing invariants of manifolds, inspired by physics. In this talk, we will discuss a particular flavour of TQFT, where we equip our manifolds with principal bundles for some finite group. After introducing TQFTs and this particular flavour, I will discuss games one can play with these TQFTs, and a possible strategy for classifying equivariant TQFTs in three dimensions. 

As parallel computers approach Exascale (10^18 floating point operations per second), processor failure and data corruption are of increasing concern. Numerical linear algebra solvers are at the heart of many scientific and engineering applications, and with the increasing failure rates, they may fail to compute a solution or produce an incorrect solution. It is therefore crucial to develop novel parallel linear algebra solvers capable of providing correct solutions on unreliable computing systems. The common way to mitigate failures in high performance computing systems consists of periodically saving data onto a reliable storage device such as a remote disk. But considering the increasing failure rate and the ever-growing volume of data involved in numerical simulations, the state-of-the-art fault-tolerant strategies are becoming time consuming, therefore unsuitable for large-scale simulations. In this talk, we will present a  novel class of fault-tolerant algorithms that do not require any additional resources. The key idea is to leverage the knowledge of numerical properties of solvers involved in a simulation to regenerate lost data due to system failures. We will also share the lessons learned and report on the numerical properties and the performance of the new resilience algorithms.

Function solutions to linear PDEs often carry rigidity properties directly associated to the equation they satsify. However, the realm of solutions covers a much larger sets of solutions. For instance, we can speak of measure solutions, as opposed to classical $C^\infty$ functions or even $L^p$ functions. It is only logical to expect that the “better” space the solution lives in, the more rigid its properties will be.

Measure solutions lie just at a comfortable half of this threshold: it is a sufficently large space which allows for a rich range of new structures; but is sufficiently rigid to preserve a meaningful geometrical pattern. For example, have you ever wondered how gradients look like in the space of measures? What about other PDE structures? In this talk I will discuss these general questions, a few examples of them, and a new theoretical approach to its understanding via PDE theory, harmonic analysis, and geometric measure theory methods.

In 1996 using techniques from model theory and intersection theory, Hrushovski obtained a generalisation of the Lang-Weil estimates. Subsequently the estimates have found applications in group theory, algebraic dynamics and algebraic geometry. We shall discuss a geometric proof of the non-uniform version of these estimates.

Property (FA) is one of the `rigidity properties’ defined for groups, concerning the way a group can act on trees. We’ll take a look at why you might be interested in an action on a tree, what the property is, and then investigate which automorphism groups of free products have it.

We study the problem of super-resolution using TV norm minimisation, where we recover the locations and weights of non-negative point sources from a few samples of their convolution with a Gaussian kernel. A practical approach is to solve the dual problem. In this talk, we study the stability of solutions with respect to the solutions to the dual problem. In particular, we establish a relationship between perturbations in the dual variable and the primal variables around the optimiser. This is achieved by applying a quantitative version of the implicit function theorem in a non-trivial way.

Deformation quantization modules or DQ-modules where introduced by M. Kontsevich to study the deformation quantization of complex Poisson varieties. It has been advocated that categories of DQ-modules should provide invariants of complex symplectic varieties and in particular a sort of complex analog of the Fukaya category. Hence, it is natural to aim at describing the functors between such categories and relate them with categories appearing naturally in algebraic geometry. Relying, on methods of homotopical algebra, we obtain an analog of Orlov representation theorem for functors between categories of DQ-modules and relate these categories to deformations of the category of quasi-coherent sheaves.
 

