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.
I will describe constraints on the topology of Lefschetz fibrations with 4-dimensional fibers (in the smooth category), obtained from Seiberg—Witten theory. Our main result relates the index of a Dirac operator on the total space with the monodromy of the fibration, and is reminiscent of the Grothendieck—Riemann—Roch formula in the holomorphic setting. As topological applications, we construct closed simply-connected symplectic 4-manifolds which admit Torelli symplectic diffeomorphisms that are not isotopic to products of squared Dehn twists on Lagrangian spheres, answering a question of Donaldson in the negative. Based on joint work with Hokuto Konno, Jianfeng Lin, and Anubhav Mukherjee.
A classical result by Maz'ya states that it is possible to bound from below the bottom of the spectrum of the Dirichlet-Bilaplacian of a general open set $\Omega \subseteq \mathbb{R}^N$ in terms of its inradius (namely, the radius of the largest ball inscribed in $\Omega$), provided that $N=2,3$. In a couple of recent papers, Bianchi $\&$ Brasco obtained two novel fractional spectral inequalities linked to this object, extending some classical results. We heavily rely on the ideas and the techniques developed in the above mentioned works, to infer some new optimal spectral bounds for higher-order fractional biharmonic operators.
Based on a work in progress with Francesca Bianchi and Lorenzo Brasco.
