Abstract: The 3x+1 conjecture, also known as the Collatz problem, offers one of the best opportunities for a mathematician to become addicted. The conjecture is easily stated but apparently difficult to prove. The underlying question is simple: for a given odd positive integer x, let T(x) be the odd positive integer (3x+1)/2^v(3x+1), where v(3x+1) stands for the exponent of the largest power of 2 that divides the number 3x+1. For example, T(5)=1, T(11)=17, T(17)=13, and T(13)=5.

The 3x+1 conjecture states that, starting from any odd positive integer x, by iterating the map T, we will reach the number 1 in finitely many steps. To give an example, T^4(11) = 1.

The goal of this talk is to present a `language' to work on the 3x+1 conjecture, and to exhibit how far we have come with our approach.

Abstract: Over the last 30 years, computer simulations of QCD, the theory of strong interactions, have matured into a first principle tool for reliable post- and predictions with controllable error margins. Their success is based upon Importance Sampling using Monte-Carlo Markov chains. After a brief introduction to these computer simulations including some highlights in relation to QCD, I will address Quantum Field Theories of cold, but dense matter. Since more than 30 years, these theories still await first principle computer results due to the notorious sign-problem, which makes Important Sampling impossible. In my talk, I will review the remarkable successes of the recent years to simulate those theories. I will then focus, in particular, on two promising methods: dualisation and the density-of-states approach.

Abstract: In this talk I will provide a very quick overview of my research interests, mostly to do with the three above areas. I'll explain what is meta-analysis, what are the main statistical problems in meta-analysis, what it has to do with actuarial problems and why the methods developed for meta-analysis are applicable to Big Data.

Abstract: Fourier analysis has proved to be a very useful tool in problems related to additive number theory: orthogonality is a way of detecting linear relations, large Fourier coefficients imply linear structures and Fourier decay can be seen as a measure of uniformity. Gowers' proof of Szemeredi's theorem, Ruzsa's proof of Freiman's theorem and Green-Tao-Ziegler's results on polynomial structures in primes rely on these principles. In this talk I will illustrate these ideas with a simple example: solving a linear relation in either integers or real numbers sets.

Abstract: We define a new knot energy (different from the previously studied Coulomb and Moebius energies) and study the evolution of the knot as the value of the energy functional F decreases by gradient descent to its minimum. It turns out that the evolution is actually an isotopy (so that the knot type does not change during the gradient descent) and the shape of the knot minimizing the value of F (we call it the normal form of the knot) allows to recognize the knot type. Thus our algorithm, which is implemented in animations that will be shown during the talk, gives a very visual approach to the knot recognition problem. The functional that we use is the sum of two functionals, $F = E + R$. The first is the Euler functional $E$ (the integral of the square of the curvature along the knot), or more precisely, its discretization, and the second is a very simple repulsive functional $R$ which forbids self-intersections during the evolution. There will be no theorems in this talk, only results of computer experiments, and in conclusion we will discuss to what extent our algorithm gives a practical solution of the knot recognition problem.

Abstract: A link between martingale representation and solutions of the Skorokhod embedding problem has been established by R.Bass. A generalization of his approach to FBSDE leads us to solutions of the Skorokhod embedding problem for diffusion processes with deterministic drift. Techniques related to decoupling fields that describe the backward process as a functional of the forward one are essential in the algorithm (Joint work with Alexander Fromm and David P\"oromel).

Abstract: We consider the problem of invisibility for bodies with mirror surface within the scope of geometrical optics. The problem amounts to studying billiards in the exterior of bounded regions. Examples of bodies invisible in 1, 2, and 3 directions and bodies invisible from 1 and 2 points are provided in the talk. It is proved that there do not exist bodies invisible in all directions. The question of maximum number of directions and/or points of invisibility of a body remains open. The duality between invisibility (unperturbed billiard trajectories outside a bounded domain) and  periodic billiard trajectories inside the domain is also discussed. Further, we consider retro-reflecting bodies with mirror surface. A body is called a perfect retroreflector, if the direction of any beam of light incident on it is changed to the opposite. We provide several examples  of asymptotically retro-reflecting sequences of bodies. On the other hand, it is not known if there exist perfect billiard retroreflectors.

