Past

Certain elastic structures and growing tissues (leaves, flowers or marine invertebrates) exhibit residual strain at free equilibria. We intend to study this phenomena through an elastic growth variational model. We will first discuss this model from a differential geometric point of view: the growth seems to change the intrinsic metric of the tissue to a new target non-flat metric. The non-vanishing curvature is the cause of the non-zero stress at equilibria.

We further discuss the scaling laws and $\Gamma$-limits of the introduced 3d functional on thin plates in the limit of vanishing thickness. Among others, given special forms of growth tensors, we rigorously derive the non-Euclidean versions of Kirchhoff and von Karman models for elastic non-Euclidean plates. Sobolev spaces of isometries and infinitesimal isometries of 2d Riemannian manifolds appear as the natural space of admissible mappings in this context. In particular, as a side result, we obtain an equivalent condition for existence of a $W^{2,2}$ isometric immersion of a given $2$d metric on a bounded domain into $\mathbb R3$.

Lattices in semisimple Lie groups have been studied from the point of view of number theory, algebraic groups, topology and geometry, and geometric group theory. The Fragestellung of one line of investigation is to what extent the properties of the lattice determine, and are determined by, the properties of the group. This talk reviews a number of results about lattices, and in particular looks at Mostow--Margulis rigidity.

We address the effect of extreme geometry on a non-convex variational problem motivated by recent investigations of magnetic domain walls trapped by sharp thin necks. We prove the existence of local minimizers representing geometrically constrained walls under suitable symmetry assumptions on the domains and provide an asymptotic characterization of the wall profile. The asymptotic behavior, which depends critically on the scaling of length and width of the neck, turns out to be qualitatively different from the higher-dimensional case and a richer variety of regimes is shown to exist.

Abstract: Cell migration and growth are essential components of the development of multicellular organisms. The role of various cues in directing cell migration is widespread, in particular, the role of signals in the environment in the control of cell motility and directional guidance. In many cases, especially in developmental biology, growth of the domain also plays a large role in the distribution of cells and, in some cases, cell or signal distribution may actually drive domain growth. There is a ubiquitous use of partial differential equations (PDEs) for modelling the time evolution of cellular density and environmental cues. In the last twenty years, a lot of attention has been devoted to connecting macroscopic PDEs with more detailed microscopic models of cellular motility, including models of directional sensing and signal transduction pathways. However, domain growth is largely omitted in the literature. In this paper, individual-based models describing cell movement and domain growth are studied, and correspondence with a macroscopic-level PDE describing the evolution of cell density is demonstrated. The individual-based models are formulated in terms of random walkers on a lattice. Domain growth provides an extra mathematical challenge by making the lattice size variable over time. A reaction-diffusion master equation formalism is generalised to the case of growing lattices and used in the derivation of the macroscopic PDEs.

Understanding the critical parameters controlling the stability of solidification interfaces in colloidal systems is a necessary step in many domains were the freezing of colloids is present, such as materials science or geophysics. What we understand so far of the solidification of colloidal suspensions is derived primarily from the analogies with dilute alloys systems, or the investigated behaviour of single particles in front of a moving interface and is still a subject of intense work. A more realistic, multi-particles model should account for the particles movement, the various possible interactions between the particles and the multiple interactions between the particles and the solid/liquid cellular interface. In order to bring new experimental observations, we choose to investigate the stability of a cellular interface during directional solidification of colloidal suspensions by using X-ray radiography and tomography. I will present recent experimental results of ice growth (ice lenses) and particle redistribution observations, their implications, and open the discussion regarding the limitations of the technique and the potential for further progress in the field using this approach.

The spectrum of the integers is an affine scheme which number theorists would like to complete to a projective scheme, adding a point at infinity. We will list some reasons for wanting to do this, then gather some hints about what properties the completed object might have. In particular it seems that the desired object can only exist in some setting extending traditional algebraic geometry. We will then present the proposals of Durov and Shai Haran for such extended settings and the compactifications they construct. We will explain the close relationship between both and, if time remains, relate them to a third compactification in a third setting, proposed by Toen and Vaquie.

We will present different ideas leading to and evolving around geometry over the field with one element. After a brief summary of the so-called numbers-functions correspondence we will discuss some aspects of Weil's proof of the Riemann hypothesis for function fields. We will see then how lambda geometry can be thought of as a model for geometry over $\mathbb{F}_\mathrm{un}$ and what some familiar objects should look like there. If time permits, we will

explain a link with stable homotopy theory.

One of the main aims in the theory of percolation is to find the `critical probability' above which long range connections emerge from random local connections with a given pattern and certain individual probabilities. The quintessential example is Kesten's result from 1980 that if the edges of the square lattice are selected independently with probability $p$, then long range connections appear if and only if $p>1/2$.  The starting point is a certain self-duality property, observed already in the early 60s; the difficulty is not in this observation, but in proving that self-duality does imply criticality in this setting.

Since Kesten's result, more complicated duality properties have been used to determine a variety of other critical probabilities. Recently, Scullard and Ziff have described a very general class of self-dual percolation models; we show that for the entire class (in fact, a larger class), self-duality does imply criticality.

