Past

Tendons are vital connective tissues that anchor muscle to bone to allow the transfer of forces to the skeleton. They exhibit highly non-linear viscoelastic mechanical behaviour that arises due to their complex, hierarchical microstructure, which consists of fibrous subunits made of the protein collagen. Collagen molecules aggregate to form fibrils with diameters of tens to hundreds of nanometres, which in turn assemble into larger fibres called fascicles with diameters of tens to hundreds of microns. In this talk, I will discuss the relationship between the three-dimensional organisation of the fibrils and fascicles and the macroscale mechanical behaviour of the tendon. In particular, I will show that very simple constitutive behaviour at the microscale can give rise to highly non-linear behaviour at the macroscale when combined with geometrical effects.

I will review the equivalence of categories of a Bernstein component of a p-adic classical group with the category of right modules over a certain affine Hecke algebra (with parameters) that I obtained previously. The parameters can be made explicit by the parametrization of supercuspidal representations of classical groups obtained by C. Moeglin, using methods of J. Arthur. Via this equivalence, I can show that the category of smooth complex representations of a quasisplit $p$-adic classical group and its pure inner forms is naturally decomposed into subcategories that are equivalent to the tensor product of categories of unipotent representations of classical groups (in the sense of G. Lusztig). All classical groups (general linear, orthogonal, symplectic and unitary groups) appear in this context.
 

The Landau-DeGennes Q-model of uniaxial nematic liquid crystals seeks a rank-one

traceless tensor Q that minimizes a Frank-type energy plus a double well potential

that confines the eigenvalues of Q to lie between -1/2 and 1. We propose a finite

element method (FEM) which preserves this basic structure and satisfies a discrete

form of the fundamental energy estimates. We prove that the discrete problem Gamma

converges to the continuous one as the meshsize tends to zero, and propose a discrete

gradient flow to compute discrete minimizers. Numerical experiments confirm the ability

of the scheme to approximate configurations with half-integer defects, and to deal with

colloidal and electric field effects. This work, joint with J.P. Borthagaray and S.

Walker, builds on our previous work for the Ericksen's model which we review briefly.

We study the small Deborah number limit of the Doi-Onsager equation for the dynamics of nematic liquid crystals. This is a Smoluchowski-type equation that characterizes the evolution of a number density function, depending upon both particle position and its orientation vector, which lies on the unit sphere. We prove that, in the low temperature regime, when the Deborah number tends to zero, the family of solutions with rough initial data near local equilibria will converge to a local equilibrium distribution prescribed by a weak solution of the harmonic map heat flow into the sphere. This flow is a special case of the gradient flow to the Oseen-Frank energy functional for nematic liquid crystals and the existence of its global weak solution was first obtained by Y.M Chen, using Ginzburg-Landau approximation.  The key ingredient of our result is to show the strong compactness of the family of number density functions and the proof relies on the strong compactness of the corresponding second moment (or the Q-tensor), a spectral decomposition of the linearized operator near the limiting local equilibrium distribution, as well as the energy dissipation estimates.  This is a joint work with Wei Wang in Zhejiang university.
 

I will present a survey of commonly considered notions of negative curvature for groups, focused on generalising properties of Gromov hyperbolic groups.

In this paper we define a new object, the momentum amplituhedron, which is the long sought-after positive geometry for tree-level scattering amplitudes in N=4 super Yang-Mills theory in spinor helicity space. Inspired by the construction of the ordinary amplituhedron, we introduce bosonized spinor helicity variables to represent our external kinematical data, and restrict them to a particular positive region. The momentum amplituhedron Mn,k is then the image of the positive Grassmannian via a map determined by such kinematics. The scattering amplitudes are extracted from the canonical form with logarithmic singularities on the boundaries of this geometry.

For a finite subgroup $G$ of $SL(2,C)$ and for $n \geq 1$,  the Hilbert scheme $X=Hilb^{[n]}(S)$ of $n$ points on the minimal resolution $S$ of the Kleinian singularity $C^2/G$ provides a crepant resolution of the symplectic quotient $C^{2n}/G_n$, where $G_n$ is the wreath product of $G$ with $S_n$. I'll explain why every projective, crepant resolution of $C^{2n}/G_n$ is a quiver variety, and why the movable cone of $X$ can be described in terms of an extended Catalan hyperplane arrangement of the root system associated to $G$ by John McKay. These results extend the algebro-geometric aspects of Kronheimer's hyperkahler description of $S$ to higher dimensions. This is joint work with Gwyn Bellamy.

In this talk I discuss how we can simulate stochastic chemical kinetics when there is a memory component. This can occur when there is spatial crowding within a cell or part of a cell, which acts to constrain the motion of the molecules which then in turn changes the dynamics of the chemistry. The counterpart of the Law of Mass Action in this setting is through replacing the first derivative in the ODE description of the Law of Mass Action by a time-­fractional derivative, where the time-­fractional index is between 0 and 1. There has been much discussion in the literature, some of it wrong, as to how we model and simulate stochastic chemical kinetics in the setting of a spatially-­constrained domain – this is sometimes called anomalous diffusion kinetics.