Abstract: Many small organisms possess flagella, slender whiplike appendages which are actuated in a periodic fashion in fluids and  allow the cells to self-propel. In particular, most motile bacteria are equipped with multiple helical rotating flagella which interact hydrodynamically, synchronise, and can form a tight helical bundle behind a swimming cell. We consider here the problem of bundling and unbundling of these flagellar filaments. Most past theoretical work has approached the problem of bundling using numerical computations. Here, we present an asymptotic treatment of the interactions between elastic rotating filaments. We first show how to asymptotically compute the hydrodynamic kernels governing hydrodynamic interactions in the case of long filaments, and we then use these results to derive the nonlocal, nonlinear, equations of motions of each filament. We finally apply our results to a few simple configurations.

Abstract: Localized bulging of an inflated elastic tube is a prototypical localization phenomenon in continuum mechanics, and its understanding can help shed light on a variety of other localization problems. Almost all previous studies of this phenomenon has been based on the membrane assumption that assumes that the tube has no bending stiffness. In this talk I will discuss what happens if we abandon the membrane assumption and consider instead an elastic tube of arbitrary thickness. Our research is motivated by possible applications to the mathematical modelling of aneurysm initiation in human arteries that have noticeable bending stiffness compared with party balloons.

Abstract: I give an introduction to studies of non-equilibirum dynamics in isolated many-particle quantum systems. These have recently attracted a lot of theoretical attention motivated by experiments on systems of ultra-cold trapped atoms. I focus on how such isolated systems relax and eventually can be described by quantum statistical mechanics. Time permitting I will discuss the time evolution of observables, which displays interesting phenomena related to the spreading of information out of equilibrium.

Abstract: Molecular modelling and simulation of the molecular dynamics are essential for much current research in physics, chemistry, materials and biomedical sciences. Molecular systems are embedded into an environment that fixes the thermodynamic conditions including temperature and pressure. To control the corresponding statistics during molecular simulations, a special mathematical tool called a `thermostat', is included in the dynamical equations. Thermostats are used to model nonequilibrium steady states as well. I will discuss the dynamic principle for derivation of various dynamical systems, stochastic as well deterministic, that sample the canonical ensemble. Among others, this dynamic principle elucidates the physical origin of thermostats that are based on extending of the phase space of the system in such a way as to preserve the invariant phase space distribution.

Abstract: The future Solvency 2 regulation for insurance introduces a risk metric taking into account all the risks involved in an insurance contract. But the risk measurement is essentially based on a one year analysis; for long term life insurance products, this methodology can imply important distortions and induce non optimal strategies. In order to take into account this maturity aspect, we consider in this paper, mixtures of conditional risk measures in order to obtain time-consistent dynamic risk measures and determine the solvency capital for long term guarantees.

Abstract:  While the longstanding problem of characterising  generically rigid bar-joint frameworks in three dimensions remains open, the area of Geometric Rigidity has nevertheless been developing rapidly in new directions. I will outline some of the subject's history and current momentum and I will present some results involving new connections with analysis, combinatorics and crystals.

Abstract: In the past few years, experiments on control of Lamb waves in plates have conclusively shown the possibility to achieve a flat lens, Maxwell's fisheye and an invisibility cloak over a range of KHz frequencies. Using analogies between flexural waves in structured plates and Rayleigh waves in structured soils, scientists at the Fresnel Institute of Aix-Marseille University and Menard's civil engineering company have designed and experimentally demonstrated that cloaking and lensing can be achieved for seismic waves around 10 Hz. Seismic metamaterials might pave the way towards a new generation of passive earthquake protection.

Abstract: Yau-Tian-Donaldson conjecture, recently proved by Chen, Donaldson and Sun, says that a Fano manifold is Kahler-Einstein if and only if it is K-stable. Its stronger form, still open, says that a polarized manifold (M,L) is K-stable if and only if M admits a constant scalar curvature with Kahler class in L. In this talk, I will describe K-stability of ample line bundles on smooth del Pezzo surfaces. I will show how to apply recent result of Dervan to prove K-stability and how to use flop-version of Ross and Thomas's obstruction to prove instability. The talk is based on my joint work with Jesus Martinez-Garcia (Johns Hopkins University, Baltimore, USA).

Abstract: Cluster algebras were introduced by Fomin and Zelevinsky in 2002 for studying total positivity of Grassmannians. Very soon after that  it turned out that the notion is connected to numerous different fields in mathematics (such as Poisson geometry, representation theory, integrable systems, combinatorics of polytopes, Teihmuller theory, dilogorithm identities and many others). In this talk we will introduce cluster algebras and show some connections to Coxeter groups. More precisely, we will see how an experience in classification of Coxeter hyperbolic polytopes helped to solve some classification problem in cluster algebras. The work is joint with Pavel Tumarkin and Michael Shapiro.

