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.
A rigorous understanding of the dynamical nature of spacelike singularities remains an open problem in mathematical cosmology. Since the heuristic work of Belinski–Khalatnikov–Lifshitz and Misner's Mixmaster construction, vacuum spatially homogeneous cosmological models are expected to play a key role for generic singularities. We therefore focus on this class of models. The most general cases are the Bianchi type VIII, type IX, and type VI$_{-1/9}$, each with a four-dimensional Hubble-normalized state space.
On one hand, we embed the types VIII and IX models into modified gravity theories and show that general relativity (GR) arises as a bifurcation point where chaotic dynamics become generic, suggesting a new approximation scheme for GR. On the other hand, we analyze the type VI$_{-1/9}$ oscillatory regime and show that only a subset of its structure is dynamically relevant.
The Hopfield model stands as a paradigm at the intersection of statistical physics, theoretical neuroscience, and machine learning. Originally introduced as a biologically inspired model of associative memory, it has since evolved into a foundational framework for understanding a wide range of complex systems.
On the one hand, its roots in neuroscience enable a fruitful cross-fertilization: biologically grounded mechanisms continue to inspire algorithmic refinements and performance improvements in modern associative memory models. On the other hand, its formal connection with Boltzmann machines provides a bridge to contemporary machine learning techniques, including strategies such as dropout, pre-training, and the optimization of activation functions.
From the perspective of statistical mechanics, the Hopfield model remains a cornerstone for the analytical study of high-dimensional systems with disorder and frustration. This viewpoint naturally extends to the investigation of structured datasets, where the model offers a tractable yet expressive starting point for developing analytical insights.
In this talk, after a gentle introduction to the model, we will highlight some of these current research directions, while keeping the presentation accessible to a non-technical audience.
