Past

We present a new method to establish the rotational symmetry

of solutions to overdetermined elliptic boundary value

problems. We illustrate this approach through a couple of

classical examples arising in potential theory, in both the

exterior and the interior punctured domain. We discuss how

some of the known results can be recovered and we introduce

some new geometric overdetermining conditions, involving the

mean curvature of the boundary and the Neumann data.

The Balog-Szemeredi-Gowers theorem states that, given any finite subset of an abelian group with large additive energy, we can find its large subset with small doubling constant. We can ask how this constant depends on the initial additive energy. In the talk, I will give an upper bound, mention the best existing lower bound and, if time permits, present an approach that gives a hope to improve the lower bound and make it asymptotically equal to the upper bound from the beginning of the talk.

Knot Floer homology (introduced by Ozsvath-Szabo and independently by

Rasmussen) is a powerful tool for studying knots and links in the 3-sphere. In

particular, it gives rise to a numerical invariant, which provides a

nontrivial lower bound on the 4-dimensional genus of the knot. By deforming

the definition of knot Floer homology by a real number t from [0,2], we define

a family of homologies, and derive a family of numerical invariants with

similar properties. The resulting invariants provide a family of

homomorphisms on the concordance group. One of these homomorphisms can be

used to estimate the unoriented 4-dimensional genus of the knot. We will

review the basic constructions for knot Floer homology and the deformed

theories and discuss some of the applications. This is joint work with

P. Ozsvath and Z. Szabo.

In this talk, I will introduce and analyse an elastic

surface energy recently introduced by G. Napoli and

L. Vergori to model thin films of nematic liquid crystals.

As it will be clear, the topology and the geometry of

the surface will play a fundamental role in understanding

the behavior of thin films of liquid crystals.

In particular, our results regards the existence of

minimizers, the existence of the gradient flow

of the energy and, in the case of an axisymmetric

toroidal particle, a detailed characterization of global and local minimizers.

This last item is supplemented with numerical experiments.

This is a joint work with M. Snarski (Brown) and M. Veneroni (Pavia).

A 1-dimensional asymptotic class (Macpherson-Steinhorn) is a class of finite structures which satisfies the theorem of Chatzidakis-van den Dries-Macintyre about finite fields: definable sets are assigned a measure and dimension which gives the cardinality of the set asymptotically, and there are only finitely many dimensions and measures in any definable family. There are many examples of these classes, and they all have reasonably tame theories. Non-principal ultraproducts of these classes are supersimple of finite rank.

Recently this definition has been generalised to `Multidimensional Asymptotic Class' (joint work with Macpherson-Steinhorn-Wood). This is a much more flexible framework, suitable for multi-sorted structures. Examples are not necessarily simple. I will give conditions which imply simplicity/supersimplicity of non-principal ultraproducts.

An interesting example is the family of vector spaces over finite fields with a non-degenerate bilinear form (either alternating or symmetric). If there's time, I will explain some joint work with Kestner in which we look in detail at this class.

We consider a contracting problem in which a principal hires an agent to manage a risky project. When the agent chooses volatility components of the output process and the principal observes the output continuously, the principal can compute the quadratic variation of the output, but not the individual components. This leads to moral hazard with respect to the risk choices of the agent. Using a very recent theory of singular changes of measures for Ito processes, we formulate the principal-agent problem in this context, and solve it in the case of CARA preferences. In that case, the optimal contract is linear in these factors: the contractible sources of risk, including the output, the quadratic variation of the output and the cross-variations between the output and the contractible risk sources. Thus, path-dependent contracts naturally arise when there is moral hazard with respect to risk management. This is a joint work with Nizar Touzi (CMAP, Ecole Polytechnique) and Jaksa Cvitanic (Caltech).

A perturbative method is introduced to analyze shear bands formation and

development in ductile solids subject to large strain.

Experiments on discrete systems made up of highly-deformable elements [1]

confirm the validity of the method and suggest that an elastic structure

can be realized buckling for dead, tensile loads. This structure has been

calculated, realized and tested and provides the first example of an

elastic structure buckling without elements subject to compression [2].

The perturbative method introduced for the analysis of shear bands can be

successfuly employed to investigate other material instabilities, such as

for instance flutter in a frictional, continuum medium [3]. In this

context, an experiment on an elastic structure subject to a frictional

contact shows for the first time that a follower load can be generated

using dry friction and that this load can induce flutter instability [4].

The perturbative approach may be used to investigate the strain state near

a dislocation nucleated in a metal subject to a high stress level [5].

Eshelby forces, similar to those driving dislocations in solids, are

analyzed on elastic structures designed to produce an energy release and

therefore to evidence configurational forces. These structures have been

realized and they have shown unexpected behaviours, which opens new

perspectives in the design of flexible mechanisms, like for instance, the

realization of an elastic deformable scale [6].

