Past

Molecular modelling has become a valuable tool and is increasingly part of the standard methodology of chemistry, physics, engineering and biology. The power of molecular modelling lies in its versatility: as potential energy functions improve, a broader range of more complex phenomena become accessible to simulation, but much of the underlying methodology can be re-used. For example, the Verlet method is still the most popular molecular dynamics scheme for constant energy molecular dynamics simulations despite it being one of the first to be proposed for the purpose.

One of the most important challenges in molecular modelling remains the computation of averages with respect to the canonical Gibbs (constant temperature) distribution, for which the Verlet method is not appropriate. Whereas constant energy molecular dynamics prescribes a set of equations (Newton's equations), there are many alternatives for canonical sampling with quite different properties. The challenge is therefore to identify formulations and numerical methods that are robust and maximally efficient in the computational setting.

One of the simplest and most effective methods for sampling is based on Langevin dynamics which mimics coupling to a heat bath by the incorporation of random forces and an associated dissipative term. Schemes for Langevin dynamics simulation may be developed based on the familiar principle of splitting. I will show that the invariant measure ('long term') approximation may be strongly affected by a simple re-ordering of the terms of the splitting. I will describe a transition in weak numerical order of accuracy that occurs (in one case) in the t->infty limit.

I will also entertain some more radical suggestions for canonical sampling, including stochastic isokinetic methods that enable the use of greatly enlarged timesteps for expensive but slowly-varying force field components.

We will define the Ran space as well as Ran space versions of some of the prestacks we've already seen, and explain what is meant by the homology of a prestack. Following Gaitsgory and possibly Drinfeld, we'll show how the Ran space machinery can be used to prove that the space of rational maps is homologically contractible.

For many questions of scientific interest, all-atom molecular simulations are still out of reach, in particular in materials engineering where large numbers of atoms, and often expensive force fields, are required. A long standing challenge has been to construct concurrent atomistic/continuum coupling schemes that use atomistic models in regions of space where high accuracy is required, with computationally efficient continuum models (e.g., FEM) of the elastic far-fields.

Many different mechanisms for achieving such atomistic/continuum couplings have been developed. To assess their relative merits, in particular accuracy/cost ratio, I will present a numerical analysis framework. I will use this framework to analyse in more detail a particularly popular scheme (the BQCE scheme), identifying key approximation parameters which can then be balanced (in a surprising way) to obtain an optimised formulation.

Finally, I will demonstrate that this analysis shows how to remove a severe bottlenecks in the BQCE scheme, leading to a new scheme with optimal convergence rate.

Algebraically closed fields, and in general varieties are among the first examples
of Zariski Geometries.
I will consider specialisations of algebraically closed fields and varieties.
In the case of an algebraically closed field K, I will show that a specialisation
is essentially a residue map, res from K to a residue field k.  
In both cases I will show universality of the specialisation is controlled by the
transcendence degree of K over k.  

In the late 70s -- early 80s Makanin came up with a very simple, but very powerful idea to approach solving equations in free groups. This simplicity makes Makanin-like procedures ubiquitous in mathematics: in dynamical systems, geometric group theory, 3-dimensional topology etc. In this talk I will explain loosely how Makanin's algorithm works.

In 1878, Jordan showed that if $G$ is a finite group of complex $n \times n$ matrices, then $G$ has a normal subgroup whose index in $G$ is bounded by a function of $n$ alone. He showed only existence, and early actual bounds on this index were far from optimal. In 1985, Weisfeiler used the classification of finite simple groups to obtain far better bounds, but his work remained incomplete when he disappeared. About eight years ago, I obtained the optimal bounds, and this work has now been extended to subgroups of all (complex) classical groups. I will discuss this topic at a “colloquium” level – i.e., only a rudimentary knowledge of finite group theory will be assumed.

Auctioneers may wish to sell related but different indivisible goods in

a single process. To develop such techniques, we study the geometry of

how an agent's demanded bundle changes as prices change. This object

is the convex-geometric object known as a `tropical hypersurface'.

