Seminários, para a disseminação informal de resultados de investigação, trabalho exploratório de equipas de investigação, actividades de difusão, etc., constituem a forma mais simples de encontros num centro de investigação de matemática.
O CAMGSD regista e publica o calendário dos seus seminários há bastante tempo, servindo páginas como esta não só como um método de anúncio dessas actividades mas também como um registo histórico.
We investigate the Stochastic Heat Equation (SHE) in the critical space dimension two, where it is ill-defined. A non-trivial solution, known as the critical 2D Stochastic Heat Flow (SHF), can be constructed through regularisation and renormalisation. We investigate the SHF in the strong-disorder regime, showing that it vanishes locally and identifying the spatial scale governing the transition from extinction to an averaged behavior. Corresponding results are established for the partition functions of 2D directed polymers, which shed light into the SHE regularised via space-time discretisation: when the disorder strength is kept fixed, the solution exhibits fluctuations on a superdiffusive scale. Based on joint works with Quentin Berger and Nicola Turchi.
Este mini-curso, dividido em duas sessões de duas horas, tem por objetivo dar uma breve introdução aos números $p$-ádicos. A primeira sessão é dedicada à construção dos números $p$-ádicos e alguns aspetos algébricos, nomeadamente:
A analogia de Hensel: um dicionário aritmético-geométrico. Valuação e valor absoluto $p$-ádico.
O anel $\mathbb{Z}_p$ dos inteiros $p$-ádicos.
O corpo $\mathbb{Q}_p$ dos números $p$-ádicos.
Topologia $p$-ádica.
Breve referência ao teorema de Ostrowski e classificação dos corpos locais.
Lemma de Hensel e aplicações.
Extensões finitas e infinitas de $\mathbb{Q}_p$. O corpo $\mathbb{C}_p$.
Na segunda sessão serão abordados os seguintes tópicos introdutórios de análise $p$-ádica:
Funções contínuas em $\mathbb{Z}_p$.
Séries de Mahler e teorema de Mahler.
Funções localmente constantes.
Derivação $p$-ádica.
Derivação da série de Mahler.
Teorema do valor médio.
Destaco, entre a literatura sobre números $p$-ádicos, três referências bibliográficas, pelo seu caráter mais elementar e por abordarem também aspetos analíticos.
Os livros de Fernando Gouvêa e Svetlana Katok são mais elementares, enquanto que o livro de Alain Robert é mais avançado, contendo uma introdução à análise funcional $p$-ádica.
Fernando Gouvêa, p-adic Numbers: An Introduction, Springer, 2020.
Svetlana Katok, p-adic Analysis Compared with Real, AMS, 2007.
Alain Robert, A Course in p-adic Analysis, Springer, 2000.
The Vaidya spacetime is a spherically symmetric solution of the Einstein equations with a null dust source. This can be used to model the gravitational collapse of a thick shell of radiation: a flat interior region is matched at an inner boundary to the null dust filled region, which is then matched at an outer boundary to Schwarzschild spacetime. A central singularity inevitably forms, and depending on the profile of the energy density of the null dust, this singularity can be globally naked. Motivated by the cosmic censorship hypothesis, we consider perturbations of this configuration. We review previous work, and describe recent work where the perturbation of the inner boundary — the past null cone of the central singularity — is analysed using a framework for studying perturbations of general hypersurfaces. This sets boundary conditions for perturbations at the past null cone, and we then consider the 3+1 evolutionary problem, focussing on the question of the stability of the Cauchy horizon of the naked singularity.
We will concentrate on the problem of characterising the minimal speed of monotone fronts in monostable scalar reaction diffusion (and advection) equations. The emphasis will be on variational principles and the puzzling situation of explicitly solvable equations. If time permits, I will also cover voting models and the connection to renormalisation group theory.
We will concentrate on the problem of characterising the minimal speed of monotone fronts in monostable scalar reaction diffusion (and advection) equations. The emphasis will be on variational principles and the puzzling situation of explicitly solvable equations. If time permits, I will also cover voting models and the connection to renormalisation group theory.
We will concentrate on the problem of characterising the minimal speed of monotone fronts in monostable scalar reaction diffusion (and advection) equations. The emphasis will be on variational principles and the puzzling situation of explicitly solvable equations. If time permits, I will also cover voting models and the connection to renormalisation group theory.
Given any open, bounded set $\Omega$, we consider suitable combinations, via a reference function $\Phi$, of the first $p$-eigenvalue of the Dirichlet Laplacian of partitions of $\Omega$. We give two different formulations of the problem, one geometrical and one functional. We prove relations among the two formulations, existence and regularity of optimal partitions, convergence, and stability with respect to $p$ and to $\Phi$. Based on a joint work with G. Stefani (Padova).
Existing machine learning frameworks operate over the field of real numbers ($\mathbb{R}$) and learn representations in real (Euclidean or Hilbert) vector spaces (e.g., $\mathbb{R}^d$). Their underlying geometric properties align well with intuitive concepts such as linear separability, minimum enclosing balls, and subspace projection; and basic calculus provides a toolbox for learning through gradient-based optimization.
But is this the only possible choice? In this seminar, we study the suitability of a radically different field as an alternative to $\mathbb{R}$ — the ultrametric and non-archimedean space of $p$-adic numbers, $\mathbb{Q}_p$. The hierarchical structure of the $p$-adics and their interpretation as infinite strings make them an appealing tool for code theory and hierarchical representation learning. Our exploratory theoretical work establishes the building blocks for classification, regression, and representation learning with the $p$-adics, providing learning models and algorithms. We illustrate how simple Quillian semantic networks can be represented as a compact $p$-adic linear network, a construction which is not possible with the field of reals. We finish by discussing open problems and opportunities for future research enabled by this new framework.
