University of Manchester

The infamous Ramanujan tau-function is the starting point for many mysterious conjectures and difficult open problems within the realm of modular forms. In this talk, I will discuss some of our recent results pertaining to odd values of the Ramanujan tau-function. We use a combination of tools which include the Primitive Divisor Theorem of Bilu, Hanrot and Voutier, bounds for solutions to Thue–Mahler equations due to Bugeaud and Gyory, and the modular approach via Galois representations of Frey-Hellegouarch elliptic curves. This is joint work with Mike Bennett (UBC), Adela Gherga (Warwick) and Samir Siksek (Warwick).

Spectral sequences are computational tools to find the (co-)homology of mathematical objects and are used across various fields. In this talk I will focus on the LHS spectral sequence, which we associate to an extension of groups to compute group cohomology. The first part of the talk will serve as introduction to both group cohomology and general spectral sequences, where I hope to provide and intuition and some reduced formalism. As main example, and core of this talk, we will look at the LHS spectral sequence associated to the group extension $(\mathbb{Z}/3\mathbb{Z})^3 \rightarrow S \rightarrow \mathbb{Z}/3\mathbb{Z}$, where $S$ is a Sylow-3-subgroup of $SL_2(\mathbb{Z}/9\mathbb{Z})$. In particular I will present arguments that all differentials on the $E^2$ page vanish.

The famous Birch & Swinnerton-Dyer conjecture predicts that the (algebraic) rank of an elliptic curve is equal to the so-called analytic rank, which is the order of vanishing of the associated L-functions at the central point. In this talk, we shall discuss the analytic rank of automorphic L-functions in an "alternate universe". This is joint work with Kyle Pratt and Alexandru Zaharescu.

Transformation theory has long been known to be a mechanism for 
the design of metamaterials. It gives rise to the required properties of the 
material in order to direct waves in the manner desired.  This talk will 
focus on the mathematical theory underpinning the design of acoustic and 
elastodynamic metamaterials based on transformation theory and aspects of 
the experimental confirmation of these designs. In the acoustics context it 
is well-known that the governing equations are transformation invariant and 
therefore a whole range of microstructural options are available for design, 
although designing materials that can harness incoming acoustic energy in 
air is difficult due to the usual sharp impedance contrast between air and 
the metamaterial in question. In the elastodynamic context matters become 
even worse in the sense that the governing equations are not transformation 
invariant and therefore we generally require a whole new class of materials.

In the acoustics context we will describe a new microstructure that consists 
of rigid rods that is (i) closely impedance matched to air and (ii) slows 
down sound in air. This is shown to be useful in a number of configurations 
and in particular it can be employed to half the resonant frequency of the 
standard quarter-wavelength resonator (or alternatively it can half the size 
of the resonator for a specified resonant frequency) [1].

In the elastodynamics context we will show that although the equations are 
not transformation invariant one can employ the theory of waves in 
pre-stressed hyperelastic materials in order to create natural elastodynamic 
metamaterials whose inhomogeneous anisotropic material properties are 
generated naturally by an appropriate pre-stress. In particular it is shown 
that a certain class of hyperelastic materials exhibit this so-called 
“invariance property” permitting the creation of e.g. hyperelastic cloaks 
[2,3] and invariant metamaterials. This has significant consequences for the 
design of e.g. phononic media: it is a well-known and frequently exploited 
fact that pre-stress and large deformation of hyperelastic materials 
modifies the linear elastic wave speed in the deformed medium. In the 
context of periodic materials this renders materials whose dynamic 
properties are “tunable” under pre-stress and in particular this permits 
tunable band gaps in periodic media [4]. However the invariant hyperelastic 
materials described above can be employed in order to design a class of 
phononic media whose band-gaps are invariant to deformation [5]. We also 
describe the concept of an elastodynamic ground cloak created via pre-stress 
[6].

[1] Rowley, W.D., Parnell, W.J., Abrahams, I.D., Voisey, S.R. and Etaix, N. 
(2018) “Deepening subwavelength acoustic resonance via metamaterials with 
universal broadband elliptical microstructure”. Applied Physics Letters 112, 
251902.
[2] Parnell, W.J. (2012) “Nonlinear pre-stress for cloaking from antiplane 
elastic waves”. Proc Roy Soc A 468 (2138) 563-580.
[3] Norris, A.N. and Parnell, W.J. (2012) “Hyperelastic cloaking theory: 
transformation elasticity with pre-stressed solids”. Proc Roy Soc A 468 
(2146) 2881-2903
[4] Bertoldi, K. and Boyce, M.C. (2008)  “Mechanically triggered 
transformations of phononic band gaps in periodic elastomeric structures”. 
Phys Rev B 77, 052105.
[5] Zhang, P. and Parnell, W.J. (2017) “Soft phononic crystals with 
deformation-independent band gaps” Proc Roy Soc A 473, 20160865.
[6] Zhang, P. and Parnell, W.J. (2018) “Hyperelastic antiplane ground 
cloaking” J Acoust Soc America 143 (5)

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

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.

Blisters form when a thin surface layer of a solid body separates/delaminates from the underlying bulk material over a finite, bounded region. It is ubiquitous in a range of industrial applications, e.g. blister test is applied to assess the strength of adhesion between thin elastic films and their solid substrates, and during natural processes, such as formation and spreading of laccoliths or retinal detachment.

We study a special case of blistering, in which a thin elastic membrane is adhered to the substrate by a thin layer of viscous fluid. In this scenario, the expansion of the newly formed blister by fluid injection occurs via a displacement flow, which peels apart the adhered surfaces through a two-way interaction between flow and deformation. If the injected fluid is less viscous than the fluid already occupying the gap, patterns of short and stubby fingers form on the propagating fluid interface in a radial geometry. This process is regulated by membrane compliance, which if increased delays the onset of fingering to higher flow rates and reduces finger amplitude. We find that the morphological features of the fingers are selected in a simple way by the local geometry of the compliant cell. In contrast, the local geometry itself is determined from a complex fluid–solid interaction, particularly in the case of rectangular blisters. Furthermore, changes to the geometry of the channel cross-section in the latter case lead to a rich variety of possible interfacial patterns. Our experiments provide a link between studies of airway reopening, Saffman-Taylor fingering and printer’s instability.