# Forthcoming Seminars

Please note that the list below only shows forthcoming events, which may not include regular events that have not yet been entered for the forthcoming term. Please see the past events page for a list of all seminar series that the department has on offer.

It is known that every continuum X is a weakly confluent image of a continuum Y which is hereditarily indecomposable and of covering dimension one. We use the ultracoproduct construction to gain information about the number of composants of Y. For example, in ZFC, we can ensure that this number is arbitrarily large. And if we assume the GCH, we can arrange for Y to have as many composants at it has points.

## Further Information:

Stochastic control problems are closely related to free boundary problems, where both the underlying fully nonlinear PDEs and the boundaries separating the action and waiting regions are integral parts of the problems. In this talk, we will propose a class of stochastic N-player games and show how the free boundary problems involve moving boundaries due to the additional game nature. We will provide explicit Nash equilibria by solving a sequence of Skorokhod problems.

For the special cases of resource allocation problems, we will show how players change their strategies based on different network structures between players and resources. We will also talk about the insights from a sharing economy perspective.

This talk is based on a joint work with Xin Guo (UC Berkeley) and Wenpin Tang (UCLA).

Small block overlapping, and non-overlapping, Schwarz methods are theoretically highly attractive as multilevel smoothers for a wide variety of problems that are not amenable to point relaxation methods. Examples include monolithic Vanka smoothers for Stokes, overlapping vertex-patch decompositions for $H(\text{div})$ and $H(\text{curl})$ problems, along with nearly incompressible elasticity, and augmented Lagrangian schemes.

While it is possible to manually program these different schemes, their use in general purpose libraries has been held back by a lack of generic, composable interfaces. We present a new approach to the specification and development such additive Schwarz methods in PETSc that cleanly separates the topological space decomposition from the discretisation and assembly of the equations. Our preconditioner is flexible enough to support overlapping and non-overlapping additive Schwarz methods, and can be used to formulate line, and plane smoothers, Vanka iterations, amongst others. I will illustrate these new features with some examples utilising the Firedrake finite element library, in particular how the design of an approriate computational interface enables these schemes to be used as building blocks inside block preconditioners.

This is joint work with Patrick Farrell and Florian Wechsung (Oxford), and Matt Knepley (Buffalo).

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)