We consider a sparse multi-task regression framework for fitting a collection of related sparse models. Representing models as nodes in a graph with edges between related models, a framework that fuses lasso regressions with the total variation penalty is investigated. Under a form of generalised restricted eigenvalue assumption, bounds on prediction and squared error are given that depend upon the sparsity of each model and the differences between related models. This assumption relates to the smallest eigenvalue restricted to the intersection of two cone sets of the covariance matrix constructed from each of the agents' covariances. In the case of a grid topology high-probability bounds are given that match, up to log factors, the no-communication setting of fitting a lasso on each model, divided by the number of agents.  A decentralised dual method that exploits a convex-concave formulation of the penalised problem is proposed to fit the models and its effectiveness demonstrated on simulations. (Joint work with Sahand Negahban and Patrick Rebeschini)

Multiple scales analysis is a powerful asymptotic technique for problems where the solution depends on two scales of widely different sizes. Standard multiple scales involves the introduction of a macroscale and microscale which are assumed to be independent. A common (and usually acceptable) assumption is that when considering behaviour on the microscale, the macroscale variable can be taken as constant, however there are instances where this assumption is not valid. In this talk, I will explain one such situation, that is, when considering conductive-radiative thermal transfer within a solid matrix with spherical perforations and discuss the appropriate measures when converting the radiative boundary condition into multiple-scales form.
 

Quantum fields can sometimes have negative energy density.  In gravitational contexts, this threatens to permit both causality violations (such as traversable wormholes, warp drives, and time machines) and violations of the Second Law for black holes.  I will discuss the thermodynamic principles that rule out such pathological situations.  These principles have led us to an interesting lower bound on the energy flux, even for field theories in flat spacetime! This Quantum Null Energy Condition has now been proven for all relativistic field theories.  I will give an intuitive argument explaining why such ``quantum energy conditions'' ought to hold. 
 

Complexity Science aims to understand what is that makes some systems to be "more than the sum of their parts". A natural first step to address this issue is to study networks of pairwise interactions, which have been done with great success in many disciplines -- to the extend that many people today identify Complexity Science with network analysis. In contrast, multivariate complexity provides a vast and mostly unexplored territory. As a matter of fact, the "modes of interdependency" that can exist between three or more variables are often nontrivial, poorly understood and, yet, are paramount for our understanding of complex systems in general, and emergence in particular. 
In this talk we present an information-theoretic framework to analyse high-order correlations, i.e. statistical dependencies that exist between groups of variables that cannot be reduced to pairwise interactions. Following the spirit of information theory, our approach is data-driven and model-agnostic, being applicable to discrete, continuous, and categorical data. We review the evolution of related ideas in the context of theoretical neuroscience, and discuss the most prominent extensions of information-theoretic metrics to multivariate settings. Then, we introduce the O-information, a novel metric that quantify various structural (i.e. synchronous) high-order effects. Finally, we provide a critical discussion on the framework of Integrated Information Theory (IIT), which suggests an approach to extend the analysis to dynamical settings. To illustrate the presented methods, we show how the analysis of high-order correlations can reveal critical structures in various scenarios, including cellular automata, Baroque music scores, and various EEG datasets.

References:
[1] F. Rosas, P.A. Mediano, M. Gastpar and H.J. Jensen, ``Quantifying High-order Interdependencies via Multivariate Extensions of the Mutual Information'', submitted to PRE, under review.
https://arxiv.org/abs/1902.11239
[2] F. Rosas, P.A. Mediano, M. Ugarte and H.J. Jensen, ``An information-theoretic approach to self-organisation: Emergence of complex interdependencies in coupled dynamical systems'', in Entropy, vol. 20 no. 10: 793, pp.1-25, Sept. 2018.
https://www.mdpi.com/1099-4300/20/10/793

 

The principal ideal theorem (1930) ascertained that any number field K embeds into a finite extension, called the Hilbert class field of K, in which every ideal of the original field became principal -- however the Hilbert class field itself will not necessarily have class number 1. The class field tower problem asked whether iteratively taking Hilbert class fields must stabilize after finitely many steps. In 1964, it was finally answered in the negative by Golod and Shafarevich who produced infinitely many examples and pioneered the framework that is still the most common setting for deciding when a number field will have an infinite class field tower.

In this talk, I will sketch the proof of their cohomological result and explain how it settled the class field tower problem.

