Collective neural crest (NC) cell migration determines the formation of peripheral tissues during vertebrate development. If NC cells fail to reach a target or populate an incorrect location, improper cell differentiation or uncontrolled cell proliferation can occur. Therefore, knowledge of embryonic cell migration is important for understanding birth defects and tumour formation. However, the response of NC cells to different stimuli, and their ability to migrate to distant targets, are still poorly understood. Recently, experimental and computational studies have provided evidence that there are at least two subpopulations of NC cells, namely “leading” and “trailing” cells, with potential further differentiation between the cells in these subpopulations [1,2]. The main difference between these two cell types is the mechanism driving motility and invasion: the leaders follow the gradient of a chemoattractant, while the trailing cells follow “gradients” of the leaders. The precise mechanisms underlying these leader-follower interactions are still unclear.

We develop and apply innovative multi-scale modelling frameworks to analyse signalling effects on NC cell dynamics. We consider different potential scenarios and investigate them using an individual-based model for the cell motility and reaction-diffusion model to describe chemoattractant dynamics. More specifically, we use a discrete self-propelled particle model [3] to capture the interactions between the cells and incorporate volume exclusion. Streaming migration is represented using an off-lattice model to generate realistic cell arrangements and incorporate nonlinear behaviour of the system, for example the coattraction between cells at various distances. The simulations are performed using Aboria, which is a C++ library for the implementation of particle-based numerical methods [4]. The source of chemoattractant, the characteristics of domain growth, and types of boundary conditions are some other important factors that affect migration. We present results on how robust/sensitive cells invasion is to these key biological processes and suggest further avenues of experimental research.

[1] R. McLennan, L. Dyson, K. W. Prather, J. A. Morrison, R.E. Baker, P. K. Maini and P. M. Kulesa. (2012). Multiscale mechanisms of cell migration during development: theory and experiment, Development, 139, 2935-2944.

[2] R. McLennan, L. J. Schumacher, J. A. Morrison, J. M. Teddy, D. A. Ridenour, A. C. Box, C. L. Semerad, H. Li, W. McDowell, D. Kay, P. K. Maini, R. E. Baker and P. M. Kulesa. (2015). Neural crest migration is driven by a few trailblazer cells with a unique molecular signature narrowly confined to the invasive front, Development, 142, 2014-2025.

[3] G. Grégoire, H. Chaté and Y Tu. (2003). Moving and staying together without a leader, Physica D: Nonlinear Phenomena, 181, 157-170.

[4] M. Robinson and M. Bruna. (2017). Particle-based and meshless methods with Aboria, SoftwareX, 6, 172-178. Online documentation https://github.com/martinjrobins/Aboria.

The Simonyi Lecture is an annual lecture under the auspices of the Charles Simonyi Professor for the Public Understanding of Science, Marcus du Sautoy. It is not part of the Oxford Mathematics Public Lectures series but its themes and topics touch not only on mathematics but the wider natural sciences and beyond. All are very welcome and there is no need to register.

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In this year’s Simonyi Lecture Geoffrey West discusses universal laws that govern everything from growth to mortality in plants, animals, cities and companies. These remarkable laws originate in the networks that sustain life from circulatory to social systems and help us address big, urgent questions from population explosion, urbanization, lifespan and cancer, to the accelerating pace of life and global sustainability. Why do we stop growing and live about 100 years rather than 1000, or just two like mice? Why do we sleep eight hours a day and not three like elephants? Why do all companies and people die whereas cities keep growing? How are these related to innovation, wealth creation, and “singularities”? And is any of this sustainable? 

Geoffrey West is a theoretical physicist whose primary interests have been in fundamental questions in physics, biology and social organizations  West is a distinguished professor at the Sante Fe Institute, where he served as the president from 2004-2008. He is author of the recent best-selling book 'Scale'.

Answering a question of Milnor, Grigorchuk constructed in the early eighties the
first examples of groups of intermediate growth, that is, finitely generated
groups with growth strictly between polynomial and exponential.
In  joint work with Laurent Bartholdi, we show that under a mild regularity assumption, any function greater than exp(n^a), where `a' is a solution of the equation
  2^(3-3/x)+ 2^(2-2/x)+2^(1-1/x)=2,
is a growth function of some group. These are the first examples of groups
of intermediate growth where the asymptotic of  the growth function is known.
Among applications of our results is the fact that any group of locally subexponential growth
can be embedded as a subgroup of some group of intermediate growth (some of these latter groups cannot be  subgroups in Grigorchuk groups).