In this presentation, I discuss some of these issues and then present two (equivalent) ways of simulating fractional stochastic chemical kinetics. The key here is to either replace the exponential waiting time used in Gillespie’s SSA by Mittag-­Leffler waiting times (MacNamara et al. [2]), which have longer tails than in the exponential case. The other approach is to use some theory developed by Jahnke and Huisinga [1] who are able towrite down the underlying probability density function for any set of mono-­molecular chemical reactions (under the standard Law of Mass Action) as a convolution of either binomial probability density functions or binomial and Poisson probability density functions). We can then extend the Jahnke and Huisinga formulation through the concept of iterated Brownian Motion paths to produce exact simulations of the underlying fractional stochastic chemical process. We demonstrate the equivalence of these two approaches through simulations and also by computing the probability density function of the underlying fractional stochastic process, as described by the fractional chemical master equation whose solution is the Mittag-­Lefflermatrix function. This is computed based on a clever algorithm for computing matrix functions by Cauchy contours (Weideman and Trefethen [3]).

This is joint work with Manuel Barrio (University of Vallodolid, Spain), Kevin Burrage (QUT), Andre Leier (University of Alabama), Shev MacNamara(University of Technology Sydney)and T. Marquez-­Lago (University of Alabama).

[1]T. Jahnke and W. Huisinga, 2007, Solving the chemical master equation for monomolecular reaction systems analytically, J. Math. Biology 54, 1, 1—26.[2]S. MacNamara, B. Henry and W. McLean, 2017, Fractional Euler limits and their applications, SIAM J. Appl. Math. 77, 2, 447—469.[3]J.A.C. Weideman and L.N. Trefethen, 2007, Parabolic and hyperbolic contours for computing the Bromwich integral, Math. Comp. 76, 1341—1356.

ACORN generators represents an approach to generating uniformly distributed pseudo-random numbers which is straightforward to implement for arbitrarily large order $k$ and modulus $M=2^{30t}$ (integer $t$). They give long period sequences which can be proven theoretically to approximate to uniformity in up to $k$ dimensions, while empirical statistical testing demonstrates that (with a few very simple constraints on the choice of parameters and the initialisation) the resulting sequences can be expected to pass all the current standard tests .

The standard TestU01 Crush and BigCrush Statistical Test Suites are used to demonstrate for ACORN generators with order $8≤k≤25$ that the statistical performance improves as the modulus increases from $2^{60}$ to $2^{120}$. With $M=2^{120}$ and $k≥9$, it appears that ACORN generators pass all the current TestU01 tests over a wide range of initialisations; results are presented that demonstrate the remarkable consistency of these results, and explore the limits of this behaviour.

This contrasts with corresponding results obtained for the widely-used Mersenne Twister MT19937 generator, which consistently failed on two of the tests in both the Crush and BigCrush test suites.

There are other pseudo-random number generators available which will also pass all the TestU01 tests. However, for the ACORN generators it is possible to go further: we assert that an ACORN generator might also be expected to pass any more demanding tests for $p$-dimensional uniformity that may be required in the future, simply by choosing the order $k>p$, the modulus $M=2^{30t}$ for sufficiently large $t$, together with any odd value for the seed and an arbitrary set of initial values. We note that there will be $M/2$ possible odd values for the seed, with each such choice of seed giving rise to a different $k$-th order ACORN sequence satisfying all the required tests.

This talk builds on and extends results presented at the recent discussion meeting on “Numerical algorithms for high-performance computational science” at the Royal Society London, 8-9 April 2019, see download link at bottom of web page http://acorn.wikramaratna.org/references.html.

The comparison of graphs is a vitally important, yet difficult task which arises across a number of diverse research areas including biological and social networks. There have been a number of approaches to define graph distance however often these are not metrics (rendering standard data-mining techniques infeasible), or are computationally infeasible for large graphs. In this work, we define a new metric based on the spectrum of the non-backtracking graph operator and show that it can not only be used to compare graphs generated through different mechanisms but can reliably compare graphs of varying size. We observe that the family of Watts-Strogatz graphs lie on a manifold in the non-backtracking spectral embedding and show how this metric can be used in a standard classification problem of empirical graphs.

The renormalized expectation value of the stress energy tensor (RSET) is an object of central importance in quantum field theory in curved space-time, but calculating this on black hole space-times is far from trivial.  The vacuum polarization (VP) of a quantum scalar field is computationally simpler and shares some features with the RSET.  In this talk we consider the properties of the VP for a massless, conformally coupled scalar field on asymptotically anti-de Sitter black holes with spherical, flat and hyperbolic horizons.  We focus on the effect of the different horizon curvature on the VP, and the role played by the boundary conditions far from the black hole.     

This is a report on work in progress with Jingyin Huang. The complement of an arrangement of linear hyperplanes in a complex vector space has a natural “Borel-Serre bordification” as a smooth manifold with corners. Its universal cover is analogous to the Borel-Serre bordification of an arithmetic lattice acting on a symmetric space as well as to the Harvey bordification of Teichmuller space. In the first case the boundary of this bordification is homotopy equivalent to a spherical building; in the second case it is homotopy equivalent to curve complex of the surface. In the case of a reflection arrangement the boundary of its universal cover is the “curve complex” of the corresponding spherical Artin group. By definition this is the simplicial complex of all conjugates of proper, irreducible, spherical parabolic subgroups in the Artin group. A cohomological method is used to show that the curve complex of a spherical Artin group has the homotopy type of a wedge of spheres.

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.