CHaNGe — Categorical, Homological, and Non-commutative methods in Geometry

This proposal is primarily interested in algebraic geometry, a field that underpins geometric intuition with the precision brought about by calculation from algebra. The algebraic information of an algebraic variety - the set of common zeros of polynomial equations – is packaged in sheaves and organized in categories, which makes it possible to study these objects by importing the tools of homological and (non-)commutative algebra. This categorical approach is natural for many of the current leading questions in mirror symmetry, birational geometry, representation theory, and theoretical physics because it serves as a bridge between the geometric and algebraic language. It is also fundamental, as many properties and even classification results are not possible without it. The proposal is deeply interdisciplinary, and it connects broad areas of algebra and geometry. It naturally splits into three parts. The first aims to investigate one of the most fundamental properties of curves, namely contractibility, with non-commutative deformation theory. It will generalize classical results using modern language, at the same time producing many new explicit examples of rational curves in threefolds. Linking contractibility and non-commutative algebra will bring deep consequences for the birational and enumerative geometry of Calabi-Yau varieties, advancing areas of string theory. The second part studies surfaces with mild positive curvature (Fano, but with ineffective anticanonical bundle). The main objective is to prove a mirror theorem, packaging physical information with tools of homological algebra. The new heuristics, definitions, and techniques will open up an avenue to investigate higher dimensional settings. The third part involves reconstructing varieties from associated categorical data. I will tackle some hard open cases, developing new tools and combining them with classical methods.