# 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.

This talk is based on joint work with Dominic Joyce and Markus Upmeier. Issues we'd like to talk about are a) the orientability of moduli spaces that

appear in various gauge-theoretic problems; and b) how to orient those moduli spaces if they are orientable. We begin with briefly mentioning backgrounds and motivation, and recall basics in gauge theory such as the Atiyah-Hitchin-Singer complex and the Kuranishi model by taking the anti-self-dual instanton moduli space as an example. We then describe the orientability and canonical orientations of the anti-self-dual instanton moduli space, and other

gauge-theoretic moduli spaces which turn up in current research interests.

The Euler equation describing motion of ideal fluids goes back to 1755.

The analysis of the equation is challenging since it is nonlinear and nonlocal. Its solutions are often unstable and spontaneously generate small scales. The fundamental question of global regularity vs finite time singularity formation

remains open for the Euler equation in three spatial dimensions. In this lecture, I will review the history of this question and its connection with the arguably greatest unsolved problem of classical physics, turbulence. Recent results on small scale and singularity formation in two dimensions and for a number of related models will also be presented.

In this seminar, I first discuss a paper by Aslak et al. on the detection of intermittent communities with the Infomap algorithm. Second, I present own work on the detection of intermittent communities with modularity-maximisation methods.

Many real-world networks represent dynamic systems with interactions that change over time, often in uncoordinated ways and at irregular intervals. For example, university students connect in intermittent groups that repeatedly form and dissolve based on multiple factors, including their lectures, interests, and friends. Such dynamic systems can be represented as multilayer networks where each layer represents a snapshot of the temporal network. In this representation, it is crucial that the links between layers accurately capture real dependencies between those layers. Often, however, these dependencies are unknown. Therefore, current methods connect layers based on simplistic assumptions that do not capture node-level layer dependencies. For example, connecting every node to itself in other layers with the same weight can wipe out dependencies between intermittent groups, making it difficult or even impossible to identify them. In this paper, we present a principled approach to estimating node-level layer dependencies based on the network structure within each layer. We implement our node-level coupling method in the community detection framework Infomap and demonstrate its performance compared to current methods on synthetic and real temporal networks. We show that our approach more effectively constrains information inside multilayer communities so that Infomap can better recover planted groups in multilayer benchmark networks that represent multiple modes with different groups and better identify intermittent communities in real temporal contact networks. These results suggest that node-level layer coupling can improve the modeling of information spreading in temporal networks and better capture intermittent community structure.

Aslak, Ulf, Martin Rosvall, and Sune Lehmann. "Constrained information flows in temporal networks reveal intermittent communities." Physical Review E 97.6 (2018): 062312.

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)