Fano manifolds, wall structures and torus fibrations

Polynomial equations, where one polynomial is equated to another, are fundamental to modern technology. They find applications in diverse fields, from the core sciences like chemistry and physics to economics and social sciences. The solutions to these equations take on geometric forms, which can be classified into basic building blocks. These building blocks are known as Fano manifolds, named after the Italian mathematician Gino Fano, who first identified their existence in geometry. Picture yourself as an explorer in a vast ocean that represents the universe of mathematical shapes and geometries. This ocean is dotted with islands, each unique in shape, size, and characteristics, symbolizing different mathematical objects. Fano manifolds are akin to a distinctive group of islands with intriguing features. They serve as the foundational elements for more complex islands that are more difficult to explore. Much like an island teeming with guides becomes easier to explore, Fano manifolds, due to their “ampleness” property, are more accessible for study compared to other mathematical objects. Fano manifolds can be thought of as an expansive category encompassing well understood objects. They are linked to several significant concepts in geometry and number theory, including the study of rational points, a key aspect of number theory. Ruddat’s research project aims to elevate the significance of Fano manifolds in algebra, geometry, and theoretical physics. The project seeks to classify these forms and compile a catalogue, drawing parallels with the periodic table of elements in chemistry. To achieve this goal, new techniques from mirror symmetry, logarithmic and tropical geometry will be used.

Systems of polynomial equations are extensively used in technology and science applications, ranging from basic chemistry and physics to economics and social science. The geometrical shape of the solution can be classified if one understands the fundamental building blocks. Mori Theory has identified these fundamental blocks as Fano manifolds. Building on prior work of successful ERC and EPSRC projects centered at the Imperial College London, Shape2030 will turn Stavanger in Norway into the center and knowledge base for the next imminent breakthrough in the foundational knowledge production of Fano manifolds and to complete the directory for Fanos of dimension three and four. Fano manifolds and more generally log Calabi-Yau manifolds lie at the heart of the enumerative mirror symmetry conjecture (EMS), the homological mirror symmetry conjecture (HMS) and the Strominger-Yau-Zaslow conjecture (SYZ). Using novel cutting edge techniques from logarithmic and tropical geometry, Shape2030 will make substantial progress on these conjectures and aim to solve EMS and SYZ. Unique new information about wall structures and torus fibrations will lead to groundbreaking new discoveries that enable future generations of scientists and provide benefit Norway's scientific and skilled labor landscape. The project team consist of 16 diverse team members: 5 local in Norway and 11 at strategically chosen international centers. Thesis writing students will be equiped with unique skill sets in mathematical problem solving, artificial intelligence and advanced techniques in geometry and polynomial equations. The proposal is set out to enable the achievement of the United Nation goals on economic growths, innovation and quality education and it facilitates the green transition by advancing the understanding of fundamental mathematics, e.g., about solutions to polynomial equations which underlies – among others – fields like geometric modelling, clean energy and robust engineering.