In this talk, I will present our recent progress collaborated with Prof. Gui-Qiang G. Chen and Prof. Paolo Secchi on two kinds of characteristic discontinuities: relativistic vortex sheets in three-dimensional Minkowski spacetime and multi-dimensional thermoelastic contact discontinuities.
 

A group is said to satisfy the Tits Alternative if its finitely generated subgroups exhibit a striking dichotomy: they are either "big" (they contain a non-abelian free subgroup) or "small" (they are virtually soluble). Many groups of geometric interest have been shown to satisfy the Tits Alternative: linear groups, mapping class groups of hyperbolic surfaces, etc. In this talk, I will explain how one can use ideas from group actions in negative curvature to prove such a dichotomy. In particular, I will show how one can prove a strengthening of the Tits Alternative for a large class of Artin groups. This is joint work with Piotr Przytycki.

Neural networks are undoubtedly successful in practical applications. However complete mathematical theory of why and when machine learning algorithms based on neural networks work has been elusive. Although various representation theorems ensures the existence of the ``perfect’’ parameters of the network, it has not been proved that these perfect parameters can be (efficiently) approximated by conventional algorithms, such as the stochastic gradient descent. This problem is well known, since the arising optimisation problem is non-convex. In this talk we show how the optimization problem becomes convex in the mean field limit for one-hidden layer networks and certain deep neural networks. Moreover we present optimality criteria for the distribution of the network parameters and show that the nonlinear Langevin dynamics converges to this optimal distribution. This is joint work with Kaitong Hu, Zhenjie Ren and Lukasz Szpruch. 

Partition Lie algebras are generalisations of rational differential graded Lie algebras which, by a recent result of Mathew and myself, govern the formal deformation theory of algebro-geometric objects in finite and mixed characteristic. In this talk, we will take a closer look at these new gadgets and discuss some of their applications in algebra and topology

Supergravity backgrounds are an essential ingredient in string theory or field theories via AdS/CFT. The simplest example of a 4d, N=1 background is the product of four-dimensional Minkowski space with a seven-dimensional manifold with G_2 holonomy in M-theory. For more complicated backgrounds where we allow non-zero fluxes, the supersymmetry conditions can be rephrased in terms of G-structure data. The geometry of these backgrounds is often complicated and their general features are not well understood.

In this talk, I will define the analogue of G_2 geometry for generic 4d, N=1 backgrounds with flux in both type II and eleven-dimensional supergravity. The geometry is characterised by a G-structure in 'exceptional generalised geometry' that includes G_2 structures and Hitchin's generalised geometry as subcases. Supersymmetry is then equivalent to integrability of the structures, which appears as an involutivity condition and a moment map for diffeomorphisms and gauge transformations. I will show how this works in a few simple examples and discuss how this can be used to understand general properties of supersymmetric backgrounds.

 

A panel discussion on non-academic careers for mathematicians with PhDs, featuring Katia Babbar (AI Wealth Technologies & QuantBright), Jara Imbers (Risk Management Solutions) and Tom Hawes (Smith Institute).
 

Cardiac fibrosis plays a significant role in the disruption of healthy electrical signalling in the heart, creating structural heterogeneities that induce and stabilise arrhythmia.  However, a proper understanding of the consequences of cardiac fibrosis must take into account the complex and highly variable patterns of its spatial localisation in the heart, which significantly affects the extent and manner of its impacts on cardiac wave propagation. In this work we present a methodology for the algorithmic generation of fibrotic patterns via Perlin noise, a technique for computationally efficient generation of textures in computer graphics.

Our approach works directly from image data to create populations of pattern realisations that all resemble the target image under a set of metrics. Our technique thus serves as a type of data enrichment, enabling analysis of how variability in the precise placement of fibrotic structures modulates their electrophysiological impact. We demonstrate our method, and the types of analysis it can enable, using a widely referenced histological image of four different types of microfibrotic structure. Our generator and Bayesian tuning method prove flexible enough to successfully capture each of these very distinct patterns.

