Past

First of all, I will give an overview of what the phenomenon of homological stability is and why it's useful, with plenty of examples. I will then introduce configuration spaces -- of various different kinds -- and give an overview of what is known about their homological stability properties. A "configuration" here can be more than just a finite collection of points in a background space: in particular, the points may be equipped with a certain non-local structure (an "orientation"), or one can consider unlinked embedded copies of a fixed manifold instead of just points. If by some miracle time permits, I may also say something about homological stability with local coefficients, in general and in particular for configuration spaces.

 We consider the emergence of classical correlations in macroscopic quantum systems, and its connection to monogamy relations for violation of Bell-type inequalities. We work within the framework of Abramsky and Brandenburger [1], which provides a unified treatment of non-locality and contextuality in the general setting of no-signalling empirical models. General measurement scenarios are represented by simplicial complexes that capture the notion of compatibility of measurements. Monogamy and locality/noncontextuality of macroscopic correlations are revealed by our analysis as two sides of the same coin: macroscopic correlations are obtained by averaging along a symmetry (group action) on the simplicial complex of measurements, while monogamy relations are exactly the inequalities that are invariant with respect to that symmetry. Our results exhibit a structural reason for monogamy relations and consequently for the classicality of macroscopic correlations in the case of multipartite scenarios, shedding light on and generalising the results in [2,3].More specifically, we show that, however entangled the microscopic state of the system, and provided the number of particles in each site is large enough (with respect to the number of allowed measurements), only classical (local realistic) correlations will be observed macroscopically. The result depends only on the compatibility structure of the measurements (the simplicial complex), hence it applies generally to any no-signalling empirical model. The macroscopic correlations can be defined on the quotient of the simplicial complex by the symmetry that lumps together like microscopic measurements into macroscopic measurements. Given enough microscopic particles, the resulting complex satisfies a structural condition due to Vorob'ev [4] that is necessary and sufficient for any probabilistic model to be classical.  The generality of our scheme suggests a number of promising directions. In particular, they can be applied in more general scenarios to yield monogamy relations for contextuality inequalities and to study macroscopic non-contextuality.

[1] Samson Abramsky and Adam Brandenburger, The sheaf-theoretic structure of non-locality and contextuality, New Journal of Physics 13 (2011), no. 113036.
[2] MarcinPawłowski and Caslav Brukner, Monogamy of Bell’s inequality violations in nonsignaling theories, Phys. Rev. Lett. 102 (2009), no. 3, 030403.
[3] R. Ramanathan, T. Paterek, A. Kay, P. Kurzynski, and D. Kaszlikowski, Local realism of macroscopic correlations, Phys. Rev. Lett. 107 (2011), no. 6, 060405.
[4] N.N.Vorob’ev, Consistent families of measures and their extensions, Theory of Probability and its Applications VII (1962), no. 2, 147–163, (translated by N. Greenleaf, Russian original published in Teoriya Veroyatnostei i ee Primeneniya).

Several recent works by D. Tamarkin, D. Nadler, E. Zaslow make use of the microlocal theory of sheaves of M. Kashiwara and P. Schapira to obtain results in symplectic geometry. The link between sheaves on a manifold $M$ and the symplectic geometry of the cotangent bundle of $M$ is given by the microsupport of a sheaf, which is a conic co-isotropic subset of the cotangent bundle. In the above mentioned works properties of a given Lagrangian submanifold $\Lambda$ are deduced from the existence of a sheaf with microsupport $\Lambda$, which we call a quantization of $\Lambda$.

In the second talk we will introduce a stack on $\Lambda$ by localization of the category of sheaves on $M$. We deduce topological obstructions on $\Lambda$ for the existence of a quantization.

A number is called transcendental if it is not algebraic, that is it does not satisfy a polynomial equation with rational coefficients. It is easy to see that the algebraic numbers are countable, hence the transcendental numbers are uncountable. Despite this fact, it turns out to be very difficult to determine whether a given number is transcendental. In this talk I will discuss some famous examples and the theorems which allow one to construct many different transcendental numbers. I will also give an outline of some of the many open problems in the field.

