Seminars (GFS)

Growth and form in plant, fungal, and bacterial cells is achieved with two complementary processes: the deposition of new wall material at the cell surface and the mechanical deformation of this material by forces developed within the cell.  To understand how these two processes contribute to cell growth, we have undertaken a biophysical analysis of one major mode of cell morphogenesis called tip growth; whichrevealed that a few basic biophysical principles can explain all essential features of tip growth.  We first showed that the seemingly obscure surface expansion in these cells can be accounted for by the dissipation of elastic energy in the wall.  We also showed that a simple force balance at the cell surface explains the striking ability of these cells to penetrate stiff environments.  Finally, recent experiments suggest that tip-growing cells regulate their shape by controlling the spatial distribution of wall deposition.

What sets the size and shape of an organism? How can robust shapes emerge from stochastic cells? We are addressing these questions by combining experimental and theoretical approches. Plants appear as active viscoelastoplastic materials with spatiotemporally variable mechanical properties.

I will discuss two examples from our recent work in animal and robot locomotion (i.e. self-propulsion via self-deformation). First, our studies of snake locomotion in heterogeneous environments have revealed  new collisional dynamics; we have used these dynamics to infer neuromechanical control templates in desert specialist snakes. That is, we have discovered that when transiting a linear array of posts, certain snakes and snake-like robots passively “scatter” into preferred directions, the extent of which is inversely related to the post spacing; these systems thus mimic diffraction dynamics of subatomic particles. A minimal model predicts that the animal operates in an open-loop scheme, whereby perturbation rejection via hypothesized fixed muscle activation patterns and passive body properties can generate the observed scattering patterns. Second, I will discuss how, inspired by the fact that all metazoans are composed of hierarchically organized living systems (cells), we have begun to construct a robot which is made of other robots. We developed a phototaxing stochastic locomotor composed of simple non-motile robots called “smarticles”. Although no single smarticle can locomote, when confined into a ring, the collective (the “supersmarticle”) diffuses randomly through collisions among continuously self-deforming smarticles and the ring; directed self-propulsion can be effected if a smarticle at the edge becomes inactive via light or sound cues.

Fossil record indicates that the emergence of arches in human ancestral feet coincided with a transition from an arboreal to a terrestrial lifestyle. Propulsive forces exerted during walking and running load the foot under bending, which is distinct from those experienced during arboreal locomotion. I will present mathematical models with varying levels of detail, accompanied by data from human subject experiments and fossilized human ancestral feet, to illustrate a simple function of the transverse arch. Just as we curve a dollar bill in the transverse direction to stiffen it while inserting it in a vending machine, the transverse arch of the human foot stiffens it for bending
deformations. A fundamental interplay of geometry and mechanics underlies this stiffening -- curvature couples the soft out-of-plane bending mode to the stiff in-plane stretching deformation.

Cathal Cummins1, 2, 3 ,a)
Ignazio Maria Viola1, b)
Maddy Seale2,3,4
Daniele Certini1
Alice Macente2
Enrico Mastropaolo4
Naomi Nakayama2, 3, 5, c)

1)
Institute for Energy Systems
School of Engineering
University of Edinburgh, EH9 3DW

2)
Institute of Molecular Plant Sciences
School of Biological Sciences
University of Edinburgh, EH9 3BF

3)
SynthSys Centre for Systems and Synthetic Biology
School of Biological Sciences
University of Edinburgh, EH9 3BF

4)
Institute for Integrated Micro and Nano Systems
Scottish Microelectronics Centre
School of Engineering
University of Edinburgh, EH9 3FF

5)
Centre for Science at Extreme Conditions
School of Biological Sciences
University of Edinburgh, EH9 3BF



a) Electronic mail: Cathal.Cummins[at]ed.ac[dot]uk
b) Electronic mail: I.M.Viola[at]ed.ac[dot]uk
c)Electronic mail: Naomi.Nakayama[at]ed.ac[dot]uk

The fluid mechanical principles that allow a passenger jet to lift off the ground are not applicable to the flight of small plant fruit (the seed-bearing structure in flowering plants). The reason for this is scaling: human flight requires very large Reynolds numbers, while plant fruit have comparatively small Reynolds numbers. At this small scale, there are a variety of modes of flight available to fruit: from parachuting to gliding and autorotation. In this talk, I will focus on the aerodynamics of small plumed fruit (dandelions) that utilise the parachuting mode of flight. If a parachute-type fruit is picked up by the breeze, it can be carried over formidable distances.

Incredibly, these parachutes are mostly empty space, yet they are effectively impervious to the airflow as they descend. In addition, the fruit can become more or less streamlined depending on the environmental conditions. In this talk, I will present results from our numerical and physical modelling that demonstrate how these parachutes achieve such impermeability despite their high porosity. We explore the form and function of the filamentous building blocks of this parachute, which confer the fruit's incredible flight capacity. 

The mammalian airway branches prenatally in a limited number of stereotyped modes. It is well established that various experimental interventions can change the nature of the branching. We have developed several  hypotheses for the mechanisms governing this mode selection, including the potential roles of geometry, mechanics, and transport, and interactions between the three. We developed a suite of models testing the implications of these hypotheses. Our models rule out some hypotheses and support others. 

