Seminars, for informal dissemination of research results, exploratory work by research teams, outreach activities, etc., constitute the simplest form of meetings at a Mathematics research centre.
CAMGSD has recorded the calendar of its seminars for a long time, this page serving both as a means of public announcement of forthcoming activities but also as a historic record.
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.
