Seminars from until

A estatística bayesiana tem sido cada vez mais utilizada em ensaios clínicos, oferecendo maior flexibilidade e eficiência no desenvolvimento de novos fármacos.

Neste seminário abordaremos este tópico utilizando como exemplo base num grande ensaio clínico muito conhecido mas que poucos sabem que utilizou métodos bayesianos. Vamos explorar em detalhe a metodologia utilizada no ensaio e em como é aplicável a outros ensaios. Será também abordado o tema de escolha do tipo de distribuições a priori e como escolher parâmetros de uma distribuição.

In this presentation we study the limiting distribution for the joint-law of the largest and the smallest singular values for random circulant matrices with generating sequence given by independent and identically distributed random elements satisfying the so-called Lyapunov condition.

Under an appropriated normalization, the joint-law of the extremal singular values converges in distribution, as the matrix dimension tends to infinity, to an independent product of Rayleigh and Gumbel laws.

The latter implies that a normalized condition number converges in distribution to a Fréchet law as the dimension of the matrix increases. Roughly speaking, the condition number measures how much the output value of a linear system can change by a small perturbation in the input argument.

The proof relies on the celebrated Einmahl--Komlós--Major--Tusnády coupling.

This is based in a paper with Paulo Manrique, Extremes 2022.

Suppose we are given a globally hyperbolic spacetime (M,g) solving the Einstein vacuum equations, and a timelike geodesic in M. I will explain how to construct, on any compact subset of M, a solution g_eps of the Einstein vacuum equations which is approximately equal to g far from the geodesic but near any point along the geodesic approximately equal to the metric of a Kerr black hole with mass m_eps. As an application, we can construct spacetimes which describe the merger of a very light black hole with a unit mass black hole, followed by the relaxation of the resulting single black hole to its equilibrium (Kerr or Kerr-de Sitter) state.

We present a random interface model on the one-dimensional torus of size $N$ with a weak perturbation, i.e. an asymmetry $\sim N^{-\gamma}$ of the direction of growth that switches from up to down based on the sign of the area underneath. The evolution of the interface can be studied in terms of the density field of an underlying, non-Markovian exclusion process. We compute the order of the correlation functions of this process for the invariant measure of the interface model, and investigate the stationary fluctuations of the density field: we establish the convergence to an Ornstein-Uhlenbeck equation for $\gamma>\frac{8}{9}$, and discuss the limit for $\frac{1}{2}\leq \gamma<\frac{8}{9}$. Based on joint work with Martin Hairer and Patrícia Gonçalves.

Fourier Restriction Theory, i.e., the study of the interaction between the Fourier transform and the curvature of surfaces, is a central topic in harmonic analysis. In this area, there are some particular cases in which an optimal inequality can be achieved.

In this talk, we will discuss some of these cases and explore some discrete analogues. This is based on a joint work with Diogo Oliveira e Silva.

Reaction-diffusion equations arise naturally when modelling multi-component systems of interacting populations. These equations are widely employed to describe pattern formation phenomena across various biological, chemical and physical processes. The kinetic theory of statical mechanics provides a powerful framework to describe different types of interactions at multiple spatial or temporal scales. Through appropriate hydrodynamic limits of the kinetic systems, macroscopic equations can be derived, describing observable quantities and explaining how macroscopic phenomena emerge from the underlying microscopic dynamics. In this talk, I will apply these tools to study the evolution and interactions of competing bacterial populations on a leaf surface. Specifically, I will consider self and cross diffusion effects and investigate Turing instability properties leading to the formation and persistence of stationary spatial patterns.

This work is a collaboration with D. Cusseddu (University of Minho), M. Bisi and R. Travaglini (University of Parma, Italy).

In this talk, we explore étale groupoids $G$ with a locally compact Hausdorff unit space $X$, where $G$ itself may not be globally Hausdorff. For such groupoids, the essential $C^*$-algebra $C_{\textrm{ess}}^*(G)$ offers a more suitable framework than the reduced $C^*$-algebra $C_r^*(G)$, as it captures additional structural nuances. Specifically, $C_{\textrm{ess}}^*(G)$ arises as a proper quotient of $C_r^*(G)$.

We introduce the concept of essential amenability for groupoids, a condition that is strictly weaker than (topological) amenability yet sufficient to guarantee the nuclearity of $C_{\textrm{ess}}^*(G)$. To establish this, we define a maximal version of the essential $C^*$-algebra and show that any function with dense cosupport must be supported within the set of "dangerous arrows”, that is, arrows that cannot be topologically separated.

This essential amenability framework characterizes the nuclearity of $C_{\textrm{ess}}^*(G)$ and establishes its isomorphism to the maximal essential $C^*$-algebra. Our results offer new insights into the interplay between groupoid structure and operator algebras, extending the utility of $C_{\textrm{ess}}^*(G)$ in non-Hausdorff settings. This is based on joint work with Diego Martinez.

3d mirror symmetry is a mysterious duality for certain pairs of hyperkähler manifolds, or more generally complex symplectic manifolds/stacks. In this talk, we will describe its relationships with 2d mirror symmetry. This could be regarded as a 3d analog of the paper Mirror Symmetry is T-Duality by Strominger, Yau and Zaslow which described 2d mirror symmetry via 1d dualities.

I will discuss some recent results obtained in collaboration with A. Figalli, S. Kim and H. Shahgholian. We consider minimizers of the Dirichlet energy among maps constrained to take values outside a smooth domain $O$ in $\mathbb{R}^m$. These minimizers can be thought of either as solutions of a vectorial obstacle problem, or as harmonic maps into the manifold-with-boundary given by the complement of $O$. I will discuss results concerning the regularity of the minimizers, the location of their singularities, and the structure of the free boundary.