The indentation of a layer of plastic material by a solid object is a classical problem in plasticity theory. Using the method of sliplines (characteristics) Prandtl provided two key solutions suitable for the indentation of either a relatively shallow or deep layer by a flat plate. In this talk I will summarize how both solutions, and their generalizations, apply in three rather different problems:
1) locomotion through a yield-stress fluid (the viscoplastic version of Taylor's 1951 viscous problem),
2) the formation of footprints, and
3) the washboarding instability of a towed plate above a deformable layer

Dynamic order is observed in natural systems at all length scales, from the schooling of fish to the coordinated beating of cilia and flagella. These systems, including flagella and cilia from the same organism and cell type, often transition between different modes of coordinated motions. For example, Chlamydomonas biflagellates switch from in-phase to anti-phase beating and ependymal cilia periodically change their collective beat orientation. While there is a substantial body of evidence supporting the hypothesis that hydrodynamic interactions provide a robust mechanism for synchrony, little is known about the mechanisms responsible for the transition between multiple synchronization modes. Here, I will present a series of models of increasing level of complexity that examine the emergence of collective coordination in microfluidic systems of swimmers and rotors. I will report new findings on flow-mediated synchrony in chains of swimmers and rotors and I will focus on the existence of bi-stable synchronization modes in systems of rotors. These results could have important implications on understanding the biophysical mechanisms underlying transitions between multiple synchronization modes, as well as on the design and self-assembly of active materials. 

While the genotype provides an instruction set for morphogenesis, how those instructions are converted to shape involves physical patterning as cells change their relative number, size, shape and position in space and time, in conjunction with chemical gradients that they are driven by and in turn drive. We study one of the simplest geometrical motifs in morphogenesis, the elongation of the vertebrate embryo, and show how spatially modulated expression of a specific signaling molecule Fgf8 leads to variations in motility and density. Our experiments and theory show how just a few cellular parameters that control activity and mechanics allow us to quantify the characteristic scale over which elongation occurs and also determines a typical velocity of the elongation of the body.

This work was done in collaboration with Karine Guevorkian, Olivier Pourqui'e and L. Mahadevan.

Soft robotics is an emerging field of research and development. Its goal is to design robots with flexible structure that can deform and change their shape and dimensions continuously. The structure and actuation of soft robotics is greatly inspired by biology, where living creatures across a wide span of scales use soft appendages or a flexible body for manipulation or locomotion - from elephant's trunk and octopus' arm to jellyfish and caterpillar. While the mechanism of biological motion is based on muscle actuation, artificial soft robots require some sort of flexible actuation. A promising approach of soft robotics is actuation by pressurization of embedded fluidic networks. While common, currently, the effects of viscosity are not examined in such configurations, thus limiting the available deformation patterns possible by such actuation.

The aim of the presented work is to analytically and experimentally examine steady and transient deformation of soft actuators by internal viscous flow. We specifically focus on interaction between elastic deflection of a slender beam and viscous flow in a long serpentine channel, embedded within the beam. The embedded network is positioned asymmetrically with regard to the neutral plane, and thus pressure within the channel creates a local moment deforming the beam. We show that by setting appropriate time-varying inlet pressure signal, viscosity enables to increase the possible deformation patterns available to a given actuator geometry and limit the deformation to a section of the actuator. This work connects fluid dynamics to soft robotics research.

Swelling-induced deformations of slender structures occur in many biological and industrial environments, and the shapes and patterns that emerge can vary across many length scales. The dynamics of fluid movement within elastic networks, and the interplay between a structure's geometry and its boundary conditions, play a crucial role in the morphology of growing tissues, the shrinkage of mud and moss, and the curling of cartilage, leaves, and pine cones. We aim to utilize swelling-induced deformations of soft mechanical structures to dynamically shape materials. Adaptive structures that can bend and fold in an origami-like manner provide advanced engineering opportunities for deployable structures, soft robotic arms, mechanical sensors, and rapid-prototyping of 3D elastomers. Swelling is a robust approach to structural change as it occurs naturally in humid environments and can easily be adapted into industrial design. Small volumes of fluid that interact favorably with a material can induce large, dramatic, and geometrically nonlinear deformations. This talk will examine the geometric nonlinearities that occur as slender structures are swollen – surfaces will crease, beams will bend and snap, circular plates will warp and twist, and fibers will coalesce and detach. I will describe the intricate connection between materials and geometry, and present a straightforward means to permanently morph 2D sheets into 3D shapes.

Co-author: Vakhtang PUTKARADZE (University of Alberta)

We derive a geometrically exact theory for flexible tubes conveying fluid. The theory also incorporates the change of the cross section available to the fluid motion during the dynamics. We consider both compressible and incompressible fluids. We proceed to the linear stability analysis and show that our theory introduces important corrections to previously derived results. Based on this theory, we derive a variational discretization of the dynamics based on the appropriate discretization of the fluid’s back-to-labels map, coupled with a variational discretization of the elastic part of the Lagrangian.

