Abstract: Pleated surfaces are an important tool introduced by Thurston to study hyperbolic 3-manifolds and can be described as piecewise totally geodesic surfaces, bent along a geodesic lamination lambda. Bonahon generalized this notion to representations of surface groups in PSL_2(C) and described a holomorphic parametrization of the resulting open charts of the character variety in term of shear-bend cocycles.
In this talk I will discuss joint work with Martone, Mazzoli and Zhang, where we generalize this theory to representations in PSL_d(C). In particular, I will discuss the notion of d-pleated surfaces, and their holomorphic parametrization.
Abstract: The study of animal movement has exploded with the development of GPS and lithium-ion battery technologies, and the Movebank data repository alone records millions of new animal locations per day. In general, data obtained by tracking animals are irregularly sampled time-series, subject to temporal autocorrelation, measurement error, and various seasonal and non-stationary behaviors. Therefore, the natural mathematical framework for these data are continuous-time stochastic process models. Here, I discuss a number of stochastic process movement models that are both biologically useful and mathematically interesting.
Abstract: In magnetic confinement fusion devices, the equilibrium configuration of a plasma is determined by the balance between the hydrostatic pressure in the fluid and the magnetic forces generated by an array of external coils and the plasma itself. The location of the plasma is not known a priori and must be obtained as the solution to a free boundary problem. The partial differential equation that determines the behavior of the combined magnetic field depends on a set of physical parameters (location of the coils, intensity of the electric currents going through them, magnetic permeability, etc.) that are subject to uncertainty and variability. The confinement region is in turn a function of these stochastic parameters as well. In this work, we consider variations on the current intensities running through the external coils as the dominant source of uncertainty. This leads to a parameter space of dimension equal to the number of coils in the reactor. With the aid of a surrogate function built on a sparse grid in parameter space, a Monte Carlo strategy is used to explore the effect that stochasticity in the parameters has on important features of the plasma boundary such as the location of the x-point, the strike points, and shaping attributes such as triangularity and elongation. The use of the surrogate function reduces the time required for the Monte Carlo simulations by factors that range between 7 and over 30. In addition, as a remedy for the large computational cost of Monte Carlo simulation, we will also investigate the Multilevel Monte Carlo, using both a set of hierarchical uniform grids and adaptive grid refinement.
Abstract: We study the existence of unstable behavior in the Restricted 3 Body Problem (R3BP), which models the motion of a massless body under gravitational interaction with two massive bodies. In particular, we are interested in the existence of orbits which connect certain arbitrarily far regions of the phase space, in the spirit of what is usually referred to as Arnold Diffusion. The occurence of this kind of unstable behavior has been conjectured by Arnold himself to be "typical" in the complement of integrable systems. Despite presenting strong degeneracies, we construct diffusive orbits in the R3BP: more concretely, we build orbits along which the angular momentum of the massless body (a conserved quantity for the 2 Body Problem) experiments arbitrarily large variations. This is joint work with Marcel Guardia (UB) and Tere M. Seara (UPC).
Abstract: Modern machine learning algorithms are compelling in prediction problems. However, regarding to features of black boxes, the performance of machine learning algorithms is hard to statistically evaluate and can vary across datasets and underlying setups. This phenomenon is even more astonishing when multiple machine learning algorithms are evaluated, including penalized regression, random forest, gradient boosting, etc. Among these algorithms, it is notoriously challenging to determine the most appropriate one to use in practice, especially in the context of causal inference. In this talk, I will cover two topics: the first one covers a robust causal machine learner in the context of mean estimation. The proposed learner enables valid statistical inference and has the property of multiple robustness, which allows multiple machine learning algorithms and has shown to be robust as long as one of candidate algorithms works well. The second topic covers an advanced scheme of integrating extra information from auxiliary data into the casual machine learner, which can substantially boost the estimation efficiency. Extensive numerical studies demonstrate the superior of our method over competing methods, regarding to smaller estimation bias and variability. In addition, the validity of the proposed method is assessed in real applications by using UK Biobank data.