In a recent work with Tianyi Zheng, we  provide  near optimal lower bounds
for Grigorchuk torsion groups, including the first Grigorchuk group. Our argument is by a construction of random walks with non-trivial Poisson boundary, defined by 
a measure with power law decay.

Recent tools make it possible to partition the space of rational Dehn 
surgery slopes for a knot (or in some cases a link) in a 3-manifold into 
domains over which the Heegaard Floer homology of the surgered manifolds 
behaves continuously as a function of slope. I will describe some 
techniques for determining the walls of discontinuity separating these 
domains, along with efforts to interpret some aspects of this structure 
in terms of the behaviour of co-oriented taut foliations. This talk 
draws on a combination of independent work, previous joint work with 
Jake Rasmussen, and work in progress with Rachel Roberts.

The problem of "unbounded rank expanders" asks 
whether we can endow a system of generators with a sequence of 
special linear groups whose degrees tend to infinity over quotient rings 
of Z such that the resulting Cayley graphs form an expander family.
Kassabov answered this question in the affirmative. Furthermore, the 
completely satisfactory solution to this question was given by 
Ershov and Jaikin--Zapirain (Invent. Math., 2010);  they proved
Kazhdan's property (T) for elementary groups over non-commutative 
rings. (T) is equivalent to the fixed point property with respect to 
actions on Hilbert spaces by isometries.

We provide a new framework to "upgrade" relative fixed point 
properties for small subgroups to the fixed point property for the 
whole group. It is inspired by work of Shalom (ICM, 2006). Our 
main criterion is stated only in terms of intrinsic group structure 
(but *without* employing any form of bounded generation). 
This, in particular, supplies a simpler (but not quantitative) 
alternative proof of the aforementioned result of Ershov and 
Jaikin--Zapirain.  

If time permits, we will discuss other applications of our result.

I will prove that the knot Floer homology group
HFK-hat(K, g-1) for a genus g fibered knot K is isomorphic to a
variant of the fixed points Floer homology of an area-preserving
representative of its monodromy. This is a joint work with Gilberto
Spano.
 

(joint work with Françoise Dal'Bo and Andrea Sambusetti)

Given a finitely generated group G acting properly on a metric space X, the exponential growth rate of G with respect to X measures "how big" the orbits of G are. If H is a subgroup of G, its exponential growth rate is bounded above by the one of G. In this work we are interested in the following question: what can we say if H and G have the same exponential growth rate? This problem has both a combinatorial and a geometric origin. For the combinatorial part, Grigorchuck and Cohen proved in the 80's that a group Q = F/N (written as a quotient of the free group) is amenable if and only if N and F have the same exponential growth rate (with respect to the word length). About the same time, Brooks gave a geometric interpretation of Kesten's amenability criterion in terms of the bottom of the spectrum of the Laplace operator. He obtained in this way a statement analogue to the one of Grigorchuck and Cohen for the deck automorphism group of the cover of certain compact hyperbolic manifolds. These works initiated many fruitful developments in geometry, dynamics and group theory. We focus here one the class of Gromov hyperbolic groups and propose a framework that encompasses both the combinatorial and the geometric point of view. More precisely we prove that if G is a hyperbolic group acting properly co-compactly on a metric space X which is either a Cayley graph of G or a CAT(-1) space, then the growth rate of H and G coincide if and only if H is co-amenable in G.  In addition if G has Kazhdan property (T) we prove that there is a gap between the growth rate of G and the one of its infinite index subgroups.

This talk is concerned with quantitative periodic homogenization in domains with boundaries. The quantitative analysis near boundaries leads to the study of boundary layers correctors, which have in general a nonperiodic structure. The interaction between the boundary and the microstructure creates geometric resonances, making the study of the asymptotics or continuity properties particularly challenging. The talk is based on work with S. Armstrong, T. Kuusi and J.-C. Mourrat, as well as work by Z. Shen and J. Zhuge