Liquid crystals (LCs) are anisotropic, viscoelastic fluids that can be used to direct colloids into organized assemblies with unusual optical, mechanical, and electrical properties. In past studies, the colloids have been sufficiently rigid that their individual shapes and properties have not been strongly coupled to elastic stresses imposed by the LCs. We will discuss how soft colloids (micrometer-sized shells) behave in LCs. We reveal a sharing of strain between the LC and shells, resulting in formation of spindle-like shells and other complex shapes. These results hint at previously unidentified designs of reconfigurable soft materials with applications in sensing and biology. Also to be discussed: related efforts relevant to biolocomotion and an immersed boundary method for computing fluid-structure interactions in a nematic liquid crystal.

Plants and brown algae do not belong to the same lineage. Phylogenetic analysis demonstrates that the divergence between these two clades occurred approximately 1800 million years ago. Their most recent common ancestors were unicellular eukaryotic organisms and the transition to multi-cellularity occurred independently in the two lineages. It is therefore remarkable that similar global morphologies can be observed in both clades. The role of physical laws and evolution in these convergences will be discussed using the Fibonacci spiral organization as a test case. 

In every seashell there is a story. It is the story of the creature – a mollusc – that lived in and built the shell. Through an incremental growth process, the mollusc builds its own house, one layer at a time. It is a process that generates a shell surface with geometrical precision, regularity, and self-similarity, properties that have been observed and appreciated by palaeontologists and geometers alike for centuries, and formed a focus point in D'Arcy Thompson's famous book. In that process, there is a mechanical story as well: the form of the shell is driven by the mechanical interaction of a soft body and the rigid shell which it is itself secreting. We hypothesise that this interaction underlies a wide array of secondary patterns termed ornamentations, including ribs, needle sharp spines, travelling waves, and fractal-like structures. With such an abundance of shapes generated through a relatively simple growth process, the mollusc shell thus provides an excellent case study for morphomechanical pattern formation. And with a fossil record over 500 million years old and 100,000 extant species of shell building mollusc, mollusc shells all together tell a story of change and increasing complexity, making an excellent case study for evolution and the physical processes that govern it. In this talk I will present several chapters of the mollusc’s story and progress we have made in trying to understand the role of solid mechanics in their unique form.

I will discuss two fluid-mechanics problems exploited by biological systems.

First, animals that drink must transport water into the mouth using either a pressure-driven (suction) or inertia-driven (lapping) mechanism. Previous work on cats shows that these mammals lap using a fast motion of the tongue with relatively small acceleration (~1g), in which gravity is balanced with steady inertia in the liquid. Do dogs employ the same physical mechanism to lap? To answer this question, we recorded 19 dogs while lapping and conducted physical modeling of the tongue's ejection mechanism. In contrast to cats, dogs accelerate the tongue upward quickly (~1-4g) to pinch off the liquid column. The amount of liquid extracted from the column depends on whether the dog closes the jaw before or after the pinch-off. Our recordings show that dogs close the jaw at the moment of pinch-off time, enabling them to maximize volume per lap.

Second, several seabirds (e.g. Gannets and Boobies) dive into water at up to 24 m/s as a hunting mechanism. We studied how diving birds survive water impacts because of their beak shape, neck muscles even with a long slender neck. The birds’ slender necks appear fragile but do not crumble under the compression due to high-speed impact. First of all, we use a salvaged bird to resolve plunge-diving phases and the skull and neck anatomical features to generate a 3D-printed skull and to quantify the effect of the neck’s musculature to provide the necessary stability. Then, physical experiments of an elastic beam as a model for the neck attached to a skull-like cone revealed the limits for the stability of the neck during the bird’s dive as a function of impact velocity and geometric factors. We find that the small angle of the bird's beak and the muscles in the neck predominantly reduce the likelihood of injury during a high-speed plunge-dive. Finally, we di scuss maximum diving speeds for humans using our results to elucidate injury avoidance. 

In this talk I will describe our research involving laser-matter interaction in surfactant modified fluid droplets using theory, experiment and simulations. 

Co-authors: Sebastian Reuther (TU Dresden), Sebastian Aland (HTW Dresden), Ingo Nitschke (TU Dresden), Simon Praetorius (TU Dresden), Michael Nestler (TU Dresden)

We consider continuum models for positional and orientational order on curved surfaces. They include surface phase field crystal models in the first case [4,6] and surface Navier-Stokes [2,3,5], surface Frank-Oseen [1] and surface Landau-deGenne models for the second case. We demonstrate the emergence of topological defects in these models and show the strong interplay between topology, geometry, dynamics and defect type and position. We comment on the derivation of these models and their numerical solution. To couple these surface models with an evolution equation for the shape of the surface is work in progress and leads to defect mediated morphologies [6].