Abstract: Cancer progression in humans is difficult to infer because we do not routinely sample patients at multiple stages of their disease. The identification of cancer stem cell (CSC) subpopulations inside tumors opens a new perspective on cancer development, since it implies that tumors can only be eradicated by targeting CSCs. Several markers have been proposed in the literature to identify CSCs both in breast and melanoma but no consensus has been reached, leading to the hypothesis that the CSC phenotype might be dynamically switched. Herein we provide quantitative evidence of CSCs in melanoma discussing the complex network regulating their biological functions.

Abstract: The goal is to review and contextualize the evolution of my thoughts and interactions with tropical geometry. This talk is based on collaborative work always with Hannah Markwig, and at different times with each one of Aaron Bertram, Paul Johnson and Dhruv Ranganathan. Back in 2007, Hannah Markwig approached me after being told by Paul Johnson that her tropical covers smelled like cut and join. Deciphering Paul’s oracle was the beginning of a fruitful and ongoing collaboration, that is pulling me closer and closer to the tropical world. Over the course of the years, we have been studying Hurwitz theory and Gromov-Witten theory, first using tropical geometry as a powerful combinatorial tool, and then trying to understand what is the conceptual reason for the remarkably tight connection between the boundary geometry of moduli spaces of curves and maps and the piecewise linear objects in tropical geometry. The introduction of the analityc point of view, brought to the moduli space of curves by Abramovich, Caporaso and Payne, offered not only a much sought for conceptual perspective, but also opened up the way for further investigation.

Abstract: Davis, Obloj and Raval (2013) developed a theory of robust pricing and hedging of weighted variance swaps given market prices of co-maturing put options. They make use of F\"ollmer’s  quadratic variation for continuous paths, and of an analogous notion of local time. Here we develop a theory of pathwise local time, defined as a limit of suitable discrete quantities along a general sequence of partitions of the time interval. Our approach agrees with the usual (stochastic) local times for a.e. path of a continuous semimartingale. We establish pathwise versions of the Ito-Tanaka, change of variables and change of time formulae. We provide equivalent conditions for existence of pathwise local time. Finally, we study in detail how the limiting objects, the quadratic variation and the local time, depend on the choice of partitions. In particular, we show that an arbitrary given non-decreasing process can be achieved a.s. by the pathwise quadratic variation of a standard Brownian motion for a suitable sequence of (random) partitions; however, such degenerate behavior is excluded when the partitions are constructed from stopping times.

Abstract: Let $\alpha$ be a real number and  set $J_\alpha(N) = N \Vert N\alpha \Vert$ where $\Vert \alpha \Vert $ denotes the distance from $\alpha$ to the nearest integer. We provide a new lower estimate on the fifths asymptotic term of $J_e(N)$ for all big enough integers $N$ and an upper estimate for infinitely many values of $N$. Similarly we obtain results for  certain powers of $e$, $\tanh 1$ and a ratio of modified Bessel functions.

Abstract: The possibility that dark matter at large scales admits a description as a viscous fluid is discussed. The effects of shear viscosity are examined in two scenarios: 1) Dark matter with shear viscosity arising from its fundamental interactions: The back-reaction of fluctuations on the average energy density could accelerate the cosmological expansion. 2) Conventional dark matter in a coarse-grained description: The effect of short-distance (galactic-scale) fluctuations can be mapped onto pressure and viscosity terms in the effective theory of long-distance perturbations. The two-loop nonlinear power spectrum of the long-distance modes within the effective theory agrees well with N-body simulations and displays better convergence properties than standard perturbation theory.

Abstract: Estimating high level quantiles of aggregated variables (mainly sums) is crucial in risk management for many application fields such as finance, insurance, environment... This question has been widely treated but new efficient methods are always welcome; especially if they apply in (relatively) high dimension. We propose an estimation procedure based on the checkerboard copula. It allows to get good estimations  from a (quite) small sample of the multivariate law and a full knowledge of the marginal laws. This situation is realistic for many applications. Estimations may be improved by including in the checkerboard copula some additional information (on the law of a sub-vector or on extreme probabilities).This is a joined work with Andrés Cuberos (SCOR) and Esterina Masiello.