Mirror Symmetry and Related Topics

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This research project concerns algebraic geometry, a branch of mathematics that studies the structure of the solutions of systems of polynomial equations. In the last twenty years, the pace of development of algebraic geometry has accelerated due to its mutually beneficial connections with mathematical physics, more specifically to string theory. The first two parts of this project aim to contribute to this growing area of research by verifying some of the mathematical predictions that are inspired by string theory. A third part of the project is expected to involve research by undergraduate students. In addition to its intrinsic scientific value in algebraic geometry, this part of the project is aimed at expanding the opportunities for undergraduate students to participate in cutting-edge mathematical research in a way that makes them cognizant of deep connections between different areas of mathematics. This project focuses on the double mirror phenomenon in algebraic geometry as well as some combinatorial aspects of the Eisenbud-Goto conjecture. The research involves three main directions of study. The first part of the project will further investigate the Pfaffian-Grassmannian double mirror example, with the goal of transferring ideas and techniques developed in the context of Calabi-Yau complete intersections in toric varieties to this new, more sophisticated setting. The second part of the project continues research in the direction of the Kawamata's conjecture, which states that birational Calabi-Yau varieties are derived equivalent. This is a fifteen-year-old conjecture of key importance to the area of homological mirror symmetry. The third part of the project tackles the longstanding Eisenbud-Goto conjecture in the particular case of toric varieties, which can be reformulated as a question about the structure of the sets of integer points in certain integer polytopes.