[1] M. Nestler, I. Nitschke, S. Praetorius, A. Voigt: Orientational order on surfaces - the coupling of topology, geometry and dynamics. Journal of Nonlinear Science DOI:10.1007/s00332-017-9405-2 [2] I. Nitschke, S. Reuther, A. Voigt: Discrete exterior calculus (DEC) for the surface Navier-Stokes equation. In Transport Processes at Fluidic Interfaces. Birkhäuser, Eds. D. Bothe, A.Reusken, (2017), 177 - 197 [3] S. Reuther, A. Voigt: The interplay of curvature and vortices in flow on curved surfaces. Multiscale Model. Simul., 13 (2), (2015), 632-643 [4] V. Schmid, A. Voigt: Crystalline order and topological charges on capillary bridges. Soft Matter, 10 (26), (2014), 4694-4699 [5] I. Nitschke, A. Voigt, J. Wensch: A finite element approach to incomressible two-phase flow on manifolds. J. Fluid Mech., 708 (2012), 418-438 [6] S. Aland, A. Rätz, M. Röger, A. Voigt: Buckling instability of viral capsides - a continuum approach. Multiscale Model. Simul., 10 (2012), 82-110

Co-author: Renzo Ricca (University of Milano-Bicocca)

Reconnection is a fundamental event in many areas of science, from the interaction of vortices in classical and quantum fluids, and magnetic flux tubes in magnetohydrodynamics and plasma physics, to recombination in polymer physics and DNA biology. By using fundamental results in topological fluid mechanics, the total helicity of a linked configuration of flux tubes can be calculated in terms of linking, writhe and twist contributions. We prove that writhe helicity is conserved under anti-parallel reconnection [1]. We discuss the Seifert framing (isophase surfaces in GPE models) for a link. We give necessary and sufficient conditions for the existence of a Seifert surface for a framed link. We give a rigorous topological proof of the result that total helicity is zero for linked vortices with Seifert framing. We will discuss parallels between the links in the Belusov-Zhabotinsky reaction and links in fluid dynamics. This is joint work with Renzo Ricca.

[1] Laing, C.E., Ricca, R.L. & Sumners, De W. L. (2015) Conservation of writhe helicity under anti-parallel reconnection. Nature Sci. Rep. 5, 9224.

Co-author: Simone Zuccher (U. Verona)

Time evolution and interaction of filamentary structures is often studied by analysing dynamics in terms of local forces. An alternative route is to investigate physical or biological properties by focussing on geometric and topological properties of the surface swept out by filament motion. In this work we present new results on the evolution, interaction and decay of a Hopf link of quantum vortices governed by the Gross-Pitaevskii equation by analysing physical information in terms of the iso-phase Seifert surface swept out during the process [1]. We interpret the surface local twist as an axial flow acting along the vortex filament [2] and the Seifert surface of minimal area in terms of linear momentum of the system. We show that GP evolution is associated with a continuous minimisation of this surface in agreement with the physical cascade process observed. This approach sheds new light on filament dynamics and bears similarities to the study of fluid membranes in biological and chemical systems.

[1] Zuccher, S. & Ricca, R.L. (2017) Relaxation of twist helicity in the cascade process of linked quantum vortices. Phys. Rev. E 95, 053109.
[2] Zuccher, S. & Ricca, R.L. (2017) Twist effects in quantum vortices and phase defects. Fluid Dyn. Res., doi.org/10.1088/1873-7005/aa8164.

The swimming of a simple vertebrate, the lamprey, can shed light on the coupling of neural signals to muscle mechanics and passive body dynamics in animal locomotion. We will present recent progress in the development of a computational model of a lamprey with sensory feedback and examine the emergent swimming behavior of the coupled fluid-neural-muscle-body system. A long-term goal of this research is to investigate the relation of spinal chord injuries to movement in a very simple system. In addition, even though the fish's material properties most likely have a strong effect on swimming performance, it is extremely difficult to use animal experiments to identify its role. While one species of fish may be stiffer than another, they also typically differ in numerous other ways, such as the anatomy of the muscle and skeleton and the way they activate their muscles during swimming. We will discuss how computational and robotic models offer a more controlled way to probe the effects of body stiffness.

Some of the most beautiful and easy-to-produce instabilities in fluid mechanics
are those that occur when a thin stream of viscous fluid like honey falls steadily
from a certain height onto a solid surface. In addition to the familiar 'liquid rope
coiling' effect, one can observe periodic folding with or without rotation of the
folding plane; periodic collapse and rebuilding of the hollow cylinder formed by a
primary coiling instability; and 'liquid supercoiling', in which the cylinder as a
whole undergoes steady secondary folding and rotation. Using a combination of
laboratory experiments, analytical theory, and numerical simulation, I and my
colleagues have determined a phase diagram for these states in the space of
dimensionless fall height and flow rate, and have identified the dimensionless
parameter that controls which state or states are observed under given
conditions. We have also studied pattern formation in the closely related 'fluid
mechanical sewing machine’ (FMSM), wherein a viscous thread falling onto a
moving belt generates a wealth of complicated 'stitch' patterns including
meanders, alternating loops, and doubly periodic patterns. We have determined
experimentally and numerically the phase diagram for these patterns in the
space of dimensionless fall height and belt speed, and have formulated a simple
reduced (three degrees of freedom) model that successfully predicts the patterns
in the limit of negligible inertia. In closing, I shall compare the observed FMSM
patterns with those of the ‘elastic sewing machine’ in which a normal elastic
thread falls onto a moving belt.

