Past

The subject of $p$-adic cohomologies is over fifty years old. Many new developments have recently occurred. I will mostly limit myself to discussing some pertaining to the de Rham-Witt complex. After recalling the historical background and the basic results, I will give an overview of the new approach of Bhatt, Lurie and Mathew.

One of the key properties of 1-parameter persistent homology is that its output can entirely encoded in a purely combinatorial way via persistence diagrams or barcodes.  However, many applications of topological data analysis naturally present themselves with more than 1 parameter. Multiparameter persistence suggests itself as the natural invariant to use, but the problem here is that the moduli space of multiparameter persistence diagrams has a much more complicated structure and we lack a combinatorial diagrammatic description.  An alternative approach was suggested by work of Giansiracusa-Moon-Lazar, where they investigated calculating a series of 1-parameter persistence diagrams as the other parameter is varied. In this talk I will discuss work in progress to produce a refinement of their perspective, making use the Algebraic Stability Theorem for persistent homology and work of Bauer-Lesnick on induced matchings.

Cellular migration can be affected by short-range interactions between cells such as volume exclusion, long-range forces such as chemotaxis, or reactions such as phenotypic switching. In this talk I will discuss how to incorporate these processes into a discrete or continuum modelling frameworks. In particular, we consider a system with two types of diffusing hard spheres that can react (switch type) upon colliding. We use the method of matched asymptotic expansions to obtain a systematic model reduction, consisting of a nonlinear reaction-diffusion system of equations. Finally, we demonstrate how this approach can be used to study the effects of excluded volume on cellular chemotaxis. This is joint work with Dan Wilson and Helen Byrne.
 

 A central task in modeling, which has to be performed each day in banks and financial institutions, is to calibrate models to market and historical data. So far the choice which models should be used was not only driven by their capacity of capturing empirically the observed market features well, but rather by computational tractability considerations. Due to recent work in the context of machine learning, this notion of tractability has changed significantly. In this work, we show how a neural network approach can be applied to the calibration of (multivariate) local stochastic volatility models. We will see how an efficient calibration is possible without the need of interpolation methods for the financial data. Joint work with Christa Cuchiero and Josef Teichmann.

The disruptive drone activity at airports requires an early warning system and Aveillant make a radar system that can do the job. The main problem is telling the difference between birds and drones where there may be one or two drones and 10s or 100s of birds. There is plenty of data including time series for how the targets move and the aim is to improve the discrimination capability of tracker using machine learning.

Specifically, the challenge is to understand whether there can be sufficient separability between birds and drones based on different features, such as flight profiles, length of the track, their states, and their dominance/correlation in the overall discrimination. Along with conventional machine learning techniques, the challenge is to consider how different techniques, such as deep neural networks, may perform in the discrimination task.

We compute the asymptotics of Arakelov functions if smooth curves degenerate to semistable singular curves. The motivation was to determine whether the delta function defines a metric on the boundary of moduli space. In fact things are slightly more complicated. The main result states that the asymptotics is mostly governed by the graph associated to the degeneration, with some subleties. The topic has been also treated by R. deJong and my student R. Wilms.

A cohomology class on the diffeomorphism group Diff(M) of a manifold M

can be thought of as a characteristic class for smooth M-bundles.
I will survey a technique for producing examples of such classes,
and then explain how the signature (of 4-manifolds) provides an
obstruction to this technique in dimension 3.

I will define Miller-Morita-Mumford classes and explain how we can
think of them as coming from classes on the cobordism category.
Madsen and Weiss showed that for a surface S of genus g all cohomology
classes
of the mapping class group MCG(S) (of degree < 2(g-2)/3) are MMM-classes.
This technique has been successfully ported to higher even dimensions d= 2n,
but it cannot possibly work in odd dimensions:
a theorem of Ebert says that for d=3 all MMM-classes are trivial.
In the second part of my talk I will sketch a new proof of (a part of)
Ebert's theorem.
I first recall the definition of the signature sign(W) of a 4 manifold W,
and some of its properties, such as additivity with respect to gluing.
Using the signature and an idea from the world of 1-2-3-TQFTs,
I then go on to define a 'central extension' of the three dimensional
cobordism category.
This central extension corresponds to a 2-cocycle on the 3d cobordism
category,
and we will see that the construction implies that the associated MMM-class
has to vanish on all 3-dimensional manifold bundles.

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.