Past

We investigate the problem of optimizing the shape and

location of actuators or sensors for evolution systems

driven by a partial differential equation, like for

instance a wave equation, a Schrödinger equation, or a

parabolic system, on an arbitrary domain Omega, in

arbitrary dimension, with boundary conditions if there

is a boundary, which can be of Dirichlet, Neumann,

mixed or Robin. This kind of problem is frequently

encountered in applications where one aims, for

instance, at maximizing the quality of reconstruction

of the solution, using only a partial observation. From

the mathematical point of view, using probabilistic

considerations we model this problem as the problem of

maximizing what we call a randomized observability

constant, over all possible subdomains of Omega having

a prescribed measure. The spectral analysis of this

problem reveals intimate connections with the theory of

quantum chaos. More precisely, if the domain Omega

satisfies some quantum ergodic assumptions then we

provide a solution to this problem.



These works are in collaboration with Emmanuel Trélat

(Univ. Paris 6) and Enrique Zuazua (BCAM Bilbao, Spain).

NOTE CHANGE OF TIME AND PLACE

It is known by results of Macintyre and Chatzidakis-Hrushovski that the theory ACFA of existentially closed difference fields is decidable. By developing techniques of difference algebraic geometry, we view quantifier elimination as an instance of a direct image theorem for Galois formulae on difference schemes. In a context where we restrict ourselves to directly presented difference schemes whose definition only involves algebraic correspondences, we develop a coarser yet effective procedure, resulting in a primitive recursive quantifier elimination. We shall discuss various algebraic applications of Galois stratification and connections to fields with Frobenius.

 Voevodsky asked what the topology of the universe is in a 
continuous interpretation of type theory, such as Johnstone's 
topological topos. We can actually give a model-independent answer: it 
is indiscrete. I will briefly introduce "intensional Martin-Loef type 
theory" (MLTT) and formulate and prove this in type theory (as opposed 
to as a meta-theorem about type theory). As an application or corollary, 
I will also deduce an analogue of Rice's Theorem for the universe: the 
universe (the large type of all small types) has no non-trivial 
extensional, decidable properties. Topologically this is the fact that 
it doesn't have any clopens other than the trivial ones.

A group is called residually finite if every non-trivial element can be homomorphically mapped to a finite group such that the image is again non-trivial. Residually finite groups are interesting because quite a lot of information about them can be reconstructed from their finite quotients. Baumslag showed that if G is a finitely generated residually finite group then Aut(G) is also residually finite. Using a similar method Grossman showed that if G is a finitely generated conjugacy separable group with "nice" automorphisms then Out(G) is residually finite. The graph product is a group theoretic construction naturally generalising free and direct products in the category of groups. We show that if G is a finite graph product of finitely generated residually finite groups then Out(G) is residually finite (modulo some technical conditions)

Soluble groups, and other classes of groups that can be built from simpler groups, are useful test cases for studying group properties. I will talk about techniques for building profinite groups from simpler ones, and how  to use these to investigate the cohomology of such groups and recover information about the group structure.

Given a Lagrangian submanifold invariant under a Hamiltonian loop, we partially compute the image of the loop's Seidel element under the closed-open string map into the Hochschild cohomology of the Lagrangian. This piece captures the homology class of the loop's orbits on the Lagrangian and can help to prove that the closed-open map is injective in some examples. As a corollary we prove that $\mathbb{RP}^n$ split-generates the Fukaya category of $\mathbb{CP}^n$ over a field of characteristic 2, and the same for real loci of some other toric  varieties.

Suppose that we have a tile $T$ in say $\mathbb{Z}^2$, meaning a finite subset of $\mathbb{Z}^2$. It may or may not be the case that $T$ tiles $\mathbb{Z}^2$, in the sense that $\mathbb{Z}^2$ can be partitioned into copies of $T$. But is there always some higher dimension $\mathbb{Z}^d$ that can be tiled with copies of $T$? We prove that this is the case: for any tile in $\mathbb{Z}^2$ (or in $\mathbb{Z}^n$, any $n$) there is a $d$ such that $\mathbb{Z}^d$ can be tiled with copies of it. This proves a conjecture of Chalcraft.

The first FFT-based algorithm for numerical homogenization from high-resolution images was proposed by Moulinec and Suquet in 1994 as an alternative to finite elements and twenty years later, it is still widely used in computational micromechanics of materials. The method is based on an iterative solution to an integral equation of the Lippmann-Schwinger type, whose kernel can be explicitly expressed in the Fourier domain. Only recently, it has been recognized that the algorithm has a variational structure arising from a Fourier-Galerkin method. In this talk, I will show how this insight can be used to significantly improve the performance of the original Moulinec-Suquet solver. In particular, I will focus on (i) influence of an iterative solver used to solve the system of linear algebraic equations, (ii) effects of numerical integration of the Galerkin weak form, and (iii) convergence of an a-posteriori bound on the solution during iterations.

