Porto - January 2025

January 17, 2025, Department of Mathematics / CMUP Universidade do Porto

Speaker: Giosuè Muratore (Universidade de Lisboa)
Title: Counts of lines with tangency conditions in A1-homotopy
Abstract: A¹-homotopy theory, introduced by Morel and Voevodsky, provides a powerful motivic framework that bridges algebraic geometry and the methods of classical topology. By extending the toolkit of algebraic geometry with concepts from homotopy theory, this approach has opened the door to a wide range of applications across the field. In this talk, we will outline the fundamental ideas behind A¹-homotopy theory and explore its relevance in enumerative geometry, highlighting recent developments and results. 

Speaker: Raquel Caseiro (Universidade de Coimbra)
Title: Modular Class of a Loday algebroid
Abstract: We review Loday algebroids, their cohomology and characteristic classes. The modular class of this structure, a special characteristic class, is defined using nonlinear connections, and we try to recover the modular class of well-known examples such as Nambu-Poisson manifolds, Courant algebroids, and Grassman-Dorfmann algebroids.  This is a joint work in progress with Fani Petalidou (University of Thessaloniki).

Speaker: Bruno Mera (Universidade de Lisboa)
Title: From ideal bands to generalized Landau levels: quantum geometry of Bloch bands in the holomorphic setting
Abstract: This talk aims to provide an introduction to the notion of Generalized Landau Levels (GLLs) from a geometric point of view. GLLs are Bloch bands that generalize the standard notion of Landau levels of a charged particle in a uniform magnetic field. Beginning with a foundational introduction to quantum geometry---the differential geometry of families of quantum states---we delve into the specific case of Bloch bands, unraveling the inequalities that emerge relating the quantum metric and the Berry curvature, the saturation of which implies holomorphicity and gives rise to the concept of Kähler band. A Kähler band can then be understood as a regular holomorphic curve in complex projective space. The geometry of holomorphic curves shares many properties with that of real curves in Euclidean space. In particular, there is a distinguished moving frame along the curve, the Frenet-Serret frame (unique up to a global phase), whose elements are the GLLs. The frame satisfies the so-called Frenet-Serret equations which, together with the Maurer-Cartan structure equation, allow us not only to derive the quantum geometry of each GLL but also to establish geometric recursion relations among them. The content of these recursion relations is a manifestation of Calabi's rigidity theorem for Kähler immersions into projective space that, in this language, not only establishes the uniqueness, up to a momentum-independent unitary transformation, of a Kähler band with a given Berry curvature profile, but also completely determines the quantum geometry of the GLLs. As a natural consequence, the quantum volume of the quantum metric of the nth GLL is exactly quantized to 2n+1. The discussion finds direct applications to moiré materials, where the 0th GLL, the Kähler band, and the 1st GLL are bands which can stabilize fractional Abelian and non-Abelian, respectively, fractional Chern insulating phases.

Speaker: Ivan Beschastnyi (Université Côte d'Azur)
Title: Geometry and analysis on Grushin manifolds
Abstract: Grushin manifolds form a class of singular Riemannian manifolds, where the metric degenerates or explodes in a uniform way when approaching a given submanifold of codimension one. It turns out to be a surprisingly rich class with a variety of geometric and analytic phenomena. For example, one construct Grushin manifolds with exploding metric and finite distance or incomplete manifolds with self-adjoint Laplace-Beltrami operator. I will give an overview of known results concerning those manifolds and discuss some open problems.