We demonstrate the importance of this tool, by presenting 2D simulations overlayed on the generated images that highlight the effects of microscopic variability on the electrophysiological impact of fibrosis. Finally, we discuss the application of our methodology to the increasingly available imaging data of fibrotic patterning on a more macroscopic scale, and indeed to other areas of science underpinned by image based modelling and simulation.    

Dimension reduction is an overarching theme in data science: we enjoy finding informative patterns, features or substructures in large, complex data sets. Within the field of network science, an important problem of this nature is to identify core-periphery structure. Given a network, our task is to assign each node to either the core or periphery. Core nodes should be strongly connected across the whole network whereas peripheral nodes should be strongly connected only to core nodes. More generally, we may wish to assign a non-negative value to each node, with a larger value indicating greater "coreness." This type of problem is related to, but distinct from, commumnity detection (finding clusters) and centrality assignment (finding key players), and it arises naturally in the study of networks in social science and finance. We derive and analyse a new iterative algorithm for detecting network core-periphery structure.

Using techniques in nonlinear Perron-Frobenius theory we prove global convergence to the unique solution of a relaxed version of a natural discrete optimization problem. On sparse networks, the cost of each iteration scales linearly with the number of nodes, making the algorithm feasible for large-scale problems. We give an alternative interpretation of the algorithm from the perspective of maximum likelihood reordering of a new logistic core--periphery random graph model. This viewpoint also gives a new basis for quantitatively judging a core--periphery detection algorithm. We illustrate the algorithm on a range of synthetic and real networks, and show that it offers advantages over the current state-of-the-art.

This is joint work with Francesco Tudisco (Strathclyde)

In Formula 1 engineers strive to produce the fastest car possible for their drivers. A lap simulation provides an objective evaluation of the performance of the car and the subsequent lap time achieved. Using this information, engineers aim to test new car concepts, determine performance limitations or compromises, and identify the sensitivity of performance to car setup parameters.

The latest state of the art lap simulation techniques use optimal control approaches. Optimisation methods are employed to derive the optimal control inputs of the car that achieve the fastest lap time within the constraints of the system. The resulting state trajectories define the complete behaviour of the car. Such approaches aim to create more robust, realistic and powerful simulation output compared to traditional methods.

In this talk we discuss our latest work in this area. A dynamic vehicle model is used within a free-trajectory solver based on direct optimal control methods. We discuss the reasons behind our design choices, our progress to date, and the issues we have faced during development. Further, we look at the short and long term aims of our project and how we wish to develop our mathematical methods in the future.

I will discuss the arithmetic significance of Fourier expansions of modular forms at cusps. I will talk about joint work with F. Brunault, where we determine the number field generated by Fourier coefficients of newforms at a cusp. Then I will discuss joint work with A. Saha and K. Česnavičius where we find denominator bounds for Fourier expansions at cusps and apply these bounds to a conjecture on the Manin constants of elliptic curves.

The problem of model uncertainty in financial mathematics has received considerable attention in the last years. In this talk I will follow a non-parametric point of view, and argue that an insightful approach to model uncertainty should not be based on the familiar Wasserstein distances. I will then provide evidence supporting the better suitability of the recent notion of adapted Wasserstein distances (also known as Nested Distances in the literature). Unlike their more familiar counterparts, these transport metrics take the role of information/filtrations explicitly into account. Based on joint work with M. Beiglböck, D. Bartl and M. Eder.

Likely instabilities in stochastic hyperelastic solids

L. Angela Mihai

School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF24 4AG, UK

E-mail: @email.uk

Nonlinear elasticity has been an active topic of fundamental and applied research for several decades. However, despite numerous developments and considerable attention it has received, there are important issues that remain unresolved, and many aspects still elude us. In particular, the quantification of uncertainties in material parameters and responses resulting from incomplete information remain largely unexplored. Nowadays, it is becoming increasingly apparent that deterministic approaches, which are based on average data values, can greatly underestimate, or overestimate, mechanical properties of many materials. Thus, stochastic representations, accounting for data dispersion, are needed to improve assessment and predictions. In this talk, I will consider stochastic hyperelastic material models described by a strain-energy density where the parameters are characterised by probability distributions. These models, which are constructed through a Bayesian identification procedure, rely on the maximum entropy principle and enable the propagation of uncertainties from input data to output quantities of interest. Similar modelling approaches can be developed for other mechanical systems. To demonstrate the effect of probabilistic model parameters on large strain elastic responses, specific case studies include the classic problem of the Rivlin cube, the radial oscillatory motion of cylindrical and spherical shells, and the cavitation and finite amplitude oscillations of spheres.