In 1917, the Scottish scientist D’Arcy Wentworth Thompson published “On Growth and Form”. In this book, d’Arcy Thompson, through many beautifully illustrated examples, studies the variety of forms appearing in nature from a mechanistic and mathematical perspective. Hailed as “the finest work of literature in all the annals of science that have been recorded in the English tongue” and criticised as “a work widely praised, but seldom used”, it still stands, a hundred years later, as a masterpiece despite some obvious shortcomings. What is the real influence and impact of this book? How did it shape questions and methodologies in modern science? In this talk, I will delve into “On Growth and Form,” extract its main themes, and show how d’Arcy Thompson’s ideas are still relevant today to unravel the physical basis of morphogenesis.

The word “shape" denotes the external form, contour or outline of something. Shape is a basic physical property of objects and plays an important role in applications, from evolutionary biology to medical image analysis. In this talk, I will give an overview of the ideas underlying shape analysis and computational analysis, the mathematical methods and the questions one tries to answer.

Based on collaborations with M.Bauer, M.Bruveris, P.Harms. For a compact manifold $M^m$ equipped with a smooth fixed background Riemannian metric $\hat g$ we consider the space $\operatorname{Met}_{H^s(\hat g)}(M)$ of all Riemannian metrics of Sobolev class $H^s$ for real $s>\frac m2$ with respect to $\hat g$. The $L^2$-metric on $\operatorname{Met}_{C^\infty}(M)$ was considered by DeWitt, Ebin, Freed and Groisser, Gil-Medrano and Michor, Clarke. Sobolev metrics of integer order on $\operatorname{Met}_{C^\infty}(M)$ were considered in [M.Bauer, P.Harms, and P.W. Michor: Sobolev metrics on the manifold of all Riemannian metrics. J. Differential Geom., 94(2):187-208, 2013.] In this talk we consider variants of these Sobolev metrics which include Sobolev metrics of any positive real (not integer) order $s$. We derive the geodesic equations and show that they are well-posed under some conditions and induce a locally diffeomorphic geodesic exponential mapping.

Joint work with Stefan Sommer Alexis Arnaudon and Line Kuhnel. Performing statistical inference of non-linear Manifold valued data has wide ranging applications in wide ranging fields including bioinformatics, shape analysis, medical imaging, computational anatomy, computer vision, and information geometry. Most common existing statistical inference techniques assume that the Manifold is a Riemannian Manifold with a pre defined canonical metric. In this talk I will present some of our recent work in estimating the Metric structure of the manifold.

We consider mathematical models for tissue growth. These models describes the dynamics of the density of cells due to pressure forces and proliferation. It is known that some cell population models of this kind converge at the incompressible limit towards a Hele-Shaw type free boundary problem. The first model introduce a non-overlapping constraint of a population choosing a singular pressure law. The second model represents two interacting populations of cells which avoid mixing. Following earlier works, we show that the models approximate a free boundary Hele Shaw type model that we characterise using both analytical and numerical argument.

Co-authors: Richard Brown (Massey University), Robert McLachlan (Massey University)

A classic problem in image processing is to recognise the similarity or planar objects (point sets, curves, or images) up to transformations from a local planar group such as the Euclidean, similarity, and projective groups. Building on Cartan’s solution to the equivalence problem, an influential new paradigm for this problem was introduced by Calabi et al., the differential invariant signature. The general theory has been developed extensively and many examples computed for planar curves, including the Euclidean, equi-affine, and projective groups. In this talk we demonstrate how to develop and apply differential invariant signatures for planar images.

In the field of computational anatomy, the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework has proved to be highly efficient for addressing the problem of modeling and analysis of the variability of populations of shapes, allowing for the direct comparison and quantization of diffeomorphic morphometric changes. However, the analysis of medical imaging data also requires the processing of more complex changes, which especially appear during growth or aging phenomena. The observed organisms are subject to transformations over time that are no longer diffeomorphic, at least in a biological sense. One reason might be a gradual creation of new material uncorrelated to the preexisting one. The evolution of the shape can then be described by the joint action of a deformation process and a creation process.
For this purpose, we offer to extend the LDDMM framework to address the problem of non diffeomorphic structural variations in longitudinal data. We keep the geometric central concept of a group of deformations acting on embedded shapes. The necessity for partial mappings leads to a time-varying dynamic that modifies the action of the group of deformations. Ultimately, growth priors are integrated into a new optimal control problem for assimilation of time-varying surface data, leading to an interesting problem in the field of the calculus of variations where the choice of the attachment term on the data, current or varifold, plays an unexpected role.

The underlying minimization problem requires an adapted framework to consider a new set of cost functions (penalization term on the deformation). This new model is inspired by the deployment of animal horns and will be applied to it. 

Keywords: computational anatomy, growth model, shape spaces, Riemannian metrics, group of diffeomorphisms, large deformations, variational methods, optimal control. 

