Seminars from until

Understanding the distribution of animals over space, as well as how that distribution is influenced by environmental covariates, is a fundamental requirement for the effective management of animal populations. This is especially the case for populations which are harvested. The sardine is one of the most important fisheries species, both for its economic, sociologic, antropologic and cultural values.

Here we intend to understand the spatial distribution of the average number of sardine eggs by $m^3$. Our main objectives are to identify the environmental variables that better explain the spatial variation in sardine eggs density and to make predictions in spatial points that were not observed.

The data structure presents an excess of zeros and extreme values. To deal with this, we propose a point-referenced zero-inflated model to model the probability of presence together with the positive sardine eggs density and a point-referenced generalized Pareto model for the extremes. Finally, we combine the results of these two models to get the spatial predictions of the variable of interest. We follow a Bayesian approach and the inference is made using the package R-INLA in the software R.

One of the main goal of extreme value theory is to infer probabilities of extreme events for which only limited observations are available and require extrapolation of the tail distribution of the observations. One major result is Balkema-de Haan-Pickands theorem that provides an approximation of the distribution of exceedances above high threshold by a Generalized Pareto distribution. We revisit these results with coupling arguments and provide quantitative estimates for the Wasserstein distance between the empirical distribution of exceedances and the limit Pareto model. In a second part of the talk, we extend the results to the analysis of a proportional tail model for quantile regression closely related to the heteroscedastic extremes framework developed by Einmahl et al. (JRSSB 2016). We introduce coupling arguments relying on total variation and Wasserstein distances for the analysis of the asymptotic behavior of estimators of the extreme value index and integrated skedasis function.

Joint work with B. Bobbia and D. Varron (Université de Franche Comté).

In 1983, Grothendieck wrote a letter to Faltings in which he formulated a conjecture for hyperbolic curves over fields which are finitely generated over the rationals. Remaining open to date, it carries the study of rational points on an algebraic variety to the realm of profinite groups. Assuming only a working knowledge of basic Algebraic Geometry, we formulate and motivate the Section Conjecture and outline some modern attempts to tackle it.

We prove the following theorem: axisymmetric, stationary solutions of the Einstein field equations formed from classical gravitational collapse of matter obeying the null energy condition, that are everywhere smooth and ultracompact (i.e., they have a light ring) must have at least two light rings, and one of them is stable. It has been argued that stable light rings generally lead to nonlinear spacetime instabilities. Our result implies that smooth, physically and dynamically reasonable ultracompact objects are not viable as observational alternatives to black holes whenever these instabilities occur on astrophysically short time scales. The proof of the theorem has two parts: (i) We show that light rings always come in pairs, one being a saddle point and the other a local extremum of an effective potential. This result follows from a topological argument based on the Brouwer degree of a continuous map, with no assumptions on the spacetime dynamics, and hence it is applicable to any metric gravity theory where photons follow null geodesics. (ii) Assuming Einstein's equations, we show that the extremum is a local minimum of the potential (i.e., a stable light ring) if the energy-momentum tensor satisfies the null energy condition.

We show that several definitions of algebras of continuous Fourier multipliers on variable Lebesgue spaces over the real line are equivalent under some natural assumptions on variable exponents. Some of our results are new even in the case of standard Lebesgue spaces and give answers on two questions about algebras of continuous Fourier multipliers on Lebesgue spaces over the real line posed by H. Mascarenhas, P. Santos and M. Seidel. The preprint is available at https://arxiv.org/abs/1903.09696.

Non-relativistic string theory is described by a sigma model that maps a two dimensional string worldsheet to a non-relativistic spacetime geometry. We discuss recent developments in understanding the spacetime geometry of non-relativistic string theory trying to provide several new insights. We show that the non-relativistic string action admits a surprisingly large number of symmetries. We introduce a non-relativistic limit to obtain the non-relativistic string action which also provides us the non-relativistic T-duality transformation rules and spacetime equations of motion.

We classify framed and oriented $2-1-0$-extended TQFTs with values in the bicategories of Landau-Ginzburg models.

The Einstein constraint equations describe the space of initial data for the evolution equations, dictating how space should curve within spacetime. Under certain assumptions, the constraints reduce to a scalar quasilinear parabolic equation on the sphere with various singularities and nonlinearity being the prescribed scalar curvature of space. We focus on self-similar Schwarzschild solutions. Those describe, for example, the initial data for the interior of black holes. We construct the space of initial data for such solutions and show that the metric at the event horizon is constrained to the global attractors of such parabolic equations. Lastly, some properties of those attractors and its solutions are explored.

Random Walks in Cooling Random Environments (RWCRE), a model introduced by L. Avena, F. den Hollander, is a dynamic version of Random Walk in Random Environment (RWRE) in which the environment is fully resampled along a sequence of deterministic times, called refreshing times. In this talk I will consider effects of the ressampling map on the fluctuations associated with the annealed law and the Large Deviation principle under the quenched measure. I conclude clarifying the paradox of different fluctuations and identical LDP for RWCRE and RWRE. This is a joint work with L. Avena, Y. Chino, and F. den Hollander.

An important conjecture in knot theory relates the large-$N$, double scaling limit of the colored Jones polynomial $J_{K,N}(q)$ of a knot $K$ to the hyperbolic volume of the knot complement, $\operatorname{Vol}(K)$. A less studied question is whether $\operatorname{Vol}(K)$ can be recovered directly from the original Jones polynomial ($N=1$). In this report we use a deep neural network to approximate $\operatorname{Vol}(K)$ from the Jones polynomial.