One of the major steps in the adaptive finite element methods (AFEM) is the adaptive selection of the next partition. The process is usually governed by a strategy based on carefully chosen local error indicators and aims at convergence results with optimal rates. One can formally relate the refinement of the partitions with growing an oriented graph or a tree. Then each node of the tree/graph corresponds to a cell of a partition and the approximation of a function on adaptive partitions can be expressed trough the local errors related to the cell, i.e., the node. The total approximation error is then calculated as the sum of the errors on the leaves (the terminal nodes) of the tree/graph and the problem of finding an optimal error for a given budget of nodes is known as tree approximation. Establishing a near-best tree approximation result is a key ingredient in proving optimal convergence rates for AFEM.

The classical tree approximation problems are usually related to the so-called h-adaptive approximation in which the improvements a due to reducing the size of the cells in the partition. This talk will consider also an extension of this framework to hp-adaptive approximation allowing different polynomial spaces to be used for the local approximations at different cells while maintaining the near-optimality in terms of the combined number of degrees of freedom used in the approximation.

The problem of conformity of the resulting partition will be discussed as well. Typically in AFEM, certain elements of the current partition are marked and subdivided together with some additional ones to maintain desired properties of the partition like conformity. This strategy is often described as “mark → subdivide → complete”. The process is very well understood for triangulations received via newest vertex bisection procedure. In particular, it is proven that the number of elements in the final partition is limited by constant times the number of marked cells. This hints at the possibility to design a marking procedure that is limited only to cells of the partition whose subdivision will result in a conforming partition and therefore no completion step would be necessary. This talk will present such a strategy together with theoretical results about its near-optimal performance.

It is nowadays well understood that the multidimensional isentropic Euler system is desperately ill–posed. Even certain smooth initial data give rise to infinitely many solutions and all available selection criteria fail to ensure both global existence and uniqueness. We propose a different approach to well–posedness of this system based on ideas from the theory of Markov semigroups: we show the existence of a Borel measurable solution semiflow. To this end, we introduce a notion of dissipative solution which is understood as time dependent trajectories of the basic state variables - the mass density, the linear momentum, and the energy - in a suitable phase space. The underlying system of PDEs is satisfied in a generalized sense. The solution semiflow enjoys the standard semigroup property and the solutions coincide with the strong solutions as long as the latter exist. Moreover, they minimize the energy (maximize the energy dissipation) among all dissipative solutions.

Leighton's Theorem states that if two finite graphs have a common universal cover then they have a common finite cover. I will present a new proof of this using groupoids, and then talk about two generalisations of the theorem that can also be tackled with this groupoid approach: one gives us control over the local structure of the common finite cover, and the other deals with graphs of spaces.

Hilbert's fifth problem asks informally what is the difference between Lie groups and topological groups. In 1950s this problem was solved by Andrew Gleason, Deane Montgomery, Leo Zippin and Hidehiko Yamabe concluding that every locally compact topological group is "essentially" a Lie group. In this talk we will show the complete proof of this theorem.

 Massless scattering amplitudes in four-dimensional Minkowski spacetime can be Mellin transformed to correlation functions on the celestial sphere at null infinity called celestial amplitudes. We study various properties of massless four-point scalar and gluon celestial amplitudes such as conformal partial wave decomposition, crossing relations and optical theorem. As a byproduct, we derive the analog of the single and double soft limits for all gluon celestial amplitudes.