Moreover, simple geometric properties translate directly to economic

properties, providing a new taxonomy for economic valuations. When

considering multiple agents, we study the unions and intersections of

the corresponding tropical hypersurfaces; in particular, properties of

the intersection are deeply related to whether competitive equilibrium

exists or fails. This leads us to new results and generalisations of

existing results on equilibrium existence. The talk will provide an

introductory tour to relevant economics to show the context of these

applications of tropical geometry. This is joint work with Paul

Klemperer.

The common convention when dealing with hyperbolic groups is that such groups are finitely

generated and equipped with the word length metric relative to a finite symmetric generating

subset. Gromov's original work on hyperbolicity already contained ideas that extend beyond the

finitely generated setting. We study the class of locally compact hyperbolic groups and elaborate

on the similarities and differences between the discrete and non-discrete setting.

Hamiltonian reduction arose as a mechanism for reducing complexity of systems in mechanics, but it also provides a tool for constructing complicated but interesting objects from simpler ones. I will illustrate how this works in representation theory and algebraic geometry via examples. I will describe a new structure theory, motivated by Hamiltonian reduction (and in particular the Morse theory that results), for some categories (of D-modules) of interest to representation theorists. I will then explain how this implies a modified form of "hyperkahler Kirwan surjectivity" for the cohomology of certain Hamiltonian reductions. The talk will not assume that members of the audience know the meaning of any of the above-mentioned terms. The talk is based on joint work with K. McGerty.

We study the behaviour of orthogonal polynomials on triangles and their coefficients in the context of spectral approximations of partial differential equations.  For spectral approximation we consider series expansions $u=\sum_{k=0}^{\infty} \hat{u}_k \phi_k$ in terms of orthogonal polynomials $\phi_k$. We show that for any function $u \in C^{\infty}$ the series expansion converges faster than with any polynomial order.  With these result we are able to employ the polynomials $\phi_k$ in the spectral difference method in order to solve hyperbolic conservation laws.

It is a well known fact that discontinuities can arise leading to oscillatory numerical solutions. We compare standard filtering and the super spectral vanishing viscosity methods, which uses exponential filters build from the differential operator of the respective orthogonal polynomials.  We will extend the spectral difference method for unstructured grids by using 
 classical orthogonal polynomials and exponential filters. Finally we consider some numerical test cases.

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.

After Gromov's foundational work in 1985, problems of symplectic embeddings lie in the heart of symplectic geometry. The Gromov width of a symplectic manifold

$(M, \omega)$ is a symplectic invariant that measures, roughly speaking, the size of the biggest ball we can symplectically embed in it. I will discuss tecniques to compute the Gromov width of a special family of symplectic manifolds, the moduli spaces of polygons in real $3$-space. Under some genericity assumptions on the edge lengths, the polygon space is a symplectic manifold; in fact, it is a symplectic reduction of Grassmannian of 2-planes in complex $n$-space. After introducing this family of manifolds we will concentrate on the spaces of 5-gons and calculate for their Gromov width. This is joint work with Milena Pabiniak, IST Lisbon.

A fast method for computing eigenpairs of positive definite matrices using GPUs is presented. The method uses Chebyshev polynomial spectral transformations to map the desired eigenvalues of the original matrix $A$ to exterior eigenvalues of the transformed matrix $p(A)$, making them easily computable using existing Krylov methods. The construction of the transforming polynomial $p(z)$ can be done efficiently and only requires knowledge of the spectral radius of $A$. Computing $p(A)v$ can be done using only the action of $Av$. This requires no extra memory and is typically easy to parallelize. The method is implemented using the highly parallel GPU architecture and for specific problems, has a factor of 10 speedup over current GPU methods and a factor of 100 speedup over traditional shift and invert strategies on a CPU.

