Singularities and String Theory

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This project concerns research at the interface of algebraic geometry in mathematics and string theory in theoretical physics. Algebraic geometry is the investigation of algebraic solutions of polynomial equations, the geometry of the graphs of these solutions, and the theoretical structure describing their properties. It has numerous applications in science and engineering, including statistics, biology, and geometric modeling, as well as the application to string theory being advanced in this project. String theory is a physical theoretical framework for a unified field theory incorporating all the forces and particles found in nature. String theory has found wide application beyond unified field theory, including condensed matter physics, black hole physics, and the application to mathematics being advanced in this project. The Principal Investigator will involve graduate students and undergraduate students in aspects of the project, assisting their professional development and contributing to the development of the scientific workforce.More specifically, a collection of interrelated questions at the interface of string theory and singularity theory in algebraic geometry will be explored. Within algebraic geometry, there are two major sets of problems. The first pursues some open directions in the theory of canonical singularities of threefolds, their crepant resolutions, and enumerative invariants of these resolutions. The other direction is the study of the singularities of elliptically fibered Calabi-Yau threefolds and their connection to Hodge theory via the decomposition theorem. Within string theory, there are also two major sets of problems. The first pursues the geometry of 5-dimensional superconformal field theories and their moduli. The other direction is the development of the properties of F-theory directly from its definition as Type IIB string theory with varying axiodilaton without invoking duality. Connections between canonical singularities and 5-dimensional superconformal field theories will be explored and developed as well as the connection between Hodge theory and F-theory. The project also includes investigation of additional topics in string theory.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.