Upcoming Talks

Seminar webpage:  https://guests.mpim-bonn.mpg.de/bianchi/wiqi.html

This talk, based on joint work with Francesco Cattafi, presents an extension of the classical correspondence between vector bundles and principal bundles to the Lie groupoid setting. To achieve this, we introduced the notion of a PB-groupoid with a structural Lie 2-groupoid, establishing a correspondence between VB-groupoids and PB-groupoids. This framework enhances our understanding of VB-groupoids, which play a fundamental role in Poisson geometry, and provides a natural perspective on the tangent space of smooth symmetries.

Mirror symmetry predicts that for any Fano manifold $X$ there should be a Landau-Ginzburg model $(X^{\vee},W)$ such that the quantum $D$-module of $X$ is isomorphic to the Gauss-Manin system of $(X^{\vee},W)$. In addition, the natural lattice structures on the spaces of flat sections of these $D$-modules, one coming from the image of the Chern character of $X$ and one from certain integral relative homology of $X^{\vee}$, should match, after the former is twisted by the Gamma class. These predictions have been verified for toric Fano manifolds.

We begin by briefly introducing the subconvexity problem for $L$-functions and the delta method, which has proven to be a powerful line of attack in this context. As an application, for a $SL(2,\mathbb{Z})$ form $f$, we obtain the sub-Weyl bound:
$$L(1/2+it,f)\ll_{f,\varepsilon} t^{1/3-\delta+\varepsilon}$$ for some explicit $\delta>0$, thereby crossing the Weyl barrier for the first time beyond $GL(1)$. The proof uses a refinement of the 'trivial' delta method.