I propose to present modeling and experimental data about the organization of telomeres (ends of the chromosomes): the 32 telomeres in Yeast form few local aggregates. We built a model of diffusion-aggregation-dissociation for a finite number of particles to estimate the number of cluster and the number of telomere/cluster and other quantities. We will further present based on eingenvalue expansion for the Fokker-Planck operator, asymptotic estimation for the mean time a piece of a polymer (DNA) finds a small target in the nucleus.

We discuss the problem to what extend a group action determines geometry of the space. 
More precisely, we show that for a large class of actions measurable isomorphisms must preserve 
the geometric structure as well. This is a joint work with Bader, Furman, and Weiss.

Several recent works by D. Tamarkin, D. Nadler, E. Zaslow make use of the microlocal theory of sheaves of M. Kashiwara and P. Schapira to obtain results in symplectic geometry. The link between sheaves on a manifold $M$ and the symplectic geometry of the cotangent bundle of $M$ is given by the microsupport of a sheaf, which is a conic co-isotropic subset of the cotangent bundle. In the above mentioned works properties of a given Lagrangian submanifold $\Lambda$ are deduced from the existence of a sheaf with microsupport $\Lambda$, which we call a quantization of $\Lambda$. In the first talk we will see that the graph of a Hamiltonian isotopy admits a canonical quantization and we deduce a new proof of Arnold's non-displaceability conjecture.

Several recent works by D. Tamarkin, D. Nadler, E. Zaslow make use of the microlocal theory of sheaves of M. Kashiwara and P. Schapira to obtain results in symplectic geometry. The link between sheaves on a manifold $M$ and the symplectic geometry of the cotangent bundle of $M$ is given by the microsupport of a sheaf, which is a conic co-isotropic subset of the cotangent bundle. In the above mentioned works properties of a given Lagrangian submanifold $\Lambda$ are deduced from the existence of a sheaf with microsupport $\Lambda$, which we call a quantization of $\Lambda$.

In the first talk we will see that the graph of a Hamiltonian isotopy admits a canonical quantization and we deduce a new proof of Arnold's non-displaceability conjecture.

I will describe how a search for the answer to an old question about the existence of monotone arithmetic progressions in permutations of positive integers led to the study of infinite words with bounded additive complexity. The additive complexity of a word on a finite subset of integers is defined as the function that, for a positive integer $n$, counts the maximum number of factors of length $n$, no two of which have the same sum.

Defect measures have successfully been used, in a variety of

contexts, as a tool to quantify the lack of compactness of bounded

sequences of square-integrable functions due to concentration and

oscillation effects. In this talk we shall present some results on the

structure of the set of possible defect measures arising from sequences

of solutions to the linear Schrödinger equation on a compact manifold.

This is motivated by questions related to understanding the effect of

geometry on dynamical aspects of the Schrödinger flow, such as

dispersive effects and unique continuation.

It turns out that the answer to these questions depends strongly on

global properties of the geodesic flow on the manifold under

consideration: this will be illustrated by discussing with a certain

detail the examples of the the sphere and the (flat) torus.

In 2005, Bonk and Kleiner showed that a hyperbolic group admits a

quasi-isometrically embedded copy of the hyperbolic plane if and only if the

group is not virtually free. This answered a question of Papasoglu. I will

discuss a generalisation of their result to certain relatively hyperbolic

groups (joint work with Alessandro Sisto). Key tools involved are new

existence results for quasi-circles, and a better understanding of the

geometry of boundaries of relatively hyperbolic groups.

(Joint work with P.E. Chaudru de Raynal and F. Delarue)

Several problems in financial mathematics, in particular that of contingent claim pricing, may be casted as decoupled Forward Backward Stochastic Differential Equations (FBSDEs). It is then of a practical interest to find efficient algorithms to solve these equations numerically with a reasonable complexity.