In this joint work with Daniela Giachetti (Rome) and Pedro J. Martinez Aparicio (Cartagena, Spain) we consider the problem

$$ - div A(x) Du = F (x, u) \; {\rm in} \; \Omega,$$

$$ u = 0 \; {\rm on} \; \partial \Omega,$$

(namely an elliptic semilinear equation with homogeneous Dirichlet boundary condition),

where the non\-linearity $F (x, u)$ is singular in $u = 0$, with a singularity of the type

$$F (x, u) = {f(x) \over u^\gamma} + g(x)$$

with $\gamma > 0$ and $f$ and $g$ non negative (which implies that also $u$ is non negative).

The main difficulty is to give a convenient definition of the solution of the problem, in particular when $\gamma > 1$. We give such a definition and prove the existence and stability of a solution, as well as its uniqueness when $F(x, u)$ is non increasing en $u$.

We also consider the homogenization problem where $\Omega$ is replaced by $\Omega^\varepsilon$, with $\Omega^\varepsilon$ obtained from $\Omega$ by removing many very

small holes in such a way that passing to the limit when $\varepsilon$ tends to zero the Dirichlet boundary condition leads to an homogenized problem where a ''strange term" $\mu u$ appears.

At the end of the 70s, Gross and Deligne conjectured that periods of geometric Hodge structures with multiplication by an abelian number field are always products of values of the gamma function at rational numbers, with exponents determined by the Hodge decomposition. I will explain a proof of an alternating variant of this conjecture for the cohomology groups of smooth, projective varieties over the algebraic numbers acted upon by a finite order automorphism.

We consider three nonparametric hypothesis testing problems: (1) Given samples from distributions p and q, a homogeneity test determines whether to accept or reject p=q; (2) Given a joint distribution p_xy over random variables x and y, an independence test investigates whether p_xy = p_x p_y, (3) Given a joint distribution over several variables, we may test for whether there exist a factorization (e.g., P_xyz = P_xyP_z, or for the case of total independence, P_xyz=P_xP_yP_z).

We present nonparametric tests for the three cases above, based on distances between embeddings of probability measures to reproducing kernel Hilbert spaces (RKHS), which constitute the test statistics (eg for independence, the distance is between the embedding of the joint, and that of the product of the marginals). The tests benefit from years of machine research on kernels for various domains, and thus apply to distributions on high dimensional vectors, images, strings, graphs, groups, and semigroups, among others. The energy distance and distance covariance statistics are also shown to fall within the RKHS family, when semimetrics of negative type are used. The final test (3) is of particular interest, as it may be used in detecting cases where two independent causes individually have weak influence on a third dependent variable, but their combined effect has a strong influence, even when these variables have high dimension.

An invariant random subgroup in a (finitely generated) group is a

probability measure on the space of subgroups of the group invariant under

the inner automorphisms of the group. It is a natural generalization of the

the notion of normal subgroup. I’ll give an introduction into this actively

developing subject and then discuss in more detail examples of invariant

random subgrous in groups of intermediate growth. The last part of the talk

will be based on a recent joint work with Mustafa Benli and Rostislav

Grigorchuk.

 I will discuss joint work with Daniel Greb and Matei Toma in which we introduce a notion of Gieseker-stability that depends on several polarisations.  We use this to study the change in the moduli space of Giesker semistable sheaves on manifolds giving new results in dimensions at least three, and to study the notion of Gieseker-semistability for sheaves taken with respect to an irrational Kahler class.

We will introduce a particle system interacting through a mean-field term that models the behavior of a network of excitatory neurons. The novel feature of the system is that the it features a threshold dynamic: when a single particle reaches a threshold, it is reset while all the others receive an instantaneous kick. We show that in the limit when the size of the system becomes infinite, the resulting non-standard equation of McKean Vlasov type has a solution that may exhibit a blow-up phenomenon depending on the strength of the interaction, whereby a single particle reaching the threshold may cause a macroscopic cascade. We moreover show that the particle system does indeed exhibit propagation of chaos, and propose a new way to give sense to a solution after a blow-up.

This is based on joint research with F. Delarue (Nice), E. Tanré (INRIA) and S. Rubenthaler (Nice).