[1] D. Bigoni, Nonlinear Solid Mechanics Bifurcation Theory and Material

Instability. Cambridge Univ. Press, 2012, ISBN:9781107025417.

[2] D. Zaccaria, D. Bigoni, G. Noselli and D. Misseroni Structures

buckling under tensile dead load. Proc. Roy. Soc. A, 2011, 467, 1686.

[3] A. Piccolroaz, D. Bigoni, and J.R. Willis, A dynamical interpretation

of flutter instability in a continuous medium. J. Mech. Phys. Solids,

2006, 54, 2391.

[4] D. Bigoni and G. Noselli Experimental evidence of flutter and

divergence instabilities induced by dry friction. J. Mech. Phys.

Solids,2011,59,2208.

[5] L. Argani, D. Bigoni, G. Mishuris Dislocations and inclusions in

prestressed metals. Proc. Roy. Soc. A, 2013, 469, 2154 20120752.

[6] D. Bigoni, F. Bosi, F. Dal Corso and D. Misseroni, Instability of a

penetrating blade. J. Mech. Phys. Solids, 2014, in press.

We describe discretisations of the shallow water equations on

the sphere using the framework of finite element exterior calculus. The

formulation can be viewed as an extension of the classical staggered

C-grid energy-enstrophy conserving and

energy-conserving/enstrophy-dissipating schemes which were defined on

latitude-longitude grids. This work is motivated by the need to use

pseudo-uniform grids on the sphere (such as an icosahedral grid or a

cube grid) in order to achieve good scaling on massively parallel

computers, and forms part of the multi-institutional UK “Gung Ho”

project which aims to design a next generation dynamical core for the

Met Office Unified Model climate and weather prediction system. The

rotating shallow water equations are a single layer model that is

used to benchmark the horizontal component of numerical schemes for

weather prediction models.

We show, within the finite element exterior calculus framework, that it

is possible

to build numerical schemes with horizontal velocity and layer depth that

have a con-

served diagnostic potential vorticity field, by making use of the

geometric properties of the scheme. The schemes also conserve energy and

enstrophy, which arise naturally as conserved quantities out of a

Poisson bracket formulation. We show that it is possible to modify the

discretisation, motivated by physical considerations, so that enstrophy

is dissipated, either by using the Anticipated Potential Vorticity

Method, or by inducing stabilised advection schemes for potential

vorticity such as SUPG or higher-order Taylor-Galerkin schemes. We

illustrate our results with convergence tests and numerical experiments

obtained from a FEniCS implementation on the sphere.

Given a field K and an ordered abelian group G, we can form the field K((G)) of generalised formal power series with coefficients in K and indices in G. When is this field decidable? In certain cases, decidability reduces to that of K and G. We survey some results in the area, particularly in the case char K > 0, where much is still unknown.

Given a field K and an ordered abelian group G, we can form the field K((G)) of generalised formal power series with coefficients in K and indices in G. When is this field decidable? In certain cases, decidability reduces to that of K and G. We survey some results in the area, particularly in the case char K > 0, where much is still unknown.

Subgroup separability is a group-theoretic property that has important implications for geometry and topology, because it allows us to lift immersions to embeddings in a finite sheeted covering space. I will describe how this works in the case of graphs, and go on to motivate the construction of special cube complexes as an attempt to generalise the technique to higher dimensions.

We consider degenerate elliptic systems like the p-Laplacian  system with p>1 and zero boundary data. The r.h.s. is given in  divergence from div F. We prove a pointwise estimate (in terms of the  sharp maximal function) bounding the gradient of the solution via the  function F. This recovers several known results about local regularity  estimates in L^q, BMO and C^a. Our pointwise inequality extends also  to boundary points. So these  regularity estimates hold globally as  well. The global estimates in BMO and C^a are new.

The Haagerup property is a group theoretic property which is a strong converse of Kazhdan's property (T). It implies the Baum-Connes conjecture and has connections with amenability, C*-algebras, representation theory and so on. It is thus not surprising that quite some effort was made to investigate how the Haagerup property behaves under the formation of free products, direct products, direct limits,... In joint work with Y.Antolin, we investigated the behaviour of the Haagerup property under graph products. In this talk we introduce the concept of a graph product, we give a gentle introduction to the Haagerup property and we discuss its behaviour under graph products.

A system of linear equations is called partition regular if, whenever the naturals are finitely coloured, there is a monochromatic solution of the equations. Many of the classical theorems of Ramsey Theory may be phrased as assertions that certain systems are partition regular.

What happens if we are colouring not the naturals but the (non-zero) integers, rationals, or reals instead? After some background, we will give various recent results.