An efficient numerical approach using cubature on Wiener spaces was introduced by Crisan and Manoralakis [1]. The algorithm uses an approximation scheme requiring the calculation of conditional expectations, a task achieved through the cubature kernel approximation. This algorithm features several advantages, notably the fact that it is possible to solve with the same cubature tree several decoupled FBSDEs with different boundary conditions. There is, however, a drawback of this method: an exponential growth of the algorithm's complexity.

In this talk, we introduce a modification on the cubature method based on the recombination method of Litterer and Lyons [2] (as an alternative to Tree Branch Based Algorithm proposed in [1]). The main idea of the method is to modify the nodes and edges of the cubature trees in such a way as to preserve, up to a constant, the order of convergence of the expectation and conditional expectation approximations obtained via the cubature method, while at the same time controlling the complexity growth of the algorithm.

We have obtained estimations on the order of convergence and complexity growth of the algorithm under smoothness assumptions on the coefficients of the FBSDE and uniform ellipticity of the forward equation. We discuss that, just as in the case of the plain cubature method, the order of convergence of the algorithm may be degraded as an effect of solving FBSDEs with rougher boundary conditions. Finally, we illustrate the obtained estimations with some numerical tests.

References

[1] Crisan, D., and K. Manolarakis. “Solving Backward Stochastic Differential Equations Using the Cubature Method. Application to Nonlinear Pricing.” In Progress in Analysis and Its Applications, 389–397. World Sci. Publ., Hackensack, NJ, 2010.

[2] Litterer, C., and T. Lyons. “High Order Recombination and an Application to Cubature on Wiener Space.” The Annals of Applied Probability 22, no. 4 (August 2012): 1301–1327. doi:10.1214/11-AAP786.

In this talk I will consider the Burgers equation with a homogeneous Possion process as a forcing potential. In recent years, the randomly forced Burgers equation, with forcing that is ergodic in time, received a lot of attention, especially the almost sure existence of unique global solutions with given average velocity, that at each time only depend on the history up to that time. However, in all these results compactness in the space dimension of the forcing was essential. It was even conjectured that in the non-compact setting such unique global solutions would not exist. However, we have managed to use techniques developed for first and last passage percolation models to prove that in the case of Poisson forcing, these global solutions do exist almost surely, due to the existence of semi-infinite minimizers of the Lagrangian action. In this talk I will discuss this result and explain some of the techniques we have used.

This is joined work Yuri Bakhtin and Konstantin Khanin.

We discuss sparse portfolio optimization in continuous time.

Optimization objective is to maximize an expected utility as in the

classical Merton problem but with regularizing sparsity constraints.

Such constraints aim for asset allocations that contain only few assets or

that deviate only in few coordinates from a reference benchmark allocation.

With a focus on growth optimization, we show empirical results for various

portfolio selection strategies with and without sparsity constraints,

investigating different portfolios of stock indicies, several performance

measures and adaptive methods to select the regularization parameter.

Sparse optimal portfolios are less sensitive to estimation

errors and performance is superior to portfolios without sparsity

constraints in reality, where estimation risk and model uncertainty must

not be ignored.

We are interested in finding the Probability Density Function (PDF) of high dimensional chaotic systems such as a global atmospheric circulation model. The key difficulty stems from the so called “curse of dimensionality”. Representing anything numerically in a high dimensional space seems to be just too computationally expensive. Methods applied to dodge this problem include representing the PDF analytically or applying a (typically linear) transformation to a low dimensional space. For chaotic systems these approaches often seem extremely ad-hoc with the main motivation being that we don't know what else to do.

The Lorenz 95 system is one of the simplest systems we could come up with that is both chaotic and high dimensional. So it seems like a good candidate for initial investigation. We look at two attempts to approximate the PDF of this system to an arbitrary level of accuracy, firstly using a simple Monte-Carlo method and secondly using the Fokker-Planck equation. We also describe some of the (sometimes surprising) difficulties encountered along the way.

• Jean-Charles Seguis - Simulation in chemotaxis and comparison of cell models
• Laura Kimpton (née Gallimore) - A viscoelastic two-phase flow model of a crawling cell
• Benjamin Franz - Particles and PDEs and robots