A year ago I gave a talk raising questions about Faraday shielding which stimulated discussion with John Ockendon and others and led to a collaboration with Jon Chapman and Dave Hewett.  The problem is one of harmonic functions subject to constant-potential boundary conditions.  A year later, we are happy with the solution we have found, and the paper will appear in SIAM Review.  Though many assume as we originally did that Faraday shielding must be exponentially effective, and Feynman even argues this explicitly in his Lectures, we have found that in fact, the shielding is only linear.  Along the way to explaining this we make use of Mikhlin's numerical method of series expansion, homogenization by multiple scales analysis, conformal mapping, a phase transition, Brownian motion, some ideas recollected from high school about electrostatic induction, and a constrained quadratic optimization problem solvable via a block 2x2 KKT matrix.

We will focus on vortex gradient fields of unit-length. The associated stream function solves the eikonal equation, more precisely it is the distance function to a point. We will prove a kinetic formulation characterizing such vector fields in any dimension.
 

A permutation group is called sharply n-transitive if it acts 

freely and transitively on the set of ordered n-tuples of distinct 

points. The investigation of such permutation groups is a classical 

branch of group theory; it led Emile Mathieu to the discovery of the 

smallest finite simple sporadic groups in the 1860's. In this talk I 

will discuss the case where the permutation group is assumed to be a 

locally compact transformation group, and explain how this set-up is 

related to Gromov hyperbolicity and to arithmetic lattices in products 

of trees.

We consider the problem of minimising the commute or shuttle time for a diffusion between the endpoints of an interval. The control is the scale function for the diffusion. We show that the dynamic version of the problem has the same solution as the static version if we start at an end point and consider the much harder case where the starting point is in the interior.

tba

I shall demonstrate, under some mild assumptions, that the symmetry group of  extreme, Killing, supergravity horzions contains an sl(2, R) subalgebra.  The proof requires a generalization of the  Lichnerowicz theorem for non-metric connections. The techniques developed can also be applied in the classification
of AdS and Minkowski flux backgrounds.
 

Computational models of the heart have been primarily developed to simulate, analyse and understand experimental measurements. Increasingly biophysical models are being used to understand cardiac disease and pathologies in patients. This shift from laboratory to clinical contexts requires the development of new modelling frameworks to simulate pathological states that invalidate assumptions in existing modelling frameworks, work flows to integrate multiple data sets to constrain model parameters and an understanding of the clinical questions that models can answer. We report on the development and application of biophysical modelling frameworks representing the cardiac electrical and mechanical systems, which are currently being customised for modelling cardiac pathologies.

Our study addresses the question of how an arbitrage-free semimartingale model is affected when the knowledge about a random time is added. Precisely, we focus on the No-Unbounded-Profit-with-Bounded-Risk condition, which is also known in the literature as the first kind of no arbitrage. In the general semimartingale setting, we provide a sufficient condition on the random time and price process for which the no arbitrage is preserved under filtration enlargement. Moreover we study the condition on the random time for which the no arbitrage is preserved for any process. This talk is based on a joint work with Tahir Choulli, Jun Deng and Monique Jeanblanc.

We present some recent work - joint with Arno Fehm - in which we give an `existential Ax-Kochen-Ershov principle' for equicharacteristic henselian valued fields. More precisely, we show that the existential theory of such a valued field depends only on the existential theory of the residue field. In residue characteristic zero, this result is well-known and follows from the classical Ax-Kochen-Ershov Theorems. In arbitrary (but equal) characteristic, our proof uses F-V Kuhlmann's theory of tame fields. One corollary is an unconditional proof that the existential theory of F_q((t)) is decidable. We will explain how this relates to the earlier conditional proof of this result, due to Denef and Schoutens.
 

In this talk I shall discuss some classical results on isometric embedding of positively/nonegatively curved surfaces into $\mathbb{R}^3$. 

    The isometric embedding problem has played a crucial role in the development of geometric analysis and nonlinear PDE techniques--Nash invented his Nash-Moser techniques to prove the embeddability of general manifolds; later Gromov recast the problem into his ``h-Principle", which recently led to a major breakthrough by C. De Lellis et al. in the analysis of Euler/Navier-Stokes. Moreover, Nirenberg settled (positively) the Weyl Problem: given a smooth metric with strictly positive Gaussian curvature on a closed surface, does there exist a global isometric embedding into the Euclidean space $\mathbb{R}^3$? This work is proved by the continuity method and based on the regularity theory of the Monge-Ampere Equation, which led to Cheng-Yau's renowned works on the Minkowski Problem and the Calabi Conjecture. 

    Today we shall summarise Nirenberg's original proof for the Weyl problem. Also, we shall describe Hamilton's simplified proof using Nash-Moser Inverse Function Theorem, and Guan-Li's generalisation to the case of nonnegative Gaussian curvature. We shall also mention the status-quo of the related problems.