Moduli spaces of pseudoholomorphic curves and discs with Lagrangian boundary conditions are fundamental objects in symplectic geometry, and especially in Lagrangian Floer theory. These spaces are in general highly singular and must be regularized in order to be useful. One method for doing so is to use Kuranishi structures. A particularly useful kind of a Kuranishi structure is a global Kuranishi chart, first developed by Abouzaid-McLean-Smith in 2021. The earlier constructions, first developed by Fukaya-Oh-Ohta-Ono in the 1990s, require multiple charts with complicated relationships between charts and are quite difficult to work with. A global Kuranishi chart is a Kuranishi structure with only one chart, which is significantly more practical.

We give a brief introduction to global Kuranishi charts and outline a new construction. The construction is highly analytic, much more like the original Kuranishi structure construction of FOOO than the algebro-geometric construction of AMS.

The quantum connection is a flat connection arising from genus 0 Gromov--Witten theory. They can be defined integrally for sufficiently positive symplectic manifolds, allowing one to consider their characteristic p or p-adic versions which bear similarity to Gauss--Manin connections in arithmetic geometry. I would like to survey aspects of this theory, focusing on the case of Calabi--Yau threefolds and the role of quantum power operations. This talk is mostly based on joint work with Shaoyun Bai and Daniel Pomerleano.

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