Past

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

The Outer Model Programme investigates L-like forcing  extensions of the universe, where we say that a model of Set Theory  is L-like if it satisfies properties of Goedel's constructible universe of sets L. I will introduce the Outer Model Programme, talk  about its history, motivations, recent results and applications. I  will be presenting joint work with Sy Friedman and Philipp Luecke.

I discuss models for the planar spreading of a viscous fluid between an elastic lid and an underlying rigid plane. These have application to the growth of magmatic intrusions, as well as to other industrial and biological processes; simple experiments of an inflated blister will be used for motivation. The height of the fluid layer is described by a sixth order non-linear diffusion equation, analogous to the fourth order equation that describes surface tension driven spreading. The dynamics depend sensitively on the conditions at the contact line, where the sheet is lifted from the substrate and where some form of regularization must be applied to the model. I will explore solutions with a pre-wetted film or a constant-pressure fluid lag, for flat and inclined planes, and compare with the analogous surface tension problems.

Toby Gee and I have proposed the definition of a "C-group", an extension of Langlands' notion of an L-group, and argue that for an arithmetic version of Langlands' philosophy such a notion is useful for controlling twists properly. I will give an introduction to this business, and some motivation. I'll start at the beginning by explaining what an L-group is a la Langlands, but if anyone is interested in doing some background preparation for the talk, they might want to find out for themselves what an L-group (a Langlands dual group) is e.g. by looking it up on Wikipedia!

This talk is a basic introduction to the wonderful world of Higgs bundles on a Riemann Surface, and their moduli space. We will only survey the basics of the theory focusing on the rich geometry of the moduli space of Higgs bundles, and the relation to moduli space of vector bundles. In the end we consider small applications of Higgs bundles. As this talk will be very basic we won't go into any new developments of the theory, but just mention the areas in which Higgs bundles are used today.

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And applications to problems involving pointwise constraints on the gradient of the state on non smooth polygonal domains

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In this talk we are concerned with an analysis of Moreau-Yosida regularization of pointwise state constrained optimal control problems. As recent analysis has already revealed the convergence of the primal variables is dominated by the reduction of the feasibility violation in the maximum norm.

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We will use a new method to derive convergence of the feasibility violation in the maximum norm giving improved the known convergence rates.

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Finally we will employ these techniques to analyze optimal control problems governed by elliptic PDEs with pointwise constraints on the gradient of the state on non smooth polygonal domains. For these problems, standard analysis, fails because the control to state mapping does not yield sufficient regularity for the states to be continuously differentiable on the closure of the domain. Nonetheless, these problems are well posed. In particular, the results of the first part will be used to derive convergence rates for the primal variables of the regularized problem.