The analysis of object data is becoming common, where example objects under study include functions, curves, shapes, images or trees. Although the applications can be very broad, the common ingredient in all the studies is the need to deal with geometrical invariances. For the simple example of landmark shapes, one can specify a model for the landmark co-ordinates (in the Top Space) and then consider the marginal distribution of shape after integrating out the invariance transformations of translation, rotation and scale. An alternative approach is to optimize over translation, rotation and scale, and carry out modelling and analysis in the resulting Quotient Space. We shall discuss several examples, including functional alignment of growth curves via diffeomorphisms, molecule matching, and 3D face regression where translation and rotation are removed. Bayesian inference is developed and the Top space versus Quotient space approaches are compared.

Shape analysis requires methods for measuring distances between shapes, to define summary statistics, for example, or Gaussian-like distributions. One way to construct such distances is to specify a Riemannian metric on an appropriate space of maps, and then define shape distance as geodesic distance in a quotient space. For shapes in two dimensions, the 'elastic metric' combines tractability with intuitive appeal, with special cases that dramatically simplify computations while still producing state of the art results. For shapes in three dimensions, the situation is less clear. It is unknown whether the full elastic metric admits simplifying representations, and while a reduced version of the metric does, the resulting transform is difficult to invert, and its usefulness has therefore been questionable. In this talk, I will motivate the elastic metric for shapes in three dimensions, elucidate its interesting structure and its relation to the two-dimensional case, and describe what is known about the representation used to construct it. I will then focus on the reduced metric. This admits a representation that greatly simplifies computations, but which is probably not invertible. I will describe recent work that constructs an approximate right inverse for this representation, and show how, despite the theoretical uncertainty, this leads in practice to excellent results in shape analysis problems. This is joint work with Anuj Srivastava, Sebastian Kurtek, Hamid Laga, and Qian Xie.

Computational Anatomy studies the normal and pathological variations of organs' shapes, often with respect to a mean organ shape called the template shape. Estimating the template shape is then the first step of the analysis. We use tools from geometric statistics to demonstrate the asymptotic biasedness of the “Frechet mean algorithm”, also called "Max-max algorithm", used for template shape estimation. The geometric intuition provided by this study leads us to suggest two debiasing procedures that we compare. Our results are illustrated on synthetic and real data sets. This is joint work with Dr. Xavier Pennec and Pr. Susan Holmes.

In recent years, several methods and frameworks have been developed to deform images in the aim of solving problems such as shape analysis or image registration. The application and usefulness of these methods is now well established as well as their mathematical foundation. Nevertheless, because various uncertainties remain at all stages of the image analysis procedure (from data capture to intrinsic variability within a dataset), practical extensions of these deterministic methods should be available. In this talk, I will walk you through one of them, by starting from the geometrical formulation of the theory of large deformation matching, then implementing a particular type of stochastic deformation to preserve the original geometrical structure to end with the description of practical methods for the estimation of the unknown noise parameters of the model. I will illustrate this theory with numerical solutions for a discrete representation of images, where the method of moments can easily be implemented to solve this inverse problem of estimating the noise parameters. This is joint work with Stefan Sommer and Darryl Holm.

We have seen great progress during the last ten years in the understanding of the riemaniann framework on shape spaces, also supported by the steady exponential growth of the computational resources. The outcome is that the modeling possibilities for shapes ensembles are now fairly large. However, it is not that clear how to address concretely the selection of different metric structures for the analysis of a shape ensembles. In this talk, we will discuss some simple geometrical point of view going in that direction and with interesting links with matrix completion. Joint work with David Jacobs, Benjamin Charlier and J. Feydy.

Longitudinal or cross-sectional differences in shape and volume has traditionally been measured as difference in shape, but recent methodologies use a dense deformation field as a large deformation diffeomorphic metric mapping or a stationary velocity field. For shapes properties of they changes along the finite flow can be obtained using surface integrals with numerical advantages. Vi show examples from longitudinal changes of brain MRIs.

The beautiful, intricate, and widely diverse shapes of snail shells have fascinated people for centuries. In particular, shells have been analysed by mathematicians wanting to understand their geometric properties. One of the greatest contributors to the field is D'Arcy Thompson with his book On Growth and Form. This book celebrates one hundred years this year, and has played a major role in the field of morphometric analysis.
One well established and simple way of visualising snail shells is built on Raup's growth model, using logarithmic equations with three growth parameters. However, analysing a shell to find the correct values for the growth parameters is not always a straightforward task. 

The goal of my research is to develop a method for obtaining the growth parameters from 2D images of shells. I will test my methods with a set of images of the species Littorina saxatilis. This species is biologically interesting because strong natural selection maintains different shell shapes in distinct environments. It would therefore be an advantage to have a way of describing this natural shape variation, and the shapes of laboratory hybrids, in terms of meaningful growth parameters. This is expected to give a better understanding of the operation of selection and of the underlying genetic basis of shape variation, compared to more classical PCA analysis of landmarks.

In this talk, we present the so-called Wasserstein-Fisher-Rao metric (also called Hellinger-Kantorovich) by its dynamical and static formulation. The link between these two formulations is made clear by generalizing the Riemannian submersion of Otto to this new setting. Then the link with the Camassa-Holm equation can be made with this metric, in the same way Brenier made it between optimal transport and incompressible Euler. Passing by, we prove that the Camassa-Holm equation is actually an incompressible Euler equation on a bigger space. We also show the use of this metric to interpret a particular Hele-Shaw model as a gradient flow. We then finish with some examples of use of this new metric as a similarity measure on diffeomorphic registration of shapes.

