Analytic rank of automorphic L-functions
Abstract
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.
The Annual OCIAM Dinner
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Multicellular Calculus
The lecture will take place in the Michael Dummett Lecture Theatre (Blue Boar quad, Christ Church).
Acoustic and hyperelastic metamaterials – stretching the truth?
Abstract
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)
Strategies for Multilevel Monte Carlo for Bayesian Inversion
Abstract
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.
Multiscale Modelling of Tendon Mechanics
Abstract
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.
Instabilities in Blistering
Abstract
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.
14:15
Multiplicity-free primitive ideals and W-algebras
Abstract
In my talk I will explain how to relate 1-dimensional representations of finite W-algebras with multiplicity free primitive ideals of universal enveloping algebras and representations of minimal dimension of the corresponding reduced enveloping algebras (Humphreys' conjecture). I will also mention some open problems in the field.
14:15
The Missing Meteorites of Antarctica
Abstract
The vast majority of the World's documented meteorite specimens have been collected from Antarctica. This is due to Antarctica’s ice dynamics, which allows for the significant concentration of meteorites onto ice surfaces known as Meteorite Stranding Zones. However, meteorite collection data shows a significant anomaly exists: the proportion of iron-based meteorites are under-represented compared to those found in the rest of the World. Here I explain that englacial solar warming provides a plausible explanation for this shortfall: as meteorites are transported up towards the surface of the ice they become exposed to increasing amounts of solar radiation, meaning it is possible for meteorites with a high-enough thermal conductivity (such as iron) to reach a depth at which they melt their underlying ice and sink back downwards, offsetting the upwards transportation. An enticing consequence of this mechanism is that a sparse layer of meteorites lies just beneath the surface of these Meteorite Stranding Zones...