Research at the interface of physics and mathematics has a long history of being beneficial to both fields. This project concerns string theory, one of the most vibrant areas of research at this interface. As we explore the boundaries of the regime of validity of string theory and address some of its theoretical shortcomings and the conditions of its peaceful co-existence with the theory of gravity, we rely on increasingly sophisticated mathematical models. F-theory is one of the most geometric corners of string theory and relies on elliptic fibrations to geometrically engineer models for particle physics and gravity at very high energy. This project explores the mathematics that governs the physics of F-theory in order to bring new insights to the mathematics of elliptic fibrations. Elliptic fibrations are geometric spaces that locally look like a long-exposure picture of an elliptic curve moving over another space. Elliptic curves are some of the oldest but yet most prominent objects across mathematics. They are studied in algebraic geometry and number theory and play a central role in representation theory, cryptography (where they are used to secure internet's transactions), computer modeling, and theoretical physics. This multi-disciplinary project will explore connections between these mathematical and physical concepts and will also encourage graduate students both in mathematics and physics to study problems at the interface of physics and mathematics. The PI will also reach out to students from underrepresented groups to inspire them to participate in the scientific endeavor.
Geometric engineering is the construction of physical systems by geometric methods. This is a natural approach in the context of string theory, where the geometry of extra dimensions and the dualities between different formulation and regimes of the theory allow for a geometric formulation of many physical constraints. This project focuses on the geometry of elliptic fibrations as seen from the point of view of string theory, with strong interest in their topology, arithmetic properties, and connection to gauge theories, Grand Unified Theories, and topological defects. The research will also address the problem of resolution of singularities of Weierstrass models and its connection to Mori's theory and representation theory and will explore the structure of the singular fibers of elliptic fibrations over higher dimensional spaces. The PI will also study the physics of topological defects such as cosmic string solutions using new models of elliptic fibrations. A central theme of this project is the prominent role of singularities, their resolutions, and their topological invariants.