Ancient oriental mirrors possess a property that seemed magical and was probably unintended by those who made them: the pattern embossed on the back of such a mirror appears in light reflected onto a screen from its apparently featureless front surface. In reality, the embossed pattern is reproduced on the front, in low relief invisible to direct observation, and analysis shows that the projected image results from pre-focal ray deviation. In this interesting regime of geometrical optics, the image intensity is given simply by the Laplacian of the height function of the relief. Observation confirms this ‘Laplacian image’ interpretation, and indicates that steps on the reflecting surface are about 400 nm high, explaining their invisibility. Current research aims to create the transparent analogue of the magic mirror: ‘magic windows’, in which glass sheets, flat to unaided vision but with gentle surface relief, concentrate light onto a screen with intens ity reproducing any desired image. Laplacian image theory implies that the desired surface relief is obtained by solving Poisson’s equation.

Related Links

• https://michaelberryphysics.wordpress.com

Co-authors: Shufang Lu (Zhejiang University of Technology), Xiaogang Jin (Zhejiang University), Fei Gao (Zhejiang University of Technology), Xiaoyang Mao (University of Yamanashi)

Ink marbling refers to techniques for creating intricate designs in colored inks floating on a liquid surface. If the marbling motions are executed slowly, then this layer of inks can be modeled as a two-dimensional incompressible Newtonian fluid. In this highly constrained model many common marbling techniques can be exactly represented by closed-form homeomorphisms. These homeomorphisms can be composed and compute the composite mapping at any resolution orders of magnitude faster than finite-element fluid-dynamics methods.Pictorial designs for flowers and animals use short strokes of a single stylus; presented is a closed form velocity field for Oseen fluid flow and its application to creating short stroke marbling homeomorphisms.These approaches extend to three-dimensions and allow creation of coherent marbled surfaces on any shape while avoiding the process of texture mapping entirely.

Related Links

• https://www.computer.org/csdl/mags/cg/2012/06/mcg2012060026.html - Mathematical Marbling
• https://www.youtube.com/watch?v=ZgVbIaKhC_4 - Mathematical Marbling video
• https://www.computer.org/csdl/mags/cg/2017/02/mcg2017020090.html - Solid Mathematical Marbling
• https://arxiv.org/pdf/1702.02106 - Oseen Flow in Ink Marblnig
• http://people.csail.mit.edu/jaffer/Marbling - Mathematical Marbling website

Co-author: Sandra Zetina (Universidad Nacional Autonoma de Mexico)

Painting is a fluid mechanical process. The action of covering a solid surface with a layer of a viscous fluid is one of the most common human activities; virtually all manmade surfaces are covered with a layer of fluid, which eventually cures and solidifies, to provide protection against the environment or simply for decoration. The process of applying layer of fluid of uniform thickness on a surface has been vastly studied and it is well understood. In the case of artistic painting, the objective is different. Painters learn how to manipulate the fluid, through lengthy empirical testing of the action and the physical properties of the fluids, to create textures that can be used to create patterns and compositions of aesthetic value. In other words, artists aim to create non uniform paint coatings, produced at will and in a controlled manner. It has been recently identified that, for the case of modern artistic painting, one successful way to create such patterns is by provo king hydrodynamic instabilities in a controlled manner. In this presentation we analyze several particular cases used by notable modern artists in their works: David A. Siqueiros used the Rayleigh-Taylor instability for his ‘accidental painting’ technique; Jackson Pollock learned to control the curling instability of viscous filaments in his dripping technique; Max Ernst used the Saffman-Taylor instability to paint with decalcomanias, etc. Furthermore, we analyze other modern painting techniques and their relation with modern and very active fluid mechanics areas of research. We also discuss the importance of the properties of modern materials and how their evolution could have influenced the emergence of new artistic painting techniques. The aim of this investigation is to create an explicit relation between the body of knowledge of modern fluid mechanics and those of art history and conservation.

Related Links

• http://www.iim.unam.mx/zenit/research.html - Prof. Zenit's research page.

A piece of paper falls in a seemingly erratic manner. Each fall is a solution to the Navier-Stokes equations, but why does it evoke such a poetic feelings in us? When our eyes trace the paper as it falls, following its flutter and tumble, punctuated by a sudden lift and turn, we can feel lines of musical phrases in air. Some motions have the sound of percussion, others of a flute, a string, or even a cry or laugh. Drop small ones en mass, and they become fireworks.

I started making ‘Music of Falling Paper’ a few years ago. It is an attempt to use falling paper both as a ‘music instrument’ and a visual means to convey the connection between the movement of simple objects and the movement of living organisms. They are improvisational pieces, often in collaboration with musicians, in public space and in response to the theme of the event. I drop pieces of paper from a height, the musicians improvise, and I in turn respond to their play when choosing the next sequence to drop. When constructing the sequences, I think about many things. I shall share some of these thoughts and clips at the talk.