We present an alternative viewpoint on recent studies of regularity of solutions to the Navier-Stokes equations in critical spaces. In particular, we prove that mild solutions which remain bounded in the

space $\dot H^{1/2}$ do not become singular in finite time, a result which was proved in a more general setting by L. Escauriaza, G. Seregin and V. Sverak using a different approach. We use the method of "concentration-compactness" + "rigidity theorem" which was recently developed by C. Kenig and F. Merle to treat critical dispersive equations. To the authors' knowledge, this is the first instance in which this method has been applied to a parabolic equation. This is joint work with Carlos Kenig.

We investigate the minimization problem for the variational integral

$$\int_\Omega\sqrt{1+|Dw|^2}\,dx$$

in Dirichlet classes of vector-valued functions $w$. It is well known that

the existence of minimizers can be established if the problem is formulated

in a generalized way in the space of functions of bounded variation. In

this talk we will discuss a uniqueness theorem for these generalized

minimizers. Actually, the theorem holds for a larger class of variational

integrals with linear growth and was obtained in collaboration with Lisa

Beck (SNS Pisa).

Joint work with C. Donati-Martin and D. Feral

This paper considers a portfolio optimization problem in which asset prices are represented by SDEs driven by Brownian motion and a Poisson random measure, with drifts that are functions of an auxiliary diffusion 'factor' process. The criterion, following earlier work by Bielecki, Pliska, Nagai and others, is risk-sensitive optimization (equivalent to maximizing the expected growth rate subject to a constraint on variance.) By using a change of measure technique introduced by Kuroda and Nagai we show that the problem reduces to solving a certain stochastic control problem in the factor process, which has no jumps. The main result of the paper is that the Hamilton-Jacobi-Bellman equation for this problem has a classical solution. The proof uses Bellman's "policy improvement"

method together with results on linear parabolic PDEs due to Ladyzhenskaya et al. This is joint work with Sebastien Lleo.

VC dimension and density are properties of a collection of sets which come from probability theory.  It was observed by Laskowski that there is a close tie between these notions and the model-theoretic property called NIP. This tie results in many examples of collections of sets that have finite VC dimension. In general, it is difficult to find upper bounds for the VC dimension, and known bounds are mostly very large. However, the VC density seems to be more accessible. In this talk, I will explain all of the above acronyms, and present a theorem which gives an upper bound (in some cases optimal) on the VC density of formulae in some examples of NIP theories. This represents joint work of myself with M. Aschenbrenner, A. Dolich, D. Macpherson and S.

Starchenko.

 

 

 

Frank-Read sources are among the most important mechanisms of dislocation multiplication,

and their operation signals the onset of yield in crystals. We show that the critical

stress required to initiate dislocation production falls dramatically at high elastic

anisotropy, irrespective of the mean shear modulus. We hence predict the yield stress of

crystals to fall dramatically as their anisotropy increases. This behaviour is consistent

with the severe plastic softening observed in alpha-iron and ferritic steels as the

alpha − gamma martensitic phase transition is approached, a temperature regime of crucial

importance for structural steels designed for future nuclear applications.

The stability of our Solar System has been debated since Newton devised

the laws of gravitation to explain planetary motion. Newton himself

doubted the long-term stability of the Solar System, and the question

has remained unanswered despite centuries of intense study by

generations of illustrious names such as Laplace, Langrange, Gauss, and

Poincare. Finally, in the 1990s, with the advent of computers fast

enough to accurately integrate the equations of motion of the planets

for billions of years, the question has finally been settled: for the

next 5 billion years, and barring interlopers, the shapes of the

planetary orbits will remain roughly as they are now. This is called

"practical stability": none of the known planets will collide with each

other, fall into the Sun, or be ejected from the Solar System, for the

next 5 billion years.

Although the Solar System is now known to be practically stable, it may

still be "chaotic". This means that we may---or may not---be able

precisely to predict the positions of the planets within their orbits,

for the next 5 billion years. The precise positions of the planets

effects the tilt of each planet's axis, and so can have a measurable

effect on the Earth's climate. Although the inner Solar System is

almost certainly chaotic, for the past 15 years, there has been

some debate about whether the outer Solar System exhibits chaos or not.

In particular, when performing numerical integrations of the orbits of

the outer planets, some astronomers observe chaos, and some do not. This

is particularly disturbing since it is known that inaccurate integration

can inject chaos into a numerical solution whose exact solution is known

to be stable.

In this talk I will demonstrate how I closed that 15-year debate on

chaos in the outer solar system by performing the most carefully justified

high precision integrations of the orbits of the outer planets that has

yet been done. The answer surprised even the astronomical community,

and was published in _Nature Physics_.

I will also show lots of pretty pictures demonstrating the fractal nature

of the boundary between chaos and regularity in the outer Solar System.

The talk is about the homotopy type of configuration spaces. Once upon a time there was a conjecture that it is a homotopy invariant of closed manifolds. I will discuss the strong evidence supporting this claim, together with its recent disproof by a counterexample. Then I will talk about the corrected version of the original conjecture.