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.
Conjectures hold a special status in mathematics. Good conjectures epitomise milestones in mathematical discovery, and have historically inspired new mathematics and shaped progress in theoretical physics. Hilbert’s list of 23 problems and André Weil’s conjectures oversaw major developments in mathematics for decades. Crafting conjectures can often be understood as a problem in pattern recognition, for which Machine Learning is tailor-made. In this talk, I will propose a framework that allows a principled study of a space of mathematical conjectures. Using this framework and exploiting domain knowledge and machine learning, we generate a number of conjectures in number theory and group theory. I will present evidence in support of some of the resulting conjectures and present a new theorem. I will lay out a vision for this endeavour, and conclude by posing some general questions about the pipeline.
Spectral networks are a combinatorial tool consisting of labelled lines on a Riemann surface. They have a surprising amount of applications and are intimately linked to non-Abelianization of flat connections, Fock–Goncharov cluster coordinates, exact WKB theory, etc. After reviewing this story for the $SL(2)$ and $SL(3)$ case, I will describe this is in detail for the group $G_2$. Time permitting, I will give as an application a concrete parametrization of the nonabelian Hodge correspondence for the Hitchin component of the split real form of $G_2$. This is joint work with Andy Neitzke.