Related Links

• https://gfm.aps.org/meetings/dfd-2014/5401254d69702d0771600100 - Music of Falling Paper - video

Folding is a universal principle that appears both in nature and artifacts. Folding can enable various shapes from thin sheets through self-constrained complex kinematic motion. The speaker talks about computational origami, i.e., the geometry and the algorithm of origami, and its application to design. The topics involve origamizer, a universal algorithm to fold any polyhedral shape from a sheet of paper; rigid origami, the kinematics of plate and hinges utilizing its flat and singular state; computational (and actual) hardness of folding compared to unfolding; and metamaterials exploiting the geometry of origami.

I was an art major as an undergraduate before being lured into biomechanics and biofluiddynamics by the beauty of natural forms, both living and physical.  Since I have done both art and science, I see two striking similarities between the two pursuits.   One is careful observation of the natural world  (but scientists have tools that let us "see" more than artists can see with their eyes alone).  The other is the use of abstraction to capture the essence of something (objects, processes), although the "language" used to communicate the abstraction is visual for the artist, while it is also mathematical for the scientist.  Most of this workshop focuses on how science and mathematics can inform and inspire art, but I will provide some examples from my own work of how collaboration with artists can aid scientific research.

Virtual image restoration, also called image inpainting, denotes the process whereby missing or occluded parts in images are filled in based on the information provided by the intact parts of the image. In this talk I will sketch and motivate different mathematical principles that can guide a digital restoration attempt. Digital photographs of art pieces are essentially mathematical objects, and this puts the vast toolbox of mathematics at the restorers’ fingertips. We will encounter the role of differential equations and patch-based methods for virtually restoring structure, texture and colour in images. In particular, we will show examples from the restoration of the Neidhart frescoes (Tuchlauben, Vienna), the restoration of a painting by Sebastiano Del Piombo (the Hamilton Kerr Institute, The Fitzwilliam Museum), and the unearthing of hidden structures in illuminated manuscripts revealed by infrared imaging (the MINIARE project, the Fitzwilliam Museum). After a critical discussion of restoration results I will conclude by pointing out the capabilities and limitations of digital restoration methods, and provide some hints towards applications of such mathematical approaches that go beyond the restoration of arts – such as medicine, forensics and geography.

Equations describing the rolling of a spherical ball on a horizontal surface are obtained, the motion being activated by an internal rotor driven by a battery mechanism. The rotor is modelled as a point mass mounted inside a spherical shell, and caused to move in a prescribed circular orbit relative to the shell. The system is described in terms of four independent dimensionless parameters. The equations governing the angular momentum of the ball relative to the point of contact with the plane constitute a six-dimensional, non-holonomic, non-autonomous dynamical system with cubic nonlinearity. This system is decoupled from a subsidiary system that describes the trajectories of the center of the ball. Numerical integration of these equations for prescribed values of the parameters and initial conditions reveals a tendency towards chaotic behaviour as the radius of the circular orbit of the point mass increases (other parameters being held constant). It is further shown that there is a range of values of the initial angular velocity of the shell for which chaotic trajectories are realised while contact between the shell and the plane is maintained. The predicted behaviour has been observed in our experiments.  

Work in collaboration with V.A.Vladimirov and K. Ilin

Spherical neodymium-iron-boron magnets are marketed as toys as they can be assembled into different shapes due to their high magnetic strength. In particular, we consider two simple structures, chains and cylinders of magnets. By manipulating these structures, it quickly appears that they exhibit an elastic response to small deformations. Indeed, chains buckle on their own weight, rings oscillate, and cylinders resist bending but recover their shape after poking. A natural question is then to understand the response of these structures based on the individual physical properties of the magnets and to understand to what extent they behave elastically. In this talk, I will show through illustrative experiments and simple model calculations that the idea of an effective magnetic bending stiffness is, in fact, an excellent macroscopic characterisation for the mechanical response of magnetic chains. I will then propose a more rigorous approach of the problem by considering discrete-to-continuum asymptotic analysis to derive a continuum model for the energy of a deformed chain of magnets based on the magnetostatic interactions between individual spheres.

2D flows around neutral buoyancy wedge submerged in a tank with continuously stratified fluid were calculated using conventional time-dependent governing equations set in frame of OpenFOAM codes. Fine structure of different variables fields (density or pressure and their gradients, velocity, vorticity, rate of energy dissipation) was analyzed in wide range of the problem geometry and stratification. Numerical results are compared with data of schlieren visualization of the self-moving wedge in the laboratory tank.

How do you design a coin that lands on its edge 1/3 of the time ? How might one throw accurately ? How does one walk along a straight line ? How can one visualize chance ? Each of these problems invokes geometric concepts in a probabilistic setting. I will discuss solutions to some of these problems that lie at the intersection of cognition, neuroscience and behavior -  and thus are of relevance to art and science.

Sporting activities provide great examples of physics in action. Participants and observers are often intrigued by the design or motion of a particular piece of sporting equipment. Physicists are intrigued by these same questions. Topics that will be considered in the presentation will include the behavior of a curling rock, the design of a clubhead in the game of golf, and a look at the physics behind the variety of pitches thrown in the game of baseball.